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Observation of vortex-pair dance and oscillation | 2412.06634 | Observation of vortex-pair dance and oscillation
Dadong Liu,1 Lai Chen,1 Li-Gang Wang 1,*
1 School of Physics, Zhejiang University, Hangzhou 310058, China * Corresponding author. E-mail address: [email protected] (L.-G. Wang).
Abstract
Vortex dynamics, which encompass the motion, evolution, and propagation of vortices, elicit both fascination and challenges across various domains such as fluid dynamics, atmospheric science, and physics. This study focuses on fundamental dynamics of vortex-pair fields, specifically known as vortex-pair beams (VPBs) in optics. VPBs have gained increasing attention due to their unique properties, including vortex attraction and repulsion. Here, we explore the dynamics of pure-phase VPBs (PPVPBs) and observe intriguing helical and intertwined behaviors of vortices, resembling a vortex- pair dance. We uncover the oscillation property of the intervortex distance for PPVPBs in free space. The observed dancing and oscillation phenomena are intricately tied to the initial intervortex distance and can be explained well in the hydrodynamic picture. Notably, the vortex dancing and oscillation alter the process of vortex-pair annihilation, extending the survival range for opposite vortices. This discovery enhances our understanding of vortex interactions and sheds light on the intricate dynamics of both vortex-vortex and vortex-antivortex interactions.
INTRODUCTION
Vortices are prevalent phenomena observed across a spectrum of scales in nature, ranging from water eddies and atmospheric typhoon or hurricanes to majestic spiral galaxies. Vortices are also fundamental solutions within cylindrical-symmetry resonators for electromagnetic fields (1) and are pervasive in the realm of light, where they are known as optical vortices (2, 3). Optical vortices exhibit a distinctive feature a dark core at the center, characterized by an indeterminate phase with vanishing amplitude (4, 5). These unique attributes of optical vortices give rise to a diverse array of applications, including optical micromanipulation (6-8), optical communications (9- 11), quantum information (12-15), super-resolution imaging (16-18), and optical measurements (19-21).
In the presence of multiple vortices within light fields, the topological dynamics and interaction among vortices can create unique and interesting phenomena, like vortex knots (22), vortex collisions (23, 24), and the consequential process of vortex creation, annihilation or nucleation (23-30). Among vortex interactions, a fundamental scenario involves the interaction of two vortices. Like in atmosphere, Fujiwhara effect occurs as two typhoons approach each other (31). In the domain of optics, the dynamic interplay of attraction and repulsion between two vortices has been previously observed in the dynamics of vortex-pair fields (25, 26). A vortex-pair beam (VPB) is a type of structured light fields containing a pair of vortices. It is sometime categorized into two scenarios: an isopolar vortex pair, i.e. two vortices with identical topological charges (TCs), and a vortex dipole, characterized by two vortices with opposite TCs (32). In 1993, Indebetouw discovered that the relative distance of an isopolar vortex pair remains constant during free space propagation, while a vortex dipole tends to exhibit mutual attraction (25). This effect was subsequently confirmed through the experiment (26). The interaction between two vortices manifests specific features, including the rotational effect observed in the isopolar vortex pair during propagation (26, 33), the reappearance of an annihilated vortex dipole in the far field (34), and an optical intrinsic orbit orbit interaction, serving as a manifestation of the attractive and repulsive interactions within
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a vortex dipole (35). Furthermore, the exploration of VPB dynamics in diverse optical systems (36-49), such as those involving a graded-index medium (40, 41), a high numerical-aperture lens (42-44), an astigmatic system (45-47), a half-plane screen (48), and a knife edge (49), has been a subject of extensive discussion.
However, the aforementioned investigations (25, 26, 32-49) primarily rely on the model of complex-amplitude VPBs (CAVPBs), in which each vortex undergoes complex-amplitude modulation, constraining our comprehensive understanding of the dynamics inherent in multiple vortices. In the exploration of fractional vortex fields (50- 55) and vortex arrays (56-60), researchers have found the intricate dynamics in the evolutions of vortex interactions among multiple vortices beyond the vortex attraction and repulsion process, like the birth or annihilation of vortex pairs. To further advance our understanding of the interaction between two vortices, here, we would like to address the dynamics of the pure-phase VPBs (PPVPBs), consisting of two pure-phase vortices. It is noteworthy that, to the best of our knowledge, the dynamics of PPVPBs have not been reported previously, despite their proposal and application in optical trapping and manipulation (61). Here, we elucidate the intriguing dynamics arising from vortex- vortex and vortex-antivortex interactions, resulting in a phenomenon reminiscent of vortex dance a helical and intertwining behaviour among vortices. The observed vortex dynamics can be well explained by using an optical hydrodynamic picture (62). Our experimental verification, employing the interference method to trace vortex trajectories in light fields, solidifies the existence of this interesting feature. Notably, the oscillation of intervortex distance in PPVPBs represents a fundamentally unique characteristic not observed in traditional CAVPBs, where no oscillation effect had been previously identified. A comprehensive investigation reveals that the observed vortex dancing and oscillation can be precisely controlled by the initial intervortex distance, reflecting the interaction strength between the vortices. Meanwhile, the vortex dance and oscillation substantially influence the process of vortex-pair annihilation. This effect enlarges the survival range of opposite vortices, with a certain similarity to Fujiwhara effect of two typhoons in atmosphere, presenting a distinct aspect not witnessed in CAVPBs.
RESULTS
Fields of PPVPBs We start by briefly reviewing the previous model of CAVPBs (25). The initial field of such CAVPBs embedded in a host Gaussian beam is usually expressed as
exp | vm ]) 1
2 w 0 | uu [(
vuMvu ),( ),(
CAVPB
| vm ]) 2
m 2
m 1
vuM i ),( , where 2u0 is the initial distance of two vortices with TCs m1 and m2, w0 is the beam width of the host Gaussian beam, and u, v refer to the transverse rectangular coordinates at the initial plane. As the magnitude of the function M(u, v) changes and deviates from unity, the two vortices in CAVPBs also undergo amplitude modulation. According to modal analysis (32, 36), these CAVPBs can be expressed as linear combinations of finite vortex modes, which, in turn, govern the dynamics of vortices. In most of works, researchers only considered the cases of m1 = m2 = 1 for an isopolar vortex pair or m1 = -m2 = 1 for a vortex dipole. The vortex trajectories, illustrating the attraction or repulsion of vortices, were demonstrated in a series of prior investigations (25, 26, 32-49). Recently, one has developed the laser hydrodynamic model to explain the vortex motions in multiple complex-amplitude vortices (63, 64). Nevertheless, unraveling the underlying physics of vortex dynamics remains a challenge in many complex optical fields, surpassing the complexity observed in CAVPBs.
uu
sgn(
i sgn(
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with
In contrast to the model of CAVPBs, one can have an alternative choice on two
pure-phase vortices, which can be written as (61)
PPVPB
vu ),(
vuF ),(
exp
2 w 0
,
(1)
m 1
with
. Here the magnitude of F(u, v)
vuF ),(
0 is always equal to unity, different from the above function M(u, v), and Eq. 1 contains 22 ime 2 being the two local 1 and . Such fields are called as PPVPBs. In Fig. 1A, it shows the azimuthal angles at phase profiles of such PPVPBs for several situations with different m1 and m2. One can
11 ime
with
and
two pure-phase vortices like
0u (
)0,
~ u 0
wu / 0 0
define the initial dimensionless relative off-axis distance as an indicator of external control on the interaction of vortex pair. It is noteworthy to reiterate that PPVPBs comprise two off-axis vortices with pure phase and without amplitude modulation. While this may appear similar to the aforementioned CAVPBs, it fundamentally differs from them. By examining the mode purities of PPVPBs and comparing them with CAVPBs, we ascertain that the orbital angular momentum (OAM) spectra of PPVPBs and CAVPBs are inherently distinct (refer to Section A of the Supplemental Materials). When m1 = m2, the OAM spectra of PPVPBs consist of infinite even OAM modes, whereas for CAVPBs their OAM spectra comprise finite even OAM modes. Conversely, when m1 = -m2, both PPVPBs and CAVPBs exhibit symmetrical OAM modes, with PPVPBs having an infinite set and CAVPBs having a finite set. These distinctions contribute to varied interactions between vortex-vortex and vortex- antivortex in PPVPBs, resulting in distinct behaviors of vortex dynamics.
The evolutions of optical fields in free space or an optical system can be well predicted by using theory of matrix optics (65, 66) and the detail description of theoretical equations can be found in the section of Materials and Methods. Once the field evolution is achieved, the locations of vortex centers can be determined either from the phase distributions or by identifying the dark cores through taking the logarithm of the light intensities, offering an intuitive display. The comparative movies between PPVPBs and CAVPBs are available in Section B of the Supplemental Materials. In the case of PPVPBs, their intensity distributions result in the formation of ripples in the light fields, akin to ripples on water caused by two falling stones, highlighting the interaction among vortices. In contrast, CAVPBs exhibit more stable and tranquil evolutions of intensity distributions during propagation, devoid of such ripples. We attribute these pronounced differences between PPVPBs and CAVPBs to the distinct dynamics of vortices, which could be seen from the vortex trajectories. We also show that the role of the host beam in PPVPBs is less important than that in CAVPBs, see the detail discussion in Section I of the Supplementary Materials.
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Fig. 1. Schematic diagrams of phase distributions of PPVPBs and experimental setup. (A) The initial phase distributions of PPVPBs with the TC values m1 and m2 marked on the top of subfigures. The phase singularities circled by green and yellow arrows represent positive and negative vortices, respectively. Other parameters here are 0 = 0.4 and w0 = 1.63 mm. (B) Experimental setup for generating PPVPBs and measuring the positions of phase singularities by the interference method. The position of z=0 is the generating plane of PPVPBs, which is also called as the input (or initial) plane for the subsequent optical system. Inset in (B) represents a specific focusing lens system with the lens position located at z=f from the input plane. Using this focusing system, at the back focal position of the lens (i.e., z=2f here), the system becomes a 2-f lens system and optical properties at z=2f here is similar to the situation of the far-field or Fraunhofer region in free space. Notations are: HWP, half-wave plate; PBS, polarized beam splitter; BE, beam expander; BS, beam splitter; SLM, spatial light modulator; L1 and L2, the focusing lens with f1 = f2 = 300 mm; AP, aperture; MR, mirror reflector; BLK, block; RAPM, right-angle prism mirror. Here, the RAPM is movable for realizing the change of the propagation distance z by using the electrically-controlled motorized system.
Experimental setup To demonstrate the vortex dynamics, we experimentally generated PPVPBs by using a phase-only SLM (Holoeye PLUTO-2-NIR-015). Figure 1B depicts the schematic of our experimental setup designed to produce PPVPBs and detect vortex locations in free space using the interference method. We use a linearly-polarized He-Ne laser with a wavelength of 632.8 nm as the light source. The half-wave plate and the polarized beam splitter are used to control the horizontal polarization of the transmission light and adjust its light intensity. The beam was then expanded via a beam expander, increasing the beam waist (w0) to approximately 1.63 mm. Subsequently, the beam underwent splitting by a beam splitter, with the reflected light serving as a reference beam for interference experiments, and the transmitted light is incident on the SLM to generate various orders of PPVPBs. Phase diagrams, as illustrated in Fig. 1A, were loaded onto the SLM. The modulated first-order diffraction beam, representing the generated PPVPB, was isolated using a suitable aperture. A 4-f lens system, composed of lenses L1 and L2 with focal lengths f1 = f2 = 300 mm, imaged the SLM plane onto the back focal plane of lens L2. Consequently, PPVPBs were created on the rear focal plane of lens L2, establishing the initial plane at z = 0 for studying the evolution of light fields in the subsequent optical
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system. A right-angle prism mirror, positioned on a motorized system, precisely adjusted the propagation distance (z) of the PPVPBs. Finally, the interference patterns between PPVPBs and the reference beam were captured by a camera with 12-bit depth. Experimentally obtained fringe patterns facilitated the reconstruction of PPVPB phase distributions using the Fourier-transform method (67). Based on this information, the vortex locations of the PPVPBs were determined through the application of the phase singularity search algorithm (68), leveraging the high-frequency characteristics of phase singularities. The methods to obtain the information of phase distributions and singularities are also introduced in the section of Materials and Methods.
Dynamics of vortices in free space Now let us discuss the dynamics of vortex pair in the fields of PPVPBs. Figure 2 shows the trajectories of vortices for PPVPBs with m1 = m2 = 1 and m1 = -m2 = 1 in free space. When m1 = m2 = 1, the two positive vortices rotate individually, gradually repelling each other. Their trajectories exhibit central symmetry about the origin of the transverse plane, see their projection on the x-y plane. This centrosymmetric characteristic is independent of the propagation distance z as shown in Fig. 2A and holds for all PPVPBs with equal TCs (i.e., m1 = m2). In the case of m1 = -m2 = 1, both positive and negative vortices rotate themselves during propagation, but their trajectories exhibit symmetry about the y-axis. This reflectionally symmetric property remains unchanged across different propagation distance (z) as depicted in Fig. 2B. It is valid for all PPVPBs with opposite TCs (i.e., m1 = -m2). These symmetries align with the symmetry properties of the initial phase distributions of such PPVPBs, which are symmetric about the origin or the y-axis, as displayed in Fig. 1A with m1 = m2 = 1 or Fig. 1A with m1 = -m2 = 1.
The evolution of each vortex within PPVPBs manifests more intriguing effects than those observed in CAVPBs. As depicted in Fig. 2 (A and B), the trajectory of each vortex in PPVPBs follows a helicoidal motion in free space. Simultaneously, the interplay among vortices induces oscillating changes in the intervortex distance, as evident in both experimental and theoretical results presented in Fig. 2 (C and D). This unique evolutionary pattern, involving simultaneous rotation and oscillation, resembles a dance and represents a interesting characteristic that has never previously observed in CAVPBs with linear polarization. Interestingly, the relative off-axis distance ( 0) plays a crucial role in the observed vortex oscillation phenomena. In Fig. 2 (C and D), it is noted that vortex oscillation persists for a longer propagation distance as 0 increases. It is noteworthy that when 0 is smaller than a certain value, vortices with opposite TCs can mutually annihilate each other after propagating a specific distance, see Fig. 2D. The critical value of 0 for the annihilation feature in PPVPBs differs from that in CAVPBs. More information on the theoretical prediction on the intervortex distance can be found in Section D of the Supplementary Materials, and further instances of vortex annihilation phenomena in PPVPBs with opposite TC vortices also can refer to the videos in Section B of the Supplemental Materials. A quantitative comparison between PPVPBs and CAVPBs will be addressed in subsequent discussions.
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Fig. 2. Experimental measurements of vortex trajectories and intervortex distance for PPVPBs in free space. (A and B) Experimental vortex trajectories for (A) m1 = m2 = 1, and (B) m1 = -m2 = 1 with 0 = 0.4. The blue and red dots denote, respectively, the evolution of positive and negative vortices. The corresponding solid lines are theoretical predictions and their projections are shown by the green lines in the x-y planes. (C and D) Evolution of the intervortex distance along the propagation distance for (C) m1 = m2 = 1 and (D) m1 = -m2 = 1, respectively, under different 0. The corresponding theoretical predications are also displayed with the same-colour curves. The experimental parameter w0 = 1.63 mm is taken for theoretical calculations.
Figure 3 further presents both experimental and theoretical trajectories of vortices in the fields of PPVPBs with equal TCs (m1 = m2 = 2) and opposite TCs (m1 = -m2 = 2) in free space. As shown in Fig. 1A, the PPVPB with m1 = m2 = 2 showcases a central symmetry of two vortices bearing +2 TC each, which progressively split into four distinct vortices with individual TCs of +1 during propagation. From Fig. 3A, an interesting interplay among vortices emerges, entwining them in a helicoidal dance as the propagation distance z extends. Intriguingly, although no oppositely signed vortices are present in the initial light field with m1 = m2 = 2, the evolving z engenders the generation of multiple pairs of positive and negative vortices, engaging in a mesmerizing alternation of intertwining, nucleation, and annihilation.
For PPVPBs with opposite TCs (m1 = -m2 = 2), as depicted in Fig. 3B, even when a pair of vortices with 2 TCs is initially present on the plane (refer to Fig. 1A), these vortices gracefully split into two pairs, one carrying +1 TCs and the other -1 TCs. Evidently, vortices boasting high-order TCs in PPVPBs exhibit instability, consistently fragmenting into vortices with +1 or -1 TC. Analogy to the phenomena observed in PPVPBs with equal TCs (m1 = m2 = 2), the entanglement, helical dynamics, and the intriguing nucleation and annihilation of vortices are also observed in PPVPBs with opposite TCs (m1 = -m2 = 2).
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Based on the theory of optical hydrodynamics in the recent studies (62-64), the motion of vortices, the splitting of higher-charge optical vortices, and the dynamics of vortices have been explained in complex-amplitude modulation vortex pairs, in which no helicoidal, intertwined and oscillating vortex dynamics have been observed. Here, we employ the same argument to explain the vortex dynamics, especially in PPVPBs. The initial total light field consisting of vortex pairs embedded in a host Gaussian beam can be written as a product of two fields: one for the initial tested field of one vortex under consideration, another for the initial background for the rest field that comprises the fields of other vortices and the host beam. The evolutions of the transverse velocity fields of the background field at any propagation distance z are demonstrated in the supplementary movies S3 and S4. In the movies, the tested vortex moves along the direction of the velocity field. In the PPVPBs with m1=m2=1 or m1=-m2=1, the tested vortex surfs in the diffraction waves from the other vortices in the background fields that induces the helical and oscillating motions (that explains the trajectories in Fig. 2). In the PPVPBs with higher-order TCs, the presence of the circulation flow from the vortex near the tested one further alters the local background velocity field. Under the diffraction ripples of the background field during propagation, the tested vortices experience not only the helical and oscillating motions but also the vortex nucleation and annihilation phenomena. In contrast, there are no complex velocity flows in CAVPBs (lacking the diffraction waves from other vortices), so there are no helical and oscillating motions of vortices. More information on the theoretical consideration of the optical hydrodynamic model and detail discussion on the supplementary movies S3 and S4 can be found in Section C of the Supplementary Materials.
From the above, the demonstrated helical and intertwined behaviors among vortex pairs not only induce the oscillation of the intervortex distance but also change the dynamics of vortex interactions like prolong the survival range of opposite vortices as discussed later. The characteristics of the dynamic behaviors within these PPVPBs intensifies with the augmentation of the relative off-axis distance ( 0). We can image that when 0 is large, the tested vortex is immersed for more time in the diffracted waves of other vortices and oscillates in a spiral form. Sections E and F of the Supplementary Materials provide further results, confirming the influential role of 0 in regulating vortex behaviors. Consequently, the relative off-axis distance ( 0) emerges as a pivotal control parameter for modulating the dynamic behaviors exhibited by these PPVPBs.
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Fig. 3. Experimental and theoretical trajectories of vortices in PPVPBs propagating in free space. (A) The vortex trajectories for the PPVPB with equal TCs of m1 = m2 = 2 and (B) the vortex trajectories for the PPVPB with opposite TCs of m1 = -m2 = 2. Here the relative off-axis distance parameter is taken as 0 = 0.4. The blue and red dots denote, respectively, data for the trajectories of positive and negative vortices, and their projections in the x-y plane are presented by the green dots. The experimental parameter w0 = 1.63 mm is also taken for theoretical calculations.
Dynamics of vortices in a focusing system It is widely recognized that the light field in far-field regions closely resembles that found at the back focal plane of a 2-f lens system (50). Therefore, to thoroughly explore the annihilation process of vortices for the PPVPBs in the far field, it is more convenient
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to examine the evolution within the 2-f lens system illustrated in the inset of Fig. 1B. Through changing the value of z in this focusing system, one can achieve the key dynamics of vortex pairs from the near-field region (i.e., the lens plane) to the far-field region (i.e., the focal plane). The only difference is that the beam profile of light in free space is spread out or divergent while it becomes to be condensed or convergent in the focusing system.
Fig. 4. Evolution of vortex trajectories for different-order PPVPBs with opposite TCs in the 2- f focusing system. (A) The influence of the relative off-axis distance 0 on the vortex-trajectory evolution, where the opposite vortices happen to merge and annihilate each other at the focusing plane when 0 = 0.358. (B, C, and D) The vortex-trajectory evolutions at various critical values of 0 for different-order PPVPBs, in which each plot corresponds to the situation that one pair of opposite vortices annihilate each other at the focusing plane. Here, the focal length of the 2-f focusing system is f = 500 mm and the beam parameter is also taken to be w0 = 1.63 mm.
In Fig. 4A, the influence of the relative off-axis distance 0 on the trajectories of vortices, when m1 = -m2 = 1, is depicted as they evolve from the lens (z=500 mm) to the back focal plane (z=2f=1000 mm). When the value of 0 is sufficiently large, such as when 0 = 0.55, the positive and negative vortices undergo oscillations and persist at the back focal plane. This observation suggests that they do not annihilate each other in free space. Conversely, when 0 = 0.358, the positive and negative vortex pair coincidentally merge at the focal plane, signifying their annihilation in the infinity of free space. This
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particular value is termed the critical value for the occurrence of vortex annihilation in PPVPBs when m1 = -m2 = 1. As 0 decreases further, the annihilation phenomenon happens before the focal plane, indicating that this effect can be observed at a suitable distance in free space. In the Section H of the Supplementary Materials, we also provide the corresponding change of the intervortex distance of the vortex pair in free space. Notably, the critical value of 0 in PPVPBs is much smaller than that in cases of CAVPBs with m1 = -m2 = 1. This distinction implies that the dynamic properties of opposite vortices in PPVPBs during evolution surpass those of CAVPBs under identical conditions. From a propagation perspective, owing to the oscillatory or dancing behaviors between opposite vortices in PPVPBs, the annihilation process becomes much slower, allowing vortices to survive over longer distances. Consequently, a slight increase in 0 results in the disappearance of the annihilation process compared to that in CAVPBs. This effect in PPVPBs bears similarity to the Fujiwhara effect between two typhoons, which often prolongs the lifespan of typhoons (69).
For m1 = -m2 = 2, the initial light field processes a pair of opposite vortices with 2 TCs, but due to the unstable properties of high-order vortex pair during propagation, they rapidly split into two pairs of opposite vortices with 1 TCs, thus there are complex dynamics as shown in Fig. 3B. In Fig. 4B, two critical situations for vortex annihilation at the focal plane are shown. When 0 = 0.569, one pair undergoes annihilation at the focal plane, while the other pair survives. When 0 further reduces to 0 = 0.218, the second pair also undergoes annihilation at the focal plane. Interestingly, at this situation the first pair of vortices actually annihilate each other at a shorter distance or earlier. Note that the phenomenon of vortex dancing here appears before the lens plane. Thus, for cases when m1 = -m2 = 2, there are two critical values of 0, each corresponding to the critical points of annihilation processes for the respective pairs of vortices.
Similarly, as demonstrated in Fig. 4(C and D), for the instances of m1 = -m2 = 3 and m1 = -m2 = 4, three and four pairs of opposing vortices with 1 TCs are observed, respectively. Each scenario is associated with a distinct critical value corresponding to the annihilation of a vortex pair at the focal plane. Additionally, within Fig. 4, the focal fields of PPVPBs reveal the oscillation and dancing of vortex trajectories, featuring vortex intertwining and helical behaviors, phenomena hitherto unobserved in CAVPBs.
Table 1. Comparison of critical values of the relative off-axis distance 0 between PPVPBs and CAVPBs under different-order opposite TCs for occurring vortex-pair annihilation at the focal plane of a 2-f lens system.
PPVPBs CAVPBs
1 TCs 0.358 0.500
2 TCs
3 TCs
0.569 0.218 0.681 0.404 0.155 0.925 0.383 1.254 0.758 0.323
0.734 1.533
4 TCs
0.529 1.066
0.313 0.661
Table 1 enumerates critical values of 0 for observing the annihilation effect of each pair of opposite vortices with 1 TCs within PPVPBs at the focal plane in the cases of m1 = -m2. For comparison, corresponding critical values for observing annihilation effects in CAVPBs at the focal plane are also included. Interestingly, each critical value of 0 for PPVPBs is smaller than the corresponding critical value for CAVPBs. In other words, under identical conditions (for example, with the same value of 0 and beam parameters), the annihilation effect occurs at a greater distance for PPVPBs than for CAVPBs. This delay is attributed to the vortex oscillation and dancing effects, which prolong the annihilation process of vortex pairs. Additional details regarding the annihilation processes of vortices in CAVPBs with opposite TCs in a 2-f lens system are provided in Section G of the Supplemental Materials.
DISCUSSION
Our study unveils the dynamic behaviors of PPVPBs in both free space and the focusing system. For PPVPBs featuring unit vortices, vortex trajectories form helical structures,
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0.119 0.285
accompanied by oscillating intervortex distances between vortices. Our experimental results are well demonstrated and confirm the theoretical predictions. In the case of PPVPBs with high-order vortices, the initial high-order vortices undergo a dynamic process, splitting into multiple unit vortices during propagation. This evolution is marked by intricate vortex intertwining and helical behaviors including vortex nucleation and annihilation. Such fast helical and intertwining behaviors resemble a dance of vortex pairs and are very common in the fields of PPVPBs. Interestingly, the light fields of PPVPBs with high-order TCs exhibit the nucleation, evolution, and annihilation of positive and negative vortex pairs during propagation. The vortex intertwined, and helical dance of vortices evoke a visual effect, resembling multiple pairs of vortex dancing. The observed vortex dynamics are explained physically from the hydrodynamics of light fluids. Notably, the intervortex distance between the vortex pair at the initial plane serves as a control parameter for orchestrating vortex dancing. This vortex dancing, in turn, emerges as the primary interaction driving the prolongation of the annihilation process for vortices with opposite TCs. These results underscore the distinctiveness of vortex dynamics in PPVPBs compared to CAVPBs. Our findings offer deeper insights into vortex interactions and hold potential applications in optical micromanipulation and the transportation of optical vortex information.
MATERIALS AND METHODS
Evolutions of optical fields in paraxial systems Here, we employ theory of light diffraction in the paraxial approximation. The evolution of PPVPBs through a linear ABCD optical system, such as free space or a lens system, can be theoretically predicted from the Collins formula (65, 66)
,( zyxE ),
exp( Bi
) ikz
),( vuE
exp
ik 2 B
([ uA
(2)
) yv
( xD
dd)] vu
(2)
where A, B, and D denote the elements of the ray transfer matrix
for a linear
optical system, is the wave number, is the wavelength, and z is the propagation distance along the propagation axis. The light field of PPVPBs at the output plane (i.e., the observation plane) is obtained by substituting Eq. 1 into Eq. 2. In free
/2 k
where z denotes the propagation distance in free space. For the beam propagation in a 2- with f denoting the focal length of the 2-f lens system, z as the propagation distance from the input to the output plane, and the lens located at z = f. In the 2-f lens system, it is well known that Eq. 2 becomes the two-dimensional (2D) Fourier transformation, since both A and D are equal to zero and B=f at the back focal position of the 2-f lens system. This 2D Fourier transformation is similar to the Fraunhofer diffraction equation of light at the far-field region of free space (50). The purpose of using the 2-f lens system here is to conveniently investigate the far-field behavior. To visually represent the intensity evolution of the light fields, Eq. 2 provides a theoretical basis.
space propagation, the ray transfer matrix is represented as (52)
(1
z
f lens system, the ray transfer matrix is expressed as (66)
1 z
Methods of achieving the phase distributions and phase singularities Accurate positioning of vortices in PPVPBs requires the phase information of optical fields. According to the Collins formula (i.e., Eq. 2) and the ray transfer matrix, one can theoretically obtain the phase distributions of PPVPBs in free space or the 2-f lens system. On that basis, one can attain the theoretical locations of vortices for the PPVPBs during propagation by using the vortex location algorithm in Ref. (68), which is based
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on the high-frequency characteristics of vortices. Thus, theoretical vortex positions can be achieved by using the matrix-optics theory and the vortex search approach. In experiments, accurate measurement of vortices is more complex, since we can only directly record the intensity distribution of the light field, rather than the phase distribution. Although it is possible to determine the vortex location through the dark region of light intensity, the accuracy of this method is relatively low compared to the vortex search algorithm based on phase information (68). In order to locate accurately the position of vortices in the experiment, we should firstly attain the experimental phase information of the light field. To reconstruct the experimental phase distribution of PPVPBs, we can use phase recovery methods (67). The principle of the phase recovery method in Ref. (67) is mainly based on the Fourier-transform of interference patterns. On the basis of the reconstructed phase distributions and the vortex search algorithm in Ref. (68), we can obtain the experimental vortex locations of PPVPBs.
Supplementary Materials This PDF file includes: Supplementary Text Figs. S1 to S18 Legends for movies S1 to S4
Other Supplementary Material for this manuscript includes the following: Movies S1, S2, S3 and S4 (please download from: https://github.com/wangligangZJU/video )
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Acknowledgments: L.-G. W. thanks Prof. C. T. Chan and Prof. Zhaoqing Zhang at Hong Kong University of Science and Technology for valuable discussions.
Funding: This work was supported by the National Natural Science Foundation of China (nos. 62375241 and 11974309).
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Author contributions: Conceptualization: L.G.W. Investigation: D.L., L.C., L.G.W. Formal analysis: D.L, L.C., L.G.W. Visualization: D.L., L.C. Validation: D.L., L.G.W. Writing-original draft: D.L. Writing-review & editing: D.L., L.C., L.G.W.
Competing interests: The authors declare that they have no competing interests.
Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.
Page 15 of 15
Supplementary Materials for Observation of vortex-pair dance and oscillation
Dadong Liu et al.
Corresponding author: Li-Gang Wang, [email protected]
This PDF file includes: Supplementary Text Figs. S1 to S18 Legends for movies S1 to S4
Other Supplementary Material for this manuscript includes the following:
Movies S1, S2, S3 and S4 (please download from: https://github.com/wangligangZJU/video )
A. Orbital angular momentum (OAM) spectra of pure-phase vortex-pair beams (PPVPBs) and complex-amplitude vortex-pair beams (CAVPBs) at the initial plane
The helical harmonic exp(il ) is the eigenfunction of orbital angular momentum (OAM), where l is the topological charge (TC), and denotes the azimuthal coordinate (70). A light field E(r, ,z) can be expanded through helical harmonics exp(il ) as (71, 72)
where the expansion coefficients
zral ),(
rE ,(
z ),
2 are given by
zra ),(
exp(
,) il
(S1)
zral ),(
1 2
2
rE ,(
z ),
exp(
il .d)
(S2)
The intensity of the l-th order helical harmonic, which is usually independent of the parameter z, can be expressed as
zra ),(
rr .d
(S3)
Thus, the intensity weight of such helical harmonic is determined by
R l
C
(S4)
which can be seen as the OAM spectra or mode purities of the light field E(r, ,z) (73, 74).
The light field of the PPVPBs at the initial plane is given by
vu ),(
vuF ),(
PPVPB
exp
2 w 0
(S5)
with
vuF ),(
two vortices are located at
located at
0 vu , ( 0
and
2
) ( 0u )0, u 0 v , 0
m 1
and
( u 0u
0 )0,
2
, where m1 and m2 are the TC values for vortices. The
, respectively. In general, these two vortices can be, respectively,
. Due to the rotational symmetry, one can make any angle of rotation to align
these two vortices along the u axis. Therefore, for the sake of simplicity, we only consider the situation of two vortices aligned along the u axis without loss of generality.
The light field of the CAVPBs at the initial plane is expressed as
with
vuM ),(
uu
i sgn(
CAVPB
m | | vm ]) 1 1
uu
exp for CAVPBs with equal or opposite TCs respectively
,
v 2 w 0
(S6)
vuMvu ),( ),(
m | vm ]) 2 2
i sgn(
(25).
Clearly, the amplitude of M(u, v) in CAVPBs is not normalized, while the amplitude of F(u, v) in PPVPBs is normalized. From Eq. (S6), the two vortices in the CAVPBs undergo not only the phase modulation but also the amplitude modulation. There are a series of prior investigations (25, 26, 32-49) on such fields of CAVPBs, demonstrating the attraction or repulsion of two vortices. However, it is hard to know the role of pure phase-only modulation on the dynamics of vortices. In Eq. (S5), for the fields of PPVPBs, it can be rewritten as
vuF ),(
im e
im 11
22
with the two local azimuthal angles
1
arctan(
v uu
and
2
arctan(
v uu
centered at
. These two vortices suffer purely phase modulations. Under this model, one can explore the role of two pure
( 0u phase-modulated vortices on their dynamic behavior. One may think that Eqs. (S5) and (S6) look to be similar each other, but this study shows that the different forms of these two kinds of vortex pairs result in distinct behaviors in their dynamics. For example, the helical and intertwined behaviors exhibited in the dynamics of PPVPBs, resembling a specific dance of vortex pairs, have never noticed in the previous studies on CAVPBs. Meanwhile, the impact of the host Gaussian beam on PPVPBs and CAVPBs is provided in Section I.
)0,
It is not difficult to rewrite, respectively, both Eq. (1) (or Eq. (S5)) and Eq. (S6) into the cylindrical coordinate
system as
PPVPB
r ,(
)
rF ,(
)
exp
r 2 w 0
(S7)
exp
r 2 w 0
im e 11
22
and
CAVPB
r ,(
)
rM ,(
)
exp
r 2 w 0
(S8)
where
1
arctan(
v uu
azimuthal angles centered at
sin r cos r and )0,
arctan(
( 0u
2 0
u
0 0u
m 1 2
m 2 2
exp sin
e
r 2 w 0
2 0
11
22
ru 2 0
ru 2 0
cos
cos
)
)
u
r cos
v uu
and
are the two local
2
arctan(
arctan(
r , respectively. We can drop the variable z in Eqs. (S1)-(S3) since we
)0,
only consider the initial fields. According to Eq. (S2), from the mathematical point of view, the expansion coefficient and )(ral
for the helical harmonic
is strongly dependent on their initial field expressions
exp( il )
E )(ral
PPVPB r ,( ) is real for the
. We have confirmed via the numerical calculations that the expansion coefficient
CAVPB r ,( )
E initial fields of both Eqs. (S7) and (S8).
Fig. S1. The expansion coefficients al(r) of helical harmonics exp(il ) within 4 l 4 for both PPVPBs and CAVPBs. (A and B) The PPVPBs with m1 = m2 = 1 and m1 = -m2 = 1 and (C and D) the CAVPBs with m1 = m2 = 1 and m1 = -m2 = 1. Note that in (A), 2 l the odd helical harmonics disappear since al(r)=0 for respectively; in (C), only two helical harmonics exist and other modes are zero; in (D), only three helical modes appear and and other modes are absent. Here we emphasize that there are other higher order modes for the PPVPBs since the range of modal orders is limited within
; in (B), the even helical harmonics overlap each other for
3,1 l
,4 l
for illustration. The parameters are 0 = 0.4 and w0 = 1.63 mm.
In Fig. S1, we plot the distributions of different
)(ral
within
for both PPVPBs and CAVPBs in
mm , see Fig. S1A, the odd modes have no contribution while the even modes have significant contributions
m
m 1
and
the cases of
. It demonstrates clearly that for the field of the PPVPB with
mm and constitute the field of the PPVPB; for the CAVPB with
mm
as shown in Fig. S1C, it is only composed
of two modes with
2,0 l
and all other modes are not present. In the case of
m 1
m
, all the modes have
contributions for the PPVPB while there are only the contributions from the three modes with
1,0 l
for the
CAVPB. Obviously, the additional amplitude modulation in
CAVPB r ,( ) but also strongly suppresses the contributions of other helical harmonics.
not only changes the distributions of
)(ral
Figure S2 shows OAM spectra of the PPVPBs and CAVPBs with equal TCs m1 = m2 = +1 under different relative off-axis distances 0 at the initial plane. As the value of 0 increases, the intensity weights of the zero-order OAM spectra for both PPVPBs and CAVPBs increase. However, for equal TCs +1, the OAM spectra of PPVPBs contains infinite even OAM modes, while CAVPBs consists of finite even OAM modes (i.e. the zero- and second-order OAM modes). To better understand the properties of OAM spectra of vortex pairs, we plot Fig. S3 to demonstrate the distributions of OAM spectra for PPVPBs and CAVPBs with different TCs. For high-order equal TCs, such as m1 = m2 = +2 and m1 = m2 = +3, the properties of the OAM spectra for PPVPBs and CAVPBs composed of infinite and finite even OAM modes, respectively, are similar to those of PPVPBs and CAVPBs with equal TCs +1. Interestingly, for opposite TCs, the distributions of OAM spectra for both PPVPBs and CAVPBs are symmetric about TC l = 0, but infinite OAM modes for PPVPBs and finite OAM modes for CAVPBs.
Through the detail analysis of theoretical calculations, we find that the fields of CAVPBs consist of finite helical in the cases of ), while the fields of PPVPBs always comprise a series of infinite helical harmonic
,2,0 2 0 m ,4,2,0
harmonic modes with
in the cases of
mmm 2
,2,1,0
, or
(here
mm m 1 modes with
,3,2,1,0
l
l
the cases of
mmm 2
, or
the cases of
Since the only difference between PPVPBs and CAVPBs is whether there is amplitude modulation or not, we argue that the mode difference between PPVPBs and CAVPBs is probably the main reason for inducing the different dynamics of PPVPBs from that of CAVPBs. However, there are still some mysterious and deep questions that remain unresolved, such as how amplitude modulation affects the OAM spectra of PPVPBs and CAVPBs, and how the OAM spectra correlate with vortex dynamics, which should be further investigated in the future.
m 1
mm
Fig. S2. OAM spectra for the PPVPBs and CAVPBs with equal TCs +1 under different 0 at the initial plane. The parameter is w0 = 1.63 mm.
Fig. S3. OAM spectra of the PPVPBs and CAVPBs with different TCs at the initial plane. The parameters are 0 = 0.4 and w0 = 1.63 mm.
Fig. S4. The evolution of vortex trajectories in the fields of the CAVPBs with equal and opposite TCs under different 0 upon
free space propagation. (A, B and C) m1 = m2 = 1 and (D, E and F) m1 = -m2 = 1, and the values of 0 are denoted in each subfigure.
The blue and red lines denote, respectively, the trajectories of positive and negative vortices, and their projections in the x-y plane are
presented by the green lines. The parameter is w0 = 1.63 mm.
B. The difference between PPVPBs and CAVPBs about intensity and vortex evolutions in free space
In Fig. S4, it demonstrates the evolution of two vortices in the fields of CAVPBs. As in most of previous studies (25, 32, 35, 36, 38), the dynamics of the vortex pair in the fields of CAVPBs are simple for both the cases of
mm
and
m 1
m
, compared to the dynamics of PPVPBs in Fig. 2. For CAVPBs, in the case of
mm
intervortex distance between two vortices increases as propagation, demonstrating the repulsion process. While in the , the intervortex distance between two vortices can decrease and it shows the attraction situation of
m 1
m
process when the initial distance is smaller than a critical value, and contrary when the initial distance is larger than the critical value the intervortex distance can also increase so that it shows the repulsion process. Meanwhile, we also plot for Fig. S5 to demonstrate the similar evolutions of vortex pairs in the cases of
mm
and
m 1
m
the fields of CAVPBs. However, in the fields of CAVPBs, from both Fig.S4 and Fig. S5, we can see that there are no fast helical and intertwined behaviors exhibited by the interaction between vortices. While, in Figs. 2 and 3 in our manuscript, in the fields of PPVPBs, such fast helical and intertwined behaviors resemble an interesting dance of vortex pairs and are very common. Such helical and intertwined behaviors induce the oscillation of the intervortex distance of the vortex pair in the dynamics of PPVPBs.
Fig. S5. The evolution of vortex trajectories for the high-order CAVPBs in free space. (A) m1 = m2 = 2, and (B) m1 = -m2 = 2.
The blue and red lines denote, respectively, the evolution of positive and negative vortices, and their projections in the x-y plane are
presented by the green lines. Other parameters are 0 = 0.4 and w0 = 1.63 mm.
In Movie S1, we present typical intensity evolutions of the PPVPBs and CAVPBs with unit TCs at different propagation distances z in free space. For the same TC +1, the intensity distributions of the PPVPBs and CAVPBs are centrosymmetric, and the intensity patterns gradually rotate anticlockwise as the value of z increases, see Movie S1(A and B). However, there are some differences between the intensity evolutions of the PPVPBs and those of the CAVPBs. For the PPVPBs with the same TC +1, as z increases, the central light intensity of the beam gradually becomes strong and an intensity peak appears, with a long rod intensity structure in the central area, see Movie S1A. For the CAVPBs with the same TC +1, with the increase of z, the intensity distributions of the beam remain unchanged, besides the anticlockwise rotation of the intensity pattern, as shown in Movie S1B. A vortex possesses a dark core at the center, where the intensity is zero (4, 5). Therefore, the dynamical behaviors of vortices can be inferred from the evolution of dark regions in the light field. For the same TC +1, the two positive vortices in the PPVPBs and CAVPBs all rotate anticlockwise as a whole, as demonstrated in Movie S1(A and B). Interestingly, in the light field of the PPVPBs with the same TC +1, the positive vortex itself rotates clockwise, except for the anticlockwise rotation behavior of the vortex pair, as clearly shown in Movie S1A. For opposite TCs 1, the intensity patterns of the PPVPBs and CAVPBs are horizontally symmetric, with a vertical line of symmetry, see Movie S1(C and D). Similarly, the positions for that pair of positive and negative vortices in the PPVPBs and CAVPBs with opposite TCs 1 are horizontally symmetric, and this symmetry property is independent of z, as displayed in Movie S1(C and D). From Movie S1(C and D), it can be observed that the vortices in the PPVPBs behave in a very different way from those in the CAVPBs. For the PPVPBs with 1 TCs, the positive vortex itself rotates clockwise, and the negative vortex itself rotates anticlockwise. While for the CAVPBs with 1 TCs, the rotation behaviors of vortices cannot be seen, and vortices of 1 TCs in this case only approach each other.
Movie S2 demonstrates typical intensity evolutions of the PPVPBs and CAVPBs with equal TCs +2 and opposite TCs 2 under different propagation distances z in free space. Similar to the PPVPBs and CAVPBs with unit TCs, the
, the
centrosymmetric and anticlockwise rotation properties or the horizontally symmetric property of the intensity patterns can be also found in the PPVPBs and CAVPBs with equal TCs +2 or opposite TCs 2. In Movie S2(A and B), the split of high-order vortices, the rotation of vortices themselves, and the birth and annihilation of vortices, which cannot be observed in the CAVPBs with equal TCs +2, can be clearly seen in the PPVPBs with equal TCs +2. In Movie S2D, the split of high-order vortices, and the phenomenon of opposite TCs vortices attracting each other can be seen in the CAVPBs with opposite TCs 2. Note that the rotation of vortices themselves, which cannot be found in the CAVPBs with opposite TCs, can be observed in the PPVPBs with opposite TCs, as shown in Movie S2C. These results imply that the vortex behaviors in the PPVPBs are quite different from those in the CAVPBs.
Movie S1. Typical intensity evolutions of the PPVPBs and CAVPBs under different propagation distances z in free space. (A) The PPVPBs with equal TCs +1, (B) the CAVPBs with equal TCs +1, (C) the PPVPBs with opposite TCs 1, and (D) the CAVPBs with opposite TCs 1. The left and middle columns are in normal and logarithmic scales, respectively. The regions marked by the black-dashed square boxes in the middle column are magnified in the corresponding figures of the right column. All the intensities are normalized. The parameters are 0 = 0.4 and w0 = 1.63 mm. The white scale bar in the left-top subfigure denotes 1.5 mm. (see https://github.com/wangligangZJU/video/blob/main/MovieS1.gif )
Movie S2. Typical intensity evolutions of the high-order PPVPBs and CAVPBs at different propagation distances z in free space. (A) The PPVPBs with equal TCs +2, (B) the CAVPBs with equal TCs +2, (C) the PPVPBs with opposite TCs 2, and (D) the CAVPBs with opposite TCs 2. The left and middle columns are in normal and logarithmic scales, respectively. The regions marked by the black-dashed square boxes in the middle column are magnified in the corresponding figures of the right column. All the intensities are normalized. The other parameters are same as those in Movie S1. The white scale bar represents 1.5 mm. (see https://github.com/wangligangZJU/video/blob/main/MovieS2.gif )
C. Hydrodynamics explanation for the motions of vortices in PPVPBs and CAVPBs upon free space propagation
Here we explain the different dynamics of vortices in PPVPBs and CAVPBs with the optical hydrodynamic picture. Using the Madelung transform on optical fields, the paraxial Helmholtz equation becomes into two coupled equations transformation (75,
76).
Following
the Madelung
assuming
the
electric
field
zyxE ), ,(
zyxA ), ,(
i exp[
zyx , ,(
with amplitude
zyxA
0),
and phase
zyx ,( ),
and substituting it into the
scalar paraxial wave equation
2
kiE 2
E z
free space, one obtains
z
1 k 2
2 A 2 kA
and
2 A z
( 2 A )
0
, where
and
are the symbols of the transverse Laplace and gradient operators,
is the transverse velocity field of the whole beam (it is analogous to the velocity field of a fluid), and
2AI
is the intensity of an optical field (like the density of a fluid). These two equations can be seen as the optical Bernoulli and the optical continuity equations. One can prove that the transverse velocity field in the Madelung picture is similar to the transverse Poynting vector for the linear polarization of light (76). From the hydrodynamic point of view, the dynamics of optical vortices is driven by two important terms: the phase gradient (i.e., the velocity field) and the intensity gradient (or the amplitude gradient). Meanwhile, one can also define the quantum potential Q and the
quantum force F
for optical fields as follows,
2 A 2 kA
and
F
. One has used these concepts to
successfully explain optical properties of Airy beams (77).
According to the recent studies (62-64), researchers have employed the two-dimensional hydrodynamics model to physically explain the motion of vortices, the splitting of higher-charge optical vortices, and the dynamics of vortices in complex-amplitude modulation vortex pairs. Here, we employ the same argument to explain the vortex dynamics,
especially in PPVPBs, which have never considered before. According to the hydrodynamic model of an optical fluid
(62, 64), the initial total light field,
zvuEi ,(
, consisting of multiple vortices embedded in a host Gaussian beam,
can be separated into a product of two fields:
zvuE ,(
i bg
i t
vuEvuE ),(
),(
, where
vuE i ),(
is the initial tested
field of one vortex under consideration, and
E i bg
vu ),(
is the initial background field for the rest field that comprises the
fields of other vortices and the host beam. In general, the total light field
zyxE ), ,(
at any propagation distance z cannot
be written as a product form, since the propagation of the background field at any propagation distance will couple to the propagation of the test field at z. Following the procedure of Ref. (62), here we use the evolution of the initial
background field in a paraxial system approximating as the background field
Ebg
zyx ,( ),
at any propagation distance
z. As stated in Ref. (62), this approximation is reasonable at early propagation stages where the vortices are expected to
be mostly circular. Thus, one can also write
zyx ,( ),
zyx ,( ),
i exp[ bg
zyx , ,(
with
Abg
zyx ,( ),
the
background amplitude and
bg
zyx ,( ),
the background phase, and the evolution of the background field obeys the
paraxial diffraction equation (66). For example, in the cases of PPVPBs with m1=1 and m2=-1, the initial background
field is given by
E i bg
vu ),(
exp
2 w 0
u (
1
, and in the cases of PPVPBs with m1=2 and m2=-2,
the initial field of the background field is consisting of one -2 TC vortex on the left side and one +1 TC vortex on the by right
side
that
the
expression
for
the
initial
background
field
now
given
E i bg
vu ),(
exp
2 w 0
u (
. From the paraxial diffraction equation, one can
obtain the evolution distribution of the background field
Ebg
zyx ), ,(
at the propagation distance z. The transverse
velocity field from the background field acting on a positive unit-charge vortex in the right side can be calculated by (62)
k
, where k is the wavenumber of the laser field and k
is the wavevector. Here k
bg
ln
k
simply taken as
k
zk
and this approximation is valid only when the transverse wavevector can be neglected.
In Movie S3, we show the numerical motions of the right-side tested vortices in the background velocity fields. At mm), in both the cases of PPVPBs with (A) m1=m2=1 and (B) m1=-m2=1, the tested vortex on the right side surfs in the diffraction waves from the left-side vortex, and the velocity fields (i.e, the circulation flow) in the background optical fluid drive quickly the local rotation and oscillation of the tested vortex during propagation and the vortex trajectories are well matched with the transverse velocity fields. At longer propagation distances, there appear some deviation between the vortex trajectories and velocity fields since we have not included the vortex tilt effect as described in Ref. (63), which involves the more complex calculations. However, since the oscillation and helical phenomena of vortices in PPVPBs mainly happen at short propagation distances, thus the
short propagation distances (roughly
1000
velocity field by
k
bg
k
ln
at the vortex locations is enough to explain the current vortex behavior.
After the diffraction wave from the left-side vortex dissipates or becomes stable, the vortex motion will back to simply
repel or attract each other. Thus, such helical and oscillation behaviors increase the repulsion effect or delay the annihilation process. As a comparison, in Movie S3(C and D) for the corresponding CAVPBs, the background velocity fields are very simple, so that there are no oscillation and helical effects in vortex motions at short propagation distances and their trajectories at far-field regions are well explained recently in Ref. (63) by including the vortex tilt effect.
In Movie S4, we further show the numerical motions of the right-side tested vortices in the background velocity fields. There are two +1 TC vortices coincidently located at the same initial position, these two vortices form a higher TC vortex with a +2 charge. When they split to multiple unit-charge vortices, we should plot out the locations of all these vortices since their initial conditions are the same as identical particles . From Movie S4, in the cases of PPVPBs with (A) m1=m2=2 and (B) m1=-m2=2, there are possibly four vortices co-existing on the right-side region due to the vortex nucleation and vortex annihilation phenomena. As demonstrated in Movie S4, at short propagation distances, all the trajectories for positive vortices are considerably well matched with the transverse velocity field. The presence of the circulation flow from the right-side positive vortex further alters the local background velocity field near the right-side vortex, compared with Movie S3(A and B). Under the diffraction ripples of the background field during propagation, the tested vortices experience not only the helical and oscillating motions but also the vortex nucleation and vortex annihilation phenomena. Thus, such helical and oscillation motions in PPVPBs with opposite TCs further delay or slow down the merger process, which is reflected in smaller critical values of occurring the vortex-pair annihilation at the far-field region (or the focal plane) in Table 1. In the cases of CAVPBs, the velocity field of the background field has no ripples, so that no helical and complex motions happen there. In the case of the CAVPB with m1=m2=2, the motion of the tested vortex is coincident with the motion of the background vortex on the right side. While in the case of the CAVPB with m1=-m2=2, one of the vortices on the right side moves very slowly, and another one moves quickly along the direction of the velocity field and it will annihilate with the opposite vortex from the left side. These results for the CAVPBs agree with those in Ref. (64). Note that in our calculations, there is also deviation in the directions of the velocity field during the vortex nucleation or vortex annihilation, where the vortex tilt should be included (63) (the current calculation is already well explained the observed phenomena), and also the velocity field in is given by Movie S4
k
bg
is not suitable for generated negative TC vortices because k
for the negative-charge tested vortices.
ln
A bg
the velocity field
In turn, one can also take the initial background field as a combination of the right-side positive unit-charge vortex and host Gaussian beam to investigate the motion of the left-side negative unit-charge vortex, or take the background field consisting of the right-side +2 TC vortex and one left-side -1 TC vortex embedded in the host beam for searching the dynamics of the left-side negative vortices. Therefore, the interesting vortex dancing and oscillation in PPVPBs can be well explained in the optical hydrodynamical picture and the interactions among vortices can be presented through the circulation flow of each background field other than the tested vortex.
Movie S3. Motions of right-side tested vortices in the background velocity fields of PPVPBs and CAVPBs under different propagation distances z in free space. (A) PPVPBs with m1 = m2 = 1, (B) PPVPBs with m1 = -m2 = 1, (C) CAVPBs with m1 = m2 = 1, and (D) CAVPBs with m1 = -m2 = 1. The green dots denote the locations of positive vortices, and the red arrows denote the velocity fields of the background fields. Brightness is the light intensity of background fields. All the intensities are normalized. The parameters are 0 = 0.4 and w0 = 1.63 mm. (See https://github.com/wangligangZJU/video/blob/main/MovieS3.gif )
Movie S4. Motions of right-side tested vortices in the background velocity fields of high-order PPVPBs and CAVPBs under different propagation distances z in free space. (A) PPVPBs with m1 = m2 = 2, (B) PPVPBs with m1 = -m2 = 2, (C) CAVPBs with m1 = m2 = 2, and (D) CAVPBs with m1 = -m2 = 2. The green and yellow dots denote, respectively, the locations of positive and negative vortices. The red arrows denote the velocity fields of the background fields. Note that the velocity field in the empty area is not shown for better displaying the velocity field in other area since it is divergent near the right-side vortex contained in the background field. Brightness is the light intensity of background fields. All the intensities are normalized. The other parameters are same as those in Movie S3. (See https://github.com/wangligangZJU/video/blob/main/MovieS4.gif )
D. The evolution of intervortex distance with different values of 0 for the PPVPBs and CAVPBs with unit vortices in free space
Fig. S6(A and B) theoretically shows the full picture on the evolution of the intervortex distance d between vortices in PPVPBs under different 0 in free space. In Fig. S6(A and B), when m1 = m2 = 1, in the near-field region, the amplitude and intervortex distance of the vortex oscillation increase with the increase of 0, and the amplitude of the vortex dance also increases as the propagation distance z increases. In the far-field region, there are no vortex oscillation phenomena and the relative intervortex distances between the vortices tend to be stable, comparing with the expansion of the beam width w(z). For m1 = m2 = 1, there is a fluctuation in the intervortex distances in the near field, and then the vortex spacing gradually increases with the increase of the propagation distance, and the vortex spacing gradually stabilizes in the far field, as shown in Fig. S6A. The similar dynamical behaviors can be also found in vortices of opposite TCs, as demonstrated in Fig. S6B. Interestingly, for the fields of PPVPBs with opposite TCs m1 = -m2 = 1, the intervortex spacing can not only oscillate but also decrease and finally becomes zero within one Rayleigh length under these small values of 0 (such as 0=0.2 and 0=0.3), see the black and red lines in Fig. S6B, indicating the vortex helical, intertwined behaviors and the vortex annihilation of positive and negative vortices in PPVPBs with opposite TCs. While in the cases of 0 = 0.4, 0 = 0.5 and 0 = 0.6, the positive and negative vortices survive in the far-field region and always remain the non-zero vortex spacing, see the blue, green and purple lines in Fig. S6B. These results mean that there is a critical value of 0 between 0.3 and 0.4, at which the opposite TCs vortices can annihilate each other in the far field.
For the detail comparison, both Fig. S6C and Fig. S6D present the intervortex distances for vortices in CAVPBs upon free space propagation. For the CAVPBs with m1 = m2 = 1, the relative distance d/ w(z) between the two vortices maintains invariant under different values of 0 as seen in Fig. S6C, although the absolute value of d increases as the host beam width w(z) increases. When m1 = -m2 = 1, the two vortices in the CAVPBs may approach each other, leading to the gradually decreasing vortex spacing as shown in Fig. S6D. In this case, if the value of 0 is small enough (such as 0=0.2, 0.3 and 0.4), the vortex spacing can be reduced to zero over a finite distance, indicating the attraction process of two vortices. From Fig. S6D, one can also find that when 0 = 0.5, the intervortex distances between two vortices will tend to be zero as z goes to the infinity of free space. As the value of 0 is larger than 0.5, the absolute value of d increases since w(z) increases, and this also indicates the repulsion process of two vortices. However, comparing Fig. S6(A and B) with Fig. S6(C and D), one can see an essential difference that the value of d/w(z) has no oscillation effect for the cases of CAVPBs, and the oscillating behaviors in the near-field regions tells us the rich interaction between two vortices in PPVPBs that is absent in CAVPBs. Through a careful comparison between Fig. S6A and Fig. S6C, one can find that in the case of m1 = m2 = 1 and under the same 0, the vortex spacing in the far fields of PPVPBs is always greater than that of CAVPBs, which indicates the stronger repulsion process in the fields of PPVPBs. Comparing Fig. S6B and Fig. S6D, we also observe that for m1 = -m2 = 1 and under the same 0, the opposite vortices in PPVPBs can survive (i.e. keeping the non-zero vortex spacing) over longer distances than the corresponding cases in CAVPBs. Obviously, the oscillation and intertwined behaviors between two vortices significantly influence the process of vortex-pair annihilation and prolong the survival range of opposite vortices in PPVPBs.
Fig. S6. The intervortex distance d between the two vortices as a function of the propagation distance z under different relative off-axis distances 0 in free space. (A and B) The PPVPBs with m1 = m2 = 1 and m1 = -m2 = 1, respectively; (C and D) the CAVPBs with m1 = m2 = 1 and m1 = -m2 = 1, respectively. Note that the propagation distance z is normalized by the Rayleigh length zR of the host Gaussian beam and the value of the intervortex distance d is also rescaled by w(z) the beam width of the host beam at z.
E. Additional theoretical results of the vortex trajectories for the PPVPBs with m1 = m2 = 2 in free space
Figure S7 shows the trajectories of vortices for the PPVPBs with equal TCs m1 = m2 = 2 in the case of free space. From Fig. S7, it can be found that there are entanglement behaviors between four separate positive vortices, and these vortices do helicoidal motions with the increase of the propagation distance z. Interestingly, there are the generation and annihilation processes for pairs of positive and negative vortices in the fields of the PPVPBs upon free space propagation. As the value of 0 increases, the vortex entanglement phenomena, the vortex helical behaviors, and the generation and annihilation processes of vortices become obvious. Thus, the parameter 0 can be used to control the dynamics of the PPVPBs.
Fig. S7. Theoretical evolution of phase singularities in the fields of the PPVPBs with the same TC m1 = m2 = 2 upon free space propagation under different relative off-axis distance parameters. (A) 0 = 0.2, (B) 0 = 0.3, (C) 0 = 0.4, and (D) 0 = 0.5. The middle and the right subfigures are the enlarging parts and the projections in the x-z plane of the left subfigures, respectively. The blue and red lines denote, respectively, the trajectories of positive and negative vortices, and their projections in the x-y plane are presented by the green lines.
F. Theoretical evolution of vortices for the PPVPBs with m1 = -m2 = 2 upon free space propagation
Figure S8 demonstrates the vortex trajectories for the PPVPBs with opposite TCs m1 = -m2 = 2 in free space. It is observed that there are two pairs of vortices with 1 TCs, and their helical behaviors and entanglement phenomena become significant with the increase of the relative off-axis distance 0. Therefore, the relative off-axis distance parameter can be used as a control parameter for controlling the dynamical behaviors of the PPVPBs. In addition, when the parameter 0 is small enough, such as 0 = 0.2, there are annihilation phenomena between one pair of positive and negative unit vortices related to the initial light field in near-field regions, as displayed in Fig. S8A. It can be inferred that a smaller 0 can contribute to the annihilation behaviors of opposite TCs vortices.
Fig. S8. Theoretical trajectories of phase singularities for the PPVPBs with opposite TCs m1 = -m2 = 2 in free space. The other explanations and parameters are same as those in Fig. S7.
G. The vortex trajectories of the CAVPBs with different opposite TCs and relative off-axis distances in a 2-f lens system
Fig. S9. Evolution of vortex trajectories in different-order CAVPBs with opposite TCs in the 2-f focusing system. (A) The influence of the relative off-axis distance 0 on the vortex-trajectory evolution, where the opposite vortices happen to annihilate each other at the focusing plane when 0 = 0.5. (B, C, and D) The vortex-trajectory evolutions at various critical values of 0 in different-order CAVPBs, in which each plot corresponds to the situation that one pair of opposite vortices annihilate each other at the focusing plane. Here, the focal length of the 2-f focusing system is f = 500 mm and the beam parameter is also taken to be w0 = 1.63 mm.
Figure S9 shows the annihilation processes of vortices for the CAVPBs with opposite TCs in a 2-f lens system with the focal length f = 500 mm. For the CAVPBs with m1 = -m2 = 1, when the relative off-axis distance ( 0) is big enough, such as 0 = 0.800, the vortex-pair can survive to the back focal plane of the 2-f lens system, as displayed in Fig. S9A. As 0 decreases, the vortex-pair tends to meet and annihilate. When 0 = 0.500, the vortex pair undergoes annihilation upon reaching the back focal plane. If 0 = 0.200, the vortex-pair will annihilate prior to reaching the back focal plane. These results indicate that 0 = 0.500 is the critical value of the vortex-pair annihilation in CAVPBs for m1 = -m2 = 1: when 0 < 0.500, the annihilation behavior of the vortex-pair can be seen; while 0 > 0.500, the vortex-pair will survive all the way to the back focal plane or the far field. Due to the unstable properties of high-order vortices in propagation, for m1 = -m2 = 2, 3 and 4, the pair of vortices with 2, 3 and 4 TCs at the initial light field for the CAVPBs will split into two, three and four pairs of opposite unit vortices upon propagation, respectively. Therefore, as demonstrated in Fig. S9(B, C and D), there are two, three and four critical values about 0 of the vortex-pair annihilation for m1 = -m2 = 2, 3 and 4, respectively. From Fig. S9, it is worth noting that the vortex-pair dancing behaviors, which can be found in the PPVPBs as clearly demonstrated in Fig. 4, have never been observed in the CAVPBs.
H. The change of the intervortex distance for the fields of PPVPBs with m1 = -m2 = 1 in free space under different relative off-axis distances 0, which correspond to the different situations in Fig. 4A.
Fig. S10. The intervortex distance d between the two vortices as a function of the propagation distance z under different relative off-axis distances 0 in free space for the fields of PPVPBs with m1 = -m2 = 1. Note that the propagation distance z is rescaled by the Rayleigh length zR of the host Gaussian beam and the value of d is rescaled by w(z), which is the beam width of the host beam at z.
Here we illustrate the similarity of light propagations in the 2-f lens system and the far-field region of free space. For the PPVPBs with m1 = -m2 = 1, when 0 = 0.55, the positive and negative vortices undergo oscillations and survive at the back focal plane, see the leftmost subfigure in Fig. 4A. This feature is consistent with the non-zero intervortex spacing in free space, see the blue line in Fig. S10. When 0 = 0.358, the positive and negative vortices coincidentally merge and annihilate each other at the back focal plane, see the middle subgraph of Fig. 4A, and this predicts that the intervortex spacing d tends to be zero at the infinity of free space as shown by the red line in Fig. S10, or it indicates that the vortex pair will merge together at the infinity of free space. When 0 =0.150, in this situation the vortex-pair can merge prior to reaching the focal plane, as seen in the rightmost subgraph of Fig. 4A, and this shows that the two vortices with opposite TCs merge and annihilate each other at a certain propagation distance in free space and the separation distance between the vortices goes to be zero at a certain distance as demonstrated by the black line in Fig. S10. Thus, the dynamics of vortices in the 2-f focusing system can effectively and conveniently demonstrate the key characteristics of vortex interactions among vortex pairs within the limited propagation distance.
I. The impact of different widths of the host Gaussian beam on the evolutions of a single off-axial pure-phase vortex, a single off-axial complex-amplitude vortex, PPVPBs, and CAVPBs
To demonstrate the amplitude modulation of the host Gaussian beam on PPVPBs and CAVPBs, we can first investigate the Gaussian modulation of the host beam on a single pure-phase vortex and a single complex-amplitude vortex for the sake of simplicity. In this situation, we can set m2 to zero and take different values of m1 in Eqs. (S5) and (S6). Figures S11-S14 show the typical intensity and phase evolutions and the distance from the origin for a single vortex in free space. For the case of a single unit vortex embedded in a host Gaussian beam, as the propagation distance increases, the vortex rotates around the origin of the transverse plane. Interestingly, a single positive vortex rotates counterclockwise, while a single negative vortex rotates clockwise, with a rotation angle of arctan(z/zR) and a distance
of 0w(z) from the origin, as shown in Figs. S11 and S12. Here,
zw )(
w 0
/(1
Rzz
is the host beam width at the
propagation distance z. When we read out the coordinate data of these vortex locations, we find that these vortex
locations in fact move along the y direction (i.e., the straight line of
x
04.0 w
in these cases (25, 78)). There is the
same law for a single high-order vortex in both pure-phase and complex-complex cases, see Fig. S13. This can be explained since both a single pure-phase vortex and a single complex-amplitude vortex are affected only by the same background field (i.e., the same host Gaussian beam). From the perspective of vortex dynamics, the behaviors of a single pure-phase vortex and a single complex-amplitude vortex are similar, and there is no oscillation in the distance between the origin and the vortex position, see Fig. S12. In the case of a single high-order vortex, such as +2 or -2 TC,
the rotation properties of the intensity and phase patterns, and the distance between the origin and the vortex location of the single pure-phase vortex are also similar to those of the single complex-amplitude vortex, as illustrated in Figs. S13 and S14, which are consistent with the single unit vortex. These results indicate that the host Gaussian beam plays the same role in controlling the dynamics of a single embedded vortex for both pure-phase and complex-amplitude cases.
However, for the evolution of light intensity in Figs. S11 and S13, they are different for the pure-phase and complex-amplitude cases. Considering the expansion of the host beam width w(z), it can be observed that the light intensity profile of a single pure-phase vortex not only rotates but also diffracts/spreads out during propagation, while for a single complex-amplitude vortex it rotates rigidly and does not diffract/spread out, see Figs. S11 and S13. The larger the TC values, the more evident the diffraction effect in the pure phase cases. Such diffraction patterns in the pure-phase vortex situation as the additional background fields will influence the vortex behaviors of other vortices in the pure-phase vortex pairs.
To better reveal the amplitude modulation of the host Gaussian beam on the vortex dynamics, we can introduce an expansion parameter b of the Gaussian beam waist to adjust the initial light field of VPBs. In this situation, the light fields of PPVPBs and CAVPBs at the initial plane now can be given by
E b
PPVPB
vu ),(
vuF ),(
exp
bw ( 0
(S9)
and
E b
CAVPB
vuMvu ),( ),(
exp
bw ( 0
v 2
(S10)
where b is a positive real number and represents the amplification factor of the waist w0 for the host Gaussian beam. The larger the value of b, the more the Gaussian beam approximates a plane wave (i.e, its amplitude of the host beam is more approximately normalized for both kinds of VPBs). Figures S15-S18 present the vortex trajectories and intervortex distance for PPVPBs and CAVPBs under different b in free space. In Fig. S15, for PPVPBs with unit TCs, the vortex spacing slightly decreases as the value of b increases. A large b can contribute to the annihilation of opposite vortices, see Fig. S15F. From Fig. S15(B, C, E and F), it is found that the oscillation effect of the intervortex distance at short distances is nearly overlapped for different values of b. Clearly, the behaviour of vortex dynamics at short distances is dominated mainly due to the interaction of two vortices. The large difference at longer distances is due to the contribution of the host beam. When the value of 0 is small (such as 0 = 0.2 and 0.4), the evolution trajectory of vortex spacing under b = 10 almost completely overlaps with that under b = 5. There are also similar phenomena observed in Fig. S16 for CAVPBs with unit TCs. Based on these results, the host Gaussian beam with b = 10 can be regarded as a plane wave in the situations of small 0, and the amplitude of the host beam is approximately normalized. Interestingly, in Figs. S15-S18, with the increase of b, vortex oscillation and helical behaviours, which cannot be observed in the CAVPBs, can be clearly seen in the PPVPBs. From Figs. S15 and S17, we can see that the host Gaussian beam has less affected on the oscillation dynamics of pure-phase vortex pairs as the value of b increases. While in the cases of CAVPBs with equal TCs, the smaller b can drive the faster motion of vortices (see Fig. S16(A, B and C) and Fig. S18(A and B)); in contrast, in the cases of CAVPBs with opposite TCs, the smaller b slows down the annihilation process. In fact, the dynamics of vortex pair in the complex-amplitude cases has been physically explained from the optical hydrodynamics (63). According to the above discussion and Ref. (63), we can see that the host Gaussian beam play a limited role in controlling the dynamic behaviors of embedded vortices.
Fig. S11. Typical intensity (upper) and phase (lower) evolutions of a single unit vortex for both PPVPBs and CAVPBs under different propagation distances z in free space. (A) PPVPBs with m1 = 1, m2 = 0, (B) PPVPBs with m1 = 1, m2 = 0, (C) CAVPBs with m1 = 1, m2 = 0, and (D) CAVPBs with m1 = 1, m2 = 0. Here, the blue solid and dashed lines to indicate the azimuthal positions of the vortices at the current and previous positions, respectively, for better showing the rotation phenomenon of intensity and phase patterns at different z. The notation z on the top of each column indicates the same propagation distance, where zR denotes the Rayleigh length of the host Gaussian beam. Other parameters are 0 = 0.4 and w0 = 1.63 mm. The white scale bar represents 1 w(z), where w(z) is the beam width of the host beam at z. In the situations of z = 0, 0.1zR, 0.5zR, 1.0zR and 5.0zR, the vortex locations (x, y) in both (A) and (C) are (0.40w0, 0), (0.40w0, 0.04w0), (0.40w0, 0.20w0), (0.40w0, 0.40w0) and (0.40w0, 2.00w0), respectively. In the cases of z = 0, 0.1zR, 0.5zR, 1.0zR and 5.0zR, the vortex locations (x, y) in both (B) and (D) are (0.40w0, 0), (0.40w0, -0.04w0), (0.40w0, -0.20w0), (0.40w0, -0.40w0) and (0.40w0, -2.00w0), respectively.
Fig. S12. The distance d between the origin and the unit vortex location as a function of the propagation distance z under different relative off-axis distances 0 in free space. (A and B) The PPVPBs with m1 = 1, m2 = 0 and m1 = 1, m2 = 0, respectively; (C and D) the CAVPBs with m1 = 1, m2 = 0 and m1 = 1, m2 = 0, respectively. Note that the propagation distance z is normalized by the Rayleigh length zR of the host Gaussian beam and the value of the vortex distance d is also rescaled by w(z) the beam width of the host beam at z.
Fig. S13. Typical intensity (upper) and phase (lower) evolutions of a single high-order vortex for both PPVPBs and CAVPBs under different propagation distances z in free space. (A) PPVPBs with m1 = 2, m2 = 0, (B) PPVPBs with m1 = 2, m2 = 0, (C) CAVPBs with m1 = 2, m2 = 0, and (D) CAVPBs with m1 = 2, m2 = 0. Here, the blue solid and dashed lines to indicate the azimuthal positions of the vortices at the current and previous positions, respectively, for better showing the rotation phenomenon of intensity and phase patterns at different z. The notation z on the top of each column indicates the same propagation distance, where zR denotes the Rayleigh length of the host Gaussian beam. Other parameters are 0 = 0.4 and w0 = 1.63 mm. The white scale bar represents 1 w(z), where w(z) is the beam width of the host beam at z. In the situations of z = 0, 0.1zR, 0.5zR, 1.0zR and 5.0zR, the vortex locations (x, y) in both (A) and (C) are (0.40w0, 0), (0.40w0, 0.04w0), (0.40w0, 0.20w0), (0.40w0, 0.40w0) and (0.40w0, 2.00w0), respectively. In the cases of z = 0, 0.1zR, 0.5zR, 1.0zR and 5.0zR, the vortex locations (x, y) in both (B) and (D) are (0.40w0, 0), (0.40w0, -0.04w0), (0.40w0, -0.20w0), (0.40w0, -0.40w0) and (0.40w0, -2.00w0), respectively.
Fig. S14. The distance d between the origin and the high-order vortex location as a function of the propagation distance z under different relative off-axis distances 0 in free space. (A and B) The PPVPBs with m1 = 2, m2 = 0 and m1 = -2, m2 = 0, respectively; (C and D) the CAVPBs with m1 = 2, m2 = 0 and m1 = -2, m2 = 0, respectively. Note that the propagation distance z is normalized by the Rayleigh length zR of the host Gaussian beam and the value of the vortex distance d is also rescaled by w(z) the beam width of the host beam at z.
Fig. S15. The vortex trajectories and intervortex distance for PPVPBs with unit TCs under different 0 and b upon free space propagation. (A and D) The vortex trajectories for (A) m1 = m2 = 1, and (D) m1 = m2 = 1 with 0 = 0.4, b =10. The blue and red dots denote, respectively, the evolution of positive and negative vortices and their projections are shown by the green dots in the x-y planes. The corresponding thin solid lines are the cases of b =1. (B, C, E and F) Evolution of the intervortex distance along the propagation distance for (B) m1 = m2 = 1, 0 = 0.4, (C) m1 = m2 = 1, 0 = 0.2, (E) m1 = m2 = 1, 0 = 0.4, and (F) m1 = m2 = 1, 0 = 0.2, respectively, under different b. The parameter is w0 = 1.63 mm.
Fig. S16. The vortex trajectories and intervortex distance for CAVPBs with unit TCs under different 0 and b upon free space propagation. (A and D) The vortex trajectories for (A) m1 = m2 = 1, and (D) m1 = m2 = 1 with 0 = 0.4, b =10. The blue and red dots denote, respectively, the evolution of positive and negative vortices and their projections are shown by the green dots in the x-y planes. The corresponding thin solid lines are the cases of b =1. (B, C, E and F) Evolution of the intervortex distance along the propagation distance for (B) m1 = m2 = 1, 0 = 0.4, (C) m1 = m2 = 1, 0 = 0.2, (E) m1 = m2 = 1, 0 = 0.4, and (F) m1 = m2 = 1, 0 = 0.2, respectively, under different b. The parameter is w0 = 1.63 mm.
Fig. S17. The evolution of vortex trajectories in the fields of PPVPBs with high-order TCs under different b upon free space propagation. (A and B) m1 = m2 = 2 and (C and D) m1 = m2 = 2, and the values of b are denoted in each subfigure. The blue and red lines denote, respectively, the trajectories of positive and negative vortices, and their projections in the x-y plane are presented by the green lines. Other parameters are 0 = 0.4 and w0 = 1.63 mm.
Fig. S18. The evolution of vortex trajectories in the fields of CAVPBs with high-order TCs under different b upon free space propagation. (A and B) m1 = m2 = 2 and (C and D) m1 = -m2 = 2, and the values of b are denoted in each subfigure. The blue and red lines denote, respectively, the trajectories of positive and negative vortices, and their projections in the x-y plane are presented by the green lines. Other parameters are 0 = 0.4 and w0 = 1.63 mm.
| The article "Observation of vortex-pair dance and oscillation" by Dadong Liu, Lai Chen, and Li-Gang Wang explores the dynamics of vortex-pair beams (VPBs) in the context of optical vortices. The study investigates pure-phase vortex-pair beams (PPVPBs), revealing unique behaviors such as helical and intertwined motions, described as a "vortex-pair dance." The authors observe oscillation in the distance between vortices, which is influenced by the initial distance between them, and find that this interaction can extend the survival range of opposite vortices, analogous to the Fujiwhara effect observed in meteorology. The research emphasizes the difference between PPVPBs and complex-amplitude vortex-pair beams (CAVPBs), highlighting that the former exhibits richer dynamics, including vortex nucleation and annihilation phenomena. Experimental results confirm theoretical predictions, suggesting potential applications in optical manipulation and quantum information. The findings enhance the understanding of vortex interactions in various scientific fields. | vortex dynamics,fluid dynamics,optics,vortex-pair beams,helical behavior,oscillation,intervortex distance,physics |
PaliGemma 2: A Family of Versatile VLMs for Transfer | 2412.03555 | 4 2 0 2 c e D 4
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1 v 5 5 5 3 0 . 2 1 4 2 : v i X r a
December 2024
PaliGemma 2: A Family of Versatile VLMs for Transfer Andreas Steiner*, , Andr Susano Pinto*, Michael Tschannen*, Daniel Keysers, Xiao Wang, Yonatan Bitton, Alexey Gritsenko, Matthias Minderer, Anthony Sherbondy, Shangbang Long, Siyang Qin, Reeve Ingle, Emanuele Bugliarello, Sahar Kazemzadeh, Thomas Mesnard, Ibrahim Alabdulmohsin, Lucas Beyer and Xiaohua Zhai Google DeepMind, *Core team, Project lead
PaliGemma 2 is an upgrade of the PaliGemma open Vision-Language Model (VLM) based on the Gemma 2 family of language models. We combine the SigLIP-So400m vision encoder that was also used by PaliGemma with the whole range of Gemma 2 models, from the 2B one all the way up to the 27B model. We train these models at three resolutions (224px2, 448px2 and 896px2) in multiple stages to equip them with broad knowledge for transfer via fine-tuning. The resulting family of base models covering different model sizes and resolutions allows us to investigate factors impacting transfer performance (such as learning rate) and to analyze the interplay between the type of task, model size, and resolution. We further increase the number and breadth of transfer tasks beyond the scope of PaliGemma including different OCR-related tasks such as table structure recognition, molecular structure recognition, music score recognition, as well as long fine-grained captioning and radiography report generation, on which PaliGemma 2 obtains state-of-the-art results.
1. Introduction
PaliGemma [9] is a 3B vision-language model (VLM) for transfer combining the SigLIP [108] vision encoder and the 2B Gemma language model [21]. It matches the performance of much larger prior VLMs consisting of a range of different vision encoders and language models. We now upgrade PaliGemma by replacing its language model component with the more recent and more capable language models from the Gemma 2 fam- ily [22], producing new PaliGemma 2 base VLMs at 3 different sizes (3B, 10B, 28B) and 3 different resolutions (224px2, 448px2, 896px2). To equip these VLMs with broad capabilities we use the same 3-stage training recipe as PaliGemma. The resulting models are designed to be fine-tuned, and when evaluated on the 30+ transfer tasks considered in [9] (which include common cap- tioning and VQA tasks, and some video and re- ferring expression tasks), PaliGemma 2 slightly outperforms PaliGemma at the same resolution and model size, and obtains substantial improve- ments at larger model sizes. We release the PaliGemma 2 VLMs as open-weight models which can serve as drop-in replacement for PaliGemma.
Having a family of models at hand that are all derived from comparable building blocks and are trained according to the same recipe allows us to analyze the effect of model size and resolution on the downstream performance in a controlled setting (see Sec. 4.1). For example, while almost every task benefits from added compute, we iden- tify which transfer tasks benefit more from com- pute due to increased resolutions, and which from compute due to a larger, more capable language model. We also show that larger models tend to have a lower optimal transfer learning rate.
We also explore new tasks which were not ex- plored in depth in [9], including text detection and recognition (Sec. 4.2), table structure recog- nition (Sec. 4.3), molecular structure recogni- tion (Sec. 4.4), optical music score recognition (Sec. 4.5), long caption generation (Sec. 4.6), spa- tial reasoning (Sec. 4.7), and radiography report generation (Sec. 4.8). PaliGemma 2 obtains state- of-the-art results on many of those tasks. Finally, we benchmark and analyze low-precision vari- ants of PaliGemma 2 for on-device deployment on CPU (Sec. 4.9).
Corresponding author(s): andstein,andresp,[email protected] 2024 Google DeepMind. All rights reserved
PaliGemma 2: A Family of Versatile VLMs for Transfer
896244822242linear projectionImage tokensInput text tokensOutput text tokens2B9B27BGemma 2SigLIP-400m/14
Figure 1 | PaliGemma 2 processes a 224px2/ 448px2/896px2 image with a SigLIP-400m en- coder with patch size 14px2, yielding 256/1024/ 4096 tokens. After a linear projection, the image tokens are concatenated with the input text to- kens and Gemma 2 autoregressively completes this prefix with an answer.
0.2740.2550.8460.4980.0460.6660.2270.807segment puffin in the back ; puffin in front<loc0255><loc0274><loc0846><loc0498><seg024>[...]<seg018> puffin in front ;<loc0046><loc0666><loc0227><loc0807><seg106>[...]<seg055> puffin in the backInput:Output:
Figure 2 | Referring segmentation example from our PaliGemma demoa. The model is pretrained with a vocabulary that includes localization to- kens (for detection) and segmentation tokens (to define a binary mask inside a bounding box).
2. Related work
https://huggingface.co./spaces/big-vision/paligemma
Over the last few years, VLMs evolved rapidly from simple dual-encoder (contrastive) [31, 77, 108] or encoder-decoder (captioning) [20, 93, 94, 98] designs trained from scratch, to more capable designs combining a pretrained vision encoder with a pretrained language model [4, 5, 14, 16, 48, 72, 96, 103]. Broadly, three paradigms are used to transfer these models: zero-shot, few- shot, and fine-tuning. Another recent trend is instruction tuning which aims to make the mod- els more user friendly [18, 54].
marize the most important aspects here. We use the same pretrained SigLIP-So400m vision en- coder [3, 108] and map its (sequence of) em- beddings to the Gemma 2 input space with a linear projection. The visual embeddings are com- bined with a text prompt and fed to the Gemma 2 language model (prefill). Predictions are then obtained by autoregressively sampling from the language model (see Fig. 1).
Several previous works [9, 19, 34, 35, 45, 66, 92, 109] have investigated the effect of scaling VLMs along different axes such as training data and compute, resolution, model size, and quality of components, in particular the vision encoder. However, we are not aware of prior work which jointly studies the effect of the image resolution and the size of the language models on transfer via fine-tuning. In particular, prior works rely- ing on different language model sizes often use models with different architecture and training recipes from different labs, e.g. [35, 92] (with the notable exception of [47]).
We pretrain PaliGemma 2 in three stages (with stage 0 corresponding to unimodal pretraining of the components, see [108] and [21]).
Stage 1 combines the pretrained SigLIP- So400m and Gemma 2 checkpoints (raw checkpoints, without post-training steps) and trains them jointly on a multimodal task mixture of 1 billion examples designed to enable transferability to a wide range of tasks via fine-tuning. The image resolution is 224px2; no parameters are frozen during this stage.
3. Model
We follow exactly the same modeling, training, and data setup as PaliGemma [9] and briefly sum-
Stage 2 first trains for 50 million examples at resolution 448px2 and then for 10 million examples at resolution 896px2. The task mix- ture has the same components but tasks ben- efiting from high resolution are upweighted, and the output sequence length is increased
PaliGemma 2: A Family of Versatile VLMs for Transfer
Training cost / example
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448px2
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Gemma 2 2B PaliGemma 2 PaliGemma 2 10B SigLIP-So400m Gemma 2 9B Gemma 2 27B PaliGemma 2 28B
3.0B 9.7B 27.7B
1.0 3.7 18.9
4.6 18.3 63.5
23.5 67.7 155.6
Table 1 | The vision encoder parameter count is small compared to the LLM, but the compute is dominated by the vision tokens in the LLM. The last three columns show the relative training cost per example (as measured in our pre-training setup). Models are trained on Cloud TPUv5e [24], except the 28B model at 896px2 is trained on TPUv5p, for which we assume a speed-up of 2.3 per chip.
(to promote e.g. learning of OCR for long sequences of visual text).
cial VLM as common among other open VLMs such as LLaVA [54].
Stage 3 fine-tunes the checkpoints from stage 1 or 2 (depending on the resolution) to the target task. PaliGemma considered a range of academic benchmarks, including some involving multiple images and short videos. We consider the same set of bench- marks here (exploring the same set of hyper- parameters from [9, Sec. 3.2.4]). In addition, we also explore new applications involving document-related tasks, long caption gener- ation, and medical image understanding.
Similar to PaliGemma, we train PaliGemma 2 models on Cloud TPUv5e Pod slices [24] (ex- cept TPUv5p for the 28B model at 896px2) of 256 to 1024 chips and use a fully-sharded data-parallel (FSDP [8, 110]) sharding strategy. PaliGemma 2 3B has roughly the same training cost as PaliGemma (3 days for Stage 1 using 256 chips); the cost for other variants and resolutions can be inferred from Table 1. It is worth noting that increasing resolution incurs a similar addi- tional cost as increasing the language model size.
Following [22], we apply logits soft-capping [6] to the attention and output logits in the Gemma 2 component with the same parameters as [22] in Stages 1 and 2, but not in Stage 3, as this led to worse results for some transfer tasks. Fur- ther, we use the Adam optimizer [42] with de- fault hyperparameters throughout, and adjust the learning rate based on the model size in Stages 1 and 2. Specifically, we multiply the learning rate of 2 10 5 used in Stages 1 and 2 for PaliGemma by 0.5 for PaliGemma 2 3B and by 0.25 for PaliGemma 2 10B and 28B.
4. Experiments
In addition to the broad range of transfer tasks considered in [9], we also consider new tasks in- volving text detection and recognition (Sec. 4.2), table structure recognition (Sec. 4.3), molecular structure recognition (Sec. 4.4), optical music score recognition (Sec. 4.5), long caption genera- tion (Sec. 4.6), spatial reasoning (Sec. 4.7), and radiography report generation (Sec. 4.8).
We provide examples for each new task in Ap-
For details on the training data mixture we re- fer to [9, Sec. 3.2.5] and provide a brief sum- mary here. The mixture involves captioning, grounded captioning (as in [94]), OCR, differ- ent machine generated visual question answer- ing (VQA) tasks [11, 75], detection [13] and in- stance segmentation [15]. Many of the corre- sponding labels are machine generated, mostly re- lying on publicly available specialist models (see [9, Sec. 3.2.5]), and none uses a large commer-
pendix A and transfer details in Appendix B.
4.1. Investigating model size and resolution
To study the effect of model size and reso- lution on task performance we finetune the 3 model variants (3B, 10B and 28B) in two resolutions (224px2 and 448px2) on the 30+ academic benchmarks used by [9], covering a broad range of captioning, VQA, and refer-
PaliGemma 2: A Family of Versatile VLMs for Transfer
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Figure 3 | Relative improvements of metrics after transfer, when choosing a pre-trained checkpoint with a larger LM, or with a higher resolution. The tasks are grouped into tasks sensitive to both model size and resolution ( ), sensitive to model size ( ), and sensitive to resolution ( ). Note that some benchmarks are quite saturated (e.g. ScienceQA s relative improvement of 2.2% corresponds to an error reduction of 53.8% see Figure 13). Data used to create this plot available in Table 13.
ring segmentation tasks on natural images, doc- uments, infographics, and videos. We reuse the optimal hyperparameters from the earlier PaliGemma work and only sweep the learning rate {0.03, 0.06, 0.1, 0.3, 0.6, 1.0, 3.0} 10 5 for every model size. Since for most tasks the earlier work used the same hyperparameters for 224px2 and 448px2, we only sweep at 224px2 resolution and reuse the selection for both resolutions. We select the best learning rate based on the respec- tive validation split for each model size and task, then retrain the models and report the test met- rics. Complete results are available in Table 13.
4.1.1. Effect on task performance
Increasing image resolution and increasing LM size both lead to an increase in the FLOPs spent on the prediction (and training, see Table 1) of our PaliGemma 2 models. Thus, we generally expect most tasks to benefit from both these changes. On the other hand, some tasks might benefit from more detail in the input (higher resolution) or bet- ter language understanding and increased world knowledge provided by a larger LM. To get a more fine-grained understanding of these aspects we visualize in Fig. 3 the relative improvement in transfer metrics when equipping PaliGemma 2
3B (224px2) with either the bigger 9B LM while keeping the resolution (3.7 more FLOPs), or keeping the model size but increasing the resolu- tion to 448px2 (4.6 more FLOPs).
As expected, most tasks similarly benefit from a resolution and model increase (green markers). There is a group of tasks (yellow markers) fo- cused on text, document, screen and chart under- standing which mainly benefit from a resolution increase. The images in the corresponding bench- marks often have a native resolution significantly larger than 224px2, which is aligned with this ob- servation. Another group of tasks (blue markers) mostly benefits from LM size increase. Some of these tasks involve multilingual data (XM3600 (avg35)), or require advanced visual reasoning (AI2D, CountBenchQA, NLVR2).
Fig. 4 provides additional detail on the scaling behavior as a function of resolution and model size. Compared to increasing model size from 3B to 10B, increasing it further to 28B often only leads to moderate improvements, or no improve- ments at all. Using the largest PaliGemma 2 can thus be useful if one wants to get the best possi- ble performance and has no compute or latency constraints. A possible factor related to the rela- tively worse transferability of PaliGemma 2 28B
PaliGemma 2: A Family of Versatile VLMs for Transfer
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Figure 4 | Transfer performance as a function of model size and resolution (median over 5 transfer runs). The shaded area marks standard deviation to reported value. Lighter lines correspond to higher resolution (448px2). The tasks are grouped into tasks sensitive to both model size and resolution ( ), sensitive to model size ( ), and sensitive to resolution ( ). Data for this plot is available in Table 13.
is that the underlying Gemma 2 27B model is trained from scratch, as opposed to the 2B and 9B models, which are distilled [22, Sec. 6.1].
4.1.2. Model size and transfer learning rate
Figure 5 visualizes the (normalized) task perfor- mance as a function of the transfer learning rate. As a general trend we observe that the optimal learning rate for larger models tends to be lower than for smaller models (diagonal patterns in the heat map). We thus recommend to sweep smaller learning rates when increasing model size. Addi- tionally, we found that the new PaliGemma 2 3B generally has a smaller optimal transfer learning rate when compared to PaliGemma.
4.1.3. Using Gemma 2 instead of Gemma 1
We also compare with PaliGemma in Table 15. It can be seen that for the same resolution and model size (i.e. 3B) PaliGemma 2 models perform slightly better than the corresponding PaliGemma models. On average over the 30+ aca- demic benchmarks the scores were 0.65 better for 224px2 and 0.85 for 448px2.
4.2. Text detection and recognition
We apply PaliGemma 2 to advanced OCR in- volving localization and recognition of individual words from images. Specifically, the outputs are pairs of {transcription, bounding box}. Following the HierText competition [57], we use word level precision, recall, and F1 as the metrics. A word
PaliGemma 2: A Family of Versatile VLMs for Transfer
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Figure 5 | Per-task performance as a function of model size and learning rate for several of the downstream tasks. Values are normalized for each task and model size, with darker color indicating better task performance. Larger models tend to have a lower optimal transfer learning rate. Zero-shot tasks not shown as their values were not used to select learning rates. The data used for this plot is provided in Table 14.
result is considered true positive if the IoU with the ground-truth bounding box is greater than or equal to 0.5 and the transcription matches the ground-truth. Note that the HierText protocol does not normalize letter cases, punctuation sym- bols, or filter based on text lengths but directly compares predictions against ground-truth.
We fine-tune PaliGemma 2 on a mixture of the train splits of ICDAR 15 [36], Total-Text [17], MLT17 and MLT19 [68], HierText [56], Tex- tOCR [84], IntelOCR [44] and evaluate on the ICDAR 15 and Total-Text test sets, which are the most commonly used OCR benchmarks. Table 2 shows the results: PaliGemma 2 3B at 896px2 outperforms the state of the art HTS [58]. We emphasize that this result is obtained simply by fine-tuning a general-purpose VLM which does not rely on task-specific architecture components as common in the OCR literature. This highlights PaliGemma 2 s versatile interface, and shows the benefits of OCR-related pretraining in Stages 2 and 3. We further tried reducing the resolution which led to substantially lower prediction qual- ity, while increasing the model size did not lead to improvements.
4.3. Table structure recognition
The goal of table structure recognition is to ex- tract table text content, corresponding bound- ing box coordinates, and the table structure in HTML format from document images. To transfer PaliGemma 2 to this task we finetune on (the train splits of) two popular data sets, PubTabNet [112] containing 516k images of tabular data from the PubMed Central Open Access Subset (commer- cial use collection) and FinTabNet [111], consist- ing of 113k financial report tables from annual reports of S&P 500 companies. We remove ex- amples with obviously corrupted ground truth (e.g. a bounding box extending outside the image frame) from the training data and further apply the refinements from [86] to FinTabNet. Images are resized to the target resolution while preserv- ing the aspect ratio, and padded to square size to match the target input resolution.
We assess model quality with the Tree Edit Distance Similarity (TEDS) [112] and the Grid Table Similarity (GriTS) [85], two families of metrics which measure cell text content, cell topology/structure, and bounding box quality. PaliGemma 2 sets a new state of the art for most of these metrics (Table 3). We further tried in-
PaliGemma 2: A Family of Versatile VLMs for Transfer
ICDAR 15 Incidental
Total-Text
HTS PaliGemma 2 3B 896px2
81.9 68.4 81.9 70.7
74.5 75.9
75.7 69.4 72.4 73.8 74.5 74.2
Table 2 | Text detection and recognition performance: The 896px2 PaliGemma 2 model outperforms the state-of-the-art model HTS [58] on ICDAR 15 Incidental and Total-Text, under the evaluation protocol of HierText [57].
FinTabNet
PubTabNet
S-TEDS
TEDS GriTS-Top GriTS-Con
S-TEDS
TEDS GriTS-Top GriTS-Con
SOTA PaliGemma 2 3B 896px2
98.9 99.2
98.2 98.9
99.0 99.4
98.6 99.2
97.9 97.6
96.9 97.3
98.0
97.8
Table 3 | PaliGemma 2 results for table structure recognition on FinTabNet [111] and PubTabNet [112], compared to the state of the art. The reference metrics are from [28, 38, 60, 86].
creasing the model size which did not lead to additional benefits, and using a lower image res- olution led to a small regression in quality.
4.4. Molecular structure recognition
We explore PaliGemma 2 for molecular struc- ture recognition, inferring the molecule graph structure (represented as a SMILES string [99]) from molecular drawings. As training data we use 1 million molecules from the PubChem dataset [41], rendered using the In- digo toolkit [71], and augmented with a variety of drawing styles and random perturbations, fol- lowing MolScribe [76]. We then evaluate on the same eval set as [76] consisting of 5.7k synthetic molecule images rendered with the ChemDraw library. We use exact match percentage as a met- ric, shown in Table 4. PaliGemma 2 outperforms the state of the art MolScribe when using 448px2 resolution; further increasing the resolution did not lead to a higher exact match percentage.
the task of
other common score-related information such as articulation and barlines.
We use the GrandStaff dataset [79] containing 53.7k images and employ the official train, valida- tion and test splits. During training we use both the original images and synthetically augmented versions. Evaluation is done on the original im- ages without distortion. The metrics are the same as in [80] and are based on the the normalized mean edit distance. More specifically, the Charac- ter Error Rate (CER) counts errors at the character level, the Symbol Error Rate (SER) measures er- rors at the symbol level (combining multiple char- acters), and the Line Error Rate (LER) is based on full lines in the **kern encoding.
The results are shown in Table 5 along with those of the current state of the art method [80]. The error rates decrease with increasing resolu- tion, with the best error rates obtained at 896px2 resolution. Increasing the model size from 3B to 10B did not lead to further error reduction.
4.5. Optical music score recognition
We apply PaliGemma 2 to optical music score recognition: translating images of single-line pi- anoform scores into their digital score representa- tion in the **kern format1. The **kern repre- sentation encodes pitch and duration along with
1https://www.humdrum.org/rep/kern/
4.6. Generating long, fine-grained captions
Generating long image captions with fine-grained detail has many use cases in multimodal learn- ing, for example to train text-to-image generation models with good controllability [7, 105]. To adapt PaliGemma 2 for this task we fine-tune on
PaliGemma 2: A Family of Versatile VLMs for Transfer
Full Match
#par. #char. #sent. NES
MolScribe [76] PaliGemma 2 10B 448px2
93.8 94.8
Table 4 | PaliGemma 2 performance for molecule structure recognition on ChemDraw data [76].
MiniGPT-4 mPLUG-Owl2 InstructBLIP LLaVA-1.5 VILA
7B 8B 7B 7B 7B
484 459 510 395 871
5.6 52.3 4.4 48.4 4.0 42.6 4.2 40.6 8.6 28.6
CER SER
LER
PaliGemma PaLI-5B
8.9 34.3 3B 5B 1065 11.3 32.9
535
Sheet Music Tr. [80] PaliGemma 2 3B 896px2
3.9 1.6
5.1 2.3
13.1 6.7
PaliGemma 2 448px2 3B PaliGemma 2 448px2 10B
529 521
7.7 28.4 7.5 20.3
Table 5 | PaliGemma 2 performance for music score recognition on the GrandStaff data set [80]. Character Error Rate (CER), Symbol Error Rate (SER), and Line Error Rate (LER) in [%].
Table 6 | PaliGemma 2 results for long captioning on the DOCCI data [69]. Pali* models are mod- els fine-tuned on DOCCI at 448px2; the other baselines are instruction-tuned on a broad range of tasks. Average prediction length in characters and sentences, and percentage of Non-Entailment Sentences (NES), measuring factual inaccuracies.
the DOCCI (Descriptions of Connected and Con- trasting Images) [69] data set which contains 15k images with detailed human-annotated En- glish descriptions with an average length of 7.1 sentences (639 characters, 136 words). The de- scriptions provide object spatial relations, object counting, text rendering, world knowledge, etc.
We first fine-tune PaliGemma 2 on DOCCI s train split, exploring the hyperparameter range suggested in [9, Sec. 3.2.4]. We select the most performant models by perplexity scores based on the test split, and generate image captions on the 100-image qual_dev split, with a max- imum decoding length of 192. We then con- duct human evaluations assessing whether each generated sentence is factually aligned with (en- tailed by) the image content (see Appendix B.5 for details on the evaluation protocol). Based on these evaluations we select the most factu- ally aligned models and retrain them on the union of train and test splits, followed by another round of human evaluation (on the qual_dev split). The results, shown in Table 6 indicate that the fine-tuned PaliGemma 2 model produces more factually aligned sentences than many pop- ular VLMs, which are often instruction-tuned on 10 100 larger high-quality captioning sets than PaliGemma 2. Unsurprisingly, we observe that in- creasing model size and resolution both improve factual alignment.
4.7. Spatial reasoning
VLMs like PaliGemma 2 obtain strong perfor- mance in vision-language tasks which involve ob- ject localization, such as referring expression com- prehension and segmentation [9, 15, 94, 104]. These tasks and the associated benchmarks of- ten rely on machine-generated annotations and are blind to complex failure modes, e.g. those involving negations.
The Visual Spatial Reasoning (VSR) bench- mark [53] is designed to overcome these issues and we use it here to assess the spatial reason- ing capabilities of PaliGemma 2. It is formulated as a classification task, where a model needs to determine whether a statement about the spa- tial relationship of objects in the image is correct or not. To use PaliGemma 2 s flexible text in- terface we frame this benchmark as a QA task with True / False answers. The results in Table 7 show that PaliGemma 2 outperforms prior fine- tuned models, and fine-tuning also provides a significant improvement over InstructBlip [18], a strong zero-shot model form the literature. We observe significant benefits from larger model size, indicating benefits from improved language understanding, whereas going beyond resolution 224 did not lead to improvements.
PaliGemma 2: A Family of Versatile VLMs for Transfer
zs. split
rand. split
F1
Human [53]
95.4
InstructBLIP (zs.) [18] LXMERT [89]
65.6 70.1
61.2
PaliGemma 2 3B 224px2 PaliGemma 2 10B 224px2
74.8 79.8
81.6 86.8
Table 7 | PaliGemma 2 accuracy on VSR [53] on the zeroshot and random test splits. We show a fine-tuned (LXMERT) and zero-shot (Instruct- BLIP) baseline from the literature.
Flamingo-CXR [90] Med-Gemini-2D [102]
13.8 10.1 29.7 20.5 17.5 20.5 28.3 24.4
PaliGemma 2 3B 896px2 19.9 14.6 31.9 28.8 PaliGemma 2 10B 896px2 17.4 15.0 32.4 29.5
Table 8 | PaliGemma 2 performance for radiogra- phy report generation on the on the MIMIC-CXR data [23, 33]. We report CIDEr (C), BlEU4 (B), Rouge-L (R), and RadGraph F1-scores [%] [30] (a clinical metric).
4.8. Radiography report generation
To explore the capabilities of PaliGemma 2 mod- els in the medical domain, we apply it to auto- matic chest X-ray report generation, which can be cast as a (long) captioning task on X-ray im- ages. We fine-tune PaliGemma 2 on the MIMIC- CXR dataset [23, 33] which contains 377k images (originating from 228k radiographic studies at the Beth Israel Deaconess Medical Center in Boston, MA) with free-text radiology reports. We use the same train, validation, and test splits as [90]. To improve quality, we use an LLM (Gemini 1.5 pro) to remove mentions of prior X-rays as the model does not have access to those.
We measure the RadGraph F1-score [30], which is the F1 score between the entities ex- tracted from the reference report and the gener- ated one using RadGraph. RadGraph takes into account the absence or presence of findings in the report, as well as their relationships to image features. Results are reported on test data held out during training and tuning.
Table 8 shows the performance of PaliGemma 2 models along with baselines from the litera- ture. PaliGemma 2 obtains a state-of-the-art Rad- Graph score. Increasing resolution and model size both lead to modest improvements.
4.9. CPU inference and quantization
CPUs, and briefly present experiments using the gemma.cpp2 framework here. gemma.cpp is a lightweight, portable C++ inference engine that supports 8-bit switched-floating-point quantiza- tion (alternative options for CPU inference include llama.cpp3, XNNPack4, and others).
To assess the inference speed for CPU-only in- ference, we run PaliGemma 2 inference on four different architectures with gemma.cpp. We use a checkpoint of PaliGemma 2 3B (224px2) fine- tuned on COCOcap and the example image for PaliGemma in gemma.cpp. The prompt de- scribe this image results in a prefill length of 256 + 4 = 260 tokens (for image + text). The out- put response A large building with two towers on the water consists of 11 tokens. All runs used batch size 1. The results are presented in Table 9 and give an overview of what can be expected on different processors (for this particular setting).
From evaluations on PaliGemma [9] we already know that going from 32-bit floating point (f32) to 16-bit (bf16) weights is possible without a loss of quality. Here we compare to the gemma.cpp mixed quantization. Table 10 shows a quality comparison for five of the fine-tuning datasets (chosen for coverage of various tasks). We fine- tuned PaliGemma 2 3B (224px2) once for each of these five datasets. (Noticeable differences to Table 13 for the Jax version are the result of us- ing greedy decoding for COCOcap and TextCaps.) We then evaluated the resulting checkpoints both in Jax and in gemma.cpp after quantization. The
In some cases we may want to run inference of PaliGemma 2 on devices without accelera- tors. We are interested in the resulting run- times and quality when running inference on
2https://github.com/google/gemma.cpp 3https://github.com/ggerganov/llama.cpp 4https://github.com/google/XNNPACK
PaliGemma 2: A Family of Versatile VLMs for Transfer
Walltime [s]
Tokens/sec
Processor
Threads
ViT
Prefill Extend Prefill Extend
Apple M1 Max Apple M3 Pro AMD Milan AMD Milan AMD Genoa AMD Genoa
4+1 7+1 8+1 32+1 8+1 32+1
1.6 0.8 0.82 0.39 0.36 0.17
8.2 4.4 4.9 1.8 1.8 0.8
0.9 0.5 0.64 0.34 0.29 0.27
32 59 53 144 147 323
12 22 17 32 37 41
Table 9 | CPU-only inference speed measurements with gemma.cpp-based implementation on different architectures. Inference of finetuned PaliGemma 2 3B (224px2) with greedy decoding. Prefill is done with 260 tokens and followed by 11 calls to extend during decoding.
COCOcap TextCaps AI2D OKVQA DocVQA(val)
Jax, F32, 12.1GB gemma.cpp, quantized, 4.0GB relative metric values [%]
140.0 139.8
99.9
126.3 126.6
100.2
75.4 75.6
100.1
64.0 64.1
100.1
39.8 39.8
99.9
Table 10 | Quality comparison between Jax/f32 inference on TPU and quantized gemma.cpp-based inference on CPU. Inference of one fine-tuned PaliGemma 2 3B (224px2) run. Noticeable differences to Table 13 for the Jax version are the result of using greedy decoding for COCOcap and TextCaps. Relative numbers based on metric values before rounding to one decimal.
relative quality after quantization shows no prac- tical quality difference.
[2] H. Agrawal, K. Desai, Y. Wang, X. Chen, R. Jain, M. Johnson, D. Batra, D. Parikh, S. Lee, and P. Anderson. NoCaps: Novel object captioning at scale. In ICCV, 2019.
5. Conclusion
With PaliGemma 2 we present a new family of open-weight models spanning a broad range of model sizes an input resolutions. PaliGemma 2 obtains strong transfer performance across a broad range of captioning, VQA, and video tasks. In particular, the newly added larger variants lead to significant improvements compared to PaliGemma for users with a larger compute bud- get. Furthermore, we show that PaliGemma 2 excels in applications beyond what was consid- ered in PaliGemma, including domains like music, molecules, and medical imaging.
[3] I. Alabdulmohsin, X. Zhai, A. Kolesnikov, and L. Beyer. Getting vit in shape: Scaling laws for compute-optimal model design. In NeurIPS, 2023.
[4] J.-B. Alayrac, J. Donahue, P. Luc, A. Miech, I. Barr, Y. Hasson, K. Lenc, A. Men- sch, K. Millican, M. Reynolds, R. Ring, E. Rutherford, S. Cabi, T. Han, Z. Gong, S. Samangooei, M. Monteiro, J. Menick, S. Borgeaud, A. Brock, A. Nematzadeh, S. Sharifzadeh, M. Binkowski, R. Barreira, O. Vinyals, A. Zisserman, and K. Simonyan. Flamingo: a visual language model for few-shot learning. In NeurIPS, 2022.
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PaliGemma 2: A Family of Versatile VLMs for Transfer
Contributions and Acknowledgments
Model development contributors
Marketing Glenn Cameron Natalie Dao
Core Contributors Andreas Steiner Andr Susano Pinto Michael Tschannen
Kaggle D. Sculley Nilay Chauhan Brenda Flynn Kinjal Parekh
Contributors Daniel Keysers Xiao Wang Yonatan Bitton Alexey Gritsenko Matthias Minderer Anthony Sherbondy Shangbang Long Siyang Qin Reeve Ingle Emanuele Bugliarello Sahar Kazemzadeh Thomas Mesnard Ibrahim Alabdulmohsin Lucas Beyer Xiaohua Zhai
Developer Relations Jetha Chan Joe Fernandez Ju-yeong Ji
Keras Divyashree Sreepathihalli Hongyu Chiu
Vertex AI Keelin McDonell
Lead Andreas Steiner
Ethics and Safety Antonia Paterson Pankil Botadra
Acknowledgments Jan Wassenberg Basil Mustafa
Hugging Face Partners Merve Noyan Pedro Cuenca Pablo Montalvo
Model release contributors and general support
Gemma Model Tris Warkentin Alek Andreev Armand Joulin Victor Cotruta Sanah Choudhry Nathan Byrd
Nvidia Partners Dong Meng Manoj Kilaru Shyamala Prayaga Ryan Timbrook Anna Warno
Ollama Partners Michael Chiang Jeffrey Morgan
Open Models Success Luiz Gustavo Martins Kat Black Phil Culliton Chris Perry D. Sculley Sara Smoot
Executive Sponsors Raia Hadsell Joelle Barral Jeremiah Harmsen Mat Velloso Allen Hutchison
PaliGemma 2: A Family of Versatile VLMs for Transfer
A. Tasks
This section provides one training example for the transfer tasks that were added in PaliGemma 2 in addition to the tasks considered in [9].
Figure 6 | Test set example from Total-Text [17] with PaliGemma 2 3B 896px2 predictions.
In millions
May 31, 2015 Securities in an unrealized loss position for less than twelve months Gross unreal- ized losses
Fair value
Securities in an unrealized loss position for more than twelve months Gross unreal- ized losses
Fair Value
Total
Gross unreal- ized losses
Fair Value
Type of issue: General obliga- tion municipal bonds Revenue mu- nicipal bonds Total
$(3.8)
$(3.2)
$(7.0)
$355.1
361.6
$896.7
$(0.5)
$(0.5)
$26.3
$26.3
$(4.3)
(3.2)
$(7.5)
$561.4
361.6
$923.0
Figure 7 | Original image from FinTabNet [111] with predicted cell content boxes (green), and resulting PaliGemma 2 model prediction.
PaliGemma 2: A Family of Versatile VLMs for Transfer
Figure 8 | Example of a rendered molecule with the corresponding SMILES string CC1([C@@H]([C@@H](C2=C(O1)C=CC(=C2)C(C(F)(F)F)(F)F)N3CCCCC3=O)O)C.
Figure 9 | Example of a pianoform sheet with its **kern transcription (source https://www. humdrum.org/guide/ch02/).
PaliGemma 2: A Family of Versatile VLMs for Transfer
Model
Caption
InstructBLIP
The image features a brown horse standing in a dirt area, with a saddle on its back. The horse appears to be well-trained, as it is standing still and paying attention to its surroundings. There are several people visible in the scene, including one person standing near the horse s left side, another person standing near the horse s right side, and a third person standing further away from the horse. In addition to the people, there are two cars parked in the background, one on the left side and another on the right side of the image.
mPLUG-Owl2
In this image we can see a horse on the ground. In the background there is a shed and we can see trees.
MiniGPT-4
This image is of a horse standing in the dirt near a building. The horse has a saddle on its back and is wearing a number 5 on its forehead. There is a barn or stable visible in the background, and some trees and shrubs in the foreground.
LLaVA-1.5
The image features a brown horse standing in a dirt field. The horse is wearing a saddle and bridle, indicating that it is ready for riding. The saddle is adorned with stars, adding a unique touch to the horse s appearance. The horse appears to be calmly standing in the field, possibly waiting for its rider or simply enjoying the outdoors.
VILA
In the image, a majestic brown horse stands in a fenced area, its coat gleaming under the sunlight. The horse is adorned with a black saddle and bridle, ready for a ride. The number 55 is prominently displayed on its side, perhaps indicating its racing number or a unique identifier. The horse s tail swishes gently, adding a sense of movement to the scene. The horse is positioned in front of a stone wall, which provides a rustic backdrop to the image. The wall s texture contrasts with the smoothness of the horse s coat, creating a visually interesting composition. The ground beneath the horse is covered in dirt, suggesting that this might be a training or riding area. The fence in the background further reinforces this idea, providing a safe and controlled environment for the horse. Overall, this image captures a moment of calm before what could be an exciting ride, with the horse poised and ready for action.
PaLI-5B (FT on DOCCI)
A side view of a brown horse that is standing on a dirt ground. The horse is facing to the right and has a black saddle on it. The horse has a white star on its face. The horse has a bridle around its neck. The horse is standing on a dirt ground that is covered in white rocks. There is a stone wall behind the horse. There is a metal shed behind the horse. There are trees behind the shed.
PaliGemma 3B 448px2
A side view of a brown horse that is standing on a dirt ground. The horse is facing to the left and has a black saddle on its back. The saddle is black and has a brown leather strap on it. The horse has a black rope around its neck. The horse has a white star hanging from its bridle. The horse has a white number "55" written on its back. The horse has white hooves. The horse has a black rope wrapped around its legs. The horse has a brown rope wrapped around its head. The horse has a black rope wrapped around its tail. The horse is standing on a dirt ground that has small white rocks on it. Behind the horse is a stone wall that is made up of different sized rocks. Behind the stone wall is a metal gate that is being held up by metal beams. Behind the gate is a tree line that is made up of green trees.
PaliGemma 2 3B 448px2
A brown horse is standing in a dirt area with rocks scattered around. The horse has a black saddle on its back and a white star painted on its chest. The number "55" is painted on the side of the horse in white. A stone building is behind the horse. A metal structure is in the background of the image with a wooden roof over it. Trees are in the background of the image as well.
PaliGemma 2 10B 448px2
A brown horse is standing in a dirt area with small rocks. The horse has a black saddle on its back and a white star painted on its side. The horse has "55" written on its back in white. There is a pile of horse manure in front of the horse. There is a stone wall behind the horse. There is a wooden structure with a metal roof behind the stone wall. There are trees in the background.
Figure 10 | Example DOCCI image and captions generated by PaliGemma 2 models and baselines, with non-entailment sentences highlighted in red.
PaliGemma 2: A Family of Versatile VLMs for Transfer
Indication
Radiologist report
PaliGemma 2 3B 896px2 prediction
INDICATION: Woman with cardiomy- opathy and cdiff with acute desatura- tion and dyspnea // PE, pulmonary edema, vs aspiration PE, pulmonary edema, vs aspiration.
IMPRESSION: Enlargement of the cardiac silhouette with pulmonary edema. Bilateral pleural effusions, more prominent on the left.
FINDINGS: There is substantial en- largement of the cardiac silhouette with pulmonary edema. Retrocardiac opacification is consistent with vol- ume loss in the left lower lobe and pleural effusion. In the appropriate clinical setting, superimposed pneu- monia would have to be considered.
Figure 11 | Example from the MIMIC-CXR [23, 33] validation set along with a PaliGemma 2 prediction.
PaliGemma 2: A Family of Versatile VLMs for Transfer
B. Transfer and evaluation details
B.1. Text detection and recognition
In all experiments, we fine-tune the checkpoints for 15k steps with a batch size of 256 on 256 TPU-v5e. The maximum sequence length is set to 2048. We experiment with learning rates {0.01, 0.05, 0.1, 0.5, 1.0} 10 4 and find that 10 5 gives the best results. We also found using a label-smoothing of 0.1 improves the results. The best results are obtained with resolution 896px2.
B.2. Table Structure Recognition
We use the same transfer setup and hyperparameter range as for text recognition described in Sec. B.1, except that we set maximum output length to 4096 and do not use label-smoothing. The optimal fine-tuning learning rate is 10 4.
Preprocessing The cropped table input images are padded to square shape with white pixels and resized to the target image resolution. Cell bounding boxes of non-empty table cells are encoded using four PaliGemma location tokens of the form <locDDDD>, where DDDD encodes a quantized image location in the range 0000 to 1023. Boxes are specified using a special coords="<locXMIN><locYMAX><locXMAX><locYMAX>" attribute of table cell <td> HTML tags. Training examples with invalid table structure and overlapping cell bounding boxes are skipped. Addi- tional correction of cell bounding box annotations and cell text annotations are applied to FinTabNet training examples using information from the source PDFs, following a similar approach as [86]. As is common in the literature [38], no filtering is applied to the test splits we report results on.
B.3. Molecule structure recognition
In all experiments, we fine-tune the pretrained checkpoint for 30k steps with batch size 256 using 256 TPU-v5e chips. The learning rate is set to 10 4, label smoothing to 0.1, and the maximum output length is 256. We pad the images to square shape with white pixels and resize them to the target image resolution.
B.4. Optical music score recognition
We follow the training setup described in Sec. B.3 except that we use maximum output length 1024.
B.5. Generating long, fine-grained captions (DOCCI)
We rely on the transfer protocol and hyperparameters suggested in [9, Sec. 3.2.4.].
Human evaluation protocol To evaluate the factual grounding of the generated captions, we conduct human evaluations assessing the relationship between each sentence and the corresponding image. Raters are presented with highlighted sentences and asked, What is the relationship of the highlighted sentence with respect to the image? . They then select from four options: Entailment , Neutral , Contradiction , and "Nothing to assess", categories adapted from the framework in [78] for evaluating the factual alignment of text and visual content. For example, the statement The pig has black, rounded hooves on its front and back feet and a pink nose (Fig. 12) would be rated as Contradiction , as the image clearly shows pink hooves. Figure 1 illustrates the annotation interface.
PaliGemma 2: A Family of Versatile VLMs for Transfer
Nothing to assessContradictionNeutralEntailment
UserModelA medium shot of a soft, plush pink pig stuffed animal facing forward. The pig has short ears that are beginning to droop down. The snout is very small, and the eyes are black with small dark brown pupils. The stomach is a light tan. The underbelly of the pig is not visible.
The pig has black, rounded hooves on
Describe this image in detailsSentence 5 out of 10
Assessment
its front and back feet and a pink nose
. The pig is sitting on a light brown hard wood floor with a sage green wall behind it. The wall has a horizontal groove where the baseboard is. The pig is casting a shadow on the wall behind it, angled towards the top of the shot. Indoors. The lights are on.
What is the relationship of the highlighted sentence with respect to the image?
Figure 12 | Annotation interface used for human evaluation of image description accuracy. Raters assess the relationship between generated sentences and the corresponding image.
Each sentence was rated by five individuals and the majority agreement was used as the rating result. The overall binary agreement is 0.8407, indicating the proportion where all raters agree on the Entailment category. We refer to both Contradiction and Neutral as Non-entailment . Examples of human evaluation results can be found in Table 4. We use the proportion of Non-entailment sentences to select the most factually accurate models.
B.6. Spatial reasoning
We fine-tune the pretrained checkpoint with batch size 1024 using 64 TPU-v5e chips. The the maximum output length is set to 18, which covers the training target outputs. We explore learning rates in {0.1, 0.2, 1.0, 3.0} 10 6, weight decay in {0.1, 0.3, 1.0} 10 6, dropout probability in {0.0, 0.1, 0.2} and epochs in {1, 3, 5, 10, 15, 30}.
B.7. Radiography report generation
Reports in MIMIC-CXR dataset [23, 33] typically have the format INDICATIONS: .... FINDINGS: {...}. IMPRESSIONS: {...}, where indications explain why the chest X-ray was ordered as clinical context for the radiologist, findings enumerate salient features of the image and impressions summarize the radiologist s interpretation of the findings.
We train on the full reports and during prediction emulate the clinical workflow by providing the
indications as a prefix to the model. The model then predicts findings and impressions sections.
After initial exploration based on the PaliGemma 2 at 448px2 resolution we find that fine-tuning for 8 epochs with learning rate 5 10 6 without label smoothing, dropout, and weight decay leads to good results when combined with greedy decoding. We fix these settings and sweep the learning rate again for higher resolutions and model sizes, considering learning rates in {0.03, 0.1, 0.3, 1.0, 5.0} 10 4.
C. Object detection
Object detection has been used as a pre-training task in all members of the PaLI and PaliGemma In transfers, family and improves downstream performance across a wide range of tasks [14]. PaliGemma performs at or close to the state of the art on localization tasks such as referring expression comprehension and segmentation. This raises the question of how well PaliGemma performs on
PaliGemma 2: A Family of Versatile VLMs for Transfer
224px2
448px2
896px2
PG1 3B
PG2 3B
PG2 10B
PG1 3B
PG2 3B
PG2 10B
PG1 3B
PG2 3B
PG2 10B
COCO DocLayNet
28.7 50.8
30.4 46.7
30.3 50.4
37.0 64.1
38.5 62.5
39.2 63.5
41.1 66.5
42.3 66.1
43.6 66.0
Table 11 | Mean average precision (mAP) after transfer to detection tasks. PG1 and PG2 refer to PaliGemma [9] and PaliGemma 2, respectively.
classical object detection tasks. We tested this by transferring PaliGemma to MS COCO [51] and to the DocLayNet document layout detection benchmark [74].
For both tasks, we use a transfer strategy inspired by pix2seq s sequence augmentation approach [13]. We use the prefix detect all classes\n . In the suffix (target sequence), we first provide box coordinates and class names for all annotated objects, in random order. The suffix is then filled up to the maximum sequence length with noise boxes, where each noise box consists of random coordinates and a dedicated <noise> token in place of the class name. During training, no loss is applied to the coordinate tokens of the noise boxes, while the <noise> class tokens receive a loss as usual. This augmentation trains the model to output a larger number of boxes. In addition, it provides a mechanism for the model to represent the confidence that a prediction represents a real object, in form of the probability assigned to the <noise> token. During inference, the <noise> and <EOS> tokens are excluded from sampling. The likelihood of the class tokens is used as a confidence score.
For COCO, we train for 50 epochs. Results are provided in Table 11. As expected, performance strongly depends on resolution. We also observe small but consistent improvements from better language models. Performance at 896px2 is roughly on par with prior sequence-based approaches [13], but lags behind specialized detection architectures like ViTDet [50].
For DocLayNet, we follow the same sequence augmentation approach and train for 50 epochs. Results are similar to COCO in that performance increases with resolution and Gemma 2 model size, although Gemma 1 performs on par with Gemma 2 on this task (Table 11). Similar to COCO, specialized detectors perform better on this task (e.g. YOLOv11 [32] reaches 79.5 mAP [70]).
These results show that, in contrast to many other tasks, classical detection poses a challenge to general-purpose VLMs like PaliGemma. We hypothesize that the limiting factor is not the model s intrinsic object understanding, since it performs well on visual question answering and referring expression comprehension tasks. Instead, performance may be limited by a mismatch between the Average Precision metric, which rewards large numbers of predictions and accurate confidence scores, and the language modeling objective. Fine-tuning with a task-specific reward [88]) could address this limitation, but is beyond the scope of the simple transfer approach we propose for PaliGemma.
D. Ethics and Safety
Besides quality-related metrics, we also evaluate the new PaliGemma 2 VLMs with respect to a number of categories relevant to ethics and safety. These evaluations include prompts covering child safety, content safety and representational harms, following the approach used in Gemma 2 [22], but with image captioning and visual question answering (VQA) setups.
In addition, we also follow the setup used in [15] and use the Perspective API [46] with threshold > 0.8 to detect the presence of toxicity, profanity, among other potential issues in the image captions generated by PaliGemma 2 VLMs across images sourced from the Fairface dataset [37]. We report the
PaliGemma 2: A Family of Versatile VLMs for Transfer
Metric
Perceived Gender
Ethnicity
Age Group
10B
28B
10B
28B
10B
28B
Maximum
Toxicity 0.14 0.15 0.19 0.29 0.39 0.39 0.26 0.18 0.32 Identity Attack 0.04 0.02 0.02 0.13 0.06 0.06 0.06 0.03 0.06 0.17 0.25 0.17 0.37 0.52 0.52 0.27 0.39 0.24 Insult 0.55 0.43 0.57 0.83 0.48 0.48 0.64 0.43 0.64 Threat 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Profanity
Median
Toxicity 0.13 0.10 0.18 0.07 0.07 0.14 0.12 0.08 0.12 Identity Attack 0.02 0.01 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.15 0.23 0.14 0.14 0.17 0.13 0.09 0.18 0.16 Insult 0.35 0.27 0.41 0.28 0.19 0.42 0.27 0.31 0.40 Threat 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Profanity
Table 12 | Safety statistics for captions generated by PaliGemma 2 VLMs on FairFace [37] using the Perspective API [46]. Numbers indicate the fraction of instances with thresholds 0.8 in [%], i.e. a value of e.g. 0.09 means 0.09%.
maximum and median values observed across subgroups for each of the perceived gender, ethnicity, and age attributes. Table 12 shows the overall results. Overall, we observe a low level of toxicity and profanity among others, across all slices and models. In addition, all PaliGemma 2 models perform comparably.
PaliGemma 2: A Family of Versatile VLMs for Transfer
E. Detailed results
ChartQA (aug)
224px 448px
RefCOCOg (val)
10%
OKVQA
100%
ChartQA (human)
100%
GQA
TallyQA (complex)
TallyQA (simple)
Error reduction 3B 10B
ST-VQA (val)
10%
InfoVQA (val)
RefCOCO+ (val)
100%
AOKVQA-DA (val)
VizWizVQA (val)
CountBenchQA
AI2D
100%
RefCOCO (val)
VQAv2 (minival)
Relative improvement 3B 10B
NLVR2
AOKVQA-MC (val)
10%
10%
xGQA (avg7)
TextVQA (val)
OCR-VQA
DocVQA (val)
ScienceQA
Figure 13 | Same data as in Figure 3 and Table 13. The left plot shows relative improvement when changing model size or resolution. The right plot shows the same improvements, but expressed in terms of error reduction. For saturated benchmarks, error reduction is a better metric for model improvement. Benchmarks without a clear normalization to a percentage (such as CIDEr scores) are not shown. Axes are in range [ 1, 100].
PaliGemma 2: A Family of Versatile VLMs for Transfer
224px2 10B
28B
448px2 10B
28B
AI2D [40] AOKVQA-DA (val) [81] AOKVQA-MC (val) [81] ActivityNet-CAP [43] ActivityNet-QA [107] COCO-35L (avg34) [91] COCO-35L (en) [91] COCOcap[51] ChartQA (aug) [63] ChartQA (human) [63] CountBenchQA [9] DocVQA (val) [64] GQA[29] InfoVQA (val) [65] MARVL (avg5) [52] MSRVTT-CAP [101] MSRVTT-QA [100] MSVD-QA [12] NLVR2 [87] NoCaps [2] OCR-VQA [67] OKVQA [62] RSVQA-hr (test) [55] RSVQA-hr (test2) [55] RSVQA-lr [55] RefCOCO (testA) [106] RefCOCO (testB) [106] RefCOCO (val) [106] RefCOCO+ (testA) [39] RefCOCO+ (testB) [39] RefCOCO+ (val) [39] RefCOCOg (test) [61] RefCOCOg (val) [61] ST-VQA (val) [10] SciCap [27] ScienceQA [59] Screen2Words [95] TallyQA (complex) [1] TallyQA (simple) [1] TextCaps [82] TextVQA (val) [83] VATEX [97] VQAv2 (minival) [25] VizWizVQA (val) [26] WidgetCap [49] XM3600 (avg35) [91] XM3600 (en) [91] xGQA (avg7) [73]
74.7 ( 0.5) 83.1 ( 0.4) 64.2 ( 0.5) 68.9 ( 0.3) 79.7 ( 1.0) 83.7 ( 1.1) 34.2 ( 0.3) 35.9 ( 0.5) 53.2 ( 0.4) 51.3 ( 0.2) 113.9 ( 0.2) 115.8 ( 0.0) 138.4 ( 0.2) 140.8 ( 0.3) 141.3 ( 0.5) 143.7 ( 0.2) 74.2 ( 0.8) 74.4 ( 0.7) 48.4 ( 1.1) 42.0 ( 0.3) 84.0 ( 1.4) 81.0 ( 1.0) 43.9 ( 0.6) 39.9 ( 0.3) 67.2 ( 0.2) 66.2 ( 0.3) 33.6 ( 0.2) 25.2 ( 0.2) 89.5 ( 0.2) 83.5 ( 0.2) 72.1 ( 0.5) 68.5 ( 1.3) 51.9 ( 0.1) 50.5 ( 0.1) 61.1 ( 0.2) 62.5 ( 0.2) 93.9 ( 0.2) 91.4 ( 0.1) 123.1 ( 0.3) 126.3 ( 0.4) 74.7 ( 0.1) 73.4 ( 0.0) 68.0 ( 0.1) 64.2 ( 0.1) 92.6 ( 0.0) 92.7 ( 0.1) 90.8 ( 0.1) 90.9 ( 0.1) 92.8 ( 0.6) 93.0 ( 0.4) 77.2 ( 0.1) 75.7 ( 0.2) 74.2 ( 0.3) 71.0 ( 0.3) 75.9 ( 0.1) 73.4 ( 0.1) 74.7 ( 0.2) 72.7 ( 0.2) 68.4 ( 0.3) 64.2 ( 0.2) 72.0 ( 0.2) 68.6 ( 0.1) 71.9 ( 0.1) 69.0 ( 0.2) 71.4 ( 0.2) 68.3 ( 0.3) 61.9 ( 0.1) 64.3 ( 0.4) 165.1 ( 0.5) 159.5 ( 0.7) 96.1 ( 0.3) 98.2 ( 0.2) 113.3 ( 0.8) 117.8 ( 0.7) 73.4 ( 0.1) 70.3 ( 0.3) 81.8 ( 0.1) 83.2 ( 0.1) 127.5 ( 0.3) 137.9 ( 0.3) 64.0 ( 0.3) 59.6 ( 0.3) 82.7 ( 0.5) 80.8 ( 0.4) 84.3 ( 0.2) 83.0 ( 0.2) 76.4 ( 0.4) 78.1 ( 0.4) 138.1 ( 0.7) 139.8 ( 1.0) 44.5 ( 0.1) 42.8 ( 0.1) 80.7 ( 0.3) 79.8 ( 0.7) 61.4 ( 0.1) 58.6 ( 0.2)
83.2 ( 0.7) 70.2 ( 0.2) 84.7 ( 0.8)
116.5 ( 0.1) 142.4 ( 0.4) 144.0 ( 0.3) 68.9 ( 0.6) 46.8 ( 0.6) 86.4 ( 1.6) 44.9 ( 0.4) 67.3 ( 0.2) 36.4 ( 0.1) 90.6 ( 0.2)
- -
94.2 ( 0.1) 127.1 ( 0.3) 75.3 ( 0.2) 71.2 ( 0.2) 92.7 ( 0.0) 90.9 ( 0.1) 93.5 ( 0.2) 76.8 ( 0.1) 73.9 ( 0.1) 75.0 ( 0.0) 73.6 ( 0.2) 67.1 ( 0.1) 70.3 ( 0.2) 70.7 ( 0.1) 70.5 ( 0.1) 65.1 ( 0.4) 156.9 ( 1.0) 98.2 ( 0.2) 122.8 ( 0.5) 74.2 ( 0.1) 83.4 ( 0.1) 139.9 ( 0.4) 64.7 ( 0.2)
84.5 ( 0.1) 78.7 ( 0.2) 138.8 ( 0.8) 45.2 ( 0.1) 81.0 ( 0.9) 61.1 ( 0.1)
76.0 ( 0.2) 67.9 ( 0.3) 82.5 ( 0.4)
115.8 ( 0.3) 140.4 ( 0.4) 143.4 ( 0.4) 89.2 ( 0.4) 54.0 ( 0.6) 82.0 ( 1.2) 73.6 ( 0.3) 68.1 ( 0.2) 37.5 ( 0.3) 82.7 ( 0.3)
- -
91.6 ( 0.2) 123.5 ( 0.3) 75.7 ( 0.1) 64.1 ( 0.4) 92.8 ( 0.0) 90.7 ( 0.2) 92.7 ( 0.8) 78.6 ( 0.3) 73.5 ( 0.1) 76.3 ( 0.1) 76.1 ( 0.2) 67.0 ( 0.3) 72.1 ( 0.3) 72.7 ( 0.1) 72.3 ( 0.2) 80.5 ( 0.1) 183.3 ( 0.7) 96.2 ( 0.2) 114.0 ( 0.5) 73.6 ( 0.2) 85.3 ( 0.1) 152.1 ( 0.3) 75.2 ( 0.2)
84.8 ( 0.2) 77.5 ( 0.2) 151.4 ( 0.8) 43.2 ( 0.1) 80.3 ( 0.8) 60.4 ( 0.2)
84.4 ( 0.4) 70.8 ( 0.5) 85.9 ( 0.2)
117.2 ( 0.1) 142.4 ( 0.4) 145.0 ( 0.3) 90.1 ( 0.5) 66.4 ( 0.5) 85.3 ( 1.7) 76.6 ( 0.5) 68.3 ( 0.3) 47.8 ( 0.2) 89.1 ( 0.0)
- -
93.7 ( 0.2) 126.9 ( 0.1) 76.3 ( 0.1) 68.6 ( 0.5) 92.8 ( 0.1) 90.7 ( 0.2) 93.1 ( 0.6) 79.7 ( 0.1) 76.2 ( 0.3) 78.2 ( 0.1) 77.7 ( 0.2) 71.1 ( 0.2) 74.4 ( 0.1) 74.8 ( 0.1) 74.4 ( 0.1) 82.0 ( 0.3) 177.2 ( 0.3) 98.5 ( 0.2) 119.1 ( 1.9) 76.7 ( 0.3) 86.2 ( 0.1) 157.7 ( 0.7) 76.6 ( 0.1)
85.8 ( 0.1) 78.6 ( 0.4) 151.9 ( 0.4) 44.6 ( 0.1) 81.5 ( 0.4) 62.6 ( 0.2)
84.6 ( 0.4) 71.2 ( 0.2) 87.0 ( 0.3)
117.2 ( 0.1) 142.3 ( 0.8) 145.2 ( 0.4) 85.1 ( 0.2) 61.3 ( 0.6) 87.4 ( 1.0) 76.1 ( 0.4) 68.3 ( 0.1) 46.7 ( 0.4) 89.7 ( 0.1)
- -
94.1 ( 0.2) 127.0 ( 0.2) 76.6 ( 0.1) 70.6 ( 0.2) 92.8 ( 0.1) 90.8 ( 0.1) 93.7 ( 0.4) 79.3 ( 0.1) 74.8 ( 0.1) 77.3 ( 0.1) 76.6 ( 0.1) 68.6 ( 0.1) 72.8 ( 0.1) 73.7 ( 0.1) 73.0 ( 0.1) 81.8 ( 0.1) 172.7 ( 1.5) 98.6 ( 0.2) 123.4 ( 0.8) 76.8 ( 0.2) 85.7 ( 0.1) 153.6 ( 0.5) 76.2 ( 0.1)
85.8 ( 0.2) 78.9 ( 0.5) 148.9 ( 0.7) 45.2 ( 0.1) 81.0 ( 0.2) 62.1 ( 0.3)
Table 13 | Mean and std-deviation over 5 finetuning runs of PaliGemma 3B, 10B, 28B models at 224px2 and 448px2 resolutions on over 30+ academic tasks from [9]. Tasks splits, preprocessing, metrics and hyper-parameters following the 224px2 versions according to previous work. Only the learning rate has been selected per model size based on validation splits.
PaliGemma 2: A Family of Versatile VLMs for Transfer
Table 14 | Sweep of learning rates on the various tasks and model sizes at 224px2 resolution. Although we report numbers in all metrics, learning rate selection was done based on the validation split and not on the zero-shot numbers.
3e-7
6e-7
1e-6
3e-6
6e-6
1e-5
3e-5
Task
Model
AI2D (minival)
AOKVQA-DA (val)
AOKVQA-MC (val)
ActivityNet-CAP (minival)
ActivityNet-QA (minival)
COCO-35L (avg34)
COCO-35L (en)
COCOcap (minival)
ChartQA (aug) (minival)
ChartQA (human) (minival)
CountBenchQA
DocVQA (val)
GQA (minival)
InfoVQA (val)
MARVL (avg5)
3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 3B 10B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 3B 10B
61.8 80.0 81.9 59.3 67.7 69.7 76.9 83.8 83.3 26.1 28.6 43.3 49.9 110.1 115.4 116.7 137.9 140.6 142.5 146.3 148.3 148.8 60.8 69.0 66.8 41.4 50.9 48.3 82.7 88.2 87.8 37.8 42.4 42.7 70.9 73.6 73.7 21.6 33.4 36.9 69.9 86.5 86.7 62.8 70.4 44.1 49.3
67.6 82.9 82.3 62.9 68.6 70.2 78.7 83.3 84.0 28.5 31.4 46.8 52.2 111.8 115.8 116.6 138.6 140.3 141.3 146.7 149.4 149.5 64.3 68.6 63.4 42.8 50.8 46.9 82.9 84.7 88.4 37.9 40.9 42.1 72.2 74.3 73.9 22.9 33.5 36.6 73.4 88.2 88.5 66.1 71.5 47.0 51.2
70.6 85.3 83.2 64.0 68.8 69.8 79.4 83.3 85.1 28.5 30.8 49.4 53.9 113.6 115.2 115.4 139.1 139.6 140.4 145.4 148.2 149.2 66.0 71.1 65.2 42.7 50.8 47.7 82.0 85.1 88.4 37.3 42.2 43.1 72.9 74.7 74.7 23.8 33.2 36.3 77.1 89.2 89.5 67.8 75.3 48.5 51.9
75.0 84.4 85.9 64.6 66.6 69.0 80.8 82.7 82.5 30.6 31.6 52.6 55.0 113.9 113.6 114.0 138.4 137.3 137.7 147.2 148.3 149.5 69.7 69.5 66.7 44.1 49.2 46.5 79.0 82.9 88.6 39.4 44.1 45.2 73.9 74.4 74.8 25.4 33.2 36.2 81.2 89.4 90.3 67.6 74.0 51.1 53.2
76.9 82.9 85.0 63.6 64.6 66.3 77.2 79.4 82.4 30.0 30.0 53.8 55.3 113.6 112.9 112.1 137.6 135.5 134.5 147.1 147.0 148.2 69.5 69.9 66.0 43.2 47.0 45.3 82.0 81.4 86.7 40.2 41.4 42.1 73.9 74.4 74.6 25.2 32.2 35.5 83.0 89.1 90.8 72.6 66.2 52.0 53.1
75.1 82.1 83.4 59.3 57.3 60.8 76.9 75.5 78.2 30.6 31.1 53.5 54.6 113.2 112.2 111.2 136.5 133.8 133.2 147.0 146.5 145.3 68.4 68.4 64.1 42.9 44.5 41.8 78.0 78.2 83.3 38.7 39.8 40.5 73.8 74.2 74.1 25.1 29.8 34.1 82.4 87.4 89.2 74.0 69.4 51.2 52.1
68.8 69.2 75.7 52.8 50.5 51.1 63.8 56.1 58.4 29.8 28.6 52.0 51.2 111.7 111.7 109.6 133.8 132.5 129.9 142.0 143.6 145.7 63.6 60.4 55.9 35.4 34.6 33.8 70.4 65.7 69.6 32.5 29.6 30.9 72.4 71.5 72.3 22.3 21.7 25.4 69.9 67.6 76.2 68.3 67.2 49.9 49.7
MSRVTT-CAP (minival)
MSRVTT-QA (minival)
Continued on next page
PaliGemma 2: A Family of Versatile VLMs for Transfer
Table 14 | Sweep of learning rates on the various tasks and model sizes at 224px2 resolution. Although we report numbers in all metrics, learning rate selection was done based on the validation split and not on the zero-shot numbers.
3e-7
6e-7
1e-6
3e-6
6e-6
1e-5
3e-5
Task
Model
MSVD-QA (minival)
NLVR2 (minival)
NoCaps
OCR-VQA (minival)
OKVQA (minival)
RSVQA-hr (minival)
RSVQA-lr (minival)
RefCOCO (testA)
RefCOCO (testB)
RefCOCO (val)
RefCOCO+ (testA)
RefCOCO+ (testB)
RefCOCO+ (val)
RefCOCOg (test)
RefCOCOg (val)
3B 10B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B
55.2 61.1 82.5 91.8 92.2 123.3 126.7 127.5 72.6 74.7 75.5 49.4 57.8 64.6 92.8 93.3 93.1 90.7 92.3 91.8 73.1 76.7 76.2 68.0 73.8 73.0 70.4 75.1 74.6 67.6 72.9 72.7 55.3 66.0 65.3 61.3 69.8 69.0 65.5 70.9 69.9 65.2 70.8 69.9 56.1 60.9 63.0
57.8 63.9 86.2 93.0 92.8 123.6 126.1 127.5 73.1 74.5 75.5 52.3 60.5 64.4 93.2 93.2 93.4 92.4 92.7 92.1 74.5 76.9 76.7 70.1 74.3 73.9 72.1 75.6 75.0 70.1 73.5 73.4 58.6 67.1 66.4 64.2 70.8 70.0 67.2 71.6 70.5 67.0 71.4 70.4 58.8 62.9 64.4
60.7 65.4 88.2 93.3 93.6 124.0 126.0 126.5 73.4 74.3 75.2 54.3 61.3 65.4 93.3 93.1 93.3 92.7 92.0 92.4 75.3 77.1 76.8 70.8 74.3 73.8 73.0 75.8 75.2 70.8 74.0 73.4 60.5 67.3 67.1 65.8 71.1 70.4 68.4 71.6 70.8 67.8 71.4 70.2 60.4 63.8 65.2
63.3 64.2 90.4 93.3 93.7 123.4 125.2 124.0 73.4 73.9 74.8 57.6 60.8 63.8 93.0 93.0 93.3 93.3 91.7 92.7 75.5 77.2 76.8 71.2 74.2 72.8 73.2 76.1 74.8 71.8 75.0 74.0 62.9 68.4 67.5 67.0 72.0 70.8 68.7 71.7 70.7 68.0 71.4 70.2 61.5 64.0 65.5
63.1 63.2 90.9 92.5 93.7 122.5 122.1 123.0 73.2 73.5 73.9 56.2 58.7 60.6 93.3 93.4 93.3 92.1 91.8 92.9 75.8 77.1 76.6 70.8 73.4 73.1 73.3 75.6 74.6 72.2 74.9 74.3 63.2 68.2 67.8 67.9 71.8 71.0 68.9 71.3 70.6 68.0 71.0 70.1 62.3 63.9 64.3
61.3 63.0 90.2 91.7 92.2 120.5 120.5 120.3 72.9 73.0 72.5 52.9 55.6 56.8 93.4 93.3 93.3 92.2 92.8 92.9 75.8 76.1 75.5 70.9 73.4 72.0 73.4 74.9 74.0 72.7 74.2 72.9 64.6 67.9 67.0 68.6 71.3 70.4 69.0 70.4 69.7 68.2 70.0 69.2 61.2 61.2 62.6
57.0 56.3 85.9 86.1 88.0 112.3 111.5 113.0 70.6 70.6 71.0 47.2 44.1 46.4 93.3 89.4 92.9 92.3 92.0 92.3 74.1 71.6 71.6 69.7 68.6 68.4 71.6 70.6 69.9 71.0 69.0 69.3 63.8 62.6 62.7 67.5 66.5 65.7 67.2 65.2 64.9 66.1 64.9 64.0 57.0 54.8 55.7
ST-VQA (val)
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PaliGemma 2: A Family of Versatile VLMs for Transfer
Table 14 | Sweep of learning rates on the various tasks and model sizes at 224px2 resolution. Although we report numbers in all metrics, learning rate selection was done based on the validation split and not on the zero-shot numbers.
3e-7
6e-7
1e-6
3e-6
6e-6
1e-5
3e-5
Task
Model
SciCap (minival)
ScienceQA (minival)
Screen2Words (minival)
TallyQA (complex)
TallyQA (simple)
TextCaps (minival)
TextVQA (val)
VATEX (minival)
VizWizVQA (val)
WidgetCap (minival)
XM3600 (avg35)
3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B 3B 10B 28B
55.2 78.6 80.3 87.7 96.9 96.8 95.1 110.9 113.0 66.6 72.0 73.1 80.4 83.0 82.9 122.8 140.3 150.9 57.6 63.4 64.5 84.4 91.4 80.9 83.8 83.8 72.5 76.1 76.3 137.0 146.3 144.0 44.2 45.0 45.2 83.7 82.5 80.9 51.7 58.5 58.8
67.4 92.5 94.7 92.1 97.1 97.1 104.2 115.4 119.5 67.8 72.5 73.5 81.1 83.3 83.3 131.9 145.3 149.0 58.7 64.1 64.7 87.2 93.2 81.5 84.1 84.1 74.2 77.1 77.6 141.9 148.4 147.6 43.9 44.5 44.6 83.1 80.6 79.8 54.0 60.5 59.2
76.9 106.2 104.0 94.5 97.6 97.4 109.0 118.2 120.4 68.6 73.4 73.9 81.3 83.1 83.3 136.5 145.4 150.2 59.3 63.9 65.3 89.8 93.4 82.1 84.3 84.1 74.8 77.8 78.2 141.8 150.9 145.9 43.7 43.9 44.0 82.2 78.6 79.4 55.3 61.4 60.8
109.4 128.1 125.9 95.1 97.6 97.2 109.3 118.1 118.8 70.0 73.5 74.8 81.8 83.2 83.5 136.2 145.4 145.5 59.6 63.2 64.8 90.7 93.7 82.7 83.7 83.8 76.4 78.0 78.8 142.3 148.2 147.0 42.7 42.1 42.3 79.1 75.0 76.4 58.0 61.3 62.3
130.3 136.9 136.2 95.2 97.1 96.8 113.2 114.7 116.2 70.0 72.7 73.8 81.9 82.7 83.0 133.6 144.2 144.0 59.4 61.6 63.3 90.2 90.4 82.4 83.1 82.8 76.6 77.3 77.8 141.7 144.5 144.1 41.7 40.7 41.1 78.3 73.0 73.6 58.7 61.8 61.9
138.8 143.2 140.1 94.3 96.2 96.1 112.5 113.0 114.2 70.5 72.0 73.0 81.5 82.1 82.2 132.8 141.0 142.1 58.0 58.1 59.3 90.2 89.9 81.9 82.0 82.0 76.7 77.2 76.7 140.6 140.8 143.0 40.8 39.3 39.1 76.9 72.0 71.3 57.8 60.2 61.7
148.1 143.8 141.7 91.4 93.7 94.2 110.1 110.0 106.3 66.7 65.8 68.1 79.1 79.1 79.7 126.0 125.8 126.2 51.1 48.3 49.9 86.3 84.5 79.6 79.4 79.7 74.0 73.3 72.5 129.7 133.3 133.0 37.8 36.8 35.8 70.9 69.9 66.1 49.1 38.0 49.4
xGQA (avg7)
PaliGemma 2: A Family of Versatile VLMs for Transfer
224px2
448px2
Task
PG1
PG2
PG1
PG2
AI2D AOKVQA-DA (val) AOKVQA-MC (val) ActivityNet-CAP ActivityNet-QA COCO-35L (avg34) COCO-35L (en) COCOcap ChartQA (aug) ChartQA (human) CountBenchQA DocVQA (val) GQA InfoVQA (val) MARVL (avg5) MSRVTT-CAP MSRVTT-QA MSVD-QA NLVR2 NoCaps OCR-VQA OKVQA RSVQA-hr (test) RSVQA-hr (test2) RSVQA-lr RefCOCO (testA) RefCOCO (testB) RefCOCO (val) RefCOCO+ (testA) RefCOCO+ (testB) RefCOCO+ (val) RefCOCOg (test) RefCOCOg (val) ST-VQA (val) SciCap ScienceQA Screen2Words TallyQA (complex) TallyQA (simple) TextCaps TextVQA (val) VATEX VQAv2 (minival) VizWizVQA (val) WidgetCap XM3600 (avg35) XM3600 (en) xGQA (avg7)
72.1 61.1 78.5 34.6 50.8 113.7 139.2 141.9 74.2 40.0 81.9 37.8 65.6 25.5 80.6 70.5 50.1 60.2 90.0 121.7 72.3 63.5 92.6 90.6 92.6 75.7 70.7 73.4 71.9 64.5 68.3 68.2 67.7 61.6 162.3 95.4 117.6 69.6 81.7 127.5 59.0 79.7 82.1 73.7 136.1 41.9 78.0 57.3
74.7 (+2.6) 64.2 (+3.1) 79.7 (+1.2) 34.2 ( 0.4) 51.3 (+0.5) 113.9 (+0.2) 138.4 ( 0.8) 141.3 ( 0.6) 74.4 (+0.2) 42.0 (+2.0) 81.0 ( 0.9) 39.9 (+2.1) 66.2 (+0.6) 25.2 ( 0.3) 83.5 (+2.9) 68.5 ( 2.0) 50.5 (+0.4) 61.1 (+0.9) 91.4 (+1.4) 123.1 (+1.4) 73.4 (+1.1) 64.2 (+0.7) 92.7 (+0.1) 90.9 (+0.3) 93.0 (+0.4) 75.7 (+0.0) 71.0 (+0.3) 73.4 (+0.0) 72.7 (+0.8) 64.2 ( 0.3) 68.6 (+0.3) 69.0 (+0.8) 68.3 (+0.6) 61.9 (+0.3) 165.1 (+2.8) 96.1 (+0.7) 113.3 ( 4.3) 70.3 (+0.7) 81.8 (+0.1) 127.5 (+0.0) 59.6 (+0.6) 80.8 (+1.1) 83.0 (+0.9) 76.4 (+2.7) 138.1 (+2.0) 42.8 (+0.9) 79.8 (+1.8) 58.6 (+1.3)
73.3 65.7 80.3 - -
115.8 141.2 144.6 88.5 54.2 83.1 74.1 67.0 37.0 76.8 - - - 88.9 123.6 74.6 63.2 92.8 90.5 93.1 77.9 72.4 75.6 74.2 64.5 69.8 71.0 70.1 79.7 181.5 95.9 119.6 72.3 84.9 153.9 74.6 - 84.6 75.5 148.4 42.4 80.0 57.9
76.0 (+2.7) 67.9 (+2.2) 82.5 (+2.2)
115.8 (+0.0) 140.4 ( 0.8) 143.4 ( 1.2) 89.2 (+0.7) 54.0 ( 0.2) 82.0 ( 1.1) 73.6 ( 0.5) 68.1 (+1.1) 37.5 (+0.5) 82.7 (+5.9)
- -
91.6 (+2.7) 123.5 ( 0.1) 75.7 (+1.1) 64.1 (+0.9) 92.8 (+0.0) 90.7 (+0.2) 92.7 ( 0.4) 78.6 (+0.7) 73.5 (+1.1) 76.3 (+0.7) 76.1 (+1.9) 67.0 (+2.5) 72.1 (+2.3) 72.7 (+1.7) 72.3 (+2.2) 80.5 (+0.8) 183.3 (+1.8) 96.2 (+0.3) 114.0 ( 5.6) 73.6 (+1.3) 85.3 (+0.4) 152.1 ( 1.8) 75.2 (+0.6)
84.8 (+0.2) 77.5 (+2.0) 151.4 (+3.0) 43.2 (+0.8) 80.3 (+0.3) 60.4 (+2.5)
Table 15 | Comparison of PaliGemma 3B and PaliGemma 2 3B at 224px2 and 448px2 resolutions. PG1 and PG2 refer to PaliGemma [9] and PaliGemma 2, respectively.
| The article discusses "PaliGemma 2," an enhanced version of the PaliGemma vision-language model (VLM) developed by Google DeepMind. This upgrade integrates the SigLIP-So400m vision encoder with the Gemma 2 family of language models, resulting in a versatile set of models optimized for various vision-language tasks.
Key features of PaliGemma 2 include:
- Three model sizes (3B, 10B, 28B) and three resolutions (224px², 448px², and 896px²) to cater to different computational needs and performance requirements.
- A training strategy involving three stages that equips the models for fine-tuning across over 30 transfer tasks, including advanced capabilities in Optical Character Recognition (OCR), molecular structure recognition, music score recognition, and medical report generation. PaliGemma 2 achieves state-of-the-art results in many of these tasks.
- The ability to analyze how model size and resolution impact transfer performance, with findings indicating that larger models often require lower optimal learning rates.
- A commitment to open-weight model release, allowing for broader research and application.
The study emphasizes the model's versatility and effectiveness across various domains while also addressing the importance of ethical considerations and safety in AI applications. Overall, PaliGemma 2 represents a significant advancement in the capabilities of vision-language models, offering enhanced performance and a range of applications in real-world scenarios. | PaliGemma,Vision-Language Models,VLM,Transfer Learning,Deep Learning,Language Models,Computer Vision,Fine-tuning |
The Llama 3 Herd of Models | 2407.21783 | "4 2 0 2\nv o N 3 2\n] I\nA . s c [\n3 v 3 8 7 1 2 . 7 0 4 2 : v i X r a\nThe Llama 3 Herd of Models(...TRUNCATED) | "The article presents the Llama 3 models, developed by the Llama Team at Meta, which are advanced fo(...TRUNCATED) | Llama 3,foundation models,language models,AI,multilinguality,Transformer,GPT-4,machine learning |
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