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Abstract
Vector vortex beams are a kind of special beam that simultaneously carry spin and orbital angular momentum. The generation of vector vortex beams usually requires a complex and expensive optical system, which becomes a bottleneck hindering its further application. Thus, a compact, low-cost and efficient special beam generation system is demanded. In this paper, a method that can produce vector vortex beams distributed anywhere in the equator of hybrid-order Poincaré Spheres based on polarization holography is proposed. Via changing some parameters of the device, this method can also produce the scalar vortex beams distributed at any position of the basic Poincaré Sphere and the vector beams distributed at the equator of the higher-order Poincaré Spheres. The work shows that polarization holography has the potential ability to regulate the spin and orbital angular momentum simultaneously, opening a new window for future research and applications of angular momentum space orientation.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Similar to macroscopic objects, photons also have angular momentum, which consists of two parts, namely spin angular momentum (SAM) and orbital angular momentum (OAM). The spin angular momentum of light is closely related to the polarization properties, and the orbital angular momentum originates from the complex field distribution of cross-section that characterizes the wavefront properties of the light beam. In recent years, owing to the respective intriguing properties, the vector beams, [1] known as possessing spatially inhomogeneous polarization states at different positions on the same wavefront simultaneously, and the scalar vortex beams (SVBs), [2] known as possessing spiral wavefronts and homogeneous polarization distribution, have been widely studied in various aspects. Most recently, because of possessing both vector polarization and helical phase, the vector vortex beams (VVBs) have been proposed and explored in various applications, such as vectorial optical vortex filtering, [3] particle acceleration, [4] photon entanglement, [5] beam focusing, [6] the photonic spin Hall effect, [7,8] and vector vortex coronagraph. [9] Compared with pure vector beam and SVB, VVB provides more degrees of freedom in beam manipulation. [6,10,11] Driven by these advantages, a variety of generation methods have been demonstrated by using spatial light modulator, [12] liquid-crystal-based polarization converter, [13] laser resonator configuration, [14,15] modified interference of different modes, [16,17] as well as uniaxial and biaxially induced crystals. [18–20] However, these methods usually face challenges of accuracy, cost, bulky, and difficulty fabricating higher-order beams. Therefore, to generate arbitrary VVBs, a flexible generation method with high precision and compact structure should be proposed.
In 1892, a prominent geometric representation of polarization known as the Poincaré sphere (PS) is proposed to describe the polarization state of light as a point on the surface of a unit sphere. [21] The PS unifies the fundamental polarizations, where the polarization states represented by complex Jones vectors are mapped to the sphere’s surface through the Stokes parameters in the sphere’s Cartesian coordinates. This geometric characterization not only greatly simplifies the calculations of the geometric phase, but also provides a deeper insight into physical mechanisms. As a result, the representation of the PS has become an important technique to deal with the polarization evolution in different physical systems. More recently, the higher-order PS and hybrid-order PS have been proposed to describe the evolution of both polarization and phase. [22–24] The north and south poles of the higher-order PS represent the opposite spin states and orbital states. Any state on the higher-order PS can be realized by a superposition of the two orthogonal states. [25,13,26] By replacing the states at the poles with two arbitrary optical vortex beams in the higher-order PS, the generated hybrid-order PS can represent an expanded set of vector vortex beams with general spin-to-orbital conversion [27,28].
Polarization holography is attractive because it has the unique ability to record all the amplitude, [29] phase, and polarization state of an optical wave. [30,31] Compared with that described by the Jones matrix, [32–34] the polarization holography described by the tensor method can handle the situation where the angle between the signal and the reference waves is arbitrary. In recent years, the research of polarization holography based on the tensor method has made great progress. With the satisfaction of the Bragg condition, some phenomena have been reported, such as faithful reconstruction, [35–40] null reconstruction, [41–44] and inverse polarization effect reconstruction. [45] Faithful reconstruction implies that the polarization states of reconstructed wave and signal wave are the same, which means the optical information stored in the material can be read correctly. Therefore, faithful reconstruction has important applications in the fields of holographic storage, micro-nano processing, and image display. For instance, the faithful reconstruction has been applied in multi-channel recording in holographic storage, [46,47] and vector beams, [48] as well as vortex beams generation. [49] In this work, the property of faithful reconstruction is used to generate special beams, including SVBs, vector beams, and VVBs. The characteristic of this work lies in the clever design of carrying the variable phase and polarization states in the signal wave, then the reconstructed wave is confirmed that it has the characteristics of SVBs, vector beams, and VVBs. In the experiment, phenanthrenequinone doped polymethyl methacrylate (PQ/PMMA) that has the ability to record the polarization is used as a recording material. The experimental results agree with the simulation results. The advantage of the proposed method is that three special beams can be realized by simply changing the parameters of the experimental setup, respectively. And the size of the generated unfocused special beam is small and the purity is high.
In the recording stage, the phase and polarization variation of signal wave is realized by a half-wave plate (HWP), a quarter-wave plate (QWP), and a linear polarizer (P); the continuous helical phase wavefront is recorded in the material through a rotating slit even though the signal wave is not the produced special beams at all. In the reconstruction stage, the material being illuminated by the reading wave that is identical to the reference wave, the SVBs, vector beams, or VVBs can be produced. In these stages, the faithful reconstruction of polarization holography is the key, [33] since the phase and polarization state of signal wave should be faithfully recorded and reconstructed to obtain the wanted special beams. Since the recorded signal wave is not the produced special beam, this work also shows the ability of polarization holography in polarization manipulation and micro-nano processing.
2. Theoretical analysis
2.1 Faithful reconstruction of polarization holography
Polarization-sensitive anisotropic recording materials are necessary for the experimental study of polarization holography. To better understand the relationship between the photo-induced anisotropic material and the polarized light field, a simple molecular model is introduced. The molecular model assumes that the polarization-sensitive material is composed of rod-shaped molecules. As shown in Fig. 1, before exposure, rod-shaped molecules are randomly orientated in the material, and the material is isotropic macroscopically. The dielectric coefficient of the material is constant, resulting in its refractive index ellipsoid being a spherical surface. After exposure, a part of the rod-shaped molecules fractures and becomes spherical molecules. Because the distribution of molecules in the polarization-sensitive material is spatially modulated by the light field, the material exhibits anisotropic properties under the radiation of polarized light, and its refractive index ellipsoid becomes an ellipsoidal surface.
The recording material used in the experiment is a phenanthrenequinone-doped polymethyl methacrylate (PQ/PMMA) photopolymer, [50–54] which was prepared in our laboratory. The detailed preparation procedure is mentioned in Ref. 48. The material has the advantages of easy preparation and low cost.
To record the signal wave correctly, faithful reconstruction is inevitable in the experiment. In Table 1, we summarize the condition regarding the faithful reconstruction that will be employed in this work. The detailed derivation about the condition may be found in Refs. [37,40,41,55].
Table 1. Condition about faithful reconstruction in this work.
In Table 1, a and b are coefficients of s- and p-components, δ is the phase delay of the s-component relative to the p-component, θ is the interference angle, i. e. the angle between signal and reference waves. The p-polarization is defined as the electrical field vibrating along the intersection of the x-z plane and cross-section of light, and the s-polarization is parallel to the y-axis. Faithful reconstruction means that the polarization states of reconstructed wave and signal wave are the same. With the interference angle, θ, being 90 degrees as shown in the rightmost column of the table, any polarization of signal wave can be faithfully reconstructed. It can be used to fabricate special beams in this work.
2.2 Special beams
VVBs with different mode distributions can be represented by points (2χ, 2Φ) on the spherical surface of the hybrid-order PSs. [27] The Jones vector of VVBs located at the equator of the hybrid-order PS can be expressed as
Equation (1) can be written as:
According to Eqs. (2) and (3), the Cartesian coordinates of hybrid-order PS are represented by the hybrid-order Stokes vector: [40]
The relations (Eq. (4)) indicate a simple representation: Sn,m1, Sn,m2 and Sn,m3 may be regarded as the Cartesian coordinates in hybrid-order PS of radius Sn,m0, and the sphere's spherical angles (2χ, 2Φ) are given by
A VVB of any initial state with specific p- and l-values can be geometrically mapped by a hybrid-order PS. Figure 2 depicts the hybrid-order PS with m = 0 and n=−2, where VVBs distributed on the equatorial plane are shown. According to the above instructions, the point on hybrid-order PS is the combination of |Lm〉 and |Rn〉 with various coefficients and can represent any VVBs characterized by the topological charge, l = (m + n)/2, and the polarization order, p = (m-n)/2. The VVBs mentioned in this paper are located at the equator of hybrid-order PSs. Figure 3 shows the polarization characteristics of VVB at points (2χ, 2Φ) on hybrid-order PSs.
Fig. 2. Schematic illustration of hybrid-order PS with l=−1 and p = 1, where. (2χ, 2Φ) are the spherical coordinates.
Fig. 3. Polarization characteristics of VVB at point (2χ, 2Φ) on hybrid-order PSs or vector beam at point (2χ, 2Φ) on higher-order PSs.
The VVBs can be decomposed into two parts, namely the phase part and the polarization part. It can be seen from Eq. (1) that for p = 0, Eq. (1) is simplified as the light field expression of SVBs:
When l = 0, Eq. (1) is simplified as the light field expression of vector beams:
From Eq. (4), for l = 0, the Stokes vector of vector beam, called the higher-order Stokes vector, is obtained, and it is mapped by higher-order PS. [22] As shown in Fig. 4, a higher-order PS with p = 1 is represented. The polarization characteristic of a vector beam at points (2χ, 2Φ) on higher-order PSs is shown in Fig. 4. The vector beams distributed on its equator are a linear combination of radial vector beam and angular vector beam. The difference between the higher-order PS and the hybrid-order PS is that the former cannot represent beams with helical phases.
Fig. 4. Schematic illustration of the higher-order PS (p = 1). (2χ, 2Φ) are the spherical coordinates.
When p = 0, VVB becomes SVB, and its polarization state is mapped by the well-known basic PS.
The topological charges of vortex beams, including SVB and VVB, can be measured by the forked gratings of the interference pattern between VBs and a plane wave, [56] where the plane wave may have a different polarization state. For SVBs, the plane wave may be linearly polarized; however, for VVBs, the plane wave must be circularly polarized. The absolute value of the topological charge of VBs equals the number of bifurcations. The sign of the topological charge can be determined by the fork direction. The sign is positive for the upward forked direction, and vice versa. For VVBs, the l- and p-values can be derived from the interference pattern between VVB and left-handed as well as right-handed circularly-polarized plane waves. We may assume that the number of bifurcation are M and N, respectively, then l- and p-values can be derived from the relations of l = (M + N)/2 and p = (M-N)/2. Generally, the detection of vector beam only needs to analyze its polarization distribution with a P. In this work, since VVB has the characteristics of vector beam and vortex beam simultaneously, we should detect its polarization and phase distribution.
2.3 Experimental setup
The experimental setup based on polarization holography is proposed, which can be used to fabricate SVBs at any position on the basic PS, vector beams at the equator of any higher-order PS, and VVBs at the equator of any hybrid-order PS.
During the recording stage, the phase and polarization variation of signal wave is achieved by a rotating HWP and a rotating P. As well known, the Jones matrix of wave plate (WP) is [57]
And the Jones matrix of P with an angle ζ between the optical axis and the x-axis is
Jones matrices of various devices can be obtained by substituting the corresponding parameters. Therefore, the Jones matrix of the QWP with η=π/2 and γ=45° is
And the Jones matrix of the HWP with η=π is
Once the wave passes a rotating narrow slit, its phase and polarization are recorded at the corresponding azimuthal angle, resulting in the continuous phase and polarization distribution. In the experiment, HWP, P, and the slit are rotated uniformly. The azimuthal angle of the slit is defined as φ at time t, where 0≤φ<2π. The initial time may be set as t = 0, then we have φ=ωst+φs0, where ωs represents the angular velocity of the slit and φs0 is the initial azimuth angle. Similarly, the azimuthal angles of HWP and P may be expressed as φH = ωHt+φH0 and φP = ωPt+φP0, where ωH represents the angular velocity of HWP, ωP represents the angular velocity of P, φH0 and φP0 are the initial azimuth angles of HWP and P, respectively. Hence the Jones matrix of the HWP at time t is
And the Jones matrix of the P at time t is
The normalized s-polarized wave passing through the aforementioned QWP, HWP and P consecutively gives
It can be seen that Eq. (17) is consistent with the expression of VVBs, and the corresponding relationship is:
It can be seen from Eq. (18) that the values of l and p depend on the angular velocities of P, HWP, and slit. And according to the polarization order, p, and the different initial azimuth angles of the P and the slit, the obtained initial angles, ξ0, of the VVB, SVB and vector beams are also different. Different initial angles determine the position of the beam on the equator of the hybrid-order PS.
According to the above relationship, to obtain a VVB with topological charge l and polarization order p, it needs to satisfy the condition of the speed ratio ωP: ωH: ωs = p: (l-p)/2: 1. The sign of the speed indicates the direction of rotation. A positive sign means counterclockwise rotation, otherwise clockwise rotation. For different initial angles, ξ0, VVBs at any position on the equator of the hybrid-order PS can be obtained. In addition to generating VVBs, SVBs and vector beams can also be generated through this method. for p = 0, we have ωP = 0 and l = 2ωH/ωs. To get an SVB with topological charge l, the speed ratio must be met ωH: ωs = l/2: 1 Similarly, when l = 0, at the same time p≠0, the condition of ωP=−2ωH needs to be satisfied. If a vector beam with a polarization order p is to be obtained, it needs to satisfy the speed ratio ωP: ωH: ωs = p: -p/2: 1. For different initial angles, ξ0, vector beams at any position on the equator of the higher-order PS can be obtained.
3. Experiments and results
The experimental setup is designed as shown in Fig. 5 based on the concept of faithful reconstruction. The light source is a fundamental TEM00 532 nm laser, whose waist radius is about 0.75 mm. The beam is split into an s-polarized wave and a p-polarized wave through a polarization beam splitter (PBS). The s-polarized wave is used as the incident wave on the signal optical path, while the p-polarized wave is used as the reference wave. Some components on the signal optical path are loaded for phase and polarization modulation in the recording stage. On the signal path, the polarizer (P1) is adjusted along the vertical direction to ensure that the transmitted wave is s-polarized. The angle between the fast axis of the QWP1 and the horizontal direction is 45° to ensure that the polarization state is right-handed circularly polarized after the s-polarized wave passes through QWP. The HWP, P2, and slit are successively placed on the electric rotating platforms (Thorlabs, KPRM1E/M) and controlled by software to guarantee the speed ratio of the HWP and P2 to the slit. At this stage, the signal wave interferes with the reference wave and creates a grating that cannot be observed by the naked eye in the material; it does not give a detectable result. However, when the recording stage is completed, the hologram can be reconstructed by the irradiation of the reading wave, which is called the reconstruction stage. Since the slit opening is small enough, the beam will be diffracted when it passes through the slit, resulting in an imperfect image of the slit when the wave reaches the material. Hence, we introduce a group of 4F imaging systems into the experimental optical path to improve the imaging quality of the experimental results. In the reconstruction stage, with the material being illuminated by a reading wave that is identical to the reference, special beams, such as VB, vector beam, or VVB, are generated. The cubic-shaped recording material with a size of 10mm×10mm×30mm is used, which ensures that the angle between the signal and reference waves is 90°. From Table 1, for this particular angle, the faithful reconstruction is achieved regardless of the exposure time since its expression is independent of the A/B value.
Fig. 5. Experimental setup for generating special beams, where HWP is half wave plate, QWP is quarter wave plate, P is polarizer, L is lens. The material is cubic-shaped polarization-sensitive polymer material (PQ/PMMA). The setup for the upper point is used to prepare VVBs and vector beams, and the setup in the lower-left corner is used to prepare SVBs. The main difference between them is whether P2 is rotated.
Based on the above experimental setup, four SVBs with different l values and three VVBs with different l- and p-values are prepared, which proved the feasibility of the setup. The experimental parameters are shown in Table 2 and Table 3, respectively. In the experiment regarding generating SVBs, the powers of signal and reference waves are about 660 µW and 9.6 mW, respectively. Figure 6 shows the intensity distribution of simulated and experimentally prepared SVBs as well as the interference results between the experimentally prepared SVBs and the plane wave. The results show that the experimentally prepared SVBs do have helical phases. Because the initial azimuth angle of the P2 is 45°, the polarization states of all generated SVBs correspond to the polarization state at the position of the basic PS point (π/2, 0).
Fig. 6. Simulation results, experimental results, and experimental interference patterns of l=−2, −1, +1, and +2 of SVBs at (π/2, 0) of the basic PS.
Table 2. Experimental parameters and power corresponding to different SVBs generated in the experiment.
Table 3. Experimental parameters and power corresponding to different beams generated in the experiment.
In the experiments producing vector beams or VVBs, the power of signal wave is about 600 µW, and the powers of reference wave and reading wave are about 26.3 mW. The verification of the reconstructed wave is divided into two steps. By passing through a P with a different orientational angle, the polarization distributions of the reconstructed beams are characterized and validated. In Fig. 7, we show the intensity distribution of the VVB with l=−1 and p = 1 as well as its intensity distribution after passing through the P in the directions of 0°, 30°, 60°, 90°, 120°, and 150°. Since the initial azimuth angle of the P2 and the slit in the experiment is different by π/3, that is, ξ0=π/3, the VVB located at the point (2π/3, 0) on the hybrid-order PS of l=−1 and p=+1 is prepared. The intensity distribution is compared with the simulation result. In addition, the VVB interferes with the left- and right-handed circularly polarized plane waves respectively. As stated in the second session, the l- and p-values of the reconstructed wave can be deduced according to the number and direction of the bifurcation. Thus, from the interference results shown in Fig. 7, we could obtain the l- and p-values. The observed patterns indicate that the generated beam indeed has an inhomogeneous polarization distribution and a helical wavefront. In Figs. 8 and 9, another two VVBs with different l- and p-values are shown. All these experimental results are satisfying and show the huge potential of polarization holography in polarization manipulation and micro-nano processing.
Fig. 7. Results of the VVB at (2π/3, 0) on the sphere of a hybrid-order PS (l=−1 and p = 1). Experimental and simulated results for a different orientational P. Results on the right are forked gratings of the experimental VVB interfered with the right- and left-handed circularly-polarized plane waves, respectively.
Fig. 8. Results of the vector beam at (4π/3, 0) on the sphere of a higher-order PS (p = 1). Experimental and simulated results for a different orientational P. Results on the right are forked gratings of the experimental VVB interfered with the right- and left-handed circularly-polarized plane waves, respectively.
Fig. 9. Results of the VVB at (π, 0) on the sphere of a hybrid-order PS (l=−2 and p = 1). Experimental and simulated results for a different orientational P. Results on the right are forked gratings of the experimental VVB interfered with the right- and left-handed circularly-polarized plane waves, respectively.
Note that the shooting positions of the experimental results for SVBs and VVBs in this article are different. As can be seen from the optical path diagram, the device adopts 4f imaging systems, which aims to reduce the influence of the poor beam quality caused by the deformation of the beam after free propagation in space. Likewise, a 4f imaging system is used to observe during the reconstruction process. The experimental results of the SVBs are taken at z = 75 mm from the focal plane of the 4f imaging system. The purpose is to prove that the beam with orbital angular momentum generated by this method does have a phase singularity, and the size of the singularity corresponding to different l-values is different. The results of the VVBs are taken at the focal plane of the 4f imaging system, therefore, the polarization and phase singularities are not obvious in the experimental results in Figs. 7–9; when the results are shot at the non-focal position of the imaging position, the singularities are relatively clear. Figure 10 shows the size of the corresponding singularity for the VVB with p = 1 and l = 1 at different observation positions. It can be seen that the larger the distance, the larger the singularity. When z = 0, as shown in Figs. 7–9, they also have singularities though it is difficult to observe due to the resolution.
4. Conclusion
Generating special beams via ordinary components has fundamental and technical significance for photon-related research. Polarization holography has a great prospect in optical storage and optical imaging, which opens up a broad application field for simultaneously regulating the amplitude, phase, and polarization of waves. The proposed method provides an unusual method, based on polarization holography to generate special beams carrying orbital angular momentum and having inhomogeneous polarization distribution, which will further stimulate the pursuit of new applications in polarization holography. Special beams (like vector beams, SVBs and VVBs) are obtained by setting different speeds of the rotating components. The system of fabricating special beams based on polarization holography may bring a different method in the beam generation and greatly reduce the volume and cost of the experimental setup. In theory, any l-value SVB or any l- and p-value VVB can be generated in this way. While in the preparation of VVBs with higher l and p values, the experimental results depend on proper control of the experimental parameters as well as the stability and uniformity of the material. In the preparation of VVB with higher l and p, the rotation speed must be increased, thus the exposure speed of the material should be improved, too. However, it may be somewhat difficult at present time. In the paper, some issues such as quantification of the beam quality and exposure are not fully presented or characterized. Perhaps the details of these could possibly improve the impact of this technique. In addition, polarization holography that has the ability to simultaneously regulate the OAM and SAM is demonstrated in this work. The polarization states of VVBs are a linear combination of orthogonal circularly polarized optical vortex beams of opposite topological charge, where the constituent components are eigenstates of the total optical angular momentum per photon of J in quantum mechanics. Moreover, J can be regarded as the result of spin-orbit coupling. The proposed method has the basic ability to carry out angular momentum space orientation research, which opens a new window for future research and application.
Funding
National Key Research and Development Program of China (2018YFA0701800); Project of Fujian province major science and technology (2020HZ01012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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