Polarization grating fabricated by recording a vector hologram between two orthogonally polarized vector vortex beams

Moritsugu Sakamoto; Yuki Nakamoto; Kotaro Kawai; Kohei Noda; Tomoyuki Sasaki; Nobuhiro Kawatsuki; Hiroshi Ono

doi:10.1364/JOSAB.34.000263

1. INTRODUCTION

Vector beams (VBs), which have spatially variant polarization patterns and polarization singularities [1], have attracted considerable research attention because they have several unique features, including a longitudinal electric field [2,3], a three-dimensional hollow electric field region [4,5], and optical needle [6] and optical chain properties [7]. Using these attractive features, VBs have been applied to several technologies, including particle delivery processes [7], particle acceleration [8], tight focusing [9], optical ablation [10], high-resolution microscopy [11–13], high-efficiency optical tweezers [14], and space-division multiplexed optical communications [15–17]. Recently, the vector vortex beam (VVB), which has not only a polarization singularity but also a phase singularity related to its orbital angular momentum, which is well known as an optical vortex, has also been attracting attention because it demonstrates different optical properties to those of VBs [1,18,19].

In general, in experimental studies of VBs and VVBs, the spatial polarization modulation technique is required to convert the homogeneous state of polarization (SOP) into an inhomogeneous SOP. Various approaches to generate VBs and VVBs have therefore been proposed, and these approaches can be classified into two types: active or passive methods. Active-type methods generate VBs directly from laser beams using specific cavity system designs [20–23]. In contrast, passive-type methods convert the polarization distributions of homogeneously polarized light beams using external polarization modulation elements or systems, such as spatially variant polarization elements [9,18,24,25] and interferometric systems [26–28]. As an alternative, polarization gratings (PGs), in which the optical anisotropy is spatially and periodically modulated on the grating elements, also have the potential to be used as passive-type VB/VVB generation elements/systems because PGs can multiply control the amplitudes, phases, propagation directions, and SOPs of incident light waves corresponding to an anisotropic distribution. While the SOP of a light beam that has been diffracted from a conventional PG is homogeneous, few approaches have been reported for VB/VVB generation that use PGs [29–32]. Fu et al. reported a method for the modulation of the spatial phase distribution between the orthogonal polarization components of an incident light wave using a PG system that included two spatial light modulators and demonstrated the arbitrary generation of VBs at each diffraction order [32]. Additionally, we have investigated PGs that were fabricated by recording vector holograms using radially polarized light in azo-dye-doped polymer films under various different conditions [29–31] and found that a hologram that was fabricated using an orthogonally polarized pair of radially polarized light beams can convert homogeneously polarized light into radially polarized light. Therefore, a vector hologram would be applicable to the fabrication of a new VB/VVB generator/converter by extension of the above concept into vector holograms using VVBs. Additionally, to the best of our knowledge, vector hologram fabrication using VVBs has not been investigated to date. In this paper, based on this background, we investigate the vector hologram located between two orthogonally polarized VVBs and fabricate PGs that have VB/VVB conversion properties. As a result, we determined that PGs that were fabricated by recording vector holograms of VVBs can generate VVBs, VBs, optical vortices, and ring-shaped optical lattices by simply controlling the SOP of the incident laser beam.

2. THEORY

In this section, we theoretically analyze the polarization-based holographic recording process using two orthogonally polarized VVBs. First, using the Jones matrix formula, the electric fields of the two VVBs are defined as

E_{VVB 1} = [\begin{matrix} \cos p θ \\ \sin p θ \end{matrix}] \exp [i ℓ_{1} θ],

where

θ

is the azimuth angle,

p

is the polarization rotational symmetry of the VVB around the optical axis, and

ℓ_{1}

and

ℓ_{2}

are the topological charges (TCs) of the phase singularity. We then consider the case where these two VVBs are superposed with crossed angles of

2 φ

, as shown in Fig. 1. To simplify the discussion, we first consider the case where

p = 0

, in which the two superposed optical vortices have orthogonal linear polarization states, i.e., 0 deg and 90 deg linear polarizations (LPs). In this case, the superposed electric field can be described as

E_{VVB 1}^{p = 0} + E_{VVB 2}^{p = 0} = [\begin{matrix} \exp [i (ℓ_{1} θ + k x \sin φ)] \\ \exp [i (ℓ_{2} θ - k x \sin φ)] \end{matrix}],

where

k

is the VVB wave number. Note here that both VVBs have uniform amplitude distributions. Based on Eq. (2), the normalized Stokes parameters of this electric field can be given:

S_{0} = 1,

where

δ ℓ = (ℓ_{1} - ℓ_{2})

and

ξ = δ ℓ θ + 2 k x \sin φ

. These Stokes parameters are described using

x^{'} y^{'}

coordinates, which are tilted from conventional

x y

coordinates by 45 deg. The tensor of the photoinduced phase changes can be expressed as follows using Stokes parameters [29–31]:

[\begin{matrix} α S_{0} + β S_{1} & β S_{2} \\ β S_{2} & α S_{0} - β S_{1} \end{matrix}],

where

α

and

β

are the isotropic and anisotropic susceptibilities, respectively. By substituting Eqs. (3a)–(3c) into Eq. (4), the Jones matrix of the VVB hologram can then be expressed as

T = R (- γ) [\begin{matrix} e^{i \frac{2 π}{λ} (α + β) d \cos ξ} & 0 \\ 0 & e^{i \frac{2 π}{λ} (α - β) d \cos ξ} \end{matrix}] R (γ) \propto R (- γ) [\begin{matrix} e^{i Δ ϕ \cos ξ} & 0 \\ 0 & e^{- i Δ ϕ \cos ξ} \end{matrix}] R (γ),

where

Δ ϕ = π Δ n d / λ

,

Δ n = 2 β

is the birefringence that is induced by the illuminating polarized light beam,

d

[m] is the thickness of the hologram, and

λ

is the probe beam wavelength.

R

is the rotation matrix and is given as

R (γ) = [\begin{matrix} \cos γ & \sin γ \\ - \sin γ & \cos γ \end{matrix}],

where

γ

is the rotation angle. Equation (5) indicates that the fast axis of the photoinduced anisotropy is radially distributed around the center axis because of the polarization azimuth angle distribution of the illuminating polarized field, and its birefringence is thus dependent on the phase difference distribution between the two superposed VVBs. In the case where

p = 0

, the direction of the fast axis rotates homogeneously by

γ = π / 4

because of the tilt angle from the original

x y

-coordinates. In contrast, in the case where

p \neq 0

, the direction of the fast axis rotates by

γ = p θ + π / 4

because the polarization azimuth of the recording electric field rotates around the optical axis because of the space-variant polarization distribution of the VVB. Using a Bessel function, Eq. (5) can be Fourier-expanded into

T_{m} = \sum_{m = - \infty}^{\infty} i^{m} R (- γ) [\begin{matrix} J_{m} (Δ ϕ) & 0 \\ 0 & J_{m} (- Δ ϕ) \end{matrix}] R (γ) e^{i m ξ} .

Here, we focus on the

\pm 1

st-order components, for which the respective transmission Jones matrixes are given as

T_{\pm 1} = \pm i J_{\pm 1} (Δ ϕ) [\begin{matrix} \cos 2 γ & \sin 2 γ \\ \sin 2 γ & - \cos 2 γ \end{matrix}] e^{\pm i ξ} = \mp \frac{1}{2} J_{\pm 1} (Δ ϕ) e^{i ((\pm δ ℓ + 2 p) θ \pm 2 k x \sin φ)} [\begin{matrix} 1 & - i \\ - i & - 1 \end{matrix}] \pm \frac{1}{2} J_{\pm 1} (Δ ϕ) e^{i ((\pm δ ℓ - 2 p) θ \pm 2 k x \sin φ)} [\begin{matrix} 1 & i \\ i & - 1 \end{matrix}] .

Here, we consider the case in which linearly polarized light passes through this PG. The Jones vector of the LP light, which has a polarization azimuth of

η

, can be expressed as

{| L P ⟩}_{η} = [\begin{matrix} \cos η \\ \sin η \end{matrix}] = \frac{1}{\sqrt{2}} (e^{i η} | R ⟩ + e^{- i η} | L ⟩),

where

| L ⟩ = {(1, i)}^{T} / \sqrt{2}

and

| R ⟩ = {(1, - i)}^{T} / \sqrt{2}

express the Jones vectors of the left- and right-handed circular polarizations (LCP and RCP), respectively. By substituting Eq. (9) into Eq. (8), the Jones vectors of the

\pm 1

st-order diffracted light beams can be given as

E_{+ 1}^{LP} = - \frac{1}{\sqrt{2}} J_{+ 1} (Δ ϕ) e^{i 2 k x \sin φ} \times (e^{i (δ ℓ + 2 p) θ} e^{- i η} | R ⟩ - e^{i (δ ℓ - 2 p) θ} e^{i η} | L ⟩),

From Eq. (10a), we found that the

+ 1

st-order diffracted light is formed by the superposition of the RCP optical vortex with

ℓ = δ ℓ + 2 p

and the LCP optical vortex with

ℓ = δ ℓ - 2 p

. In contrast, based on Eq. (10b), the

- 1

st-order diffracted light is formed by the superposition of the RCP optical vortex with

ℓ = - δ ℓ + 2 p

and the LCP optical vortex with

ℓ = - δ ℓ - 2 p

. In Eqs. (10a) and (10b), the helical phase terms

\exp [\pm i 2 p θ]

and

\exp [\pm i δ ℓ θ]

are classified into the geometric phase and the dynamic phase caused by the radially distributed fast axis and the birefringence distribution, respectively. Therefore, the sign of

\exp [\pm i 2 p θ]

is dependent on the handedness of the incident circular polarization, whereas the sign of

\exp [\pm i δ ℓ θ]

is independent of that parameter. Equations (10a) and (10b) can be arranged to give

E_{+ 1}^{LP} = - i J_{+ 1} (Δ ϕ) e^{i 2 k x \sin φ} [\begin{matrix} \sin (2 p θ - η) \\ - \cos (2 p θ - η) \end{matrix}] e^{i δ ℓ θ},

From Eqs. (11a) and (11b), we see that each diffracted VVB has a phase singularity of

\pm δ ℓ

. The polarization azimuths of the diffracted VVBs rotate

2 p

times around the optical axis. Therefore, variable VVBs can be generated through arrangement of a combination of

p

,

ℓ_{1}

, and

ℓ_{2}

. Additionally, the amplitude ratios of the LCP and RCP components of the diffracted VVBs can be controlled by adjusting the ellipticity angle of the polarized probe beam. The polarization pattern can also be rotated by varying the polarization azimuth angle

η

of the incident light beam. When circularly polarized light passes through the hologram, the Jones vectors of the diffracted light beams are given as

E_{+ 1}^{LCP} = - \frac{1}{\sqrt{2}} J_{+ 1} (Δ ϕ) e^{i 2 k x \sin φ} e^{i (δ ℓ + 2 p) θ} e^{- i η} | R ⟩,

where the superscripts and subscripts indicate the incident polarization and the diffraction order, respectively. From Eqs. (12a)–(12d), we found that optical vortices with

ℓ = δ ℓ + 2 p, - δ ℓ + 2 p, δ ℓ - 2 p

, and

- δ ℓ - 2 p

are generated under the incidence of light with circular polarization.

Fig. 1. Schematic of coordinates used for holographic recording of two orthogonally polarized vector vortex beams in the case where $(p, ℓ_{1}, ℓ_{2}) = (1, 0, 0)$ .

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3. EXPERIMENTAL DEMONSTRATION

To verify the feasibility of the approach described theoretically in the previous section, we performed experiments. In these experiments, we used a photo-crosslinkable polymer liquid crystal (LC) with 4-(4-methoxycinnamoyloxy)biphenyl side groups (P6CB) as a polarization-sensitive material. The P6CB was synthesized in our laboratories, and the material was described in detail in a previous paper [33]. The P6CB coating was applied to the glass substrate by spin coating. The P6CB film thickness was set at approximately 300 nm.

In the experiments, we recorded two types of vector holograms, which were the cases of $(p, ℓ_{1}, ℓ_{2}) = (2, 2, 0)$ and $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ . To generate the VVBs, we used an axially symmetric polarizer (ASP) and an axially symmetric half-wave plate (AHP) made from a photonic crystal (Photonic Lattice Inc., Japan). The ASP and AHP are spatially variant polarization elements in which both the transmission and the fast axis are radially distributed on the elements [24,34,35]. The optical setup used to record the vector hologram between two orthogonal VVBs is shown in Fig. 2. Both elements were optimized at 325 nm. In this system, a linearly polarized UV laser beam that was emitted from an He-Cd laser (IK3501R-G, Kimmon Kouha Inc., Japan) operating at 325 nm was first passed through the spatial filter, and the emerging light beam was then collimated using a convex lens. The SOP of the collimated laser beam was converted into an LCP using the linear polarizer (P) and the quarter-wave plate ( ${QWP}_{1}$ ), and the beam was then incident on the ASP. Based on the functions of the ASP [34,35], the emerging light is converted into a VVB with $p = 1$ and $ℓ = 1$ . This VVB can be considered to be the superposition of the RCP optical vortex with $ℓ = 2$ and the LCP plane wave with $ℓ = 0$ . These LCP and RCP optical vortex components are then passed through ${QWP}_{2}$ and converted into 0 deg and 90 deg LPs. These 0 deg and 90 deg LP components are then spatially separated by a polarizing beam splitter and illuminate the AHP at the same position with crossing angle $2 φ$ . Because of the functions of the AHP [24,35], these polarization components are converted into ${VVB}_{1}$ and ${VVB}_{2}$ , respectively, and their electric fields are given by $E_{VVB 1} = {(\cos 2 θ, \sin 2 θ)}^{T} \exp [i 2 θ]$ and $E_{VVB 2} = {(\sin 2 θ, - \cos 2 θ)}^{T}$ , respectively, i.e., the case where $(p, ℓ_{1}, ℓ_{2}) = (2, 2, 0)$ . These VVBs are finally imaged on the P6CB film using a 4f system. Additionally, to avoid Fresnel diffraction effects [36], we also image between the ASP plane and the AHP plane. The exposure dose was set at $600 mJ / {cm}^{2}$ . By removing the ASP and ${QWP}_{2}$ , as indicated by the dashed line, we also generate two VVBs with electric fields of $E_{VVB 1} = {(\cos 2 θ, \sin 2 θ)}^{T}$ and $E_{VVB 2} = {(\sin 2 θ, - \cos 2 θ)}^{T}$ , i.e., the case where $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ . The crossing angle between the two VVBs was set at approximately $2 φ = 6 \deg$ .

Fig. 2. Schematic of experimental setup. M: mirror; P: polarizer; QWP: quarter-wave plate; ASP: axially symmetric polarizer; AHP: axially symmetric half-wave plate; PBS: polarizing beam splitter.

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We also investigated the polarization diffraction properties of the fabricated PGs. In the experiments, an He–Ne laser operating at 632 nm was used to provide the probe beam, and its SOP before incidence on the PG was controlled using a linear polarizer and wave plates. The diffracted light beams were observed using a CMOS imaging camera (LaserCam HR; Coherent, Inc.). The normal distance between the PGs and the observation plane was set at 70 cm. To analyze the resulting polarization pattern, we also observed six polarization component images, i.e., those of the 0 deg LP, 45 deg LP, 90 deg LP, 135 deg LP, LCP, and RCP, using a quarter-wave plate and a linear polarizer.

4. RESULTS AND DISCUSSION

Figure 3 shows polarization optical microscope images of the fabricated PGs. Figures 3(a) and 3(b) show the cases where $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ and $(p, ℓ_{1}, ℓ_{2}) = (2, 2, 0)$ , respectively. In both cases, the fabricated PGs have same radial anisotropy because the fast axis rotates around the center as a result of the polarization azimuth distributions of the recorded VVBs. Additionally, in the case of Fig. 3(a), the PG has a stripe-shaped fringe pattern with a grating vector that is oriented in the transverse direction. In contrast, in the case of Fig. 3(b), the PG has a fork-shaped grating pattern for which the grating vector is oriented in the transverse direction. These fringe patterns are caused by the phase difference between the two superposed VVBs.

Fig. 3. Polarizing optical microscope images of PGs observed under crossed Nicols conditions. (a) $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ . (b) $(p, ℓ_{1}, ℓ_{2}) = (2, 2, 0)$ .

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The intensity patterns of the diffracted light beams are shown in Figs. 4 and 5. Figures 4 and 5 correspond to the cases where $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ and $(p, ℓ_{1}, ℓ_{2}) = (2, 2, 0)$ , respectively. In both figures, parts (a) and (b) correspond to the $+ 1$ st and $- 1$ st diffraction orders, respectively. The theoretical images were calculated using numerical simulations based on a combination of the Jones matrix method and Fresnel diffraction theory. All images are normalized to enhance their contrast. The experimental and theoretical images of the diffracted light beams agree well with each other. The LP components show petal-like patterns due to their spatially variant polarization distributions. The LCP and RCP components show doughnut-shaped patterns because they are optical vortices, as described by Eqs. (10a) and (10b). These results indicate that VVBs were generated for $(p, ℓ) = (4, 0)$ [Figs. 4(a) and 4(b)], $(p, ℓ) = (4, 2)$ [Fig. 5(a)], and $(p, ℓ) = (4, - 2)$ [Fig. 5(b)]. Therefore, the fabricated PGs can convert homogeneously polarized light beams into VVBs.

Fig. 4. Intensity patterns of $\pm 1$ st orders that were diffracted from the PG shown in Fig. 3(a) [ $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ ]. (a) and (b) show the results for the $+ 1$ st and $- 1$ st orders, respectively. A marker is shown in each image to indicate the polarization analyzer used, as follows: $N$ : no analyzer; $\to$ : 0 deg LP; $↗$ : 45 deg LP; $↑$ : 90 deg LP; $↖$ : 135 deg LP. The normal distance between the PG and the observation plane is 70 cm.

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Fig. 5. Intensity patterns of $\pm 1$ st orders that were diffracted from the PG shown in Fig. 3(b) [ $(p, ℓ_{1}, ℓ_{2}) = (2, 2, 0)$ ]. (a) and (b) show the results for the $+ 1$ st and $- 1$ st orders, respectively. A marker is shown in each image to indicate the polarization analyzer used, as follows: $N$ : no analyzer; $\to$ : 0 deg LP; $↗$ : 45 deg LP; $↑$ : 90 deg LP; $↖$ : 135 deg LP. The normal distance between the PG and the observation plane is 70 cm.

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Figure 6 shows the diffracted light images of the $+ 1$ st order that were observed by changing the ellipticity angle $ε$ of the probe beam before it was incident on the PG. All images were filtered using a linear polarizer. Figures 6(a) and 6(b) correspond to the cases where $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ and $(p, ℓ_{1}, ℓ_{2}) = (2, 2, 0)$ , respectively. All images are again normalized to enhance their contrast. As shown in Fig. 6, the diffraction pattern is continuously transformed between a doughnut pattern and a petal pattern, depending on the value of $ε$ , because the amplitude ratio between the LCP and RCP components is varied. In contrast, Fig. 7 shows the images that were produced by changing the azimuth angle $η$ of the probe beam before it was incident on the PG for the case where $(p, ℓ_{1}, ℓ_{2}) = (2, 0, 0)$ . A petal-like pattern is continuously rotated around the optical axis depending on $η$ , because $η$ in Eq. (11a) is varied by the rotation of the polarization azimuth of the probe beam. In addition, for the case where the PG of Fig. 3(b) is shown in the movie in Visualization 1 (the ellipticity angle of the incident probe beam is $ε = - 15 \deg$ ), an obtained image is also continuously rotated around optical axis while maintaining the petal-like pattern. Therefore, the polarization pattern of the generated VVB can be controlled by varying the ellipticity and the azimuth angle of the polarized probe beam before incidence on the PG.

Fig. 6. Ellipticity angle dependence of $+ 1$ st-order diffracted light. (a) and (b) correspond to the cases of use of the PGs shown in Figs. 3(a) and 3(b), respectively. $ε$ is the ellipticity angle of the probe beam before incidence on the PG. All images are filtered using a linear polarizer, i.e., the images obtained correspond to the 0 deg linear polarization components.

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Fig. 7. Polarization azimuth angle dependence of $+ 1$ st-order diffracted light in the case where the PG of Fig. 3(a) is used. $η$ indicates the polarization azimuth angle of the probe beam before incidence on the PG (where the ellipticity angle of the incident probe beam is $ε = 0 \deg$ ). All images are filtered using a linear polarizer, i.e., the images obtained correspond to the 0 deg linear polarization components. See Visualization 1 for the case where the PG of Fig. 3(b) is used (the ellipticity angle of the incident probe beam is $ε = - 15 \deg$ ).

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The PGs that were fabricated by recording vector holograms between two orthogonally polarized VVBs were formed with a radially distributed fast axis and a fringe-shaped birefringence pattern. These two factors act as a spatial phase shifter based on the geometric phase term $\exp [\pm i 2 p θ]$ and a dynamic phase term $\exp [\pm i δ ℓ θ]$ , respectively. The sign of the geometric phase is dependent on the handedness of the circular polarization, and thus, the polarization diffraction properties can be controlled by varying the SOP of the incident beam.

As shown in Fig. 3, there are incomplete LC orientation regions at the center of the fabricated PG. This incomplete region is caused by a misalignment between two superposed VVBs on the holographic recording process of Fig. 2. To fabricate the ideal PG, it is required that the polarization singularities of the illuminated two VVBs are accurately superposed at same position of the sample plane. However, in our experiment, there are small spatial shifts between respective polarization singularities due to the difficulty of optical alignment, and hence, the fabricated PG has an incomplete LC orientation region at the center. The distance between the respective singularities is estimated to be about 100 μm. This incomplete region affects the quality of intensity and polarization distribution of diffracted VVBs as distortion. Although, in our experimental results, this influence is quite small, as shown in Figs. 4–7, and an accurate optical setup is required to fabricate PGs for generating higher-quality VVBs.

From another perspective, because a VVB consists of LCP and RCP optical vortices with different TCs, a VVB that passes through a linear polarizer can be regarded as an interference pattern of two co-propagating optical vortices. This type of interference pattern was called a “ring-shaped optical lattice” by Arnold et al. [37] and has been applied to optical trapping [38] because its beam profile can be transformed and rotated by varying the amplitude ratio and the relative phase difference between the two interfering optical vortices. When used in combination with a linear polarizer, our PG can also generate ring lattices that can be flexibly transformed and rotated by controlling the SOP of the incident beam, as shown in Figs. 6 and 7, because the two interfering optical vortices are converted from the LCP and RCP components of the incident beam. A similar approach using a polarization-selective optical vortex converter has already been proposed by Sakamoto et al. for flexible ring-lattice generation [35]. Unlike the method of Ref. [35], our PG can generate a ring lattice using only a single PG and a linear polarizer and can thus be used as a compact ring-lattice generation system. However, we must note that the conversion efficiency of our PG is limited to approximately 17% for the generation of a ring lattice. From Eq. (8), diffraction efficiencies of $\pm 1$ order are theoretically obtained by ${| J_{1} (Δ ϕ) |}^{2}$ , so the theoretical maximum efficiency is 33.9%, which is a peak value of ${| J_{1} (Δ ϕ) |}^{2}$ . Since the parameter of $Δ ϕ$ depends on the factors of film thickness $d$ and the photo-induced birefringence $Δ n$ of PG, we can improve the diffraction efficiencies of both first-order lights to 33.9% by optimizing polarization-sensitive materials. For generating ring lattices using our PG, diffracted VVBs have to pass through the linear polarizer. Therefore, the conversion efficiency is limited to approximately 17%, whereas the conversion efficiency of the method described in Ref. [35] is 50%.

5. CONCLUSION

In this paper, we have described PGs that were fabricated by recording a vector hologram between two orthogonally polarized VVBs. Their spatial polarization diffraction properties were theoretically analyzed using Jones calculus. The feasibility of the principle was demonstrated experimentally by recording polarization holograms in photo-crosslinkable liquid crystal polymers, and the results showed good agreement with those of the numerical simulations. Our PGs can generate VVBs, vector beams, optical vortices, and ring-shaped optical lattices by simply controlling the state of polarization of the incident laser beam and would thus be applicable to optical communications and optical manipulation, in which they would act as vortex generators and/or converters.

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Polarization grating fabricated by recording a vector hologram between two orthogonally polarized vector vortex beams

Abstract

1. INTRODUCTION

2. THEORY

3. EXPERIMENTAL DEMONSTRATION

4. RESULTS AND DISCUSSION

5. CONCLUSION

REFERENCES

Supplementary Material (1)

Cited By

Figures (7)

Equations (20)

Journal of the Optical Society of America B