1. INTRODUCTION

The optical vortex (OV), which has a spiral wavefront and a phase singularity at its optical axis, has attracted attention in recent decades because of its unique optical properties, such as carrying orbital angular momentum (OAM) and dark hole formation at the phase singularity [1]. The OV spiral phase is described by $\exp [i\ell \theta ]$, where $\theta$ is the azimuth angle and $\ell$ is the topological charge (TC) that corresponds to the OV phase ramp. Various applications using the unique features of OVs have been proposed in several research fields, including biology [2], astronomy [3], spectroscopy [4], classical and quantum communications [5–7], and optical processing [8].

In general, for experimental study of OVs, an external spatial phase modulator is required to convert the wavefront of a light wave between planes into a spiral shape, and thus various approaches for OV generation have been proposed [9–16]. The liquid-crystal-on-silicon spatial light modulator (LCOS-SLM) is a well-known OV generator in which the effective refractive indices of the liquid crystals (LCs) can be locally controlled using an electrode array. The wavefront conversion properties of LCOS-SLMs are highly flexible, allowing them to be used for applications such as optical communications, which required the switching functions. Space-variant polarization elements have also attracted considerable attention in recent years [10–12]. The principle of wavefront conversion is based on the geometric phase, which is also known as the Pancharatnam phase and the Berry phase, and which is caused by the polarization conversion process when a light wave passes through a polarizing optical system [17,18]. The geometric phase is independent of the wavelength of the light waves, and thus has achromatic wavefront conversion properties. It has therefore been applied to astronomical coronagraphs and ultra-fast optics for treatment of broadband light sources [19,20]. In a recent study, however, we found that space-variant three-dimensional anisotropic structures, which differ from previously reported anisotropic structures, also show OV generation properties [21]. These anisotropic structures are passively formed in functionalized LCs by recording vector beams, which have cylindrical symmetry in their polarization distributions. In our previous report, OV generation with $\ell =\pm 2$ was demonstrated using a system consisting of a functionalized LC cell and a vector beam illuminator, and we have thus considered the use of this system to produce new types of OV generator.

Reflection on the history of OV research shows that advancement in the functions of OV generation methods also lead to evolution in OV application fields. In particular, flexibility and achromaticity are both important factors for OV generator functionality. However, because the wavefront conversion principle of the LCOS-SLM is based on phase delay, the converted wavefront is strongly dependent on the wavelength. In contrast, in the case of space-variant polarization elements, the phase distribution of the converted light is determined by the two-dimensional anisotropic distribution and the incident polarization states. Because the anisotropic distribution of the space-variant polarization elements is fixed on the elements, flexible control of the amplitude and phase distribution of the converted OVs is difficult. In contrast, while our OV generation system consisting of functionalized LCs and a vector beam illuminator has been shown to have OV generation properties, its wavefront conversion properties have only been demonstrated for generation of monochromatic OV with $\ell =\pm 2$. Moreover, its wavelength dispersion property has not been investigated in [21]. Accordingly, in this paper, focusing on the ability of flexibility and achromaticity for generating OV, we experimentally and theoretically analyze the wavefront conversion properties of our system to show that it can be used as an OV generator. As a result, we have found that our system enables generation of even-numbered OVs with high purity for a broadband spectrum whose TC can be controlled by changing a polarization pattern of the recording vector beam.

2. EXPERIMENT

Figure 1(a) shows a schematic of the configuration used for OV generation using functionalized LCs. This system consists of two crossed polarizers, a uniaxially aligned azo-dye-doped liquid crystal (ADDLC) cell, and an external vector beam illuminator. The LCs in the ADDLC cell are initially aligned by the anchoring energy of rubbed substrates. The principle for creation of three-dimensional twisted anisotropic structures in ADDLC cells is explained as follows. When a polarized electric field is incident on the uniaxially aligned ADDLC, its polarization state is continuously modulated in the propagation direction because of the initial anisotropy of the LCs [22]. Because azobenzene molecules align in the orthogonal direction of the polarization azimuth of the illuminating light because of trans-cis-trans isomerization, the directors of the LCs in the ADDLC cell are three-dimensionally reoriented by the polarization distribution of the recording light. Figure 1(b) shows a model of the LC director reorientation process [23,24]. As shown in Fig. 1(b), the director of a reorienting LC can be described by ${\mathbf{n}}_{\mathrm{re}}=(\cos ({\theta }_0+{\theta }_{\mathrm{re}})\cos ({\phi }_0+{\phi }_{\mathrm{re}}),\cos ({\theta }_0+{\theta }_{\mathrm{re}})\sin ({\phi }_0+{\phi }_{\mathrm{re}}),\sin ({\theta }_0+{\theta }_{\mathrm{re}}))$, where ${\theta }_0$ is the angle from the substrate [the $x\text{-}y$ plane indicated in Fig. 1(b)], and ${\phi }_0$ is the angle between the projection of the initial LC director to the $x\text{-}y$ plane and the $x$-axis. ${\theta }_{\mathrm{re}}$ and ${\phi }_{\mathrm{re}}$ are the photoinduced reorientation angles of the LC directors, and they are defined as

 

Fig. 1. (a) Schematic of optical vortex (OV) generation system consisting of functionalized liquid crystals (LCs) and external polarization recording system. (b) Schematic showing the coordinates for the reoriented LC director ${\mathbf{n}}_{\mathrm{re}}$.

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Figure 2 shows examples of the three-dimensional reorientation angle distributions induced by the recording vector beams, which were calculated using Eqs. (1) and (2). In this calculation, we set the susceptibilities of the reorientation angles at $C_{\theta }=0.033{\mathrm{cm}}^2/\mathrm{W}$ and $C_{\phi }=0.029{\mathrm{cm}}^2/\mathrm{W}$. These susceptibilities were obtained by preliminary experiment for the prepared ADDLC cell used in the next section. The electric field of the pump beam before incident on the ADDLC is defined as $\mathbf{E}(\phi ,z=0)={(\cos (p\phi ),\sin (p\phi ))}^T$. The pump beam wavelength and power density were set at ${\lambda }_{\mathrm{pump}}=532\mathrm{nm}$ and $I_{\mathrm{pump}}=0.9\times {10}^3\mathrm{mW}/{\mathrm{cm}}^2$, respectively. The initial directors of the LCs were set uniformly at ${\theta }_0=0\mathrm{deg}$ and ${\phi }_0=0\mathrm{deg}$, respectively. The ordinary and extraordinary refractive indices of the ADDLC were set at $n_o=1.52$ and $n_e=1.75$, respectively. The calculation range of $z$ was set at 2.3 μm, which is the optical length where the optical retardation is $2\pi \mathrm{rad}$. The images on the left side of Fig. 2 correspond to the polarization patterns of the recording beam. We found that various anisotropic structures are formed, depending on the polarization patterns. In addition, the reorientation angle distributions change continuously with propagation direction “$z$,” and thus three-dimensional anisotropic structures are formed in the ADDLC cell. In the cases of Figs. 2(a) and 2(b), the maximum amplitudes of the induced reorientation angles ${\phi }_{\mathrm{re}}$ are dependent on the azimuth positions. In contrast, in the case of Figs. 2(c) and 2(d), the maximum amplitudes of the induced reorientation angles ${\phi }_{\mathrm{re}}$ are the same at each azimuth position, and the distributions of ${\phi }_{\mathrm{re}}$ continuously rotate in clockwise or anticlockwise directions to preserve their distributions. Reorientation angle is periodically distributed around the optical axis. As shown in Figs. 1(a)–1(d), their rotation periods are every $\pi$. When a more high-order vector beam, whose polarization pattern rotates “$2p$” times around the optical axis, is used as the recording beam, the photoinduced reorientation period is $\pi /p$ due to the cylindrical symmetry of the polarization pattern, as shown in Figs. 2(e)–2(h). In our previous work, we found that the anisotropic structures shown in Figs. 2(c) and 2(d) can generate OVs with $\ell =+2$ and $\ell =-2$ under crossed-polarizer conditions, as shown in Fig. 1(a) [21]. In contrast, the anisotropic structures shown in Figs. 2(a) and 2(b) generate the superposition of OVs with $\ell =\pm 2$, with four-petalled patterns [25].

 

Fig. 2. Examples of numerically simulated spatial distributions of LC director reorientation angles. (a)–(d) and (e)–(h) are the cases that the recording vector beams have the axially symmetric polarization patterns which are rotating two times and four times around optical axis, respectively.

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Therefore, we investigated the OV generation properties of these structures experimentally. In these experiments, for the ADDLC, we used an uniaxially aligned azo-dye-doped polymer liquid crystal composite made by mixing a nematic LC mixture (E7; BDH-Merck, Inc.), a side-chain liquid crystalline polymer (SLCP), and disperse red 1 (DR1; Aldrich Co. Ltd.). A detail of sample preparation is shown in our previous paper [21]. The ADDLC layer thickness was measured to be 9 μm using a spectrometer. The susceptibilities of this ADDLC cell were preliminarily investigated by recording polarization hologram between uniform $s$ and $p$ polarizations and measuring diffraction efficiencies of $\pm 1$st-order beams conducted in our previous study [23].

Figure 3 shows the experimental setup used to record the high-order vector beams in the uniaxially aligned ADDLC. A Nb:YAG laser (operating at 532 nm, which is in a DR1 absorption band) was used as a light source. The Gaussian beam emitted from the Nb:YAG laser was initially incident on the vector beam converter, consisting of a polarizer, three quarter-wave plates (${\mathrm{QWP}}_1$, ${\mathrm{QWP}}_2$, and ${\mathrm{QWP}}_3$), the LCOS-SLM (X10468-04; Hamamatsu Photonics), a convex lens, and a reflective mirror [26]. The LCOS-SLM has two phase modulation regions that display the helical phase distributions of $\exp [i{\ell }_1\theta ]$ and $\exp [i{\ell }_2\theta ]$. The incident Gaussian beam is converted to linear polarization and is then incident on the right side of the LCOS-SLM, in which the phase singularity of ${\ell }_1$ is displayed. The reflected light is converted into a linearly polarized OV, for which the polarization azimuth and the TC are 0 deg and $-{\ell }_1$. The polarization azimuth of this OV is then converted to 45 deg by being doubly transmitted through ${\mathrm{QWP}}_1$ (which has its fast axis aligned to 22.5 deg) and is then incident on the left side region of the LCOS-SLM, in which the phase singularity of ${\ell }_2$ is displayed. A 0 deg linear polarization component is also applied to the helical phase of $\exp [i{\ell }_2\theta ]$, such that the resulting light is a superposition of two orthogonally polarized OVs with TCs of ${\ell }_1$ and ${\ell }_1+{\ell }_2$. The polarization states of these two OVs are then converted into left- and right-handed circular polarizations by passing through ${\mathrm{QWP}}_2$ (for which the fast axis is aligned at 45 deg). Superposition of these left- and right-handed circular polarizations with different TCs form a linearly polarized vector beam [27,28]. The Jones vectors of these vector beams can be described as $\mathbf{E}={(\cos (p\theta ),\sin (p\theta ))}^t$, where $p$ is a parameter representing the rotational symmetry of the polarization distribution around the optical axis. The correspondence between a number of $p$ values and their associated polarization patterns is illustrated in Fig. 3. ${\mathrm{QWP}}_3$, whose fast axis is ${\phi }_Q$, is used to convert the linearly polarized vector beams into elliptically polarized vector beams, as illustrated in Figs. 2(b)–2(d). A prepared ADDLC cell was then illuminated using the converted vector beams by imaging of the light from the SLM plane on to the sample plane, and this induced three-dimensional reorientation of the LC molecules. To investigate the wavefront modulation properties, a He–Ne laser (operating at 632.8 nm, which is outside the DR1 absorption band) was used as a probe beam. Two polarizers were located on either side of the ADDLC cell in the optical path of the probe beam. The probe beam diffraction pattern output from the ADDLC cell was observed using a complementary metal-oxide-semiconductor (CMOS) imaging camera (LASERCAM HR; Coherent, Inc.). Also, to investigate the TC of the output light, we observed interference images between the output probe beam and the Gaussian beam, which was separated using a beam splitter before incidence on the ADDLC cell. The phase distributions of the output light beam were recovered using the Fourier transform method [29,30].

 

Fig. 3. Experimental setup for OV generation using vector beam-recorded azo-dye-doped LC (ADDLC) cell. Correspondence between phase patterns and polarization patterns of conversion vector beams are illustrated on the right. These linearly polarized vector beams are converted into elliptically polarized vector beams by passing through the quarter-wave plates (${\mathrm{QWP}}_3$) with fast axes that are aligned at ${\phi }_Q=0\mathrm{deg}$ and ${\phi }_Q=\pm 45\mathrm{deg}$. The spatial light modulator (SLM) plate is imaged at the ADDLC cell to avoid Fresnel diffraction effects.

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3. RESULTS AND DISCUSSION

Figures 4 and 5 show the diffraction patterns of the probe beam emitted from the ADDLC cell system that were obtained by the CMOS imaging camera. In the case of no ${\mathrm{QWP}}_3$, which corresponds to the linearly polarized vector beams, $2p$-petalled diffraction patterns are obtained. These diffraction patterns correspond to the interference patterns of two coaxially superposed OVs with $\ell =\pm 2p$ [25]. In the case where ${\mathrm{QWP}}_3$ was inserted, corresponding to the elliptically polarized vector beams, a petalled pattern identical to the no-QWP case is obtained when the fast axis of ${\mathrm{QWP}}_3$ is ${\phi }_Q=0\mathrm{deg}$. In contrast, when the fast axis of ${\mathrm{QWP}}_3$ is ${\phi }_Q=\pm 45\mathrm{deg}$, although the diffraction pattern has a petalled fringe with low visibility, approximately doughnut-shaped images are obtained. This petalled pattern is caused by the interference between the main OV component with $\ell$ and the slightly included OV component with opposite signed TC $-\ell$. In addition, the diameters of the diffraction patterns are dependent on the number of $p$. These experimental results show good agreement with the simulated results. Figure 6 shows the phase distributions of each of the diffracted beams that were reconstructed from the interference pattern using the Fourier transform method. Figures 6(a)–6(e) and 6(f)–6(j) show the results for the cases of ${\phi }_Q=45\mathrm{deg}$ and ${\phi }_Q=-45\mathrm{deg}$, respectively. We found that each diffraction beam has a spiral phase distribution, and the TC is dependent on the number of $p$. In Figs. 6(a)–6(e), the diffracted light has a phase singularity of $\ell =2p$. However, in Figs. 6(f)–6(j), the diffracted light has a phase singularity of $\ell =-2p$. Therefore, we found that the uniaxially aligned ADDLC cell system shown in Fig. 1(a) works as an OV generator by recording the vector beams, and the TC of the generated OV is controlled by varying the polarization pattern of the recording vector beam.

 

Fig. 4. Experimentally and numerically obtained diffraction patterns of probe beam output from the system shown in Fig. 1(a). For the cases of without ${\mathrm{QWP}}_3$ and ${\phi }_Q=0\mathrm{deg}$.

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Fig. 5. Experimentally and numerically obtained diffraction patterns of probe beam output from the system shown in Fig. 1(a). For the cases of ${\phi }_Q=+45\mathrm{deg}$ and ${\phi }_Q=-45\mathrm{deg}$.

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Fig. 6. Phase distributions of output probe beam reconstructed using the Fourier transform method. (a)–(e) and (f)–(j) show the cases of ${\phi }_Q=+45$ and ${\phi }_Q=-45$, respectively. The gray-scale range is $-\pi \sim \pi$.

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We have also investigated the wavelength dependence of the proposed OV generation system. In these calculations, we assume that the ADDLC is illuminated by a vector beam with $p=1$ passing through ${\mathrm{QWP}}_3$ (${\phi }_Q=45\mathrm{deg}$), and then subject the TC distribution of the light output from the system to Fourier analysis [31]. In this analysis, the TC spectra can be calculated as the Fourier series of the complex amplitude of the electric field $E_{\mathrm{out}}(r,\theta )$ output from the ADDLC system, and this equation can be described as

 

Fig. 7. (a) Relative intensities of OV components with $\ell =2$, plotted as functions of wavelength $\lambda$. (b) Beam profiles and reconstructed phase distributions of probe beams with wavelengths of 633, 780, and 1064 nm (ADDLC thickness is 4.5 μm).

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Finally, we investigated the possibility of OV conversion switching. In our ADDLC cell, the photoinduced three-dimensional anisotropic structures can be erased by simply turning off the pump beam, because the LC director in the ADDLC cell returns to its initial uniaxial alignment state within a few tens of seconds because of the anchoring energy of the rubbed substrates. Therefore, the three-dimensional anisotropic structure can be rerecorded and changed into another structure. Figure 8 shows the experimentally observed intensity distribution of the diffracted light produced by changing the pump beam polarization pattern. In Fig. 8(a), we change the polarization pattern generated from the intensity pattern of the diffracted light continuously between ring-shaped patterns and other pattern shapes. In addition, we also monitored the diffraction pattern while varying the $p$ of the vector beam. An example result is shown in Fig. 8(b). The intensity pattern of the diffracted light has a doughnut shape and the ring diameter increases with increasing $p$ number. This result indicates that the present system has the ability to switch the TC value for OV conversion.

 

Fig. 8. Temporal change in diffraction pattern observed when switching the polarization pattern of the recording vector beam. (a) The case of changing the arrangement of ${\mathrm{QWP}}_3$ (${\phi }_Q=+45\to \mathrm{without}{\mathrm{QWP}}_3\to {\phi }_Q=+45$) while maintaining $p=1$. (b) The case of changing the number of $p$ ($1\to 2\to 3\to 4\to 5$) while maintaining the arrangement of ${\mathrm{QWP}}_3\hspace{0.17em}{\phi }_Q=+45$.

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The proposed system can modulate not only the spatial phase distribution of the probe beam but also the spatial amplitude distribution of the beam. In our OV generation system, the polarization azimuth of the probe beam when incident on the ADDLC cell is parallel with the initial alignment direction of the ADDLC, and the orthogonal linear polarization component is filtered by the analyzer, so that the local throughput at which the ADDLC reorientation angle is zero is 0%. Because the amplitude of the reorientation angle distribution depends on the power density of the recording vector beam, the spatial amplitude distribution of the probe beam can also be modulated by controlling the intensity distribution of the recording vector beam.

Based on Jones calculus and Fourier analysis, we found that the system that consists of the two crossed polarizers and the three-dimensional anisotropic structure that is induced in an ADDLC cell can achromatically generate OVs with high TC purity. Here, we consider a reason why our system can generate OV without or with low TC dispersion over a broadband spectrum. In our previous work, we found that the three-dimensional anisotropic structure fabricated by polarization holography between 90 deg and 0 deg linearly polarized Gaussian beams can diffract 0 deg and 90 deg linearly polarized Gaussian beams as 0th- and $-1$st-order directions where the case that the 90 deg linearly polarized Gaussian beam is incident as the probe beam [23]. This result indicates that a three-dimensional anisotropic structure can holographically reconstruct wavefront information between recorded two laser beams only as $-1$st order with orthogonal polarization state before incidence. Here we consider the case that a three-dimensional anisotropic structure is fabricated by the recording vector beam of Fig. 2(d). This case can be treated as collinear holography since the vector beam of Fig. 2(d) can be decomposed into 90 deg and 0 deg linearly polarized OVs with $\ell =+1$ and $-1$. Hence, based on the polarization diffraction property of the three-dimensional anisotropic structure, when the 90 deg linearly polarized Gaussian beam is incident on the ADDLC as the probe beam, the 90 deg linearly polarized Gaussian beam and the 0 deg linearly polarized OV with $\ell =+2$ are coaxially generated as 0th-order direction. Since the secondary polarizer passes only the 0 deg linear polarization component of probe beam output from the ADDLC, a pure OV with $\ell =+2$ reconstructed wavefront information between $\ell =+1$ and $-1$ can be generated with the arrangement of Fig. 1(a). However, we note that, depending on the retardation of the ADDLC, there is a condition where this polarization diffraction property has not been established. In this case, when the $p$-polarized Gaussian beam is incident on the ADDLC, two 0 deg linearly polarized OVs with $\ell =+2$ and $-2$ are coaxially generated as $-1$st- and $+1$st-order reconstructed light waves. This spurious OV component of $\ell =-2$ reduces the TC purity of $\ell =+2$, as shown in Fig. 7(a).

We note that while the bandwidth of high OAM purity increases with decreasing ADDLC cell thickness, the optical throughput is also reduced with respect to the ADDLC cell thickness. This is because the OV generated from the ADDLC is a 0 deg linear polarization component caused by the photoinduced three-dimensional anisotropic structure for the incidence of 90 deg linearly polarized probe beam. This polarization change is smaller in the case of a thinner ADDLC, so that OV conversion efficiency is decreased with thickness of the ADDLC. In addition, we need to avoid use of an operating wavelength that falls within the absorption band of the azo dye-LC mixture, because the anisotropic structure is reoriented by the probe beam. To solve this problem, we must change the azo dye according to the application requirements. However, the DR1 azo dye that was used in this study does not show absorption in the band from 600 to 1100 nm, and thus is expected to be applied to generation of ultra-short OV pulses emitted from a Ti:sapphire laser with a central wavelength of approximately 800 nm. Recently, several other methods for the achromatic generation of OVs have been reported by using patterned cholesteric liquid crystal structure [32,33]. We here note, however, that our three-dimensional anisotropic structure is different from them.

In addition, the proposed system has the potential to switch the optical modulation properties by varying the polarization and the intensity pattern of the recording light. In this work, it took a few tens of seconds to switch between the different TC values. This switching speed would be accelerated by changing the SLCP for a material with lower molecular weight, or increasing the proportion of the low molecular LC (E7). We also found that the reorientation speeds of LC molecules can accelerate under low temperature conditions, and thus faster optical switching could be realized using these approaches.

4. CONCLUSION

In this study, we proposed an OV generation concept based on a functionalized LC system driven by external polarization recording. The uniaxially aligned ADDLC, which is used for the functionalized LCs in this study, forms new types of three-dimensional anisotropic structures by recording of vector beams. This photoinduced anisotropic structure shows spatial wavefront modulation properties that enable OV generation, and the structure can be changed by varying the polarization pattern of the recording vector beam. The developed external polarization address-type OV generation system based on functionalized LCs has the advantages of both spatial light modulation flexibility and OV conversion achromaticity. These features are important for OV applications, including communications, astronomical coronagraphs, and ultra-fast spectroscopy. The purpose of this study is to propose a new concept for OV generation, and thus we have not studied performance aspects, such as throughput, rapid switching speeds, high spatial resolution, and bandwidths at available wavelengths. However, the requirements for these aspects are expected to be achieved using the approaches described above. In the future, these functionalized LCs may promote high OV generation functionality and lead to new applications of OVs.