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Abstract
Simultaneously controlling the spatial distribution of multiple parameters of a light field in a three-dimensional (3D) space is highly desirable because of its prominent applications in the areas of optical imaging, microscopy, and manipulation. Phase-only encoding techniques that use a phase-only computer-generated hologram (CGH) to reshape and efficiently reconstruct target fields have fostered substantial interests. In this paper, we propose a convenient encoding method to construct vector fields with spatially structured multiple parameters in a 3D space by integrating the Fourier phase-only encoding technique into a modified Sagnac polarization conversion system. Without spatial filtering, various vector fields are constructed instantly at the image plane. Furthermore, utilizing a macro-pixel encoding approach, we demonstrate the possibility of a simultaneous and an independent construction of multiple vector fields in a 3D space. This method can also benefit the design of a metasurface to implement a polarization hologram.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spatially modulating the polarization, phase, and amplitude parameters of a light field has produced numerous unique spatially structured light fields (SSLFs) [1], such as Airy beams, Bessel beams, vortex beams, and vector beams [2–6]. Moreover, it has introduced significantly interesting physical phenomena and effects such as transverse acceleration, non-diffraction, autofocusing, tightly focusing with a sub-diffraction limit, and the spin Hall effect of light [7–9]. As a prominent example, vector beams, which have spatially inhomogeneous polarizations, show extremely novel propagation dynamics and interactions with materials, tremendously promoting their applications in super-resolution imaging, optical micro-machining, quantum information, and optical communication [10–15]. Moreover, vector vortex beams, which simultaneously possess spatially structured polarization and phase distributions, further exhibit enhanced spin–orbital interaction [16–18]. The above in turn, promote the exploitation of the efficient and convenient generation method of vector fields, specifically those simultaneously possessing spatially structured intensity and phase parameters.
Until now, numerous vector field generation methods have been proposed, which can be classified into two groups. Active approaches output resonant higher-order modes by suppressing the intracavity fundamental modes [19–22]. Passive approaches employ polarization conversion elements and coherent superposition of orthogonal polarized modes [23–30]. For instance, coherent superposition methods employing a spatial light modulator (SLM) have been primarily developed because of their flexibility [29–35]. Based on the images uploaded onto the SLM, these methods can also be classified into two categories. In the first category, computer-generated holograms (CGHs) are uploaded to create vector fields with a complex spatial structure; however, the unavoidable higher diffraction orders will dramatically reduce the generation efficiency [26,27]. The second category uses phase-only images to efficiently generate vector fields, but it only accurately outputs vector fields with distinct polarization structures at a fixed plane [31,36]. Therefore, developing a generation method that is both efficient and flexible is still a challenging research topic.
Currently, the phase-only CGH technique, which is a promising method to reconstruct target fields, has attracted tremendous research and application interests in color display, information security, medical academia, and biomedical imaging [37–39]. Understandably, the wavefront reshaping technique has been introduced into the generation of vector fields as a potential candidate to overcome the efficiency problem, such as the Gerchberg–Saxton iterative algorithm [40,41]. Furthermore, it has been extended to metasurfaces, implementing the construction of light fields with both a spatial intensity and polarization structures [42,43]. However, it is time-consuming, which is another challenge for real-time construction.
In this work, we combine the Fourier phase-only encoding technique with a modified Sagnac polarization conversion interferometer to develop a vector field generation method that allows the convenient construction of vector fields with spatially structured multiple parameters anywhere in a three-dimensional (3D) space. Accordingly, we co-axially reconstruct two target fields with a desired amplitude, phase, and polarization structures near the image plane, and then output vector fields with spatially structured multiple parameters. Furthermore, we demonstrate the capability of this method to simultaneously and independently construct multiple target fields in a 3D space.
2. Encoding theory
Considering the spatial property, an arbitrary light field can be described by its polarization and spatial modes [44]. According to the Poincaré sphere description method, any polarization state can be divided into two orthogonal basis vectors, e.g., right- and left-handed circular polarizations, denoted as eR and eL, respectively. Hence, the polarization mode can be expressed as Ψ = aeR + $\sqrt {1 - {a^2}}$eL, where the coefficient, a, is a complex constant such that |a| ≤ 1. Supposing that the field simultaneously has a transverse spatial mode, u, in the cylindrical coordinates, (r,ϕ,z), a scalar field with a homogeneous polarization can be described as a product of the polarization and spatial modes, i.e., E = E0uΨ = E0u(aeR + $\sqrt {1 - {a^2}}$eL), where E0 is the amplitude envelope. However, for a vector field with an inhomogeneous polarization that correlates to the transverse spatial coordinate, the polarization cannot be described as a product of spatial and polarization modes. This type of a field should be expressed as E = E0Ψ = E0(auReR + $\sqrt {1 - {a^2}}$uLeL), where uR and uL denote two distinct solutions of the paraxial equation, i.e., spatial modes [44].
For a generic vector field that simultaneously has transversely structured multiple parameters, including polarization, phase, and intensity, its spatial property can be rewritten as
Here, we select the Fourier phase-only encoding technique to accurately construct spatial modes [45]. This encoding technique implements the Fourier phase reshaping at the pupil plane, and then reconstructs the targeted complex amplitude at the image plane. Figure 1 schematically shows the encoding and reconstruction processes of this technique. Here, as an example, we choose alphabet “A” and a square frame as the target amplitude and phase, respectively, as shown in Fig. 1(a). Moreover, we suppose that this target field has a complex amplitude expressed as E(x, y) = A(x, y)exp[iΦ(x, y)] in the Cartesian coordinates (x,y,z), where A and $\varPhi$ are the transversal amplitude and phase structures, respectively. Accordingly, under paraxial condition, its spatial spectrum can be expressed as T(fx,fy) =ℱ{E(x,y)} = M(fx,fy)exp[iψ(fx,fy)], where M is the normalized amplitude by the max pixel, ψ is the phase structure of the spatial spectrum, and fx = xf/λf and fy = yf/λf are the spatial frequencies in the (xf, yf) plane, depending on the focus length, f, of the Fourier transformation lens and wavelength λ. The distributions of M and ψ are shown in Fig. 1(b). To generate a phase-only CGH including the amplitude information, i.e., the phase mask in Fig. 1, a random complex amplitude encoding scheme is employed to modulate the phase function, ψ. Thus, the phase-only CGH can be expressed as [45]
Fig. 1. Fourier phase-only encoding principle. (a), (b) Amplitude and phase of the target field and its Fourier transform field; (c) simulated and experimentally measured intensity and phase distributions of the reconstructed field.
3. Experiment
Figure 2 schematically shows the experimental set-up of a modified Sagnac interferometer integrated with the Fourier phase-only encoding [32]. A collimated Gaussian beam with linear polarization from a He–Ne laser is incident onto the Sagnac interferometer consisting of a polarized beam splitter (PBS), two mirrors, and a reflected-type phase SLM (Hamamatsu X11840-07 with 792 × 600 pixels). Further, the beam is divided into horizontal and vertical components, which pass through the Sagnac interferometer along opposite directions [46]. The half-wavelength plate placed before the SLM is used to transfer the polarization to match the response of the SLM (only response to the horizontal component). The SLM is divided into two segments to upload the phase-only CGHs [256 gray-scale images with 300 × 300 pixels created according to Eqs. (2) and (3)] that independently modulate the two spatial modes, respectively, as shown in inset (a). Two constituent components are then coaxially produced by the PBS, and orderly passed through a lens and quarter-wavelength plate (QWP), which is used to transfer two orthogonally linear polarizations into opposite circular polarizations, realizing polarization mode modulation. Experimentally, first, the SLM is placed at the front focal plane of the lens, and then a CCD is positioned at the back focal plane to observe the reconstructed field. The combination of the QWP and polarizer, as depicted by the dotted rectangles in the figure, are employed to measure the polarization of the reconstructed field.
Fig. 2. Experimental set-up for the generation of the vector field. HWP: half-wavelength plate; PBS: polarized beam splitter; QWP: quarter-wavelength plate; M: mirror; SLM: reflective-type phase SLM; P: polarizer. Insets: (a) phase-only CGHs corresponding to two polarization modes; (b) experimentally detected intensities of the first-order perfect vector field.
According to Eq. (4), the reconstructed field at the back focal plane can be rewritten as
The results shown in inset (b) are the experimentally measured intensities of the right- and left-handed circularly polarization modes as well as the total field of a perfect vector field with a first-order polarization mode. For such a distinct case, AL = AR, $\varPhi$L = -$\varPhi$R = ϕ. From the results obtained, we can see that two spots inevitably appear inside the ring because of the reflected energy from the SLM (modulation efficiency is approximately 82%) and optical elements. Therefore, in practice, we attach a blazed grating onto the CGHs to remove the residual fields that are not modulated.
Initially, we demonstrate robust controlling on multiple parameters by generating perfect vector fields [32,35], which manifest as vector fields with intensity structures independent of their polarization orders. Figure 3 displays five perfect vector fields constructed from the coherent superposition of two perfect vortex modes with opposite topological charges, i.e., $\varPhi$L = -$\varPhi$R = lϕ, where l is the topological charge of the vortex mode. Figures 3(a) and 3(b) show the second- and fifth-order (i.e., l1 = 2, l2 = 5) perfect vector beams with an annular intensity distribution, respectively. Figures 3(d)–3(e) orderly show the fifth-order perfect vector fields with ellipse, square, and triangle intensity structures, respectively. The panels from top to bottom are the total intensity, Stokes parameter S3, and polarization orientation (φ) calculated from the measured Stokes parameters, respectively. The experimental results clearly show that the vector fields present the desired polarization modes. Because the locally linear polarization and cylindrically symmetric polarization structures are independent of the intensity structure, this provides potential in various polarization-mediated applications, such as particle trapping and lithography, with complex geometries. More importantly, this method enhances the generation efficiency of the perfect vector fields to approximately 20.7% [35].
Fig. 3. Constructed perfect vector fields with different intensities and polarization modes. (a) l1 = 2; (b)–(e) l2 = l3 = l4 = l5 = 5. From top to bottom: the distributions of intensity, Stokes parameter S3, and local polarization orientation (arrows), respectively.
Next, we demonstrate the flexibility of this method by constructing vector fields with complex intensities and polarization modes. Figures 4(a)–4(d) display the intensity distributions of the total field, horizontal and vertical components, and local polarization orientation of a vector LG field with l = 3 order polarization and p = 1 order radial index, i.e., a vector ${\textrm{LG}_1^{3}}$ field. As expected, a vector field with the desired spatial and polarization properties is observed. However, the above-mentioned vector fields are canonical so that the reconstructed spatial modes are the same as or approximate to the eigen solutions of the wavefunction. Besides, the phase-only CGHs obtained from Eq. (2) do not have sufficient spatial spectrum information. Therefore, to adequately verify the feasibility of this method, we further design a complex target field with sufficient spatial spectrum information.
Fig. 4. Constructed vector fields with complex intensities and polarization modes. (a) Constructed vector LG3 1 field; (b), (c) horizontal and vertical components; (d) measured polarization orientation; (e) alphabet pattern with two orthogonal polarizations; (f) constructed field; (g), (h) horizontal and vertical components; (i) measured polarization orientation.
Figure 4(e) depicts a binary logo consisting of alphabets “OITLAB,” where alphabets “LAB” are modulated by a binary grating. Concurrently, we apply a binary polarization mode to this field with, i.e., alphabets “O” and “ITLAB” represent two orthogonally linear polarizations. To achieve this purpose, we set the left and right segments of the two reconstructed target fields to have phase differences of 0 and π/2, respectively. Consequently, the constructed vector field has horizontal and vertical polarizations, which separately correspond to the left and right segments. Figures 4(f)–4(h) display the total intensity, horizontal, and vertical components of the reconstructed vector field. Figure 4(i) displays the local polarization orientation. In comparison with the target pattern shown in Fig. 4(e), we can see that the logo, “OIT,” is well reconstructed, except its corners that correspond to a relatively higher spatial spectrum. However, the alphabets, “LAB,” present a fuzzy morphology with a low resolution. Here, we measured the peak signal to noise ratio (PSNR) of these reconstructed intensity patterns corresponding to the two target fields in Fig. 4, which are 20.88 and 11.97, respectively. Clearly, the latter has a lower reconstruction quality than the former. Considering the spatial spectrum of these target fields, it is notable that the grating in the second target field generates higher spatial spectral components, which have smaller weight coefficients, i.e., amplitude M than those for the first one. This signifies that in the phase-only encoding process, these spatial spectral components will be encoded with a conical phase, and then diverge into a ring at the image plane, resulting in the distortion of the reconstructed fields. Specifically, the higher spatial spectral components that depend on the resolution are diverged because of the Fourier phase-only encoding.
From the experimental results shown in Figs. 3 and 4, we can see that this Fourier phase-only encoding method exhibits advantages in the on-axially simultaneous reconstruction of the amplitude and phase as well as the polarization mode. In comparison to traditional holography and other phase-only encoding methods that only require one calculation [47,48], this method reconstructs the target field at the image plane without a 4f filter system.
4. 3D shaping vector fields
Phase-only encoding methods have been demonstrated with robustness in 3D imaging and multiple-target reconstruction. To simultaneously construct multiple vector fields in a 3D space, here, we utilize macro-pixels to design the phase-only CGH, whose principle is schematically shown in Fig. 5(a). Considering the practical pixel of the SLM, here we construct only two target fields simultaneously at different z positions. Therefore, in Fig. 5(a), the colorized checkerboard depicts a hybrid CGH (2N × 2N pixels) composed of two constituent CGHs (N × N pixels) corresponding to different target fields 1 and 2, with each square corresponding to one pixel of the CGH. In the hybrid CGH, four adjective pixels compose a macro-pixel (the square with a purple edge), as shown in the inset of Fig. 5(a), in which two diagonal pixels correspond to the same pixel of one constituent CGH, e.g., the first (m = 1, n = 1) macro-pixel is composed of two first pixels in two constituent CGHs having the same order. Thus, the pixel number of the macro-pixel is N × N. It should be noted that to suppress the diffraction effect induced by the macro-pixel encoding, here, the relative positions of the diagonal pixels corresponding to the two target fields in a macro-pixel are also random [49], as the dichromatic checkerboard shows in the pupil plane.
Fig. 5. (a) Reconstruction principle of multiple vector fields in a 3D space based on the macro-pixel method. The dichromatic checkerboard depicts a hybrid CGH composed of two constituent CGHs corresponding to target fields 1 and 2. Inset: a macro-pixel consisting of four pixels, wherein two diagonal pixel correspond to the same pixel in a constituent CGH; (b), (c) intensity and Stokes parameter S3 distributions of a Taiji pattern-like vector field at the image plane (z = 0), respectively; (d), (e) off-focal reconstruction of two orthogonal circular polarizations with z1 = −0.1 m, z2 = 0.1 m, respectively.
Moreover, to independently reconstruct multiple vector fields in 3D space, we introduce a spherical phase into the target fields, whose phase structure is expressed as
To demonstrate the possibility of this method in 3D reconstruction, we first independently control the 3D position of the two orthogonal polarizations that compose the vector field. Figures 5(b) and 5(c) display the intensity and Stokes parameter S3 distributions of a Taiji pattern-like vector field at the image plane (N = 150). This distinct field consists of two orthogonally circular polarizations. Figures 5(d) and 5(e) show the off-focal reconstruction of the two orthogonal polarizations at z1 = −0.1 m and z2 = 0.1 m planes, respectively. It is evident that the reconstructed off-focal fields have identical intensity patterns same as those generated at the image plane.
Further, we simultaneously construct two perfect vector fields with different polarizations and intensity structures at different spatial positions. Figures 6(a) and 6(b) show the reconstructions of two annular perfect vector fields at z1 = −0.1 m and z2 = 0.075 m planes, respectively. The panels from top to bottom are the intensity distributions of total field, horizontal and vertical components, as well as local polarization orientations. As the experimental results depict, second- and fifth-order perfect vector fields are reconstructed at the z1 and z2 planes, respectively. In comparison, at the off-focal planes, two perfect vector fields individually evolve into quasi-Bessel beams. Figures 6(c) and 6(d) show the reconstruction of two fifth-order perfect vector fields with rectangular and triangular patterns at two longitudinal planes, respectively. Evidently, for this encoding method, the number of vector fields is limited by the pixels of the SLM. In practice, to individually perform the Fourier phase-only encoding of two orthogonal polarizations, an SLM with pixels of 792 × 600 is divided into two segments with pixels of 300 × 300; limited by the SLM, we only reconstruct two vector fields simultaneously.
Fig. 6. Reconstruction of multiple perfect vector fields at two off-focal planes. (a), (b) Second- and fifth-order perfect vector fields with annular patterns; (c), (d) fifth-order perfect vector fields with rectangular and triangular intensity patterns. z1 = −0.1 m, z2 = 0.075 m.
These experimental results prove the feasibility of this method in the construction of vector fields with multiple structured parameters. In comparison to coherent superposition methods using CGHs or other phase-only modulation methods, e.g. double-phase method [48], this method immediately constructs the vector fields dispensing with the 4f filter system, so that it can achieve construction with a relatively higher efficiency. Importantly, it allows the independent and simultaneous construction of multiple vector fields in a 3D space. However, the results also reveal that the pixel of the SLM plays a crucial role in the reconstruction, which not only affects the number of reconstructed fields but also determines the reconstruction quality of the target field. Moreover, the practical non-uniformity of the input beam intensity and deletion of a part of the frequency information also reduce the reconstruction quality. To overcome the distortion induced by the inhomogeneous intensity of the input light, we should select an expanded light beam to illuminate the SLM, which will also reduce the generation efficiency. Potential solutions for further exploration would be the pre-correction of phase-only CGHs.
5. Conclusion
To conclude, we integrated the Fourier phase-only encoding technique that simultaneously reconstructs the desired amplitude and phase structures with the polarization conversion system. Further, we proposed a convenient method to generate vector fields. We demonstrated the performance of this method by remarkably constructing vector fields with various spatial and polarization modes. Furthermore, we also demonstrated the possibility of independently constructing multiple vector fields in 3D spaces. Our approach may lead to the design of metasurfaces [43], which simultaneously construct vector fields with a structured amplitude, polarization, and phase in a 3D space for 3D imaging and display [50].
Funding
National Natural Science Foundation of China (11634010, 11774289, 61675168, 91850118); Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (U1630125); Fundamental Research Funds for the Central Universities (3102018zy036, 3102019JC008); National Key Research and Development Program of China Stem Cell and Translational Research (2017YFA0303800); Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (ZZ2019220).
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