Universal phase-only spatial light modulators (UP-SLMs) are proposed and demonstrated by simulation and experiment. UP-SLMs, which consist of unitarily coupled electrically controllable wave plates, convert the optically anisotropic properties of the paired two wave plates into an effective single isotropic phase plate and can be realized with electrically tunable birefringence materials such as uniaxial liquid crystals (LCs). The universal approaches are applicable to any uniaxial anisotropic materials based cells, any LC cell operation modes and any incident light polarization states. Further the UP-SLMs are experimentally demonstrated even with commercial LCD panels having low geometrical rotational symmetry. These UP-SLMs will play significant roles in wave optics, communications, information displays, digital holography, quantum optics and quantum information technology by harnessing their unique capacity of modulating the only phase of linearly, circularly, elliptically, azimuthally, radially and even randomly polarized light.
© 2017 Optical Society of America
Advanced optical and signal applications require of independently electrical manipulation of phase and amplitude light information with independently spatial controllability [1–15]. The devices controlling the amplitude, phase, and spatial location of light are called spatial light modulators (SLMs) . In general, the independent control of amplitude-only or phase-only light information is challenging due to their coupled attribute of light. Thus the independent control of phase and amplitude light information is not trivial but possible with the combination of SLMs or the utilization of several pixels on the SLMs costing the spatial resolution [1,2,16–25]. However, amplitude-only SLMs as well as independently phase and amplitude controllable SLMs can be achieved with phase-only SLMs [16,19,21,26–31].
Typically existing phase-only SLMs are only operable to a specific light polarization state due to the light polarization state dependent physical properties of the SLMs as well as the incident light polarization states. However, phase-only SLMs operable with any light polarization states are highly demanding for various applications such as display devices [3,5,6], optical communications [11,12], laser pulse shaping , three-dimensional holograms [3,4,8,9,13,14], optical angular momentum control [11,12,32–34], optical tweezers [8,10], quantum information technology [35,36] to name a few.
In this paper, we propose electrically tunable phase-only SLMs based on the electrically tunable birefringence of optically anisotropic materials such as uniaxial liquid crystals (LCs) operable with any light polarization states.
Phase-only light modulation can be easily realized using non-absorbing isotropic materials such as wide band gap amorphous materials or two dimensional crystals with square symmetry for normal incidence because the refractive index of isotropic materials does not depend on the light polarization state but its phase is varied with the optical path length. However, the electrically tunable SLMs are often realized with anisotropic materials rather than isotropic materials due to their electrical controllability at relatively low bias voltages.
However, phase-only modulation with anisotropic materials is not trivial because their anisotropic optical properties originated from the light polarization-dependent refractive index causing the modulation of phase as well as polarization state simultaneously. To overcome these problems for phase-only modulation, significant research activities have been conducted [22,23,37–43]. Especially, LC based SLMs are extensively studied because of its high spatial resolution, high switching speed, low operation bias voltages, diverse types of operation modes, and heavy applications in optical communications and information displays [3,5,6,11,12].
The typical methods for LC based phase-only SLMs include quasi-eigenstate methods [22,23,37–39], residual phase methods [40,41] and orthogonal alignment methods [42,43]. In the quasi-eigenstate methods employed for twisted-nematic (TN) LC based phase-only SLMs [22,23,37–39], only a specific input light polarization state is utilized for phase modulation. Further parallelly aligned (PA) LC based phase only modulators can hold exact-eigenstates even at varying the applied bias voltages . But these exact-eigenstate methods also are applicable to only a specific input light polarization state. In the residual phase methods [40,41], the dynamic operation range is typically limited by ~0.1 π because the phase modulation is operated over the saturation voltage region.
Unlike the above mentioned methods, the orthogonal alignment methods for the phase only light modulation are applicable for various input polarization states and have been experimentally demonstrated using the orthogonally aligned two anisotropic layers with the azimuthal angle of the slow optical axis fixed at a specific angle [42,43]. However, in general, electrically tunable wave plates or anisotropic optical cells are stacks of anisotropic layers with azimuthal angles varied with the vertical position rather than fixed at a constant azimuthal angle. The non-commuting property of the wave plates with optical axis varying in azimuthal angle indicates that the matching condition of the optical path length for preserving the input light polarization state is not the whole story.
In this study, the universal approaches for input light polarization state independent phase-only SLMs, called universal phase-only spatial light modulators (UP-SLMs), are proposed and demonstrated by accounting the generally non-commuting properties of wave plates, which can be built using any anisotropic materials, any slow axis distributions, and any operational modes. For example, PA, VA (Vertically Aligned), ECB (Electrically Controllable Birefringence), π cell, and TN modes to name a few can be utilized for UP-SLMs. For the expressional convenience in the following sections, the optically anisotropic materials will be exemplified with uniaxial LC materials. Therefore, the followings are often expressed with LC materials and terminology but they are not limited to LC materials and applicable to any anisotropic materials as one can find out.
2. Concept of universal phase-only spatial light modulators (UP-SLMs)
In this study, light of wavelength with an arbitrary polarization state placed in an x-y plane propagates from the left along the z direction through optical plates parallel to the x-y plane in a Cartesian laboratory coordinate system. The optical plates can be LC cells, which are a stack of very thin homogeneous LC layers having anisotropic electrical and optical properties. The slow optical axis of the wave plate layer or the LC director () in each layer is varied with the layer position (z) and are function of polar () and azimuthal () angles. As a result, the refractive indices in the each layer for the two orthogonal light polarization states are given by,  where and are ordinary and extraordinary refractive index of the uniaxial LC molecules, respectively. The effective extraordinary refractive index depends only on the polar angle (), but.
2.1 Unitarily coupled two wave plates
First consider a homogeneous wave plate (WP) with thickness of dM (or a single LC layer) where the slow axis of the WP (or LC director) is oriented with a specific polar angle θM and azimuthal angle ϕM in the laboratory coordinate system. Jones matrix for the WP in the laboratory coordinate system,, can be expressed as a function of θM, ϕM, and dM or common average phase (), phase retardation () and ϕM as shown below .
where is the rotation matrix from the laboratory coordinate to the principal coordinate of the WP (LC director), is the matrix representation of the WP (LC layer) in the principal coordinate system, is wave number, , is the common average phase, and is the phase retardation.
In general, as a light propagates through a wave plate (a LC layer), both the light polarization state and phase are simultaneously modulated as one can expect from the non-diagonal Jones matrix of Eq. (1). To preserve the input light polarization state while the phase change is allowed, the deformed polarization state by the WP should be recovered by an additional optical component. Thus the combination of the wave plate (M) and the coupled optical component (A) for preserving the input light polarization state with the total phase shift of can be expressed by
where I is the unit matrix.
One can see from Eq. (1), the Jones matrix of the single WP (LC layer) without the common phase factor, , is an unitary matrix, (The superscript + means the complex conjugate and transpose operation and −1 does the inverse operation). Thus the inverse matrix can be expressed by the complex conjugate and transpose of the unitary matrix and corresponds to the just shift of the azimuthal angle by ± 90° as shown below,
By multiplying the inverse matrix of M in the both right hand side of Eq. (2) and using the unitary property of matrix MU, the coupled unknown matrixcan be obtained as below
When , , and , acts like the inverse matrix of and is equivalent to another coupled WP, which is orthogonally aligned in azimuth angle () with respect to the first WP, , with common phase, and phase retardation, . Thus the total phase shift by the unitarily coupled two WPs is .
This suggests that the unitarily coupled WPs [Fig. 1(a)] act as an isotropic material phase plate [Fig. 1(b)] as shown in Fig. 1. Thus the unitarily coupled WPs allow the phase only modulation for any incident light polarization states. The projection of the effective directors of the combined WPs on an x-y plane shows square symmetry while the projection of the director of each plate is a uniaxial symmetry. The square symmetry on the plane is optically isotropic for normally incident light. Thus when light goes through the isotropic materials, the phase is changed but the polarization state is preserved [Fig. 1(b)]. The unitarily coupled anisotropic WPs can be achieved by any kinds of uniaxial systems or structures or orientations. The approach shown above can be universally applicable to any anisotropic systems.
2.2 Unitarily coupled two LC cells - stacks of unitarily coupled wave plates
Unlike homogeneous wave plates with a fixed optical axis orientation, electrically tunable WPs or LC cells can be considered as a stacking of many homogeneous, anisotropic (LC) layers with the vertical position (z) dependent slow optical axis orientations (or LC directors),, where polar angel and azimuthal angle is a function of z. Even though the unitarily coupled two anisotropic (LC) layers are fulfill the requirement for the incident light polarization state independent phase-only light modulation, in general, the unitary matrices representing WPs varied with azimuthal angle are not commutable as one can find from Eq. (1). Thus the stacking order in the stacked LC multilayers determines their optical properties. This requires a specific stacking order (manner) in the unitarily coupled LC cells for UP-SLMs.
The requirements for the stacking order depend on whether the LC directors are function of layer position dependent azimuthal angle or not. First, when the matrices representing WPs are not commutable due to the LC directors with the azimuthal angle dependency, the LC layers with an equal optical path difference from the mid-plane in both cells should satisfy the unitary coupling requirement for the incidence light polarization state independent phase-only modulation. Thus the product of two matrices representing the coupled WPs becomes the unit matrix multiplied by a common phase term as shown in Fig. 2.
When the LC cell is considered as a stacking of homogenous LC layers, the multiplication of matrices corresponding to the closest WPs from the mid-plane results in the unit matrix with a common phase term as shown in Eq. (2) and acts like an isotropic phase plate as shown in Fig. 1(b). Thus the closest layers from the mid-plane do not affect to the multiplication of the next matrices except the common phase terms as schematically shown in Fig. 2.
The next nearest neighboring WPs are represented by matrices of and , respectively, and their product becomes the unit matrix with a phase term by satisfying the unitary relationship as shown in Eq. (3). With continuation of this multiplication process as shown in Eq. (5) and schematically shown in Fig. 2(a), the final production of the matrices representing the unitarily coupled LC cells becomes the unit matrix with a global phase term () as shown in Eq. (5).
The matrices A and M come from Eq. (2). The total phase change () is given by the argument of where is the common phase change by each paired layer.
When the both cells consist of the same electro-optical material and thickness, and are biased with the common ground and the equal bias voltages (V1 = V2) as shown in Fig. 2(b), they satisfy the unitary coupling conditions for UP-SLMs even at bias voltages.
3. Simulation and simulation results
For the verification of the proposed UP-SLMs, various UP-SLMs structures for bias voltage on- and off-states are tested by simulating using MATLAB for four different input light polarization states of x, y, LHC (Left Handed Circular), and an elliptical polarization state as shown in Eq. (6).
For the simulation of the tunable phase retardation, three θ distribution models of LC directors are employed for the better description of the director distribution under the bias voltages: a typically adapted constant θ distribution model [Eq. (7)], as well as an employed sinusoidal polar angular distribution model [Eq. (8)] and another employed equal step per distancedistribution model [Eq. (9)].
The constant θ0(V) is varied with applied bias voltages (V) for all layers (or z). In the proposed sinusoidal model, the polar angle θ in Eq. (8) varying with the vertical position of z is modulated by the amplitude constant of C(V), which is varied with the applied voltages (V). As the amplitude parameter C becomes smaller to 0, the θ approaches to π/2. This corresponds to states at low applied bias voltages. However, when C becomes a high value, the θ approaches to 0 and almost all LCs are aligned along the z direction. This corresponds to the states at high bias voltages.
In the employed equal angular step per distance distribution model, the variable, the polar angle increment per unit distance (nm), is varied with applied bias voltages (V), and the polar angle θ decreases linearly with the step from π/2 (First layer) to θ = 0° (Last layer) in the lower cell and if θ values become negative, the negative values are replaced by 0°.
In the simulation, the ordinary () and extraordinary () refractive indices for the employed anisotropic materials (or uniaxial liquid crystals) are assumed as 1.2 and 1.7, respectively, for the wavelength of the input plane wave light, 532 nm.
3.1. Stacks of non-commutable wave plates for UP-SLMs
First, to universally prove and demonstrate the proposed concept of UP-SLMs, a stack of unitarily coupled non-commutable WPs (or LC cells), of which the slow optical axes (LC directors) are randomly distributed in θ and ϕ along the stacking direction as shown in Fig. 3(a), is chosen but the each pair is unitarily coupled and thus satisfying the requirement for the incidence light polarization state independent phase-only modulation as shown in Eq. (5).
The light propagation through the unitarily coupled two stacks is simulated and the results are shown in Fig. 3 (see Visualization 1). Each WP stack (Single LC cell) is 6000 nm in thickness and considered of 30 homogenous WPs (LC layers). As predicted, the final polarization states () shown in Fig. 3 are the exactly same with those of the input beams [Eq. (6)]. The light polarization states at each layer position in the lower cell are the mirror symmetry states to those at the corresponding layer position in the upper cell as shown at the left columns in Fig. 3.
Further, the red and blue markers with numbers on the Poincaré sphere are indicating the polarization states in the lower and upper cells, respectively, while the numbers are indicating the layer stacking order number from the bottom of the stacked cells. The red and blue markers are scattered on the sphere due to the randomly distributed slow optical axes of the LC layers but completely overlapped with each other. The overlapped each pair satisfy the unitary coupling condition [Eq. (4)]. This indicates that the polarization state changes by the LC layers (or divisions) in the first LC cell are exactly canceled by the counter LC layers in the second cell. The total common phase () from the unitarily coupled cells with randomly distributed directors is 58.1 and does not depend on the input polarization states as shown in Fig. 3.
3.2. Stacks of commutable wave pates for UP-SLMs
The multiplication of two matrices representing the two arbitrary WPs having the same azimuthal angle can be replaced by a single matrix representing the two WPs and is commutable as shown in Eq. (10).
In general, in a LC cell consisted of directors fixed in an azimuthal angle ϕ, the cell can be expressed by Eq. (11).
Equation (11) says that a single cell consisted of commutable WPs can be converted to an effective single wave plate. Using the commutable WPs without any further requirements in materials (ne, no), cell thickness (D) and above mentioned stacking order, Eq. (4) (or equivalently and) is enough for the unitary coupling conditions for UP-SLMs as shown in Fig. 1. Thus UP-SLMs can be implemented with two LC cells made of different LC materials (ne1, no1, ne2, no2) and different cell thickness (D1 and D2) having directors fixed in an azimuthal angle (, ) as long as the optical phase difference is . The optical properties of these LC cells do not depend on the LC layer stacking order because of the commuting properties shown in Eq. (10). Equivalently the optical path lengths and the optical path length differences for the two orthogonal light polarization states in the cell don't depend on the stacking order.
Thus the unitarily coupled cells, which are satisfying , are expressed as below.
For example, consider UP-SLM structures shown in Fig. 4. For simple illustration, three configurations of cells, which are consisted of random layer stacking order [Fig. 4(a)] or a specific stacking order with fixed director distribution at a specific azimuthal angle ϕ [Figs. 4(b) and 4(c)], are made of the same LC materials (ne, no) and the cell thickness (D). Thus, the integrated areas of the thickness-birefringence schematic shown in Fig. 4 are equal to each other. Thus as indicated by Eq. (12), three configurations shown in Fig. 4 show all equivalent optical properties to each other by preserving the input light polarization states () and retarding the same phase and they are all equivalent UP-SLMs.
This allows to facilitating the construction of UP-SLMs with the commutable LC directors, which are not varying with azimuthal angle . Since PA, VA, ECB, and π cells have constant azimuth angular director distribution and the matrices representing the LC layers in each cell are commutable, the coupled cell structures for UP-SLMs can be easily formed using these cells.
For example, overlaid two PA LC cells angled with 90° in azimuthal angle with respect to each other but the random stacking order show the distribution ofas shown in Fig. 4(a). Simulation of the UP-SLM structures for (b) x-polarization, (c) y-polarization, (d) left-handed circular polarization and (e) an elliptical polarization is shown in Fig. 5. The single cell thickness is 921.45 nm and the single cell is equally divided by 12 layers.
The commuting property allows that the each cell can be considered as a single wave plate and the coupled cells act as a unitarily coupled wave plates as shown in Fig. 1.
3.3 Tunable phase retardation of UP-SLMs
Once the coupled LC cells satisfy the unitary requirement for UP-SLM structures, they can maintain the required conditions for UP-SLM structures when the bias voltage is mirror symmetrically applied along the z-direction as shown in Fig. 2(b) and they become tunable. The θ and distributions of LC directors are changed by applying bias voltages (V). However, the output beam characteristics through the unitarily coupled cells is a just function of the polar angle (θ), layer thickness (d), birefringence () and the wavelength of incidence light (), but not of the azimuthal angle (). Once these physical parameters are determined, the polar angle distribution can be modulated by applying bias voltages to obtain the desired total relative phase change (). Thus this allows to electrically modulating only the phase for each pixel of UP-SLMs.
For , the maximum phase retardation is occurred when all LC molecules are aligned along the orthogonal direction () with respect to z direction while the minimum phase retardation is happened when all LC molecules are aligned along z direction (). For these cases, the effective refractive indices are
When the LC cell is equally divided into N layers with the thickness d of each layer, the maximum () and minimum () common phase changes for two unitarily coupled LC layers made of the same materials and the same thickness are as below,
The total maximum () and minimum () common phase changes are given by , and, respectively. Therefore, the maximum relative phase modulation () is and given by
Here, is, (birefringence) is , and (thickness of a single LC cell).
As a result, the dynamic relative phase modulation amplitude () for the two coupled wave plates can be given in a simple manner by Eq. (16).
Here, the indices 1 and 2 are for M and A, respectively. The total relative phase retardation is equal to the sum of the phase difference by the each coupled pair.
4. Experimental verification
The conceptually proposed and computationally verified UP-SLMs are experimentally demonstrated as below using the available commercial LCD displays. The employed commercial displays are two MVA (Multidomain Vertical Alignment) mode FHD (1920 x 1080) LCD monitors (PHILIPS 200V4Q) with 20 inches in diagonal while the back light units and sheet polarizers are removed. The each pixel (230 μm x 230 μm) consists of RGB subpixels with eight-domains with different alignment directions in azimuthal angle () as shown in the optical microscopic and scanning electron microscopic (SEM) images of Fig. 6.
The optical and SEM images of the bottom plate show diagonal lines as shown in Figs. 6(a) and 6(b), respectively. With combination of the snow flake like pattern on the top plate [Figs. 6(c) and 6(d)], the azimuthal angular distributions of the alignment layer in each subpixel can be considered as eight domains having alternatively alignment layers with azimuthal angles as shown in Fig. 6(e).
4.1. UP-SLM structures consisted of two commercial MVA LC displays
Based on the identified subpixel structures shown in Fig. 6, an UP-SLM can be formed by overlapping the two commercial commutable LCD panels with a specific adjustment, which is simply shifting one of the overlapped panels to the other by (one quarter of the subpixel height) along the vertical direction [Figs. 7(a)-7(c)] or shifting by (one half of the subpixel width) along the horizontal direction [Figs. 7(d)-7(f)]. In this study, the UP-SLM structure is achieved by the vertical shifting as shown in Fig. 7(c). The only central region becomes effective due to the blocking of the light by black matrix as shown in Fig. 6(c).
4.2. Characteristics of UP-SLMs consisting of two MVA LC cells
A UP-SLM is built by overlapping two commercial MVA-LCD panels as shown in Fig. 6. As shown in Fig. 8(a), the UP-SLM is tested with a normally incident linearly polarized light of 532 nm in wavelength (a DPSS laser) at several azimuthal light polarization angles of 90°, 60°, 45°, 30°, 0°, −30° and −60° with respect to the x-axis controlled by a linear polarizer. The discrete grey levels are addressed at the LCD panels to control the orientation of the LC directors and scanned with the grey levels of 0, and 8n - 1 when n is a positive integer less than 32. The each grey level is maintained for 0.1 second.
The output polarization states show almost the same polarization states with the input light polarization states with a small deviation in the azimuthal angle, the ellipticity and the amplitude as shown in Figs. 8(b)-8(d), respectively. The deviation in the azimuthal anlge, the ellipticity and the amplitude increases with the grey level. The deviation in the azimuthal anlge and the ellipticity for the 0 and 90° azimuthal angle of the incident light polarization state is almost 0 but their corresponding amplitude deviations show almost the same trends with those of the other polarization states. The highest deviation is observed when the azimuthal angle of the light polarization state is ± 60°.
The microscopic images of the transmitted green laser [Figs. 9(b) and 9(c)] and white incoherent light [Figs. 9(d) and 9(e)] through the UP-SLM between two parallel polarizers at the addressed grey level of 0 and 255 are shown in Fig. 9(a). At the gray level of 0, the relatively uniform transmitted light is observed as shown in Figs. 9(b) and 9(d). However, at the grey level of 255, the six discrete patterns per pixel are clearly observed as shown in Figs. 9(c) and 9(e). First of all, it is attributed to the disclination of the LC molecules at the domain wall . This can cause the output polarization states in azimuthal angle, ellipticity and amplitude deviated from the input polarization states. Another possible contribution to the deviation can be unwanted beam passing through the other color filtered areas as shown in Fig. 9(f), which shows a finite transmittance for 532 nm green light through the blue color filters. The other possible reason for the deviation in azimuthal angle, ellipticity and amplitude can be misalinment as schematically shown in Fig. 9(g). Further, when the coherent light is employed for the imaging, the bright spot size of each domain is varied with the grey levels as shown in Figs. 9(b) and 9(c). This is attributed to the interference of the light originated from the neighboring domains as schematically shown in Fig. 9(h).
4.3. Phase modulation by the UP-SLM consisting of two MVA-LCDs
A Mach-Zehnder interferometer configuration with a DPSS 532 nm laser is employed to measure the phase modulation driven by the addressed grey level as shown in Fig. 10(a). The employed input linear polarization states are 0°, 30°, 45°, 60°, 90°, 120°, 135° and 150° in azimuthal angle. Due to the size limits of available optical set-ups of the investigators’ laboratory, the collimated laser beam width is about 2 inches. The first polarizer is used for setting the input polarization state and the second polarizer is used to match their polarization state and brightness. To eliminate unwanted external factors such as mechanical vibration hindering the accurate phase modulation measurement, one-half of the display area is addressed with varying the grey level from 0 to 255 while the other half is addressed with a fixed black grey level (0) as a reference part [Figs. 10(b) and 10(c)]. The grey level (GL) dependent phase shift information (Δδ (GL)) is extracted from the recorded sinusoidal interference pattern shift by a CCD camera [Fig. 10(c)] when the gray scale is swept from 0 to 255. For the efficient extraction of the phase modulation, a FFT (Fast Fourier Transform) algorithm is employed as below.
The grey level dependent sinusoidal interference pattern () is transformed by Fourier transform (), and the grey level dependent phase shift (Δδ (GL)) with respect to the 0 (black) grey level is obtained as shown in Eq. (18).
The measured phase modulation for the UP-SLM for 8 different, 0° (dark dots), 30° (red dots), 45° (green dots), 60° (yellow dots), 90° (blue dots), 120° (purple dots), 135° (cyan dots), and 150° (gray dots) is shown in Fig. 11. As expected from Eqs. (16) and (17), the experimentally observed dynamic phase modulation widths are about 165° in average for all above light polarization states as shown in Fig. 11. The experimental results are well matched with the simulation result at the minimum thickness condition for each LCD panel acting as a half wave plate as shown in Fig. 11. Since the dynamic phase modulation width is proportional to the thickness of the LC cell and the birefringence, it can increase with the thickness and the birefringence.
Universal approaches for incident light polarization state independent phase-only spatial light modulation are proposed and demonstrated. Unitarily coupled any two anisotropic cells or LC cells allow modulating the only phase of light while any input light polarization states are preserved. In general, when light goes through a single anisotropic cell or a LC cell, the polarization and phase of the incident light are simultaneously changed. To preserve the incident light polarization state, the second optical component (or a LC cell) should be unitarily coupled to the first anisotropic cell (or LC cell). The unitarily coupled cell allows to recover the input light polarization state while the desirable phase change is allowed. Particularly, the proposed and demonstrated methods for incident light polarization state independent phase only modulation don't depend on the type of anisotropic materials and the LC operation modes and correspond to universal phase only-spatial light modulators (UP-SLMs). These UP-SLMs will play important roles in light modulation of linearly, circularly, elliptically, azimuthally, radially and randomly polarized light for various wave optical applications, communications, digital holography, quantum optics, quantum information technologies, to name a few.
Kyung Hee University (KHU-20121737).
References and links
1. N. Savage, “Digital spatial light modulators,” Nat. Photonics 3(3), 170–172 (2009). [CrossRef]
2. W. P. Bleha and L. A. Lei, “Advances in Liquid Crystal on Silicon (LCOS) spatial light modulator technology,” Proc. SPIE 8736, 87360A (2013). [CrossRef]
3. K. Sarma, “Recent developments in stereoscopic and holographic 3D display technologies,” Proc. SPIE 9086, 908606 (2014).
6. M. A. Karim and A. A. S. Awwal, “Electrooptic displays for optical information processing,” Proc. IEEE 84(6), 814–827 (1996). [CrossRef]
7. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]
8. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002). [CrossRef]
11. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]
12. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef] [PubMed]
15. J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006). [CrossRef]
18. H. Kim, C.-Y. Hwang, K.-S. Kim, J. Roh, W. Moon, S. Kim, B.-R. Lee, S. Oh, and J. Hahn, “Anamorphic optical transformation of an amplitude spatial light modulator to a complex spatial light modulator with square pixels [invited],” Appl. Opt. 53(27), G139–G146 (2014). [CrossRef] [PubMed]
19. A. Shibukawa, A. Okamoto, Y. Goto, S. Honma, and A. Tomita, “Digital phase conjugate mirror by parallel arrangement of two phase-only spatial light modulators,” Opt. Express 22(10), 11918–11929 (2014). [CrossRef] [PubMed]
20. V. Arrizón, G. Méndez, and D. Sánchez-de-La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express 13(20), 7913–7927 (2005). [CrossRef] [PubMed]
21. B. Ma, F. Peng, M. Kang, and J. Zhou, “Optimize the modulation response of twisted-nematic liquid crystal displays as pure phase spatial light modulators,” Proc. SPIE 9296, 929606 (2014). [CrossRef]
23. L. G. Neto, D. Roberge, and Y. Sheng, “Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions,” Appl. Opt. 35(23), 4567–4576 (1996). [CrossRef] [PubMed]
24. S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. 37(11), 1955–1957 (2012). [CrossRef] [PubMed]
25. S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22(15), 17999–18009 (2014). [CrossRef] [PubMed]
26. L. Hu, L. Xuan, Y. Liu, Z. Cao, D. Li, and Q. Mu, “Phase-only liquid crystal spatial light modulator for wavefront correction with high precision,” Opt. Express 12(26), 6403–6409 (2004). [CrossRef] [PubMed]
28. E. Frumker and Y. Silberberg, “Phase and amplitude pulse shaping with two-dimensional phase-only spatial light modulators,” J. Opt. Soc. Am. B 24(12), 2940 (2007). [CrossRef]
29. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre-Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34(1), 34–36 (2009). [CrossRef] [PubMed]
30. M. Lin, K. Nitta, O. Matoba, and Y. Awatsuji, “Parallel phase-shifting digital holography with adaptive function using phase-mode spatial light modulator,” Appl. Opt. 51(14), 2633–2637 (2012). [CrossRef] [PubMed]
31. Z. Zhang, Z. You, and D. Chu, “Fundamentals of phase-only liquid crystal on silicon (LCOS) devices,” Light Sci. Appl. 3(10), e213 (2014). [CrossRef]
32. E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3(5), e167 (2014). [CrossRef]
33. H. Yu, H. Zhang, Y. Wang, S. Han, H. Yang, X. Xu, Z. Wang, V. Petrov, and J. Wang, “Optical orbital angular momentum conservation during the transfer process from plasmonic vortex lens to light,” Sci. Rep. 3(1), 3191 (2013). [CrossRef] [PubMed]
34. Y. Wang, X. Feng, D. Zhang, P. Zhao, X. Li, K. Cui, F. Liu, and Y. Huang, “Generating optical superimposed vortex beam with tunable orbital angular momentum using integrated devices,” Sci. Rep. 5(1), 10958 (2015). [CrossRef] [PubMed]
35. J. Harris, V. Grillo, E. Mafakheri, G. C. Gazzadi, S. Frabboni, R. W. Boyd, and E. Karimi, “Structured quantum waves,” Nat. Phys. 11(8), 629–634 (2015). [CrossRef]
36. G. B. Lemos, J. O. De Almeida, S. P. Walborn, P. H. S. Ribeiro, and M. Hor-Meyll, “Characterization of a spatial light modulator as a polarization quantum channel,” Phys. Rev. A 89, 1–8 (2014).
37. J. A. Davis, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37(11), 3048 (1998). [CrossRef]
38. L. G. Neto, D. Roberge, and Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34(11), 1944–1950 (1995). [CrossRef] [PubMed]
40. H. Ren, Y.-H. Lin, Y.-H. Fan, and S.-T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005). [CrossRef]
41. H. Ren, Y. H. Lin, C. H. Wen, and S. T. Wu, “Polarization-independent phase modulation of a homeotropic liquid crystal gel,” Appl. Phys. Lett. 87(19), 191106 (2005). [CrossRef]
42. Y.-H. Lin, H. Ren, Y.-H. Wu, Y. Zhao, J. Fang, Z. Ge, and S.-T. Wu, “Polarization-independent liquid crystal phase modulator using a thin polymer-separated double-layered structure,” Opt. Express 13(22), 8746–8752 (2005). [CrossRef] [PubMed]
43. H. Ren, Y. H. Lin, and S. T. Wu, “Polarization-independent and fast-response phase modulators using double-layered liquid crystal gels,” Appl. Phys. Lett. 88(6), 061123 (2006). [CrossRef]
44. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley & Sons, 2010), Chap. 4.
45. C. C. Peng, K. C. Hsu, J. J. Wu, S. H. Fan, H. T. Lee, and Y. Shen, “Wide-viewing angle twisted-vertical alignment liquid crystal cells without disclination lines,” Displays 31(4–5), 210–215 (2010). [CrossRef]