Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display

Abstract

This chapter describes an efficient approach to generating light beams with arbitrary intensity profile and phase distribution. Accordingly, a fast method is described to characterize liquid crystal displays based on the Mach-Zehnder interferometer and fringe analysis in the Fourier domain. Then, the double-pixel hologram Arrizón’s approach is reviewed. This approach is able to generate an on-axis computer-generated hologram into a low-resolution twisted-nematic liquid crystal for encoding arbitrary complex modulations. Furthermore, a fast algorithm to map holographic cells based on the k-nearest neighbor (k-NN) classifier is introduced in order to generate computer-generated holograms faster than the conventional calculation. Finally, two beam profiles are produced with the described approach and assessed at the entrance pupil and the depth of focus of a high-NA microscope objective.

1. Introduction

Liquid crystal display devices are a kind of spatial light modulators (SLMs). SLMs are able to relate electronic data to spatially modulated coherent light. In particular, twisted nematic liquid crystal spatial light modulators (TNLC-SLMs) are kind of relatively low-cost electro-optics devices widely used in many branches of optical information processing, such as digital holography [1, 2], spatially-variant polarized beams [3], coherent diffraction imaging [4], generating vector beams [5, 6, 7], pattern recognition and optical correlators [8], Fresnel lenses [9, 10], and optical cryptosystem [11, 12, 13].

On the other side, highly focused beams and their properties have been investigated in many fields, such as nonlinear optics, super-resolution microscopy, tomography, and optical tweezers [14, 15, 16, 17, 18, 19, 20, 21]. Tightly focused beams attract much attention because of the non-neglectable component of the electric field in the direction of propagation.

Since SLMs change the properties of light, such as amplitude, phase, and polarization, it is necessary to find the proper operating conditions to control the SLM’s response. The twisted angle and the birefringence of a TNLC are two main parameters that control the modulation. There are several proposed methods and configurations to find these parameters to introduce the Jones matrix of TNLC [22, 23, 24, 25, 26]. On the contrary, Martín-Badosa et al. proposed a method to characterize liquid crystal displays (LCDs) based on the fringe analysis obtained by a Mach-Zehnder interferometer configuration, without the necessity of finding the physical properties of LCD [27]. Nevertheless, their approach is based on counting the displacement of the fringes and is time-consuming. Wang et al. proposed a faster method to characterize SLM based on extracting the phase values of the Fourier spectra of the interference pattern directly [28]. They indicated that analyzing the phase values in the frequency domain is sufficient to obtain the imposed modulated phase. Their mathematical approach is based on Fresnel diffraction considering the transfer function. However, this approach can be modified by adding a 4-f imaging system to the Mach-Zehnder interferometer.

The double-pixel hologram Arrizón’s approach is a cell-oriented computer-generated hologram (CGH) encoding. In the cell-oriented method, each encoding point is split into a couple of holographic cells [29]. He proposed a modification of the previous works regarding a double-phase CGH method with an on-axis reconstruction field [30, 31, 32]. His approach improved the signal-to-noise ratio (SNR) in the reconstruction plane using two pixels of SLM to encode one holographic cell [33]. Since then, he generalized his approach to producing a more symmetric and high-SNR-signal domain using four pixels of SLM, the so-called modified Double-Pixel Hologram (DPH) [34]. Although his approach was suggested for encoding phase-only SLMs, he adjusted the method to encoding complex modulation with a transmission TNLC as a low-resolution spatial light modulator [35, 36, 37].

This chapter aims to present a modification of the method proposed by Martín-Badosa et al. and Wang et al. to characterize LCDs and review the DPH Arrizón’s approach to generating an on-axis CGH proper for encoding TNLCs. Since the codification algorithm is time-consuming, a fast algorithm using the k-NN classifier is presented. The application of this codification is extended to provide vector beams using a high-NA microscope objective.

The following text is organized as follows: Section 2 briefly describes the physical and optical properties of a TNLC display. Section 3 describes the characterizing process based on the modified Mach-Zehnder interferometer and fringe analysis in the Fourier domain. Section 4 reviews the DPH Arrizón’s approach. Section 5 introduces a fast algorithm to generate DPHs applying the k-NN machine learning algorithm. Section 6 demonstrates the experimental setup to produce and analyze highly focused beams with arbitrary beam profiles.

2. Twisted nematic liquid crystal

A twisted nematic liquid crystal display is constructed by sandwiching a nematic liquid crystal between two transparent glass plates. Different voltages impose an external electric field through the medium using electrodes connected to each glass plate. Nematic molecules inside the medium have a helical structure parallel to their elongated direction with an optical axis. In this regard, TNLCs are a sort of birefringent medium with ordinary and extraordinary refractive indices ($n_o$,$n_e$). The birefringence of the medium is altered by applying different voltages resulting in tilting nematic molecules in the direction of the applied electric field. Hence, a TNLC can electrically be controlled to be used as optical wave retarders, modulators, and switches. In the absence of the applied electric field, nematic molecules are appropriately oriented with respect to each other in the plane, parallel to the surface of glasses. In the particular case that the input beam is linearly polarized parallel to the direction of the liquid crystal director (practically is unknown to users), the beam keeps its state of polarization traveling through TNLC but rotating as much as its twisted angle ($\alpha$), which is usually equal to 90°. In this condition, the TNLC acts as a polarization rotator. If the direction of polarization of the input beam is oriented concerning the LC director, the beam experiences phase retardation as $2\pi \left(\mathit{ne}-\mathit{no}\right)d/\lambda$, where $d$ is the thickness of LC display. In the presence of the applied electric field, the nematic molecules tilt to be aligned with the direction of the applied field. The amount of this tilt angle ($\theta$) depends on the applied voltages causing different birefringence. So, LC becomes a variable retarder with retardation $\mathrm{\Gamma }=2\pi \left[n\left(\theta \right)-n_0\right]d/\lambda$. The retardation varies monotonically from 0 (when the molecules are not tilted, $\theta =0$) to ${\mathrm{\Gamma }}_{\max }=2\pi \left[n_e-n_0\right]d/\lambda$ (when molecules are tilted 90°, $\theta =90$). In summary, based on the physical configuration of the experimental setup, an input beam traveling through nematic liquid crystal cells might face only-twisted, twisted and tilted, or only-tilted nematic molecules. The twisted angle, the birefringence, and the director of LC should be obtained to introduce the Jones matrix of TNLC display (for more information, see [38]).

3. Characterizing TNLC display

Regarding specific situations, including the state of polarization, the wavelength of input beams and applied voltages, three different modulations are mainly interesting: amplitude-only, phase-only, and complex amplitude-phase. Applying different voltages to SLM results in different degrees of birefringence that can be practically obtained by displaying different gray values on the LCD, ranging from 0 to 255. Accordingly, the phase and intensity of the input beam alter passing through the SLM differently. Hence, the effect of a TNLC display in the optical system can be described by

$$E_t=Aexp\left(-\mathit{j\phi }\right),$$E1

where $A$ stands for the transmitted amplitude and $\phi$ represents the imposed phase to the transmitted beam. In this regard, characterizing LCDs aims to find the Eq. (1) for each gray value ranging from 0 to 255.

Martín-Badosa et al. applied a Mach-Zehnder interferometer for characterizing LCDs. With the same optical setup and a different mathematical description, Wang et al. proposed a faster and more convenient way to obtain phase modulation. Since the digital holography approach used in this work is based on DPH Arrizón’s approach, the Mach-Zehnder interferometer should be accordingly modified in order to eliminate the off-axis diffraction orders caused by the codification algorithm (it is explained in Section 4). Therefore, the optical setup used in this work is modified by adding a 4-f spatial filtering system. Accordingly, the mathematical analysis based on this experimental setup is presented.

As shown in Figure 1, the optical setup consists of a Mach-Zehnder interferometer plus a 4-f spatial filtering system. The coherent beam provided by a pig-tailed laser (Thorlabs LP520-SF15A) with $\lambda =514$ nm is collimated and linearly polarized by means of the collimator lens and the linear polarizer (LP1), respectively. Then, the beam is divided into two arms of the interferometer by the first beam splitter (BS1). The right arm of the interferometer, which the object beam passes through, includes a half-wave plate (HWP1), a quarter-wave plate (QWP1), and the transmissive TNLCD (Holoeye HEO 0017 with a resolution of 1024 × 768 pixels and a pixel pitch of 32 μm).

Figure 1.

The experimental setup. LP, BS, HWP, QWP, M, SF, and CCD stand for the linear polarizer, beam splitter, the half-wave plate, the quarte-wave plate, mirror, spatial filter, and charged-coupled device, respectively.

The left arm of the interferometer, which the reference beam passes through, includes a half-wave plate (HWP2). Subsequently, the interference occurs when the two beams reach the second beam splitter (BS2). Then the interferometric pattern passing through the second linear polarizer (LP2) reaches the CCD camera’s sensor plane through the 4-f imaging system.

Regarding the experimental setup, LCD is placed at the back focal plane of lens A (LA), and the CCD camera is placed at the front focal plane of lens B (LB). Besides, the diffraction orders (except zero-order) caused by the pixelated structure of LCD and the digital holography approach are removed using the spatial filter (SF) placed at the common focal plane of LA and LB. Mirror M2 is properly tilted in such a way that the fringes are aligned in the y-direction, as shown in Figure 2a. HWP2 is used to adjust the contrast intensity of the interference pattern. The orientation of axes of linear polarizers, LP1 and LP2, and fast axes of HWP1 and QWP1 with respect to each other define the modulated characteristic of the TNLC display. Hence, the desired modulation can be obtained by some practical attempts.

Figure 2.

(a) The interferometric pattern recorded by CCD. (b) The corresponding Fourier spectra.

The fringe pattern recorded by the CCD can be expressed mathematically in the following form:

$$g\left(x,y\right)=a\left(x,y\right)+b\left(x,y\right)cos\left[2\pi f_{0}x+\varphi \left(x,y\right)+{\phi }_m\right],$$E2

where $\varphi \left(x,y\right)$ and ${\phi }_m$ are the phase of the object beam and the modulated phase imposed by SLM, respectively. Furthermore, $a\left(x,y\right)$ represents possible nonuniform background, $b\left(x,y\right)$ represents the local contrast of the pattern, and $f_0$ is the spatial-carrier frequency. As explained in [39], the fringe pattern can be rewritten in the following form:

$$g\left(x,y\right)=a\left(x,y\right)+c\left(x,y\right)exp\left(j2\pi f_{0}x\right)+c^{\ast }\left(x,y\right)exp\left(-j2\pi f_{0}x\right),$$E3

with

$$c\left(x,y\right)=\left(1/2\right)b\left(x,y\right)exp\left[\mathit{j\varphi }\left(x,y\right)\right]exp\left(j{\phi }_m\right).$$E4

The Fourier transform of Eq. (3) gives

$$G\left(f_x,f_y\right)=A\left(f_x,f_y\right)+C\left(f_x-f_0,f_y\right)+C^{\ast }\left(f_x+f_0,f_y\right),$$E5

where the capital letters denote the Fourier spectra, $f_x$ and $f_y$ are the spatial frequencies in the x- and y-direction, respectively. In fact, $A\left(f_x,f_y\right)$ is the zero-order of the interference and $C\left(f_x-f_0,f_y\right)$ and $C^{\ast }\left(f_x+f_0,f_y\right)$ are $\pm 1$ interference orders, respectively.

Since the spatial variations of $a\left(x,y\right)$, $b\left(x,y\right)$, and $\varphi \left(x,y\right)$ are slow compared with spatial frequency $f_0$, the Fourier spectra are separated by carrier frequency $f_0$, as shown in Figure 2b.

Once the configuration of the experiment, for instance, wavelength and the optical path difference between the reference and object beam, remains constant, the position of the peaks in the frequency domain remains unchanged. Thereby, the modulated phase can be obtained by analyzing either $C$ or $C^{\ast }$. Complex value $C$ can be rewritten as follows:

$$C\left(f_x,f_y\right)=\text{FT}\left\{\left(1/2\right)b\left(x,y\right)exp\left[j\right(\varphi (x,y)+2\pi f_{0}x)]exp\left(j{\phi }_m\right)\right\}=\left|C\right|exp\left(j\mathrm{\Phi }\right)exp\left(j{\phi }_m\right),$$E6

where $\text{FT}$ denotes the Fourier transform, and $\left|C\right|$ and $\mathrm{\Phi }$ are the amplitude and phase of $C$ in the frequency domain, respectively. Since the experimental parameters remain constant during the whole process of the measurement, the phase of $C\left(f_x,f_y\right)$ varies only with ${\phi }_m$, that is, the modulated phase imposed by SLM. As a result, if we split the object beam into two parts which are separated by a gray reference value (zero) and variable gray value ranging from 0 to 255, subtracting the phase of $C\left(f_x,f_y\right)$ corresponding with each part gives the phase shift as follows:

$$C{\left(f_x,f_y\right)}_{\mathit{rg}}=\left|C\right|exp\left(j\mathrm{\Phi }\right)exp\left(j{\phi }_{\mathit{rg}}\right),$$E7

$$C{\left(f_x,f_y\right)}_{\mathit{vg}}=\left|C\right|exp\left(j\mathrm{\Phi }\right)exp\left(j{\phi }_{\mathit{vg}}\right),$$E8

$$\mathrm{\Delta }\phi =arg\left[C{\left(f_x,f_y\right)}_{\mathit{vg}}\right]-arg\left[C{\left(f_x,f_y\right)}_{\mathit{rg}}\right]+{\phi }_0.$$E9

The subscripts $\mathit{rg}$, $\mathit{vg}$, and $\mathrm{\Delta }\phi$ indicate the reference, the variable gray values, and the phase shift, respectively. ${\phi }_0$ is selected in which $\mathrm{\Delta }\phi =0$ when the image loading on LCD has zero value. In the following, the experimental processes are expressed in two main stages: phase modulation and amplitude modulation.

3.1 Phase modulation

The gray-level images should be synthesized in such a way that the images are divided into two parts corresponding with the reference part (zero value) and the variable part (ranging from 0 to 255), as explained previously. These images loading on SLM cause different phase shifts between the variable and reference part resulting in a displacement of fringes. The amount of this phase shift depends directly on the orientation axes of retarders and linear polarizers and can be obtained mathematically by means of Eq. (9). The first row of Figure 3 demonstrates three examples of the gray-level images loading on the LCD, whereas the second row indicates the corresponding interference patterns recorded by the CCD camera. The fringes were experimentally obtained by setting the axes of LP1 and LP2 at 0 and 90 degrees, respectively, with respect to the x-axis. In addition, the fast axes of HWP2, HWP1, and QWP1 were rotated −20, −63, and 45 degrees, respectively, with respect to the x-axis.

Figure 3.

The first row demonstrates three examples of gray-level images loaded on the LCD. The second row indicates the corresponding interference patterns recorded by the CCD camera.

Then, two equal regions of the reference and variable part of each interference pattern should be selected for calculating the 2D fast Fourier transform, as indicated by the red rectangles in Figure 4b. As a result, the modulated phase can be obtained by applying Eq. (9) to the known frequency $C\left(f_x,f_y\right)$ for each gray value. Note that the frequencies are shown in Figure 4c that the first order of interference occurring remains constant for the variable and reference parts and all 256 interference patterns.

Figure 4.

(a) An example of synthesized gray-level image loading on LCD. (b) The recorded fringe pattern. The red selected rectangles have been used for calculating the 2D Fourier transform. (c) The Fourier domain.

In this practical case, the maximum achievable phase imposed by SLM is around 1.75$\mathrm{\pi }$. However, this value depends on the TNLC physical properties, the optical wavelength, and the polarization state of the incoming light. The obtained phase curve versus gray values is shown in Figure 5a.

Figure 5.

(a) The phase modulation. (b) The amplitude modulation. (c) The polar plot of the phase-mostly modulation.

3.2 Amplitude modulation

With the same configuration without changing the direction axes of linear polarizers or retarders, the amplitude curve can be obtained as follows: First, blocking the left arm of the Mach-Zehnder interferometer. Then, providing 256 gray-level images in which the entire images have the same gray value. Loading gray-level images on the LCD and subsequently recording the intensity pattern by the CCD camera. The modulated amplitude versus each gray value can be obtained by calculating the square root of the mean value of the recorded intensities and normalizing them to the maximum obtained amplitude. Also, this process can be done by using an intensity detector. The experimentally obtained amplitude curve versus gray values is shown in Figure 5b. Finally, Eq. (1) can be obtained for each gray value by combining the amplitude and phase responses of the SLM in consequence of the present configuration. This process is called the characterization or calibration of SLM. The polar plot shown in Figure 5c indicates the obtained phase-mostly modulation curve.

4. Computer-generated hologram: Double-pixel Arrizón’s approach

Very often, TNLCs provide a coupled amplitude-phase modulation, as indicated in Figure 5. This section aims to review the DPH Arrizón’s approach to expanding the accessible modulations beyond the restricted SLM response. Then, we review the modified DPH approach by applying 4 pixels to encode one holographic cell adapted to the experimentally obtained modulation curve. As a result, the approach is able to generate an on-axis computer-generated hologram with the optimum reconstruction efficiency, maximum signal bandwidth, and high SNR suitable for encoding arbitrary complex modulation into a low-resolution TNLC display.

Considering the obtained phase-mostly modulation curve shown in Figure 5, assume each complex modulation point belongs to the modulation curve as follows:

$$M_g=\left|M_g\right|exp\left(j{\psi }_g\right),$$E10

where subscript $g$ denotes gray values, which is an integer value between 0 and 255. In addition, the pixelated structure of the display is considered as a matrix with $N\times M$ arrays (pixels). The modulation $M_{\mathit{nm}}$ in the (n,m) the pixel can be defined as follows:

$$M_{\mathit{nm}}=\left|M_{\mathit{nm}}\right|exp\left(j{\psi }_{\mathit{nm}}\right).$$E11

To encode a desired complex modulation value

$$q_{\mathit{mn}}=\left|q_{\mathit{nm}}\right|exp\left(j{\tau }_{\mathit{nm}}\right),$$E12

Arrizón employed the holographic double pixel shown in Figure 6, whose pixels have the complex modulation $M_{\mathit{nm}}^1$ and $M_{\mathit{nm}}^2$ that belong to the modulation curve. As shown in Figure 6, the holographic cell is equal to a double pixel with the encoded modulation $q_{\mathit{nm}}$ plus an error double pixel, with the modulation values $e_{\mathit{nm}}^1$ and $e_{\mathit{nm}}^2$. The conditions required to produce an on-axis signal reconstruction with a null contribution of the error term at the zero frequency are

Figure 6.

The DPH configuration.

$$q_{\mathit{nm}}=\left(M_{\mathit{nm}}^1+M_{\mathit{nm}}^2\right)/2,$$E13

$$e_{\mathit{nm}}^2=-e_{\mathit{nm}}^1.$$E14

To explain this, considering the modulation points on the modulation curve ($M_g$) as a vector with the origin of the polar plot shown in Figure 7, encoded modulation points ($q$) are obtained by the average of the superposition of vectors $M^1$ and $M^2$. As a result, we can access the modulation points ($q$) beyond the restricted SLM responses ($M_g$). Consequently, the modulation errors are $e_{\mathit{nm}}^1=M_{\mathit{nm}}^1-q_{\mathit{nm}}$ and $e_{\mathit{nm}}^2=M_{\mathit{nm}}^2-q_{\mathit{nm}}$_, which lead to Eq. (14).

Figure 7.

The codification algorithm.

Regarding the experimental modulation curve, all possible complex values that can be obtained with this codification algorithm are shown with the green points in Figure 8. However, only those complex values that fall inside the blue circle with a radius $A_0=0.29$ can encode a complex function with an amplitude ranging from 0 to $A_0$ and a phase ranging from 0 to 2π.

Figure 8.

The red and green points show the experimental modulation curve and all possible complex values using the DPH approach. The green points inside the blue circle with the radius A₀ = 0.29 are those accessible complex values to encode a complex function.

To go through Arrizón’s approach in more detail, assume the transmittance of the CGH that can be displayed on the LCD is

$$h\left(x,y\right)=\sum _{n,m}{M}_{\mathit{nm}}w\left(x-\mathit{np},y-\mathit{mp}\right),$$E15

where $p$ is the pixel pitch and $w\left(x,y\right)=\text{rect}\left(x/a\right)\text{rect}\left(y/b\right)$. Considering the CGH is intended to encode the spatially quantized complex function

$$q\left(x,y\right)=\sum _{\mathit{nm}}{q}_{\mathit{nm}}w\left(x-\mathit{np},y-\mathit{mp}\right),$$E16

where $q_{\mathit{nm}}$ is defined in Eq. (12) in which $\left|q_{\mathit{nm}}\right|\le 1$. Assuming the spectrum of $q\left(x,y\right)$ denoted $Q\left(u,v\right)$ is centered at the zero frequency $\left(u,v\right)=\left(0,0\right)$. Hence, the CGH transmittance must be related to the encoded complex modulation $q\left(x,y\right)$ by the following expression:

$$h\left(x,y\right)=A_{0}q\left(x,y\right)+e\left(x,y\right),$$E17

then, the Fourier transform of Eq. (17) gives

$$H\left(u,v\right)=A_{0}Q\left(u,v\right)+E\left(u,v\right).$$E18

The error spectrum, $E\left(u,v\right)$, should be negligible within the largest possible band centered at the zero frequency to obtain a high SNR. So, Arrizón proposed an error function as follows:

$$e\left(x,y\right)=l\left(x,y\right)g\left(x,y\right),$$E19

$$l\left(x,y\right)=\sum _{n,m}{l}_{\mathit{nm}}w\left(x-\mathit{np},y-\mathit{np}\right),$$E20

$$g\left(x,y\right)=\sum _{n,m}{g}_{\mathit{nm}}w\left(x-\mathit{np},y-\mathit{np}\right).$$(21)E21

He demonstrated that the optimal choice $l\left(x,y\right)$ is the binary grating with discrete modulation $l_{\mathit{nm}}={\left(-1\right)}^{n+m}$. In this regard, the Fourier transform of the error function contributed by the noise field is given mainly by four off-axis replicas of the function $G\left(u,v\right)$ centered at the spatial frequency coordinates $\left(1/2p,1/2p\right)$, $\left(-1/2p,1/2p\right)$, $\left(1/2p,-1/2p\right)$, and $\left(-1/2p,-1/2p\right)$. Therefore, the reconstructed field with a zero noise contribution places on the optical axis while symmetric off-axis error contributions occur far enough from the optical axis at the Fourier plane, which should be removed using a 4-f spatial filtering system.

According to Eqs. (17) to (21), $g\left(x,y\right)$ is specified by its discrete modulation $l_{\mathit{nm}}$, which is related to the CGH modulation by the formula

$$M_{\mathit{nm}}=A_0{q}_{\mathit{nm}}+{\left(-1\right)}^{n+m}{g}_{\mathit{nm}}.$$E22

Since both function $q\left(x,y\right)$ and $g\left(x,y\right)$ have on-axis spectrum bands, their variation should be negligible when the increment ($\mathrm{\Delta }n=1$) in $x$ is of the order of the pixel pitch. To satisfy this condition, Arrizón proposed to establish the discrete function $g_{\mathit{nm}}$ such that both complex vectors,

$$M_{\mathit{nm}}^1=A_0{q}_{\mathit{nm}}+g_{\mathit{nm}},$$E23

$$M_{\mathit{nm}}^2=A_0{q}_{\mathit{nm}}-g_{\mathit{nm}},$$E24

belong to the SLM modulation curve. Note that Eqs.(23) and (24) are the general forms of Eqs. (13) and (14). The constant value $A_0$ is a maximum possible amplitude (for instance, the radius of the blue circle shown in Figure 8) to fulfill complex amplitude-phase modulation, in which the average of each pair of modulation points ($M_{\mathit{nm}}^1$,$M_{\mathit{nm}}^2$) on the modulation curve should be an interior point inside the circle. As a result, the pair of modulation points ($M_{\mathit{nm}}^1$,$M_{\mathit{nm}}^2$) always exist. Besides, the maximum CGH efficiency is related to the maximum possible value$A_0$.

Since the set of the modulation points is finite and discrete, we should find the nearest accessible complex value denoted $q_{\mathit{nm}}^a$ to the desired complex value $A_0{q}_{\mathit{nm}}$. Thus, practically we select the pair modulation points ($M_{\mathit{nm}}^1$,$M_{\mathit{nm}}^2$) in such a way that its middle point has the minimum Euclidean distance from the desired complex value corresponding with each holography cell as follows:

$${\epsilon }_{\mathit{nm}}=min\left|A_0{q}_{\mathit{nm}}-q_{\mathit{nm}}^a\right|.$$E25

The remaining issue is to select the position of $M_{\mathit{nm}}^1$ and $M_{\mathit{nm}}^2$ on the modulation curve. In this regard, there are two possibilities as follows:

• Selection 1: $M_{\mathit{nm}}^1$ performs a clockwise rotation (smaller than 180°) of the radial line containing $q_{\mathit{nm}}^a$, where $M_{\mathit{nm}}^1$ has a smaller phase than $M_{\mathit{nm}}^2$ as demonstrated in Figure 9a.

• Selection 2: $M_{\mathit{nm}}^1$ performs a counter-clockwise rotation of this radial line, where $M_{\mathit{nm}}^2$ has a smaller phase than $M_{\mathit{nm}}^1$, as indicated in Figure 9b.

Figure 9.

(a) Selection 1 (b) selection 2.

On one side, he showed that the appropriate position-selection of the pair modulation points to encode a discrete modulation $q_{\mathit{nm}}=\left|q_{\mathit{nm}}\right|exp\left(j{\tau }_{\mathit{nm}}\right)$, where both the modulus $\left|q_{\mathit{nm}}\right|$ and the phase ${\tau }_{\mathit{nm}}$ are soft or quasi-continuous functions, is a way explained in selection 1, which leads to

$$M_{\mathit{nm}}=\left\{\begin{array}{cc}{M}_{\mathit{nm}}^1& \left(n+m\right)\text{even}\\ M_{\mathit{nm}}^2& \left(n+m\right)\mathrm{odd}\end{array}\right..$$E26

This configuration is shown in Figure 10.

Figure 10.

The representation of the modified DPH at the LCD plane applying selection 1.

On the other side, he demonstrated that the appropriate position-selection for the pair modulation points to encode complex functions of the type $q_{\mathit{nm}}=r_{\mathit{nm}}exp\left(j{\tau }_{\mathit{nm}}^0\right)$, where both $r_{\mathit{nm}}$, which is a real factor, and ${\tau }_{\mathit{nm}}^0$ are quasi-continuous functions, is as follows:

$$M_{\mathit{nm}}=\left\{\begin{array}{c}\left\{\begin{array}{cc}{M}_{\mathit{nm}}^1& \left(n+m\right)\text{even}\\ M_{\mathit{nm}}^2& \left(n+m\right)\mathrm{odd}\end{array}\right.r_{\mathit{nm}}\ge 0\\ \begin{array}{cc}\left\{\begin{array}{cc}{M}_{\mathit{nm}}^2& \left(n+m\right)\text{even}\\ M_{\mathit{nm}}^1& \left(n+m\right)\mathrm{odd}\end{array}\right.& r_{\mathit{nm}}<0\end{array}\end{array}\right.$$E27

This configuration is shown in Figure 11. The display plane is divided into two areas: Area U corresponds $r\ge 0$ and area D corresponds $r<0$.

Figure 11.

The representation of the modified DPH at the LCD plane applying selections 1 and 2. The order of the distribution of values $M_{\mathit{nm}}^1$ and $M_{\mathit{nm}}^2$ in the transition area from U to D is changed.

This encoding algorithm is also appropriate for encoding continuous functions of the type $F\left(r,\theta \right)=R\left(r\right)exp\left(\mathit{jt\theta }\right)$ where $t$ is the topological charge, $\left(r,\theta \right)$ are polar coordinates, and $R\left(r\right)$ is a real function of a radial coordinate. In addition, if we intend to encode the complex function $F_{\mathit{nm}}=\left|r_{\mathit{nm}}\right|exp\left(j{\theta }_{\mathit{nm}}\right)$ the property of this function should be defined as $r_{\mathit{nm}}={\left(x_{\mathit{nm}}^2+y_{\mathit{nm}}^2\right)}^{1/2}$,$x_n=2\mathit{np}{y}_n=2\mathit{mp}$, $-N/2\le n<N/2$, and $-M/2\le m<M/2$. Note that the pixel size of the function is twice the pixel size of the applied LCD, in which four pixels of the LCD encode one holographic cell, as shown in Figures 10 and 11.

5. Fast generating DPHs using k-NN

As explained in Section 4, generating CGHs in the way that Arrizón proposed requires an extensive search of the minimum Euclidean distance between the desired complex values ($q_{\mathit{nm}}$) and the accessible ones ($q_{\mathit{nm}}^a$) defined by Eq. (25). The conventional calculation requires the use of several nested loops to satisfy Eq. (25) for each holographic cell. This conventional calculation is time-consuming. For instance, to generate a 768 × 1024 pixels CGH, and since four pixels at the SLM plane provide one holographic cell, there are (768 × 1024)/4 = 196,608 holographic cells that should be mapped among all accessible complex values. In [40], we presented a method using the k-NN classifier, which is able to generate DPHs 80 times faster than the conventional calculation.

The k-NN classifier is a type of nonparametric supervised machine learning algorithm [41]. Nonparametric models are characterized by memorizing the training datasets instead of learning them. This technique aims to train a dataset to label them into different known classes based on defined features. This algorithm can be described in three main steps:

1. To select the optimum number of k based on a distance metric.

2. To find the k nearest neighbors of the samples to be classified.

3. To predict the class label by majority vote.

In those cases where there is no majority vote, the machine predicts the class label based on the defined weight. We used the Scikit-learn Python library to implement the k-NN algorithm. Regarding the KNeighborsClassifier module, three parameters should be determined: k, weight, and metrics. Besides, a matrix with a-samples and b-features should be defined to train the machine. In this case, the number of samples is the number of all accessible complex points, while the features are chosen based on the real and imaginary value of each accessible complex point. According to the experimental modulation curve, the total number of classified samples (green points inside the blue circle shown in Figure 8) is 3540 (a = 3540) with two features (b = 2). The machine is trained based on data that come from the experimental modulation curve, while the machine will predict the nearest accessible complex values to the desired ones for each holographic cell. The optimum results are obtained by choosing k = 1, weights = distance, and metrics = Euclidean distance. On the one hand, a look-up matrix is made from all accessible complex values ($q^a$), as shown in Figure 12a. On the other hand, the machine is trained, as shown in Figure 12b.

Figure 12.

(a) The provided look-up matrix with 3540 rows and 3 columns. (b) the configuration of labeled training dataset with 3540 samples and two features.

So, the machine predicts the class label of the nearest accessible complex value to the desired one. According to the predicted label and look-up matrix, the pair of gray-level ($M_g^1$,$M_g^2$) will be distributed to the corresponding holographic cell, as shown in Figures 10 and 11.

Here, two beam profiles are considered to generate their CGHs. The first one is a (1,1)-Hermite-Gaussian (${\mathrm{HG}}_{11}$) with the wave equation given by

$${\mathrm{HG}}_{11}=\frac{4\mathit{xy}}{{w_0}^2}exp\left(-{\left(\frac{r}{w_0}\right)}^2\right)\mathit{\text{circ}}\left(\frac{r}{R}\right),$$E28

where $r=\sqrt{x^2+y^2}$, $w_0$ is the beam waist ($w_0=R/2$), and R is the radius of the circular beam support. The first row of Figure 13 indicates the amplitude and phase of ${\mathrm{HG}}_{11}$, respectively, whereas the second row demonstrates the nearest accessible values predicted by the k-NN classifier according to the experimental modulation curve.

Figure 13.

The first row shows the distribution of the amplitude and phase of HG11, respectively. The second row shows their nearest values predicted by the k-NN classifier according to the experimental modulation curve.

The second beam profile is a (0,1)-Laguerre-Gaussian (${\mathrm{LG}}_{01}$), in which the complex wave equation is given by

$${\mathrm{LG}}_{01}=\left[\left(\frac{2x}{w_0}\right)+j\left(\frac{2y}{w_0}\right)\right]exp\left(-{\left(\frac{r}{w_0}\right)}^2\right)\mathit{\text{circ}}\left(\frac{r}{R}\right)$$E29

The amplitude and phase distribution of ${\mathrm{LG}}_{01}$ and their nearest accessible values predicted by the machine are shown in the first and second rows of Figure 14. The corresponding CGHs for ${\mathrm{HG}}_{11}$ and ${\mathrm{LG}}_{01}$ are shown in Figure 15.

Figure 14.

The first row shows the numerical distribution of the amplitude and phase of ${\mathrm{LG}}_{01}$, respectively. The second row shows their nearest values predicted by the k-NN classifier.

Figure 15.

The CGHs correspond with (a) ${\mathrm{HG}}_{11}$ and (b) ${\mathrm{LG}}_{01}$. The practical part of CGHs is selected for illustration purposes.

Table 1 indicates the required time to generate CGHs corresponding with ${\mathrm{HG}}_{11}$ and ${\mathrm{LG}}_{01}$ with four different resolutions. Numerical calculations have been carried out using Python 3.7.5 and the Scikit-learn library, and a Laptop with CPU i7-4510U (2 GHz) and 6 GB RAM. Besides, the processing time is obtained by the timeit module, and the results are averages of 10 runs. Note that the k-NN classifier can be accelerated using the RAPID cuML library performing on GPUs. As reported in [42], performing k-NN using the RAPID cuML library on GPUs is 600 times faster than performing k-NN applying the Scikit-learn library on CPUs. As a result, this proposed approach can generate real-time double-pixel computer-generated holograms.

Pixels	250 × 250	512 × 512	768 × 1024	1152 × 1920
${\mathrm{HG}}_{11}$	0.16	0.64	1.92	5.47
${\mathrm{LG}}_{01}$	0.38	1.54	5.53	16.39

Table 1.

Processing time (in seconds) for generating CGHs using the k-NN classifier.

6. Experimental results

The experimental setup (sketched in Figure 1) is modified in order to generate and analyze the beam at the entrance pupil and focal plane of a highly focusing system, as shown in Figure 16.

Figure 16.

The sketch of the experimental setup to capture the beam profile at the SLM and focal plane. R stands for a retarder that can be a QWP or vortex for providing circularly or radially polarized beam, respectively. MO stands for microscope objective. MS stands for the movable stage.

The left arm of the Mach-Zehnder interferometer is blocked by an obstacle. A vortex retarder (ThorLab, WPV10L-532) or a QWP (R) is added after LP2 to provide a radially or circularly polarized beam, respectively. In order to provide a radial polarization, the fast axis of the vortex retarder is placed parallel to the y-axis. In order to provide a circular polarization, the fast axis of QWP is rotated 45° with respect to the x-axis.

The beam is separately imaged at the entrance pupil of microscope objective MO1 (Nikon Plan Fluorite N40X-PF with NA = 0.75) and at the sensor plane of CCD1 by means of the 4f-system and BS3. Microscope objective MO2 (Nikon with NA = 0.8) is mounted on a movable stage driven by a motorized device (Newport LTA-HL) with uni-directional repeatability of $\pm 100$nm. MO2 is used to scan different planes close to the focal plane of MO1 and image them to the sensor plane of the CCD2 camera. Note that MO2 has a larger NA than MO1 to collect the entire beam. Furthermore, the actual magnification of the imaging system provided by MO2 is obtained by imaging a USAF target placed in the front of MO2, resulting in a 100x and spatial sampling of 37.5 nm. LP3 and QWP2 are used to record a set of six polarimetric images.

We recently used this experimental setup to estimate the longitudinal component of a highly focused beam using a phase retrieval algorithm and Gauss’s theorem. The method is able to retrieve transverse and longitudinal components of a highly focused electromagnetic field (for more details, see [43]).

Since the intensity pattern of a beam at the focal plane strongly depends on its polarization state at the entrance pupil of MO1 [44], two different polarization states have been considered to compromise the experimental results with the numerical ones. The numerical calculations have been implemented by applying the focused field calculation method introduced in [16]. Figure 17 shows the intensity patterns recorded by CCD1 regarding ${\mathrm{HG}}_{11}$ and ${\mathrm{LG}}_{01}$ beams.

Figure 17.

The intensity patterns recorded by CCD1 correspond to (a) ${\mathrm{HG}}_{11}$ beam and (b) ${\mathrm{LG}}_{01}$ beam.

Figure 18 indicates the intensity patterns of the circularly polarized ${\mathrm{HG}}_{11}$ beam at the focal plane of MO1. The first row indicates the Stokes images, which were obtained numerically, whereas the second row shows the intensity measurement of the Stokes images, which were recorded by CCD2. The Stokes images are denoted by $\mathrm{I}\left(\theta ,\delta \right)$, where $\theta$ and $\delta$ are the rotation angles of the axis of LP3 and the phase delay introduced by means of QWP2 with respect to the x-axis, respectively. Moreover, the polarimetric images are normalized by the maximum intensity of the transverse components of the electromagnetic field.

Figure 18.

The stokes images correspond to the circularly polarized ${\mathrm{HG}}_{11}$ beam at the focal plane. The first row shows the numerical results, while the second row demonstrates the recorded intensity by CCD2. The size of each image is 3 μm.

In a similar way, Figure 19 indicates the Stokes images correspond to the radially polarized ${\mathrm{LG}}_{01}$ beam. As results show, the obtained Stokes images are in excellent agreement with the numerical ones. However, the state of polarization is altered slightly due to the imperfection of applied retarders.

Figure 19.

The stokes images correspond to the radially polarized ${\mathrm{LG}}_{01}$ beam at the focal plane. The first row shows the numerical results, while the second row demonstrates the recorded intensity by CCD2. The size of each image is 3 μm.

7. Conclusions

This chapter provided all the necessary steps to generate complex beams with arbitrary intensity and phase distribution using a translucent TNLC display in one frame. Characterizing a TNLC-SLM accompanied by the DPH Arrizon’s approach has been widely reviewed, and the k-NN classifier has been applied to generate CGHs faster than conventional calculation. Two wave functions have been experimentally assessed at the SLM and the focal plane of a high-NA microscope objective. Finally, the experimental setup has been described in order to generate focused vector beams and measure the corresponding Stokes images.

Acknowledgments

The author acknowledges support from the PredocsUB program.