Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display

*Kavan Ahmadi*

## **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.

**Keywords:** Twisted-nematic liquid crystal display, computer-generated hologram, beam shaping, highly focused beam, machine-learning algorithms

## **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 [5–7], pattern recognition and optical correlators [8], Fresnel lenses [9, 10], and optical cryptosystem [11–13].

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 [14–21]. 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 [22–26]. 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 computergenerated 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 [30–32]. 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-SNRsignal 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 [35–37].

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 (*no*,*ne*). 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

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

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 (*α*), 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*π*ð Þ *ne* � *no d=λ*, 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 (*θ*) depends on the applied voltages causing different birefringence. So, LC becomes a variable retarder with retardation Γ ¼ 2*π*½ � *n*ð Þ� *θ n*<sup>0</sup> *d=λ*. The retardation varies monotonically from 0 (when the molecules are not tilted, *θ* ¼ 0) to Γmax ¼ 2*π*½ � *ne* � *n*<sup>0</sup> *d=λ* (when molecules are tilted 90°, *θ* ¼ 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

$$E\_t = A \exp\left(-j\rho\right),\tag{1}$$

where *A* stands for the transmitted amplitude and *φ* 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 *λ* ¼ 514 nm is collimated and linearly polarized by means of the collimator lens and the linear polarizer (LP1), respectively. Then, the

**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.*

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).

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.

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

$$\mathbf{g}(\mathbf{x},\mathbf{y}) = a(\mathbf{x},\mathbf{y}) + b(\mathbf{x},\mathbf{y})\cos\left[2\pi f\_0 \mathbf{x} + \phi(\mathbf{x},\mathbf{y}) + \rho\_m\right],\tag{2}$$

where *ϕ*ð Þ *x*, *y* and *φ<sup>m</sup>* are the phase of the object beam and the modulated phase imposed by SLM, respectively. Furthermore, *a x*ð Þ , *y* represents possible nonuniform background, *b x*ð Þ , *y* represents the local contrast of the pattern, and *f* <sup>0</sup> is the spatialcarrier frequency. As explained in [39], the fringe pattern can be rewritten in the following form:

$$\mathbf{g}(\mathbf{x},\mathbf{y}) = \mathbf{a}(\mathbf{x},\mathbf{y}) + \mathbf{c}(\mathbf{x},\mathbf{y})\exp\left(j2\pi f\_0 \mathbf{x}\right) + \mathbf{c}^\*(\mathbf{x},\mathbf{y})\exp\left(-j2\pi f\_0 \mathbf{x}\right),\tag{3}$$

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

#### **Figure 2.**

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

with

$$c(\mathbf{x}, \mathbf{y}) = (\mathbf{1}/2)b(\mathbf{x}, \mathbf{y}) \exp\left[j\phi(\mathbf{x}, \mathbf{y})\right] \exp\left(j\rho\_m\right). \tag{4}$$

The Fourier transform of Eq. (3) gives

$$\mathcal{G}\left(f\_{\mathbf{x}}, f\_{\mathbf{y}}\right) = \mathcal{A}\left(f\_{\mathbf{x}}, f\_{\mathbf{y}}\right) + \mathcal{C}\left(f\_{\mathbf{x}} - f\_{\mathbf{0}}, f\_{\mathbf{y}}\right) + \mathcal{C}^\*\left(f\_{\mathbf{x}} + f\_{\mathbf{0}}, f\_{\mathbf{y}}\right), \tag{5}$$

where the capital letters denote the Fourier spectra, *f <sup>x</sup>* and *f <sup>y</sup>* are the spatial frequencies in the x- and y-direction, respectively. In fact, *A f <sup>x</sup>*, *f <sup>y</sup>* is the zero-order of the interference and *C f <sup>x</sup>* � *f* <sup>0</sup>, *f <sup>y</sup>* and *<sup>C</sup>*<sup>∗</sup> *<sup>f</sup> <sup>x</sup>* <sup>þ</sup> *<sup>f</sup>* <sup>0</sup>, *<sup>f</sup> <sup>y</sup>* are �1 interference orders, respectively.

Since the spatial variations of *a x*ð Þ , *y* , *b x*ð Þ , *y* , and *ϕ*ð Þ *x*, *y* are slow compared with spatial frequency *f* <sup>0</sup>, the Fourier spectra are separated by carrier frequency *f* <sup>0</sup>, 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*<sup>∗</sup> . Complex value *C* can be rewritten as follows:

$$\mathcal{C}\left(f\_{\mathbf{x}}f\_{\mathbf{y}}\right) = \text{FT}\{ (\mathbf{1}/2)b(\mathbf{x},\mathbf{y})\exp\left[ (\phi(\mathbf{x},\mathbf{y}) + 2\pi f\_{\mathbf{0}}\mathbf{x}) \right] \exp\left(j\rho\_m\right) \}\tag{6}$$

where FT denotes the Fourier transform, and j j *C* and Φ 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 f <sup>x</sup>*, *f <sup>y</sup>*  varies only with *φm*, 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 f <sup>x</sup>*, *f <sup>y</sup>* corresponding with each part gives the phase shift as follows:

$$\mathcal{C}\left(f\_{\mathbf{x}}, f\_{\mathbf{y}}\right)\_{\mathbf{y}} = |\mathsf{C}| \exp\left(j\Phi\right) \exp\left(j\rho\_{\mathbf{y}}\right),\tag{7}$$

$$\mathcal{C}\left(f\_{\mathbf{x}}, f\_{\mathbf{y}}\right)\_{\mathrm{vg}} = |\mathsf{C}| \exp\left(j\Phi\right) \exp\left(j\rho\_{\mathrm{vg}}\right),\tag{8}$$

$$
\Delta \boldsymbol{\rho} = \arg \left[ \mathbf{C} \left( \boldsymbol{f}\_{\times} \boldsymbol{f}\_{\times} \right)\_{\text{rg}} \right] - \arg \left[ \mathbf{C} \left( \boldsymbol{f}\_{\times} \boldsymbol{f}\_{\times} \right)\_{\text{rg}} \right] + \boldsymbol{\rho}\_0. \tag{9}
$$

The subscripts *rg*, *vg*, and Δ*φ* indicate the reference, the variable gray values, and the phase shift, respectively. *φ*<sup>0</sup> is selected in which Δ*φ* ¼ 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,

## **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.*

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

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.

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 f <sup>x</sup>*, *f <sup>y</sup>* 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.

In this practical case, the maximum achievable phase imposed by SLM is around 1.75π. 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**.

## **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

#### **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.*

**Figure 5.** *(a) The phase modulation. (b) The amplitude modulation. (c) The polar plot of the phase-mostly modulation.*

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:

$$M\_{\mathfrak{g}} = |M\_{\mathfrak{g}}| \exp\left(j\nu\_{\mathfrak{g}}\right),\tag{10}$$

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* � *M* arrays (pixels). The modulation *Mnm* in the (*n*,*m*) the pixel can be defined as follows:

$$M\_{nm} = |\mathcal{M}\_{nm}| \exp\left(j\varphi\_{nm}\right). \tag{11}$$

To encode a desired complex modulation value

$$q\_{mn} = |q\_{nm}| \exp\left(j\tau\_{nm}\right),\tag{12}$$

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

**Figure 6.** *The DPH configuration.*

**Figure 7.** *The codification algorithm.*

Arrizón employed the holographic double pixel shown in **Figure 6**, whose pixels have the complex modulation *M*<sup>1</sup> *nm* and *M*<sup>2</sup> *nm* that belong to the modulation curve. As shown in **Figure 6**, the holographic cell is equal to a double pixel with the encoded modulation *qnm* plus an error double pixel, with the modulation values *e*<sup>1</sup> *nm* and *e*<sup>2</sup> *nm*. The conditions required to produce an on-axis signal reconstruction with a null contribution of the error term at the zero frequency are

$$q\_{nm} = \left(\mathbf{M}\_{nm}^1 + \mathbf{M}\_{nm}^2\right) / 2,\tag{13}$$

$$
\varepsilon\_{nm}^2 = -\varepsilon\_{nm}^1.\tag{14}
$$

To explain this, considering the modulation points on the modulation curve (*Mg*) 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*<sup>1</sup> and *M*<sup>2</sup> . As a result, we can access the modulation points (*q*) beyond the restricted SLM responses (*Mg*). Consequently, the modulation errors are *e*<sup>1</sup> *nm* <sup>¼</sup> *<sup>M</sup>*<sup>1</sup> *nm* � *qnm* and *<sup>e</sup>*<sup>2</sup> *nm* <sup>¼</sup> *<sup>M</sup>*<sup>2</sup> *nm* � *qnm*, which lead to Eq. (14).

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

#### **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.*

radius *A*<sup>0</sup> ¼ 0*:*29 can encode a complex function with an amplitude ranging from 0 to *A*<sup>0</sup> and a phase ranging from 0 to 2*π*.

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

$$h(\mathbf{x}, \mathbf{y}) = \sum\_{n,m} \mathcal{M}\_{nm} w(\mathbf{x} - np, \mathbf{y} - mp),\tag{15}$$

where *p* is the pixel pitch and *w x*ð Þ¼ , *y* rectð Þ *x=a* rectð Þ *y=b* . Considering the CGH is intended to encode the spatially quantized complex function

$$q(\mathbf{x}, \mathbf{y}) = \sum\_{nm} q\_{nm} w(\mathbf{x} - np, \mathbf{y} - mp), \tag{16}$$

where *qnm* is defined in Eq. (12) in which *qnm* � � � �≤1. Assuming the spectrum of *q x*ð Þ , *y* denoted *Q u*ð Þ , *v* is centered at the zero frequency ð Þ¼ *u*, *v* ð Þ 0, 0 . Hence, the CGH transmittance must be related to the encoded complex modulation *q x*ð Þ , *y* by the following expression:

$$h(\mathbf{x}, \mathbf{y}) = A\_0 q(\mathbf{x}, \mathbf{y}) + e(\mathbf{x}, \mathbf{y}), \tag{17}$$

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

$$H(u,v) = A\_0 Q(u,v) + E(u,v). \tag{18}$$

The error spectrum, *E u*ð Þ , *v* , 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:

$$e(\mathbf{x}, \mathbf{y}) = l(\mathbf{x}, \mathbf{y}) \mathbf{g}(\mathbf{x}, \mathbf{y}), \tag{19}$$

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

$$l(\mathbf{x}, \mathbf{y}) = \sum\_{n,m} l\_{nm} w(\mathbf{x} - np, \mathbf{y} - np), \tag{20}$$

$$\lg(\mathbf{x}, \mathbf{y}) = \sum\_{n,m} \mathbf{g}\_{nm} w(\mathbf{x} - np, \mathbf{y} - np). \tag{21}$$

He demonstrated that the optimal choice *l x*ð Þ , *y* is the binary grating with discrete modulation *lnm* ¼ �ð Þ<sup>1</sup> *<sup>n</sup>*þ*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 u*ð Þ , *v* centered at the spatial frequency coordinates 1ð Þ *=*2*p*, 1*=*2*p* , ð Þ �1*=*2*p*, 1*=*2*p* , ð Þ 1*=*2*p*, �1*=*2*p* , and ð Þ �1*=*2*p*, �1*=*2*p* . 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 x*ð Þ , *y* is specified by its discrete modulation *lnm*, which is related to the CGH modulation by the formula

$$M\_{nm} = A\_0 q\_{nm} + \left(-\mathbf{1}\right)^{n+m} \mathbf{g}\_{nm}.\tag{22}$$

Since both function *q x*ð Þ , *y* and *g x*ð Þ , *y* have on-axis spectrum bands, their variation should be negligible when the increment (Δ*n* ¼ 1) in *x* is of the order of the pixel pitch. To satisfy this condition, Arrizón proposed to establish the discrete function *gnm* such that both complex vectors,

$$\mathbf{M}\_{nm}^{1} = \mathbf{A}\_{0}\mathbf{q}\_{nm} + \mathbf{g}\_{nm},\tag{23}$$

$$\mathbf{M}\_{nm}^2 = \mathbf{A}\_0 \mathbf{q}\_{nm} - \mathbf{g}\_{nm},\tag{24}$$

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*<sup>0</sup> 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*<sup>1</sup> *nm*,*M*<sup>2</sup> *nm*) on the modulation curve should be an interior point inside the circle. As a result, the pair of modulation points (*M*<sup>1</sup> *nm*,*M*<sup>2</sup> *nm*) 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<sup>a</sup> nm* to the desired complex value *A*0*qnm*. Thus, practically we select the pair modulation points (*M*<sup>1</sup> *nm*,*M*<sup>2</sup> *nm*) 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:

$$
\varepsilon\_{nm} = \min \left| A\_0 q\_{nm} - q\_{nm}^a \right|. \tag{25}
$$

The remaining issue is to select the position of *M*<sup>1</sup> *nm* and *M*<sup>2</sup> *nm* on the modulation curve. In this regard, there are two possibilities as follows:

• Selection 1: *M*<sup>1</sup> *nm* performs a clockwise rotation (smaller than 180°) of the radial line containing *q<sup>a</sup> nm*, where *M*<sup>1</sup> *nm* has a smaller phase than *M*<sup>2</sup> *nm* as demonstrated in **Figure 9a**.

**Figure 9.** *(a) Selection 1 (b) selection 2.*

• Selection 2: *M*<sup>1</sup> *nm* performs a counter-clockwise rotation of this radial line, where *M*<sup>2</sup> *nm* has a smaller phase than *M*<sup>1</sup> *nm*, as indicated in **Figure 9b**.

On one side, he showed that the appropriate position-selection of the pair modulation points to encode a discrete modulation *qnm* ¼ *qnm* � � � � exp ð Þ *jτnm* , where both the modulus *qnm* � � � � and the phase *τnm* are soft or quasi-continuous functions, is a way explained in selection 1, which leads to

$$M\_{nm} = \begin{cases} M\_{nm}^1 & (n+m)\text{even} \\ M\_{nm}^2 & (n+m)\text{odd} \end{cases} \tag{26}$$

This configuration is shown in **Figure 10**.

On the other side, he demonstrated that the appropriate position-selection for the pair modulation points to encode complex functions of the type *qnm* <sup>¼</sup> *rnm* exp *<sup>j</sup>τ*<sup>0</sup> *nm* � �, where both *rnm*, which is a real factor, and *τ*<sup>0</sup> *nm* are quasi-continuous functions, is as follows:

$$M\_{nm} = \begin{cases} \begin{cases} M\_{nm}^1 & (n+m)\text{even} \\ M\_{nm}^2 & (n+m)\text{odd} \end{cases} & r\_{nm} \ge 0 \\ \begin{cases} M\_{nm}^2 & (n+m)\text{even} \\ M\_{nm}^1 & r\_{nm} < 0 \end{cases} & r\_{nm} < 0 \end{cases} \tag{27}$$

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

This encoding algorithm is also appropriate for encoding continuous functions of the type *F r*ð Þ¼ , *θ R r*ð Þ exp ð Þ *jtθ* where *t* is the topological charge, ð Þ *r*, *θ* are polar coordinates, and *R r*ð Þ is a real function of a radial coordinate. In addition, if we intend to encode the complex function *Fnm* ¼ j j *rnm* exp ð Þ *jθnm* the property of this function

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

#### **Figure 10.**

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

#### **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*<sup>1</sup> *nm and M*<sup>2</sup> *nm in the transition area from U to D is changed.*

should be defined as *rnm* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> *nm* <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> *nm* <sup>1</sup>*=*<sup>2</sup> ,*xn* ¼ 2*npyn* ¼ 2*mp*, �*N=*2≤ *n*< *N=*2, and �*M=*2 ≤ *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 (*qnm*) and the accessible ones (*qa nm*) 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 bfeatures 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 lookup matrix is made from all accessible complex values (*qa*), as shown in **Figure 12a**. On the other hand, the machine is trained, as shown in **Figure 12b**.

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 graylevel (*M*<sup>1</sup> *<sup>g</sup>* ,*M*<sup>2</sup> *<sup>g</sup>*) will be distributed to the corresponding holographic cell, as shown in **Figures 10** and **11**.

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*


#### **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.*

Here, two beam profiles are considered to generate their CGHs. The first one is a (1,1)-Hermite-Gaussian (HG11) with the wave equation given by

$$\text{HG}\_{11} = \frac{4\text{xy}}{w\_0^{-2}} \exp\left(-\left(\frac{r}{w\_0}\right)^2\right) \text{circ}\left(\frac{r}{R}\right),\tag{28}$$

where *<sup>r</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>p</sup> , *<sup>w</sup>*<sup>0</sup> is the beam waist (*w*<sup>0</sup> <sup>¼</sup> *<sup>R</sup>=*2), and *<sup>R</sup>* is the radius of the circular beam support. The first row of **Figure 13** indicates the amplitude and phase of HG11, 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 (LG01), in which the complex wave equation is given by

$$\text{LG}\_{01} = \left[ \left( \frac{2r}{w\_0} \right) + j \left( \frac{2y}{w\_0} \right) \right] \exp\left( -\left( \frac{r}{w\_0} \right)^2 \right) \text{circ} \left( \frac{r}{R} \right) \tag{29}$$

The amplitude and phase distribution of LG01 and their nearest accessible values predicted by the machine are shown in the first and second rows of **Figure 14**. The corresponding CGHs for HG11 and LG01 are shown in **Figure 15**.

**Figure 14.**

*The first row shows the numerical distribution of the amplitude and phase of* LG01*, respectively. The second row shows their nearest values predicted by the k-NN classifier.*

#### **Figure 15.**

*The CGHs correspond with (a)* HG11 *and (b)* LG01*. The practical part of CGHs is selected for illustration purposes.*

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*


**Table 1.**

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

**Table 1** indicates the required time to generate CGHs corresponding with HG11 and LG01 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.

## **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**.

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

#### **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.*

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 100nm. 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 HG11 and LG01 beams.

**Figure 18** indicates the intensity patterns of the circularly polarized HG11 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

### **Figure 17.** *The intensity patterns recorded by CCD1 correspond to (a)* HG11 *beam and (b)* LG01 *beam.*

#### **Figure 18.**

*The stokes images correspond to the circularly polarized* HG11 *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.*

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

**Figure 19.**

*The stokes images correspond to the radially polarized* LG01 *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.*

the Stokes images, which were recorded by CCD2. The Stokes images are denoted by Ið Þ *θ*, *δ* , where *θ* and *δ* 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.

In a similar way, **Figure 19** indicates the Stokes images correspond to the radially polarized LG01 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.

## **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.

## **Acknowledgements**

The author acknowledges support from the PredocsUB program.

*Holography - Recent Advances and Applications*

## **Author details**

Kavan Ahmadi University of Barcelona, Barcelona, Spain

\*Address all correspondence to: k1ahmadi@ub.edu

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

## **References**

[1] Maria-Luisa C, Albertina C, Arrizón V. Phase shifting digital holography implemented with a twistednematic liquid-crystal display. Applied Optics. 2009;**48**:6907-6912

[2] Maluenda D, Juvells I, Martínez-Herrero R, Carnicer A. A digital holography technique for generating beams with arbitrary polarization and shape. Proceedings of SPIE, Optical System Design. 2012:**8550**:84403Q. DOI: 10.1117/12.979926

[3] Davis JA, McNamara DE, Cottrell DM, Sonehara T. Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator. Applied Optics. 2000;**39**:1549-1554

[4] Shen C, Tan J, Wei C, Liu Z. Coherent diffraction imaging by moving a lens. Optics Express. 2016;**24**:16520-16529

[5] Rong Z-Y, Han Y-J, Wang S-Z, Guo C-S. Generation of arbitrary vector beams with cascaded liquid crystal spatial light modulators. Optics Express. 2014;**22**:1636-1644

[6] Martínez-Herrero R, Maluenda D, Juvells I, Carnicer A. Experimental implementation of tightly focused beams with unpolarized transversal component at any plane. Optics Express. 2014;**22**: 32419-32428

[7] Martínez-Herrero R, Maluenda D, Juvells I, Carnicer A. Synthesis of highly focused fields with circular polarization at any transverse plane. Optics Express. 2014;**22**:6859-6867

[8] Moreno I, Campos J, Gorecki C, Yzuel MJ. Effects of amplitude and phase mismatching errors in the generation of a Kinoform for pattern recognition. Japanese Journal of Applied Physics. 1995;**34**:6423-6432

[9] Lin S, Li C, Kuo C, Yeh H. Fresnel lenses in 90° twisted-Nematic liquid crystals with optical and electrical controllability. IEEE Photonics Technology Letters. 2016; **28**(13):1462-1464. DOI: 10.1109/ LPT.2016.2555699

[10] Lin C, Huang H, Wang J. Polarization-independent liquid-crystal Fresnel lenses based on surface-mode switching of 90° twisted-Nematic liquid crystals. IEEE Photonics Technology Letters. 2010;**22**(3):137-139. DOI: 10.1109/LPT.2009.2036738

[11] Maluenda D, Carnicer A, Martínez-Herrero R, Juvells I, Javidi B. Optical encryption using photon-counting polarimetric imaging. Optics Express. 2015;**23**:655-666

[12] Wu CW, Thompson G, Wright SL. Multiple images viewable on twistednematic mode liquid-crystal displays. IEEE Signal Processing Letters. 2003; **10**(8):225-227. DOI: 10.1109/ LSP.2003.814394

[13] Cheng C-J, Chen M-L, Tu H-Y. Polarization encoding for multi-channel optical encryption using twisted Nematic liquid crystal displays. In: Pacific Rim Conference on Lasers and Electro-Optics. Vol. 2005. Tokyo: CLEO- Technical Digest; 11-15 Jul 2005. pp. 981-982. DOI: 10.1109/CLEOPR. 2005.1569632

[14] Dorn R, Quabis S, Leuchs G. Sharper focus for a radially polarized light beam. Physics Review Letters. 2013;**91**:233901

[15] Davidson N, Bokor N. Highnumerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens. Optics Letters. 2004;**29**:1318-1320 [16] Leutenegger M, Rao R, Leitgeb RA, Lasser T. Fast focus field calculations. Optics Express. 2006;**14**:11277-11291

[17] Kozawa Y, Sato S. Sharper focal spot formed by higher-order radially polarized laser beams. Journal of the Optical Society of America. A. 2007;**24**: 1793-1798

[18] Wang H, Shi L, Lukyanchuk B, Sheppard C, Chong CT. Creation of a needle of longitudinally polarized light in vacuum using binary optics. Nature Photonics. 2008;**2**:501-505

[19] Lerman G, Levy U. Effect of radial polarization and apodization on spot size under tight focusing conditions. Optics Express. 2008;**16**:4567-4581

[20] Hao X, Kuang C, Wang T, Liu X. Phase encoding for sharper focus of the azimuthally polarized beam. Optics Letters. 2010;**35**:3928-3930

[21] Khonina SN, Volotovsky SG. Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures. Journal of the Optical Society of America. A. 2010;**27**: 2188-2197

[22] Moreno I, Bennis N, Davis JA, Ferreira C. Twist angle determination in liquid crystal displays by location of local adiabatic points. Optics Communications. 1998;**158**:231-238

[23] Davis JA, Moreno I, Tsai P. Polarization eigenstates for twistednematic liquid-crystal displays. Applied Optics. 1998;**37**:937-945

[24] Ponce R, Serrano-Heredia A, Arrizón VM. Simplified optimum phase-only configuration for a TNLCD. Proceedings of SPIE. 2004; **5556**:206-213

[25] Pezzaniti JL, Chipman RA. Phaseonly modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states. Optics Letters. 1993;**18**:1567-1569

[26] Chiang Y, Chou T, Chen S, Chao C. Effects of low viscosity liquid on the electro-optical properties of inverse twisted Nematic liquid crystal display. IEEE Transactions on Electron Devices. 2017;**64**:1630-1634

[27] Matín-Badosa E, Carnicer A, Juvells I, Vallmitjana S. Complex modulation characterization of liquid crystal devices by interferometric data correlation. Measurement Science and Technology. 1997;**8**:764-772

[28] Wang H, Dong Z, Fan F, Feng Y, Lou Y, Jiang X. Characterization of spatial light modulator based on the phase in Fourier domain of the hologram and its applications in coherent imaging. Applied Sciences. 2018;**8**:1146. DOI: 10.3390/app8071146

[29] Wu YH, Chavel P. Cell-oriented onaxis computer-generated holograms for use in the Fresnel diffraction mode. Applied Optics. 1984;**23**:228-238

[30] Hsueh CK, Sawchuk AA. Computergenerated double-phase holograms. Applied Optics. 1987;**17**:3874-3883

[31] Florence JM, Juday RD. Full-complex spatial filtering with a phase mostly DMD. In: Wave Propagation and Scattering in Varied Media II. San Diego, CA, United States: SPIE 1558; 1991. DOI: 10.1117/12.49655

[32] Mendlovic D, Shabtay G, Levi U, Zalevsky Z, Emanuel M. Encoding technique for design of zero-order (on-axis) Fraunhofer computergenerated holograms. Applied Optics. 1997;**36**:8427-8434

*Beam Implementation with a Translucent Twisted-Nematic Liquid Crystal Display DOI: http://dx.doi.org/10.5772/intechopen.105671*

[33] Arrizón V. Improved double-phase computer-generated holograms implemented with phase-modulation devices. Optics Letters. 2002;**27**:595-597

[34] Arrizón V, Sánchez-de-la-Llave D. Double-phase holograms implemented with phase-only spatial light modulators: Performance evaluation and improvement. Applied Optics. 2002;**41**: 3436-3447

[35] Arrizón V. Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: Double-pixel approach. Optics Letters. 2003; **28**(15):1359-1361. DOI: 10.1364/ ol.28.001359

[36] Arrizón V. Optimum on-axis computer-generated hologram encoded into low-resolution phase-modulation devices. Optics Letters. 2003;**28**: 2521-2523

[37] Arrizón V, González LA, Ponce R, Serrano-Heredia A. Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquidcrystal displays. Applied Optics. 2005; **44**:1625-1634

[38] Saleh BEA, Teich MC. Electro-optics. In: Saleh BEA, Teich MC, editors. Fundamentals of Photonics. 2nd ed. Hoboken, New Jersey: Wiley & Sons; 2007. pp. 696-736

[39] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringepattern analysis for computer-based topography and interferometry. Journal of the Optical Society of America. 1982; **72**:156-160

[40] Ahmadi K, Maluenda D, Carnicer A. Fast mapping of double-pixel holograms using k-nearest neighbors. In: OSA Imaging and Applied Optics Congress. Washington, DC United States: OSA

Technical Digest: DW5E-7. 19-23 Jul 2021. DOI: 10.1364/DH.2021.DW5E.7

[41] Raschka S. Python Machine Learning. Birmingham: Packt Publishing; 2015

[42] Deotte C. Accelerating k-Nearest Neighbors 600x using RAPIDS cuML [Internet]. 2005. Available from: https:// medium.com/rapids-ai/acceleratingk-nearest-neighbors-600x-using-rapidscuml-82725d56401e

[43] Maluenda D, Aviñoá M, Ahmadi K, Martínez-Herrero R, Carnicer A. Experimental estimation of the longitudinal component of a highly focused electromagnetic field. Scientific Reports. 2021;**11**:17992. DOI: 10.1038/ s41598-021-97164-z

[44] Novotny L, Hecht B. Propagation and focusing of optical fields. In: Novotny L, Hecht B, editors. Principales of Nano-Optics. New York: Cambridge University Press; 2006. pp. 45-87

## **Chapter 18**

## Correlations in Scattered Phase Singular Beams

*Vanitha Patnala, Gangi Reddy Salla and Ravindra Pratap Singh*

## **Abstract**

We discuss about the correlations present in the scattered phase singular beams and utilize them for obtaining the corresponding mode information. We experimentally generate the coherence vortices using the cross-correlation functions between the speckle patterns and validate them with the exact analytical expressions. We also explore their propagation characteristics by considering their geometry along with their divergence. We utilize the autocorrelation measurements of speckle patterns for obtaining the mode information. Further, we study the correlations present in scattered perfect optical vortices which lead to a new class of coherence functions, Bessel-Gauss coherence functions, and utilized for generating the non-diffracting random fields, i.e. propagation invariant fields. We utilized these correlation functions, which are orderdependent although the speckle patterns are order-independent, for encrypting the information which has higher advantage than normal random optical fields.

**Keywords:** phase singular beams, scattering, speckles, correlation function, optical encryption

## **1. Introduction**

Phase singular beams or optical vortices are well known due to their applications in guiding the particles, for coding larger information per photon, for transferring spatial structure to the materials, and for enhanced accuracy in metrological measurements [1–10]. These beams have ring-shaped intensity distribution along with helical wavefronts and have phase singularity at the center [11–16]. These beams carry an orbital angular momentum (OAM) of *m*ℏ per photon due to its azimuthal phase, where *m* is the order or topological charge defined as number of helices completed in one wavelength. The propagation of these light beams through various media such as turbid media [17, 18], turbulence atmosphere [19, 20], and under water [21, 22] have attracted lot of interest in recent days for utilizing them for communication applications [23–30]. The vortices can be generated using computer generated holography [31, 32] along with the help of spatial light modulator [33, 34], spiral phase plate [35, 36], and using an astigmatic mode converter [37]. Some advanced techniques have been introduced for generating vortex beams through laser cavity and using materials [38, 39]. After including the polarization to the spatial mode of light beam,

we get the vector vortices which have been studied extensively for sensing and communication applications [40, 41]. For sensing the magnetic field, these beams will pass through the materials that have magnetic field-dependent properties [42].

The field distribution of an optical vortex beam in polar coordinates can be expressed mathematically as [43]:

$$E(r,z) = E\_0 r^m \exp\left(im\,\phi\right) \exp\left(-\frac{r^2}{\alpha(z)^2}\right); r^2 = \varkappa^2 + \jmath^2\tag{1}$$

where *E0* is the field amplitude, *ω*(*z*) is the beam width at propagation distance *z*, and *m* is topological charge. The wavefront, phase profile, and the intensity distribution of vortices have been shown in **Figure 1**.

The vortices can be observed in all the random optical fields, known as speckles, which have random temporal and spatial coherence properties [44, 45]. These patterns can be obtained upon the propagation of coherent random waves through an inhomogeneous media such as ground glass plate (GGP) [46, 47]. This speckle is due to the superposition of many scattered waves originating from the inhomogeneities of the medium [48]. The size of the speckles can be carried by changing the width and wavelength of the beam, and the distribution of speckles can be changed by varying

#### **Figure 1.**

*The intensity distribution (left), wavefronts (middle), and the phase profiles (right) of optical vortex (OV) beams with* m *= 0 (top),* m *= +1 (middle), and* �*1 (bottom).*

**Figure 2.**

*The speckle patterns generated by the scattering of an optical vortex of order +1 (left), order +2 (middle), and order +3 (right) through the ground glass plate.*

the field distribution incidenting on the rough surface [49–52]. The speckle patterns obtained by the scattering of optical vortices of orders *m* = 1–3 have been shown in **Figure 2**.

The phase singularities have also been observed in correlation functions and named as coherence vortices [47, 53–55]. The singularities have been verified both theoretically and experimentally using the interferometric techniques. The intensity correlation between the speckle patterns has attained a lot of interest due to their applications in speckle imaging and encryption applications [56–60]. These correlations have been used for finding the roughness of the surface and the effect of turbulence on the spatial modes [61, 62]. The roughness of the surface can be characterized by assuming the delta-correlated random phase screen and well described using a Gaussian correlation function.

In this chapter, we consider the correlations present in the scattered phase singular beams, normal optical vortices, and perfect optical vortices (POVs) for obtaining the information about the spatial mode. We discuss about the coherence vortices which can be obtained through the cross-correlation present in the speckle patterns corresponding to two optical vortices of different orders. We present the intensity distribution and propagation characteristics of coherence vortices by considering the cross-correlations and utilize the autocorrelation measurements for obtaining the mode information. Then we study the correlations present in scattered perfect optical vortices which lead to a new class of coherence functions, Bessel-Gauss functions, and utilized for generating the non-diffracting random fields. We utilized these correlation functions, which are order-dependent although the speckle patterns are orderindependent, for encrypting the information which has higher advantage than normal random optical fields.

## **2. Cross-correlations present in scattered optical vortices: realization of coherence vortices**

The phase singularities have been studied extensively in coherent light beams, and in recent days, partially coherent phase singularities have gained a considerable interest due to their robustness against atmospheric propagation [63–65]. The vortices present in partially coherent fields are known as coherence vortices as they can be realized in correlation functions [66–70]. These coherence vortices have been utilized for many applications such as free-space optical communication, remote sensing, and optical imaging [71–75]. The correlation between the two optical random fields plays

an important role in obtaining the various types of coherence functions and their usage in applications, such as optical communication and for producing the physical unclonable functions (PUFs) for cryptography [60, 76–78]. The coherence vortices can be observed in the intensity correlation between two speckle patterns obtained by scattering the coherent vortices of different orders [55, 79]. The coherence vortices can be formulated with mutual coherence function between two speckle patterns corresponding to the vortices of orders *m*<sup>1</sup> and *m*<sup>2</sup> and is given by [55, 80]:

$$\tilde{\Gamma}\_{m\_1,m\_2} = A \int r\_1^{|m\_1|+|m\_2|} e^{[i(m\_1-m\_2)\phi\_1]} \mathcal{e}^{\left[-2r\_1^2/a\_0^2\right]} \mathcal{e}^{\left[-2\vec{k}\cdot\vec{r}\_1\right]} d\vec{r}\_1 \tag{2}$$

One can clearly observe the phase singularity with order *m* = *m*1–*m*<sup>2</sup> where *m* is the order of the coherence vortex.

For realizing the singularities in coherence functions, we need to scatter the coherent vortex beams through a rough surface such as GGP. The coherent optical vortices can be generated using a computer-generated hologram displayed on a spatial light modulator. After selecting the required vortex beam by an aperture, we scatter these beams through the GGP, and the corresponding speckle patterns are recorded using a CCD camera. We now find the cross-correlation function between two speckle patterns corresponding to optical vortices of different orders using MATLAB software. **Figure 3** shows the speckle patterns along with the determined coherence functions for different values of *m*<sup>1</sup> and *m*2. It is clear from the figure that the autocorrelation between the speckle patterns provides the coherence function of order

**Figure 3.**

*The recorded speckle patterns and the corresponding cross-correlation functions, coherence vortices (here m* ¼ *m*<sup>2</sup> � *m*1*).*

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

0. The cross-correlation between the speckle patterns corresponding to two different orders provides the higher-order coherence functions.

The coherence vortices have been characterized through their geometry by considering similarly as that of coherent vortex beams. **Figure 4** shows the intensity distribution of an optical vortex and its line profile along the center for order *m* = 1. We characterized the optical vortices by considering them as thin annular rings and using the parameters inner and outer radii *r*1, *r*<sup>2</sup> as shown in figure. These are the nearest (inner) and farthest (outer) radial distances from center at which the intensity falls to 1∕e2 (13.6%) of the maximum intensity observed at *r*= *r*<sup>0</sup> [46].

**Figure 5** shows the variation of inner and outer radii of the coherence vortex ð Þ *m* ¼ *m*<sup>2</sup> � *m*<sup>1</sup> of order 2 obtained by considering the cross-correlation between two speckle patterns of different values of *m*<sup>1</sup> and *m*<sup>2</sup> (with constant *m*) at the propagation distance of *z* = 20 cm. We considered the combinations of (*m*1, *m*2) = (0,2), (1,3), (2,4), (3,5), (4,6), (5,7), (6,8) where the difference (*m*1-*m*2) is constant.

From the figure, we observe that the inner and outer radii for all the combinations mentioned earlier are constant and independent of the input vortex beams considered for scattering. From this, we confirm that the intensity distribution of coherence

**Figure 4.** *(a) Intensity distribution and (b) line profile through its center for an optical vortex of order 1.*

**Figure 5.**

*Variation of inner and outer radii of the coherence vortex of order 2 with different combinations of input vortex beams.*

#### **Figure 6.**

*Variation of inner (a) and outer (b) radii for coherence vortices of order* m *= 1–8 with the propagation distance* z*.*

vortices depends only on the order difference but not on the individual orders of the optical vortices considered for scattering.

Now, we study the propagation characteristics of these coherence vortices. **Figure 6a** and **6b** show the variation of inner and outer radii for different orders *m* = 1–8 with respect to the propagation distance from *z* = 10–30 cm. The speckles have been recorded from *z* = 10–30 cm at an interval of 5 cm. It is observed that the inner and outer radii from figure vary linearly with the propagation distance for all orders and increase with order as shown in **Figure 6**.

We consider the rate of change of inner and outer radii with propagation distance as divergence and can be obtained by finding the slope of the line drawn between inner or outer radius and the propagation distance [79, 81, 82]. The slope has been determined using the linear fit to the experimental data. The variation of divergence with the order by considering inner and outer radii has been shown in **Figure 7**. It is clear from the figure that the divergence increases linearly with order (*m*). One can utilize the inner and outer radii at the source plane and their divergence for

**Figure 7.** *Variation of inner and outer radii along with their divergence as a function of order of the coherence functions.*

characterizing the order of a coherence vortex. One can also find the information about the incident spatial modes using these coherence vortices.

## **3. Autocorrelation studies for scattered optical vortices**

For the applications in free-space optical communication using spatial modes, one needs to propagate these modes for longer distances. After propagating through the channel, the mode information gets disturbed, and one needs to find the mode information of these perturbed beams. Although there are many techniques to find the order of a higher-order coherent optical vortex [46], they are not suitable for partially coherent or incoherent vortices. A limited number of techniques are available for finding the order of a partially coherent vortex beam. In this section, we study the autocorrelation properties of scattered optical vortices for diagnosing the spatial mode information [83–85]. The number of zero points or dark rings present in 2D spatial correlation function provides the information about the spatial mode. The spatial autocorrelation function of a perturbed optical vortex is equivalent to the Fourier transform (FT) of its intensity in the source plane [86]. The number of dark rings presented in the spatial correlation function is equal to the topological charge of vortex beam which has also been verified by verifying the number of zero points present in Fourier transform of a coherent vortex beam which further will be discussed. Here, we show that the existence of the ring dislocations in the spatial correlation function corresponds to the scattering of the optical vortex field [87].

The theoretical background for the 2D autocorrelation function starts by assuming the field distribution of Laguerre-Gaussian beam with azimuthal index *m* and zero radial index in the source plane (*z* = 0) in cylindrical coordinates as:

$$E(\rho, \theta, \mathbf{0}) \propto \rho^{|m|} \exp\left(\frac{-\rho^2}{\rho \alpha\_0^2}\right) \exp\left(im\,\theta\right) \tag{3}$$

where *ω*<sup>0</sup> is beam waist of the input beam and ð Þ *ρ*, *θ* are the cylindrical coordinates in incident plane. The scattering of optical vortex (OV) beams through a ground glass plate (GGP) that can be well described by a random phase function exp ð Þ *i*Φ where Φ varies randomly from 0 to 2*π*. A particular way of obtaining this type of phase distribution Φ is by taking a 2D convolution between a random spatial function and a Gaussian correlation function [44]. The field *U*ð Þ *ρ*, *θ* after the GGP can be obtained from the incident field *E*ð Þ *ρ*, *θ* and can be written as:

$$U(\rho,\theta) \propto \exp\left(i\Phi\right)E(\rho,\theta) \tag{4}$$

where the autocorrelation of the phase exponential factor is a Dirac-delta function at plane ð Þ *ρ*, *θ* , which can be written mathematically as:

$$\langle \exp\left[i(\Phi(\rho\_1, \ \theta\_1) - \Phi(\rho\_2, \ \theta\_2))\right] \rangle = \delta(\rho\_1 - \rho\_2)\delta(\theta\_1 - \theta\_2) \tag{5}$$

where h i *a* denotes the ensemble average operation in *a*. The autocorrelation function between two speckle patterns of same order obtained by scattering of OV beams through GGP is given by:

$$
\Gamma(r\_1, \rho\_1; r\_2, \rho\_2) = \left\langle U\_1(r\_1, \,\rho\_1) U\_2^\* \,(r\_2, \,\rho\_2) \right\rangle \tag{6}
$$

where (*r*, φ) are the coordinates at the detection plane. The filed at the detection plane in terms field at the incident plane can be evaluated using Fresnel's diffraction integral in cylindrical coordinates as [88, 89]:

$$\mathcal{U}(r,\rho,z) = \frac{e^{ikz}}{i\lambda\overline{z}} \left\{ \rho d\rho \int d\theta \mathcal{U}(\rho,\theta) e^{\left\{\frac{\hbar}{2i}(\rho^2 + r^2 - 2\rho r \cos\left(\theta - \rho\right))\right\}}\right\} \tag{7}$$

Using Eq. (7) and Eq. (6), we have that

$$\begin{aligned} \Gamma(r\_1, \rho\_1; r\_2, \rho\_2) &= \left< U\_1(r\_1, \,\rho\_1) U\_2^\* \,(r\_2, \,\rho\_2) \right> \\ &= \left< \frac{e^{ikz}}{i\lambda z} \right> \left< \rho\_1 d\rho\_1 \int d\theta\_1 U\_1(\rho\_1, \,\theta\_1) e^{\left\{\frac{ik}{2\lambda} (\rho\_1^2 + r\_1^2 - 2\rho\_1 r\_1 \cos(\theta\_1 - \rho\_1))\right\}} \right> \\ &\times \frac{e^{-ikz}}{-i\lambda z} \left[ \rho\_2 d\rho\_2 \int d\theta\_2 U\_2^\* \,(\rho\_2, \,\theta\_2) e^{\left\{\frac{ik}{2\lambda} (\rho\_2^2 + r\_2^2 - 2\rho\_2 r\_2 \cos(\theta\_2 - \rho\_2))\right\}} \right] \\ &= \frac{e^{\left\{\frac{ik}{2\lambda} [r\_1^2 - r\_1^2] \right\}}}{\lambda^2 z^2} \left[ \rho\_1 d\rho\_1 \int d\theta\_1 \int \rho\_2 d\rho\_2 \int d\theta\_2 \langle U\_1(\rho\_1, \,\theta\_1) U\_2 \,^\*(\rho\_2, \,\theta\_2) \rangle \right. \tag{8} \\ &\times e^{\left\{\frac{ik}{2\lambda} (\rho\_1^2 - \rho\_1^2 - 2\rho\_1 r\_1 \cos(\theta\_1 - \rho\_1) + 2\rho\_2 r\_2 \cos(\theta\_2 - \rho\_2)) \right\}} \end{aligned} \tag{9} \tag{8}$$

which is a fourfold integral and includes cross-correlation of filed at the incident plane ð Þ *<sup>ρ</sup>*, *<sup>θ</sup>* namely, *<sup>U</sup>*<sup>1</sup> *<sup>ρ</sup>*<sup>1</sup> ð Þ , *<sup>θ</sup>*<sup>1</sup> *<sup>U</sup>* <sup>∗</sup> <sup>2</sup> *ρ*<sup>2</sup> ð Þ , *θ*<sup>2</sup> � �*:* Using Eq. (4) and Eq. (5), we get the cross-correlation function as

$$\begin{aligned} \left< U\_1(\rho\_1, \theta\_1) U\_2 \,^\*(\rho\_2, \,\theta\_2) \right> &= \left< E\_1(\rho\_1, \,\theta\_1) e^{i\Phi(\rho\_1, \,\theta\_1)} E\_2^\*(\rho\_2, \,\theta\_2) e^{i\Phi(\rho\_2, \,\theta\_2)} \right> \\ &= E\_1(\rho\_1, \,\theta\_1) E\_2^\*\left(\rho\_2, \,\theta\_2\right) \left< e^{i(\Phi(\rho\_1, \,\theta\_1) - \Phi(\rho\_2, \,\theta\_2))} \right> \\ &= E\_1(\rho\_1, \,\theta\_1) E\_2^\*\left(\rho\_2, \,\theta\_2\right) \times \delta(\rho\_1 - \rho\_2) \delta(\theta\_1 - \theta\_2) \end{aligned} \tag{9}$$

The ground glass plate (random phase screen) is modeled as a δ-correlated phase function. The autocorrelation function after using the same is

$$
\langle U\_1(\rho\_1, \theta\_1) U\_2 \,^\*(\rho\_2, \,\theta\_2) \rangle = E(\rho\_1, \theta\_1) E^\*(\rho\_2, \theta\_2) \tag{10}
$$

Using Eq. (10) in Eq. (9) and the properties of the Dirac-delta function, the fourfold integral of the autocorrelation is reduced to the two-fold integral as:

$$\begin{split} \Gamma(r\_1, \rho\_1; r\_2, \rho\_2) &= \left\langle U\_1(r\_1, \,\rho\_1) U\_2^\* \,(r\_2, \,\rho\_2) \right\rangle \\ &= \frac{e^{\left\{\frac{\mu}{2\pi} \left[r\_1^2 - r\_2^2\right] \right\}}}{\lambda^2 x^2} \int \rho d\rho \int d\theta E(\rho, \,\theta) E^\* \,(\rho, \,\theta) \\ &\times e^{\left\{\frac{-\mu}{x} \rho \left(r\_1 \cos\left(\theta - \rho\_1\right) - r\_2 \cos\left(\theta - \rho\_2\right)\right) \right\}} \end{split} \tag{11}$$

$$\Gamma\_{12}(\Delta r) = \frac{e^{\left\{\frac{ik}{2z}\left[r\_1^2 - r\_2^2\right]\right\}}}{\lambda^2 z^2} \iint \left| E(\rho, \ \theta) \right|^2 \exp\left[ -\frac{ik}{z} \left(\rho \Delta r \cos\left(\rho\_s - \theta\right)\right) \right] \rho d\rho d\theta \tag{12}$$

where Δ*r* cosð Þ¼ *φ<sup>s</sup>* � *θ r*<sup>1</sup> cos *φ*<sup>1</sup> ð Þ� *r*<sup>2</sup> cos *φ*<sup>2</sup> ½ð Þ ð Þ cos *θ*� þ *<sup>r</sup>*<sup>1</sup> sin *<sup>φ</sup>*<sup>1</sup> ð Þ� *<sup>r</sup>*<sup>2</sup> sin *<sup>φ</sup>*<sup>2</sup> ½ � ð Þ ð Þ sin *<sup>θ</sup>* and <sup>Δ</sup>*r*<sup>2</sup> <sup>¼</sup> *<sup>r</sup>*<sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup> <sup>2</sup> � 2*r*1*r*<sup>2</sup> cos *φ*<sup>2</sup> � *φ*<sup>1</sup> ð Þ*:* Using Eq. (3), the absolute value of the field distribution is

$$\left|E(\rho,\ \phi,\ \mathbf{0})\right|^2 = \rho^{2|m|} \exp\left(\frac{-2\rho^2}{a\_0^2}\right) \tag{13}$$

Let us calculate the integral part of the correlation as

$$\begin{split} \Gamma\_{12}(\Delta r) &= \frac{e^{\left\{\frac{ik}{2}[r\_{i}^{2}-r\_{i}^{2}]\right\}}}{\lambda^{2}z^{2}} \iint \rho^{2|m|} \exp\left(\frac{-2\rho^{2}}{\alpha\_{0}^{2}}\right) \exp\left[-\frac{ik}{z}\left(\rho\Delta r\cos\left(\varphi\_{s}-\theta\right)\right)\right] \rho d\rho d\theta \\ &= \int \rho^{2|m|+1} \exp\left(\frac{-2\rho^{2}}{\alpha\_{0}^{2}}\right) d\rho \int \exp\left[-\frac{ik}{z}\left(\rho\Delta r\cos\left(\varphi\_{s}-\theta\right)\right)\right] d\theta \end{split} \tag{14}$$

and *I* <sup>0</sup> <sup>¼</sup> <sup>Ð</sup> exp �*ik <sup>z</sup>* ð Þ *<sup>ρ</sup>*Δ*<sup>r</sup>* cosð Þ *<sup>φ</sup><sup>s</sup>* � *<sup>θ</sup>* � �*d<sup>θ</sup>* can be calculated by using Anger-Jacobi identity *<sup>e</sup>*�*iz* cos *<sup>θ</sup>* <sup>¼</sup> <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼�<sup>∞</sup>ð Þ �<sup>1</sup> *<sup>n</sup> i nJn*ð Þ*<sup>z</sup> <sup>e</sup>in <sup>θ</sup>*, we get [90]

$$I' = \int \exp\left[\frac{-ik}{z}\left(\rho \Delta r \cos\left(\rho\_r - \theta\right)\right)\right] d\theta = 2\pi I\_0 \left(\frac{k\rho}{z}\Delta r\right) \tag{15}$$

Substitute the aforementioned equation in Eq. (14), then the far-field autocorrelation function or the Fourier transform of the incident intensity in the source plane becomes

$$\Gamma\_{12}(\Delta r) = \frac{2\pi e^{\left\{\frac{k}{2r}\left[r\_1^2 - r\_2^2\right]\right\}}}{\lambda^2 x^2} \int\_0^\infty \rho^{2|m|+1} \exp\left(\frac{-2\rho^2}{\alpha\_0^2}\right) J\_0\left(\frac{k\rho}{x}\Delta r\right) d\rho \tag{16}$$

where *J*<sup>0</sup> *kρ <sup>z</sup>* Δ*r* � � represents the zeroth-order Bessel function. Using the following integral,

$$\int\_0^\infty x^\mu \exp\left(-a x^2\right) I\_\nu(\infty y) dx = \frac{y^\nu \Gamma(\frac{1}{2}(\mu-\nu+1))}{2^{\nu+1} a^{\frac{1}{2}(\mu+\nu+1)}} \exp\left(\frac{-y^2}{4a}\right) L^\nu\_{\frac{1}{2}(\mu-\nu+1)}\left(\frac{y^2}{4a}\right) \tag{17}$$

From Eqs. (16) and (17), the mutual coherence function will become

$$\Gamma\_{12}(\Delta r) = \frac{\pi \alpha\_0^{2|m|+2} e^{\left\{\frac{k}{\hbar}\left[r\_1^2 - r\_2^2\right]\right\}}}{2^{|m|+1} \lambda^2 z^2} \exp\left(\frac{-k^2 \alpha\_0^2 \Delta r^2}{8x^2}\right) L\_{|m|} \left(\frac{k^2 \alpha\_0^2 \Delta r^2}{8x^2}\right) \tag{18}$$

where *<sup>μ</sup>* <sup>¼</sup> <sup>2</sup>j j *<sup>m</sup>* <sup>þ</sup> 1,*<sup>α</sup>* <sup>¼</sup> <sup>2</sup> *ω*2 0 ,*<sup>y</sup>* <sup>¼</sup> *<sup>k</sup> <sup>z</sup>* Δ*r*,*ν* ¼ 0 and *L*j j *<sup>m</sup> k*2 *ω*2 0Δ*r*<sup>2</sup> 8*z*<sup>2</sup> � � represents the Laguerre polynomial of order *m*. The aforementioned equation represents the autocorrelation of scattered Laguerre-Gaussian (LG) beam, and it depends on the azimuthal index and propagation distance. In the spatial correlation filed, the number of dark rings or number of zero points in the Laguerre polynomial gives the information about the order or azimuthal index of the vortex beam. We verify these theoretical findings experimentally, and the details are given as follows.

**Figure 8.** *Intensity distributions of optical vortices and their corresponding speckle patterns.*

We have generated the optical vortices of orders *m* = 0–8 by displaying a computer-generated hologram on a spatial light modulator and scatter them through a GGP. We have shown the intensity distributions of optical vortices at the plane of GGP and the corresponding speckle patterns in **Figure 8** which have been recorded using a CCD camera. It is clear from the figure that the size of the speckles decreases with the increase in order, and we observe the structures in speckle distributions corresponding to higher orders.

Further, we have processed the speckle patterns for finding the autocorrelation function using MATLAB software. We found that the order or topological charge of a given spatial mode is given by the number of dark rings present in the autocorrelation function. This method is suitable for vortices with low topological charges. However, as we increase the order, we must identify the number of dark rings carefully because the adjacent two dark rings are very close to each other, and it is very difficult to distinguish them. This technique is alignment free as the autocorrelation function does not depend on the alignment. **Figure 9** shows the experimentally obtained spatial autocorrelation functions (top) for the speckle patterns corresponding to the vortices of orders *m* = 0–3 from left and right. The results are in good agreement with the theoretically obtained correlation function as shown in bottom row of the figure. It is clear from the figure that the order of vortex is equal to the number of dark rings present in the spatial autocorrelation field. One can also utilize the propagation characteristics for the better diagnosis of the information of a given spatial mode.

The generalized theory for autocorrelation functions of LG beams with nonzero radial index is provided and experimentally verified as well. The number of dark rings is equal to the sum of twice the radial index and azimuthal index [86]. The autocorrelation function of a scattered LG beam with nonzero radial and azimuthal indices is given by [91]:

$$\chi(\xi) = \frac{\pi \alpha\_0^{2|m|+2}}{2^{|m|+1}} \frac{(p+|m|)!}{p!} \exp\left(-\frac{\pi^2 \alpha\_0^2 \xi^2}{2}\right) L\_p\left(-\frac{\pi^2 \alpha\_0^2 \xi^2}{2}\right) L\_{p+|m|}\left(-\frac{\pi^2 \alpha\_0^2 \xi^2}{2}\right) \tag{19}$$

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

#### **Figure 9.**

*Experimental (top) and theoretical (bottom) 2D spatial autocorrelation function for a speckle pattern generated by scattering a vortex beams of orders* m *= 0-3 from left to right.*

where *p* is the radial index and *m* is the azimuthal index of a LG beam. We need to study the following subcases from the aforementioned expression for the better understanding of correlation function:

i. If *p* = 0, then the correlation function corresponds to the optical vortices that carry OAM and is given by:

$$\chi(\xi) = \frac{\pi \alpha\_0^{2|m|+2}}{2^{|m|+1}} |m|! \exp\left(-\frac{\pi^2 \alpha\_0^2 \xi^2}{2}\right) L\_{|m|}\left(-\frac{\pi^2 \alpha\_0^2 \xi^2}{2}\right) \tag{20}$$

The aforementioned expression is exactly matching with our equation obtained for LG beams with zero radial index.

ii. If *m* =0 (non-vortex beams), then the autocorrelation function is given by:

$$\chi(\xi) = \frac{\pi \alpha\_0^2}{2} \exp\left(-\frac{\pi^2 \alpha\_0^2 \xi^2}{2}\right) \left(L\_p \left(-\frac{\pi^2 \alpha\_0^2 \xi^2}{2}\right)\right)^2 \tag{21}$$

From Eqs. (19), (20), and (21), one can obtain the relation between number of dark rings and radial and azimuthal indices as:

$$\begin{aligned} N &= 2p + |m| \quad \text{when} \quad m \neq 0\\ &= p \quad \text{when} \quad m = 0. \end{aligned} \tag{22}$$

where *N* is the number of dark rings present in the autocorrelation function.

The numerical results for the LG beams of nonzero radial index have been shown in **Figure 10**. We can observe the number of dark rings in the far-field autocorrelation function which depends on both radial and azimuthal indices. The radial and azimuthal indices for the contour plots are (a) *m* = 1, p = 1; (b) *m* = 2, p = 1; (c) *m* = 3, p = 1; (d) *m* = 1, p = 2; (e) *m* = 2, p = 2; and (f) *m* = 3, p = 2.

**Figure 10.**

*Theoretical far-field auto-correlation function with different combinations of radial and azimuthal indices: (a) m = 1, p = 1; (b) m = 2, p = 1; (c) m = 3, p = 1; (d) m = 1, p = 2; (e) m = 2, p = 2; (f) m = 3, p = 2.*

Now, we verify Van Cittert-Zernike theorem states that the autocorrelation function of a scattered light beam is same as the Fourier transform (FT) of intensity distribution incident on the rough surface, i.e. source plane. Here, we present the results obtained for the FT of intensity distribution of a LG beam in which the number of dark rings is equal to the order of the vortex [92] as shown in **Figure 11**.

#### **Figure 11.**

*The theoretical and experimental Fourier transform contour plots for the intensity distribution of LG beams with azimuthal index* m *= 1–4 from left to right.*

## **3.1 Correlations in scattered perfect optical vortices**

From the aforementioned discussions, it is known that the field and intensity distributions of an optical vortex are strongly influenced by its topological charge which may be a drawback for controlling them while they propagate through optical channels [93]. To overcome this, another class of vortices has been introduced, known as perfect optical vortices (POVs) with order-independent intensity distribution [94]. The POV beams are known for very thin annular rings whose width and radius are independent of topological charge [95–98]. The POV beams can be generated experimentally by Fourier transforming Bessel-Gauss beams which will be generated by passing optical vortex beams through an axicon [99–101]. The radius and width of the ring of a POV beam can easily be controlled by changing the apex angle or axicon parameter [100, 102]. We study the correlations present in scattered POV beams and utilize for generating the non-diffracting optical random fields [103]. The theoretical analysis for the cross-correlation between two speckle patterns obtained by scattering POV beams of different orders is as follows:

The field distribution of a perfect optical vortex (POV) beam, described by a thin annular ring of order *m*, which can be represented mathematically as:

$$E(\rho, \theta) = \delta(\rho - \rho\_0)e^{im\,\theta} \tag{23}$$

where *ρ*<sup>0</sup> is the radius of the POV beam and *δ* represents the Dirac-delta function. In practical, the POV beams can be realized with finite ring width which can be represented mathematically as:

$$E(\rho, \theta) = \mathbf{g}(\rho - \rho\_0; \varepsilon)e^{im\,\theta} \tag{24}$$

where *g ρ* � *ρ*<sup>0</sup> ð Þ ; *ε* is a narrow function in the radial direction with a finite "width" *ε*, such as Gaussian and height proportional to 1/*ε*. The scattered field *U*ð Þ *ρ*, *θ* after the GGP can be obtained from the incident field *E*ð Þ *ρ*, *θ* on the GGP as

$$U(\rho,\theta) = e^{i\Phi(\rho,\ \theta)}E(\rho,\theta) \tag{25}$$

where the cross-correlation of the phase exponential factor is a Dirac-delta function at plane ð Þ *ρ*, *θ* , which implies the mutually independent inhomogeneities that can be expressed mathematically as:

$$\left\langle e^{i[\Phi(\rho\_1,\ \theta\_1) - \Phi(\rho\_2,\ \theta\_2)]} \right\rangle = \delta(\rho\_1 - \rho\_2)\delta(\theta\_1 - \theta\_2) \tag{26}$$

where ⟨ *a* ⟩ denotes the ensemble average operation in *a* [9]. The mutual coherence function between the two scattered POV fields at a distance of *z* from the GGP is given by:

$$
\Gamma(r\_1, \rho\_1; r\_2, \rho\_2) = \left\langle U\_1(r\_1, \,\rho\_1) U\_2^\* \,(r\_2, \,\rho\_2) \right\rangle \tag{27}
$$

where *ρ* and *θ* are source plane coordinates, and *r* and *φ* are detection plane coordinates. The field at detection plane can be obtained using Fresnel diffraction integral in cylindrical coordinates as: [88, 89]

$$U(r,\rho,z) = \frac{e^{ikz}}{i\lambda\overline{z}} \left\{ \rho d\rho \int d\theta U(\rho,\theta) e^{\left\{\frac{ik}{\lambda z} (\rho^2 + r^2 - 2\rho r \cos\left(\theta - \rho\right))\right\}} \right. \tag{28}$$

From Eq. (27) and Eq. (28), we have

$$\begin{split} \Gamma(r\_1, q\_1; r\_2, q\_2) &= \left< U\_1(r\_1, \,\,\rho\_1) U\_2^\* \begin{pmatrix} r\_2 \ \,\,\rho\_2 \end{pmatrix} \right> \\ &= \left< \frac{e^{ikx}}{i\lambda\overline{x}} \right> \rho\_1 d\rho\_1 \Big| d\theta\_1 U\_1(\rho\_1, \,\,\theta\_1) e^{\left\{\frac{\pm\mu}{\overline{x}} (\rho\_1^2 + r\_1^2 - 2\rho\_1 r\_1 \cos(\theta\_1 - \rho\_1))\right\}} \right. \\ &\times \frac{e^{-ikx}}{-i\lambda\overline{x}} \Big[ \rho\_2 d\rho\_2 \Big] d\theta\_2 U\_2 \,^\*(\rho\_2, \,\,\theta\_2) e^{\left\{\frac{-\pm\mu}{\overline{x}} (\rho\_2^2 + r\_2^2 - 2\rho\_2 r\_2 \cos(\theta\_2 - \rho\_2))\right\}} \Big] \\ &= \frac{e^{\left\{\frac{\pm\mu}{\overline{x}} (r\_1^2 - r\_2^2)\right\}}}{\lambda^2 \overline{x}^2} \Big[ \rho\_1 d\rho\_1 \Big] d\theta\_1 \Big[ \rho\_2 d\rho\_2 \Big] d\theta\_2 \langle U\_1(\rho\_1, \,\theta\_1) U\_2 \,^\*(\rho\_2, \,\,\theta\_2) \rangle \\ &\times e^{\left\{\frac{\pm\mu}{\overline{x}} (\rho\_1^2 - \rho\_2^2 - 2\rho\_1 r\_1 \cos(\theta\_1 - \rho\_1) + 2\rho\_2 r\_2 \cos(\theta\_2 - \rho\_2))\right\}} \end{split} \tag{29}$$

which is a fourfold integral, and this integral includes the cross-correlation of the field at the incident plane ð Þ *<sup>ρ</sup>*, *<sup>θ</sup>* , namely, *<sup>U</sup>*<sup>1</sup> *<sup>ρ</sup>*<sup>1</sup> ð Þ , *<sup>θ</sup>*<sup>1</sup> *<sup>U</sup>* <sup>∗</sup> <sup>2</sup> *ρ*<sup>2</sup> ð Þ , *θ*<sup>2</sup> � �. Using Eqs. (24) and (25), we can get the cross-correlation function at plane ð Þ *ρ*, *θ* as:

$$\begin{aligned} \left< U\_1(\rho\_1, \theta\_1) U\_2 \,^\*(\rho\_2, \,\theta\_2) \right> &= \left< E\_1(\rho\_1, \,\theta\_1) e^{i\Phi(\rho\_1, \,\theta\_1)} E\_2^\* \,(\rho\_2, \,\theta\_2) e^{i\Phi(\rho\_2, \,\theta\_2)} \right> \\ &= E\_1(\rho\_1, \,\theta\_1) E\_2^\* \,(\rho\_2, \,\theta\_2) \left< e^{i(\Phi(\rho\_1, \,\theta\_1) - \Phi(\rho\_2, \,\theta\_2))} \right> \\ &= E\_1(\rho\_1, \,\theta\_1) E\_2^\* \,(\rho\_2, \,\theta\_2) \times \delta(\rho\_1 - \rho\_2) \delta(\theta\_1 - \theta\_2) \end{aligned} \tag{30}$$

Using Eq. (30) in Eq. (29) and the properties of the Dirac-delta function, the fourfold integral of the cross-correlation is reduced to the twofold integral as:

$$\begin{aligned} \Gamma(r\_1, \rho\_1; r\_2, \rho\_2) &= \left\langle U\_1(r\_1, \,\rho\_1) U\_2^\* \,(r\_2, \,\rho\_2) \right\rangle \\ &= \frac{e^{\left\{\frac{kl}{x} \left[r\_1^2 - r\_2^2\right] \right\}}}{\lambda^2 x^2} \int \rho\_1 d\rho\_1 \int d\theta\_1 E\_1(\rho\_1, \theta\_1) E\_2^\* \,(\rho\_1, \theta\_1) \\ &\times e^{\left\{\frac{-kl}{x} \rho \left(r\_1 \cos\left(\theta\_1 - \rho\_1\right) - r\_2 \cos\left(\theta\_1 - \rho\_2\right)\right) \right\}} \end{aligned} \tag{31}$$

In the special case of incident POV beams, we can use Eq. (24) to write

$$E\_1(\rho\_1, \theta\_1) E\_2^\*\left(\rho\_1, \theta\_1\right) = \mathbf{g}(\rho\_1 - \rho\_{01}; \varepsilon) \mathbf{g}(\rho\_1 - \rho\_{02}; \varepsilon) \mathbf{e}^{i(m\_1\theta\_1 - m\_2\theta\_1)}\tag{32}$$

As we know that the radius of POV beams is independent of order, i.e. *ρ*<sup>01</sup> ¼ *ρ*<sup>02</sup> ¼ *ρ*0, therefore

$$E\_1(\rho\_1, \theta\_1) E\_2^\*\left(\rho\_1, \theta\_1\right) = \mathbf{g}^2(\rho\_1 - \rho\_0; \varepsilon) e^{i(m\_1 - m\_2)\theta\_1} \tag{33}$$

Under the condition *<sup>ε</sup>* ! 0, one can replace *<sup>g</sup>*<sup>2</sup> *<sup>ρ</sup>* � *<sup>ρ</sup>*<sup>0</sup> ð Þ ; *<sup>ε</sup>* with a single Dirac-delta function *δ ρ* � *ρ*<sup>0</sup> ð Þ and the aforementioned expression becomes

$$E\_1(\rho\_1, \theta\_1) E\_2^\*\left(\rho\_1, \theta\_1\right) = \delta(\rho\_1 - \rho\_0) \mathbf{e}^{i(m\_1 - m\_2)\theta\_1} \tag{34}$$

After substituting Eq. (34) in Eq. (31), we get

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

$$\Gamma(r\_1, \rho\_1; r\_2, \rho\_2) = \frac{e^{\left\{\frac{ik}{\hbar} \left[r\_1^2 - r\_2^2\right] \right\}}}{\lambda^2 z^2} \left\{\rho\_1 d\rho\_1 \left[d\theta\_1 \delta(\rho\_1 - \rho\_0) e^{i(m\_1 - m\_2)\theta\_1} e^{\left\{\frac{-ik}{\hbar}\rho (r\_1 \cos(\theta\_1 - \rho\_1) - r\_2 \cos(\theta\_1 - \rho\_2)) \right\}}\right] \right\} \tag{35}$$

The integral in the aforementioned equation can be evaluated as:

$$\begin{split} I &= \int \delta(\rho\_1 - \rho\_0) \rho\_1 d\rho\_1 \int e^{i(m\_1 - m\_2)\theta\_1} e^{\frac{-ik}{\pi}\rho(r\_1 \cos\left(\theta\_1 - \rho\_1\right) - r\_2 \cos\left(\theta\_2 - \rho\_2\right))} d\theta\_1 \\ &= \int \delta(\rho\_1 - \rho\_0) \rho\_1 d\rho\_1 \int e^{i(m\_1 - m\_2)\theta\_1} e^{\frac{-ik}{\pi}(\rho \Delta r \cos\left(\theta\_1 - \theta\_1\right))} d\theta\_1 \end{split} \tag{36}$$

For solving the integral of *θ*1, we assume that *θ*<sup>0</sup> ¼ *ϕ<sup>s</sup>* � *θ*<sup>1</sup> then we get the integral as:

$$I = e^{i(m\_1 - m\_2)\rho\_r} \int \delta(\rho\_1 - \rho\_0) \rho\_1 d\rho\_1 \int e^{i(m\_1 - m\_2)\theta'} e^{\frac{ik}{\pi}(\rho \Delta r \cos \theta')} d\theta' \tag{37}$$

where *θ*<sup>0</sup> varies from �*ϕ<sup>s</sup>* to 2*π* � *ϕs*and using Anger-Jacobi identity *<sup>e</sup>*�*iz* cos *<sup>θ</sup>* <sup>¼</sup> <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼�<sup>∞</sup>ð Þ �<sup>1</sup> *<sup>n</sup> i nJn*ð Þ*<sup>z</sup> <sup>e</sup>in <sup>θ</sup>*, we get [90]

$$I = e^{i(m\_1 - m\_2)\rho\_t} \int \delta(\rho\_1 - \rho\_0) \rho\_1 d\rho\_1 \sum\_{n = -\infty}^{\infty} (-1)^n i^n f\_n \left(\frac{k\rho}{z} \Delta r\right) \left[e^{in\,\theta} e^{i(m\_1 - m\_2)\theta'} d\theta'\right] \tag{38}$$

The aforementioned integral has nonzero value only when *n* ¼ *m*<sup>2</sup> � *m*<sup>1</sup> and the integral becomes

$$I = 2\pi e^{i(m\_1 - m\_2)\rho\_s} \int \delta(\rho\_1 - \rho\_0)\rho\_1 d\rho\_1 (-1)^{m\_2 - m\_1} i^{m\_2 - m\_1} J\_{m\_2 - m\_1} \left(\frac{k\rho}{z}\Delta r\right) \tag{39}$$

By using the integral properties of Dirac-delta function [90], we get that

$$I = 2\pi \rho\_0 (-i)^{m\_1 - m\_1} e^{i(m\_1 - m\_2)\rho\_s} I\_{m\_2 - m\_1} \left(\frac{k\rho\_0}{z} \Delta r\right) \tag{40}$$

Now, the cross-correlation function as defined in Eq. (35) becomes

$$\Gamma\_{12}(\Delta r) = \frac{2\pi\rho\_0(-i)^{m\_2 - m\_1}e^{\frac{ik}{2\pi}\left(r\_1^2 - r\_2^2\right)}}{\lambda^2 z^2} e^{i(m\_1 - m\_2)\rho\_i} J\_{m\_2 - m\_1}\left(\frac{k\rho\_0}{z}\Delta r\right) \tag{41}$$

The aforementioned equation representing the mutual coherence function of two speckle patterns is described well by the Bessel function of order *m* ¼ *m*<sup>2</sup> � *m*1*:* The corresponding cross-correlation function of two speckle patterns is given by:

$$\mathbf{C}(\Delta r) = (-i)^{m\_2 - m\_1} e^{i(m\_1 - m\_2)\rho\_r} J\_{m\_2 - m\_1} \left(\frac{k\rho\_0}{z} \Delta r\right) \tag{42}$$

Normalized intensity distribution of the coherence function can be evaluated in terms of time-averaged intensity *I*<sup>0</sup> as:

$$I(\Delta r) = I\_0^2 \left( \mathbf{1} + |\mathbf{C}(\Delta r)|^2 \right) = I\_0^2 \left( \mathbf{1} + J\_{m\_2 - m\_1}^2 \left( \frac{k \rho\_0}{z} \Delta r \right) \right) \tag{43}$$

If two speckle patterns correspond to the same order, the cross-correlation function is converted into an autocorrelation function, which can be obtained by keeping *m*<sup>1</sup> ¼ *m*<sup>2</sup> in the aforementioned equation. We obtain the autocorrelation function as:

$$I(\Delta r) = I\_0^2 \left( \mathbf{1} + J\_0^2 \left( \frac{k \rho\_0}{z} \Delta r \right) \right) \tag{44}$$

It is clear from the aforementioned analysis that the autocorrelation functions can be described with Bessel functions of order zero and cross-correlation functions can be described with Bessel functions of nonzero orders ð Þ *m* ¼ *m*<sup>2</sup> � *m*<sup>1</sup> *:*

The experimental validation of aforementioned theoretical findings has been done and the details are as follows: **Figure 12** shows the speckle patterns generated by the scattering of POV beams and the corresponding cross-correlation functions. From the figure, we confirm the Bessel-Gauss nature of coherence functions with order *m* ¼ *m*<sup>2</sup> � *m*1*:*

**Figure 12.** *The recorded speckle patterns and the corresponding cross-correlation functions, Bessel coherence functions (here m* ¼ *m*<sup>2</sup> � *m*1*).*

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

From Eq. (43), we analyze the size of the speckles under the condition *m*1=*m*2, i.e. by considering the width of the autocorrelation function. The speckle size is defined as the spatial length up to which the correlations exist in the field [103, 104]. From Eq. (44), the first zero of zeroth-order Bessel function *J*0ð Þ¼ *x* 0 can happen at *x* = 2.4, and the correlation length or speckle size can be obtained as:

$$
\Delta r = \frac{\varkappa z}{k \rho\_0} = \frac{2.4z}{k \rho\_0} \tag{45}
$$

It is clear from the aforementioned equation is that the size of near-field speckles varies linearly with propagation distance *z*, independent of order *m* and inversely proportional to the ring radius *ρ*0. We have observed that the speckle size and distribution are independent of the order, and they vary with propagation distance. The recorded speckles have been shown in **Figure 13**, and the linear variation of speckle size with propagation distance has been verified in **Figure 14**.

**Figure 13.**

*The speckle patterns obtained by the scattering of POV beam of order* m *= 0 at different propagation distances z = 20 cm, z = 45 cm, and z = 70 cm in the near field.*

#### **Figure 14.**

*Experimental (blue) and theoretical (red) results for the variation of near-field or diverging speckle size with propagation distance.*

Further, we consider the Fourier transform of near-field speckles that produce the spatially invariant optical random fields. The Fourier transform can be realized with the help of a simple convex lens (here, we consider its focal length as *f*2). The far-field autocorrelation function Γ12<sup>0</sup> Δ*r*<sup>0</sup> ð Þ of scattered POV beams is given by [89]:

$$\Gamma\_{12}{}^{\prime}(\Delta r^{\prime}) = \frac{1}{\lambda^2 f^2} \iint \left| U\_1(\rho, \ \theta) \right|^2 e^{\frac{-ik}{f\_2} (\rho \Delta r^{\prime} \cos \left( \phi\_{i1} - \theta \right))} \rho d\rho d\theta \tag{46}$$

Substituting Eq. (25) in Eq. (46) and following the same procedure, we get

$$
\Gamma\_{12}{}^{\prime}(\Delta r^{\prime}) = \frac{2\pi\rho\_0}{\lambda^2 f\_{\,\,2}{}^2} J\_0\left(\frac{k\rho\_0}{f\_{\,\,2}}\Delta r^{\prime}\right) \tag{47}
$$

From the aforementioned equation, one can easily observe that the correlation function is independent of order m as well as propagation distance *z*. As compared to the near-field diffraction, the spatial coherence function does not increase anymore with the propagation which can be utilized for communication and encryption applications.

From Eq. (47), we get the size of non-diffracting random fields as:

$$
\Delta \mathbf{r}' = \frac{\mathbf{x} f\_2}{k \rho\_0} = \frac{2.4 f\_2}{k \rho\_0} \tag{48}
$$

The speckle size is independent of propagation distance *z* and directly proportional to focal length *f*<sup>2</sup> and inversely proportional to ring radius *ρ*<sup>0</sup> . We can control the size by just varying the focal length and ring radius that can be controlled by axicon parameter [103].

**Figure 15** shows the speckle patterns recorded for different propagation distances and clear that are independent of propagation distances. **Figure 16** shows the variation of speckle for order 2 with propagation distance for different axicon parameters as mentioned. The size of the speckles decreases with the increase in axicon parameter which we attribute to the increase in area of illumination on the GGP. It is also shown that the speckle size is independent of the order.

#### **Figure 15.**

*The speckle patterns obtained by the scattering of POV beam of order* m *= 0 at different propagation distances in the far field.*

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

**Figure 16.** *The variation of speckle size with the propagation distance (left) and order (right) for different axicon parameters.*

## **4. Physical unclonable functions using the correlations of scattered POV beams**

Nowadays, securing private data, i.e. authenticating the authorized users to access the sensitive (personal) information, becomes mandatory. In the cryptographic algorithms, information that needs to be sent from a sender end is encrypted (i.e. input data are converted into an unreadable format) using secret keys. At the receiver end, by appropriately using the keys, encoded information can be retrieved (without loss), and this process is known as decryption. It is known that, depends on the cryptographic algorithm used, the keys for both the encryption and decryption process can be same or different [58, 105, 106]. Due to this reason, cryptographic algorithms are widely used in various fields, such as banking, healthcare, social medias, emails, and military communication, to name a few. However, recent developments in highperformance computers increased the vulnerability of cryptographic techniques for a number of different reasons [106]. To prevent from these attacks, a physical one-way function has been introduced in cryptographic systems which can be (physically) realized using the scattering of light beams [107]. These functions are, in general, known as physical unclonable functions (PUFs) and can be embedded into any optical systems for data authentication as this involves a scattering of light beams which results a random output, i.e. speckles [108]. Some of the advantages of PUF include (i) low cost (ii) high output complexity (iii) difficult to replicate, and (iv) high security against attacks [76, 109, 110]. Therefore, in this work, for the first time, we demonstrate an encryption system (i.e. linear canonical transform-based double random phase encoding (LCT-DRPE)) using PUFs that are generated by taking a correlation function between two speckle patterns obtained after scattering the POV beams through a ground glass plate. We wish to take the extra advantage of order-dependent correlation functions generated by the scattering of POV beams for producing the keys for encryption. Here, one should note that the speckle size and their distribution are order-independent, but the correlation between them is order-dependent [94]. We briefly describe the usage of the correlation functions as keys for encryption along with the decryption process as follows:

The LCT is a three-parameter class of linear integral transform and defined as [111]:

$$\Psi\_{a\beta\cdot\chi}\{f(\mathbf{x},\ \mathbf{y})\} = \mathcal{C}\_1 \left[ \int\_{-\infty}^{\infty} f(\mathbf{x},\ \mathbf{y}) \exp\left\{i\pi \left[a(\mathbf{x}^2 + \mathbf{y}^2) - 2\beta(\mathbf{u}\mathbf{x} + \mathbf{v}\mathbf{y}) + \gamma(\mathbf{u}^2 + \mathbf{v}^2)\right]\right\} \right] \tag{49}$$

where *α*,*β*,*γ* are the real-valued parameters that are independent of the coordinates that are applied symmetrically in both horizontally ð Þ *x* and ð Þ*y* , i.e. 2D separable LCT. The encrypted (output) image *E*ð Þ *ω*, *φ* can be expressed as [112, 113]:

$$E(o, \,\rho) = LCT\{LCT\{f(x, \,\,\,y) \times O\_1(x, \,\,y)\} \times O\_2(x, \,\,y)\}\tag{50}$$

where *f x*ð Þ , *y* is the 2D input image, *O*1ð Þ *x*, *y* and *O*2ð Þ *x*, *y* are two random phase masks (RPMs) considered as secret keys which are generated using a correlation function obtained from two scattered POV light beams, i.e. speckles. The schematic for LCT-DRPE is shown in **Figure 17**.

The resultant encrypted image resembles a white noise, i.e. speckle image. Therefore, it does not reveal any of the input information. It is therefore possible to reverse this process called decryption and get the original image back without loss. This process is given mathematically as:

$$f(\mathbf{x}, \boldsymbol{y}) = \text{ILCT}\{\text{ILCT}\{\mathbf{E}(\boldsymbol{\alpha}, \cdot \boldsymbol{\varrho})\} \times \mathcal{O}\_2^\*(\mathbf{x}, \boldsymbol{y})\} \times \mathcal{O}\_1^\*(\mathbf{x}, \boldsymbol{y})\tag{51}$$

where ILCT refers to inverse linear canonical transform and \* denotes the complex conjugate operation. The LCT parameters alpha, beta, and gamma are set as 10,100,1, respectively.

**Figure 18a** shows the input image (i.e. reconstructed hologram of a 3D object) [113, 114], and **Figure 18b** is the amplitude of the complex encrypted image, and information contained in it is very difficult to be observed. **Figure 18c** shows the decrypted image using appropriate secret keys. The decrypted image quality is the classical mean squared error (MSE) which is calculated between the input image and decrypted image.

In **Figure 19**, changes in the LCT parameter yield the fruitful results, i.e. not able to get proper decrypted images for the corresponding input data.

**Encryption using Fourier domain:** This is the one of the methods that allows to encode a primary image into a stationary white noise. We demonstrate how

**Figure 17.** *The schematic for LCT-based DRPE system.*

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

#### **Figure 18.**

*Simulation results: (a) input grayscale image, (b) encrypted image, and (c) decrypted image (MSE = 1.3685e–27).*

straightforward and reliable it is to rebuild the original image using the encoded image [115]. In fact, it is critical to have the ability to encrypt data in a way that makes it challenging to decode without a key yet simple to do so with a key but easy if one knows that key [116]. Let us consider the input signal to be encoded is a face Images, that is, since the image is a positive function and is two-dimensional, it is well known that it is possible to reconstruct an image from its Fourier magnitude [91, 117, 118]. The encoded image can be expressed as:

$$\psi(\mathbf{x}) = \{ f(\mathbf{x}) \exp\left(i2\pi n(\mathbf{x})\right) \} \* h(\mathbf{x}) \tag{52}$$

where f(x) is the input function and exp i2 ð Þ πn xð Þ is the random phase mask. Then, we convolve this image by the impulse response h(x) is the Fourier transform of the exp i2 ð Þ πbð Þν (**Figure 20**).

The encrypted image ψ(x) is optically Fourier-transformed and multiplied by the phase mask exp ð Þ �i2πbð Þν and then inverse Fourier-transformed to produce decrypted image. The decrypted image is expected for the input image with the addition of some noise u(x).

## **5. Conclusion**

In conclusion, we have briefly explained about the correlations present in scattered phase singular beams and their applications toward communication and encryption.

**Figure 20.** *Simulation results: (a) input grayscale image, (b) encrypted image, and (c) decrypted image.*

We have shown that the number of dark rings present in the autocorrelation function of speckles provides the information about the incident spatial mode, and one can utilize these results in free-space optical communication. Further we have utilized, the cross-correlations present in speckle patterns corresponding to vortices of different orders for generating the coherence vortices. We have discussed the geometry of coherence vortices along with their propagation characteristics. We further discussed about the correlations present in the scattered POV beams which produce the orderindependent speckle patterns. Finally, we utilized these cross-correlation functions for encryption applications and discussed in detail.

## **Author details**

Vanitha Patnala<sup>1</sup> , Gangi Reddy Salla<sup>1</sup> \* and Ravindra Pratap Singh<sup>2</sup>

1 Department of Physics, SRM University-AP, Amaravati, India

2 Physical Research Laboratory, Ahmedabad, India

\*Address all correspondence to: gangireddy.s@srmap.edu.in

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

## **References**

[1] Wu Y, Xu C, Qiu H, Xu D, Deng D. Guiding the optical vortex along predesigned parabolic trajectories from circular symmetric Airy-like beams. Applied Optics. 2022;**61**(8):1906-1911

[2] Song D, Wang H, Ma J, Tang L, Zheng X, Hu Y, et al. Synthetic optical vortex beams from the analogous trajectory change of an artificial satellite. Photonics Research. 2019;**7**(9):1101-1105

[3] Rode AV, Desyatnikov AS, Shvedov VG, Krolikowski W, Kivshar YS. Optical guiding of absorbing nanoclusters in air. Optics Express. 2009;**17**(7):5743-5757

[4] Liu F, Zhang Q, Cheng T, Wu X, Wei Y, Zhang Z. Photophoretic trapping of multiple particles in tapered-ring optical field. Optics Express. 2014; **22**(19):23716-23723

[5] Zhao J, Chremmos ID, Song D, Christodoulides DN, Efremidis NK, Chen Z. Curved singular beams for three-dimensional particle manipulation. Scientific Reports. 2015;**5**:12086

[6] Gong L, Lu R-D, Chen Y, Ren Y-X, Fang Z-X. Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device. Applied Optics. 2015;**54**(27):8030-8035

[7] Stoyanov L, Maleshkov G, Stefanov I, Paulus GG, Dreischuh A. Focal beam structuring by triple mixing of optical vortex lattices. Optical and Quantum Electronics. 2022;**54**:34

[8] Mao D, Zheng Y, Zeng C, Lu H, Wang C, Zhang H, et al. Generation of polarization and phase singular beams in fibers and fiber lasers. Advanced Photonics. 2021;**5**:014002

[9] Bhattacharya R. Generation of phase singular optical beams in microstructure optical fibers. Optics Communications. 2018;**428**:15-21

[10] Wang W, Yokozeki T, Ishijima R, Wada A, Miyamoto Y, Takeda M, et al. Optical vortex metrology for nanometric speckle displacement measurement. Optics Express. 2006;**14**(1):120-127

[11] Shen Y, Wang X, Xie Z, Min C, Fu X, Liu Q, et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light: Science & Applications. 2019;**8**:90

[12] Kovalev AA, Porfirev AP, Kotlyar VV. Asymmetric Gaussian optical vortex. Optics Letters. 2017; **42**(1):139-142

[13] Curtis JE, Grier DG. Structure of optical vortices. Physical Review Letters. 2003;**90**(13):133901-133904

[14] Allen L, Beijersbergen MW, Spreeuw RJC, Woerdman JP. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Physical Review A. 1992;**45**: 8185-8189

[15] Berry MV. Optical vortices evolving from helicoidal integer and fractional phase steps. Journal of Optics A: Pure and Applied Optics. 2004;**6**:259-268

[16] Franke-Arnold S, Allen L, Padgett M. Advances in optical angular momentum. Laser And Photonics Reviews. 2008;**2**(4):299-313

[17] Wang WB, Gozali R, Nguyen TA, Alfano RR. Propagation and transmission of optical vortex beams through turbid scattering wall with

orbital angular momentums. Proceedings of SPIE. 2015;**9318**:931805

[18] Shi L, Lindwasser L, Alfano RR, Gozali R, Wang WB. Deep transmission of Laguerre–Gaussian vortex beams through turbid scattering media. Optics Letters. 2016;**41**(9):2069-2072

[19] Zhu K, Li S, Tang Y, Yu Y, Tang H. Study on the propagation parameters of Bessel-Gaussian beams carrying optical vortices through atmospheric turbulence. Journal of the Optical Society of America A. 2012;**29**(3):251-257

[20] Zhou G, Tang H, Zhu K, Zheng X, Li X. Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere. Optics Express. 2008; **16**(26):21315-21320

[21] Hufnagel F, Sit A, Grenapin F, Bouchard F, Heshami K, Heshami K, et al. Characterization of an underwater channel for quantum communications in the Ottawa River. Optics Express. 2019; **27**(19):26346-26354

[22] Mobashery A, Parmoon B, Saghafifar H, Karahroudi MK, Moosavi SA. Performance evaluation of perfect optical vortices transmission in an underwater optical communication system. Applied Optics. 2018;**57**(30): 9148-9154

[23] Singh M, Atieh A, Grover A, Barukab O. Performance analysis of 40 Gb/s free space optics transmission based on orbital angular momentum multiplexed beams. Alexandria Engineering Journal. 2022;**61**(7): 5203-5212

[24] Li L, Zhang R, Zhao Z, Xie G, Liao P, Pang K, et al. High-capacity free-space optical communications between a ground transmitter and a ground receiver via a UAV using multiplexing of multiple orbital-angular-momentum beams. Scientific Reports. 2017;**7**:17427

[25] Gibson G, Courtial J, Vasnetsov M, Padgett MJ, Franke-Arnold S, Barnett SM, et al. Free-space information transfer using light beams carrying orbital angular momentum. Optics Express. 2004;**12**(22):5448-5456

[26] Qu Z, Djordjevic IB. Orbital angular momentum multiplexed free-space optical communication systems based on coded modulation. Applied Sciences. 2018;**8**(12):2179

[27] Willner AE, Pang K, Song H, Zou K, Zhou H. Orbital angular momentum of light for communications. Applied Physics Reviews. 2021;**8**:041312

[28] Eyyuboğlu HT. Optical communication system using Gaussian vortex beams. Journal of the Optical Society of America A. 2020;**37**(**10**): 1531-1538

[29] Wang J, Liu J, Li S, Zhao Y, Du J, Zhu L. Orbital angular momentum and beyond in free-space optical communications. Nano. 2022;**11**(4): 645-680

[30] Chen R, Zhou H, Moretti M, Wang X, Li J, Member S. Orbital angular momentum waves: Generation. Detection and Emerging Applications, IEEE Communications Surveys & Tutorials. 2020;**22**(2):840-868

[31] White AG, Smith CP, Heckenberg NR, McDuff R. Generation of optical phase singularities by computer-generated holograms. Optics Letters. 1992;**17**(3):221-223

[32] Li S, Wang Z. Generation of optical vortex based on computer-generated holographic gratings by

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

photolithography. Applied Physics Letters. 2013;**103**:141110

[33] Li D, Xuan L, Hu L, Mu Q, Liu Y, Cao Z. Phase-only liquid-crystal spatial light modulator for wave-front correction with high precision. Optics Express. 2004;**12**(26):6403-6409

[34] Efron U. Spatial Light Modulator Technology: Materials, Devices, and Applications. New York: Marcel Dekker; 1995

[35] Kovalev AA, Moiseev OY, Skidanov RV, Khonina SN, Soĭfer VA, Kotlyar VV. Simple optical vortices formed by a spiral phase plate. Journal of Optical Technology. 2007;**74**(10): 686-693

[36] Khonina SN, Ustinov AV, Logachev VI, Porfirev AP. Properties of vortex light fields generated by generalized spiral phase plates. Physical Review A. 2020;**101**:043829

[37] Beijersbergen MW, Allen L, van der Veen HELO, Woerdman JP. Astigmatic laser mode converters and transfer of orbital angular momentum. Optics Communications. 1993;**96**(1-3):123-132

[38] Kano K, Kozawa Y, Sato S. Generation of a purely single transverse mode vortex beam from a He-Ne laser cavity with a spot-defect mirror. International Journal of Optics. 2012; **2012**:359141

[39] Guo Y, Pu M, Zhao Z, Wang Y, Jin J, Gao P, et al. Merging geometric phase and plasmon retardation phase in continuously shaped metasurfaces for arbitrary orbital angular momentum generation. ACS Photonics. 2016;**3**(11): 2022-2029

[40] Liu C. Vortex beam and its application in optical tweezers. Journal of Physics: Conference Series. 2020; **1549**:032012

[41] D'Ambrosio V, Nagali E, Walborn SP, Aolita L, Slussarenko S, Marrucci L, et al. Complete experimental toolbox for alignment-free quantum communication. Nature Communications. 2012;**3**:961

[42] Yu S, Pang F, Liu H, Li X, Yang J, Wang T. Compositing orbital angular momentum beams in Bi4Ge3O12 crystal for magnetic field sensing. Applied Physics Letters. 2017;**111**:091107

[43] Nye J, Berry M. Dislocations in wave trains. Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences. 1974;**336**(1605): 165-190

[44] Goodman JW. Speckle Phenomena in Optics: Theory and Applications. Second Edition. Washington, USA: SPIE Press; 2013

[45] Dainty JC. Laser Speckle and Related Phenomena. London: Springer; 1976

[46] Reddy SG, Prabhakar S, Kumar A, Banerji J, Singh RP. Higher order optical vortices and formation of speckles. Optics Letters. 2014;**39**(15):4364-4367

[47] Wang W, Hanson SG, Miyamoto Y, Takeda M. Experimental investigation of local properties and statistics of optical vortices in random wave fields. Physical Review Letters. 2005;**94**:103902

[48] Vinu RV. Shaping and analysis of laser speckle for imaging applications, 308210515 [Accessed: June 30, 2022]

[49] Hu XB, Dong MX, Zhu ZH, Gao W, Rosales-Guzmán C. Does the structure of light influence the speckle size? Scientific Reports. 2020;**10**:199

[50] Facchin M, Dholakia K, Bruce GD. Wavelength sensitivity of the speckle patterns produced by an integrating sphere. Journal of Physics: Photonics. 2021;**3**:035005

[51] Wang S, Fan X, Wan Y, Zhang Z, He Z. High-resolution wavemeter using Rayleigh speckle obtained by optical time domain reflectometry. Optics Letters. 2020;**45**(4):799-802

[52] Bruce GD, Dholakia K, O'Donnell L, Chen M, Facchin M. Femtometer-resolved simultaneous measurement of multiple laser wavelengths in a speckle wavemeter. Optics Letters. 2020;**45**(7):1926-1929

[53] Gbur G, Visser TD. Phase singularities and coherence vortices in linear optical systems. Optics Communications. 2006;**259**(2):428-435

[54] Wang W, Duan Z, Hanson SG, Miyamoto Y, Takeda M. Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function. Physical Review Letters. 2006;**96**:073902

[55] Jesus-Silva AJ, Alves CR, Fonseca EJS. Characterizing coherence vortices through geometry. Optics Letters. 2015;**40**(12):2747-2750

[56] Heeman W, Steenbergen W, van Dam GM, Boerma EC, Heeman W. Clinical applications of laser speckle contrast imaging: A review. Journal of Biomedical Optics. 2019;**24**(8):080901

[57] Leibov L, Ismagilov A, Zalipaev V, Nasedkin B, Grachev Y, Petrov N, et al. Speckle patterns formed by broadband terahertz radiation and their applications for ghost imaging. Scientific Reports. 2021;**11**:20071

[58] Stallings W. Cryptography and Network Security Principles and Practice. New York: Prentice Hall; 2022 [59] Muniraj I, Sheridan JT. Optical Encryption and Decryption. SPIE Press Book; 2019

[60] Vanitha P, Manupati B, Reddy SG, Singh RP, Muniraj I, Anamalamudi S. Augmenting data security: Physical Unclonable Functions for digital holography based quadratic phase cryptography. 2022. DOI: 10.21203/rs.3. rs-1509081/v1

[61] Léger D, Perrin JC. Real-time measurement of surface roughness by correlation of speckle patterns. Journal of the Optical Society of America. 1976; **66**(11):1210-1217

[62] Chen C, Yang H. Correlation between turbulence-impacted optical signals collected via a pair of adjacent spatial-mode receivers. Optics Express. 2020;**28**(10):14280-14299

[63] Dong K, Cheng M, Lavery MPJ, Geng S, Wang P, Guo L. Scattering of partially coherent vortex beam by rough surface in atmospheric turbulence. Optics Express. 2022;**30**(3):4165-4178

[64] Liu X, Liu L, Chen Y, Cai Y. Partially coherent vortex beam: From theory to experiment, vortex dynamics and optical vortices. London: Springer; 2017

[65] Salem M, Shirai T, Dogariu A, Wolf E. Long-distance propagation of partially coherent beams through atmospheric turbulence. Optics Communications. 2003;**216**(4-6): 261-265

[66] Dong M, Yang Y. Coherent vortices properties of partially coherent Elegant Laguerre-Gaussian beams in the free space. Optics and Photonics Journal. 2020;**10**(6):159-166

[67] Gbur G, Visser TD. Coherence vortices in partially coherent beams. *Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

Optics Communications. 2003;**222**(1-6): 117-125

[68] Yadav BK, Kandpal HC, Joshi S. Experimental observation of the effect of generic singularities in polychromatic dark hollow beams. Optics Letters. 2014; **39**(16):4966-4969

[69] Fel'de CV, Wolf E, Bogatyryova GV, Soskin MS, Polyanskii PV, Ponomarenko SA. Partially coherent vortex beams with a separable phase. Optics Letters. 2003;**28**(11):878-880

[70] Jesus-Silva AJ, Neto APS, Alves CR, Amaral JP, Neto JGMN. Measuring the topological charge of coherence vortices through the geometry of the far-field cross-correlation function. Applied Optics. 2020;**59**(6):1553-1557

[71] Paterson C. Atmospheric turbulence and orbital angular momentum of single photons for optical communication. Physical Review Letters. 2005;**94**:153901

[72] Davidson FM, Ricklin JC. Atmospheric turbulence effects on a partially coherent Gaussian beam: Implications for free-space laser communication. Journal of the Optical Society of America A. 2002;**19**(9): 1794-1802

[73] Liu X, Peng X, Liu L, Wu G, Zhao C, Wang F, et al. Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle. Applied Physics Letters. 2017;**110**(18):181104

[74] Wu G, Cai Y. Detection of a semirough target in turbulent atmosphere by a partially coherent beam. Optics Letters. 2011;**36**(10): 1939-1941

[75] Kermisch D. Partially coherent image processing by laser scanning. Journal of the Optical Society of America. 1975;**65**(8):887-891

[76] Wang P, Chen F, Li D, Sun S, Huang F, Zhang T, et al. Authentication of optical physical unclonable functions based on single-pixel detection. Physical Review Applied. 2021;**16**:054025

[77] Mandel L, Wolf E. Optical Coherence and Quantum Optics. New York: Cambridge University Press; 1995

[78] Wolf E, Gbur G, Schouten HF, Visser TD. Phase singularities of the coherence functions in Young's interference pattern. Optics Letters. 2003;**28**(12):968-970

[79] Vanitha P, Lal N, Rani A, Das BK, Salla GR, Singh RP. Correlations in scattered perfect optical vortices. Journal of Optics. 2021;**23**:095601

[80] Jesus-Silva AJ, Hickmann JM, Fonseca EJS, Allen L, Beijersbergen MW, Spreeuw RJ, et al. Strong correlations between incoherent vortices. Optics Express. 2012;**20**(18):19708-19713

[81] Anwar A, Permangatt C, Banerji J, Singh RP, Reddy SG, Prabhakar S. Divergence of optical vortex beams. Applied Optics. 2015;**54**(22):6690-6693

[82] Vallone G, Parisi G, Spinello F, Mari E, Tamburini F, Villoresi P. General theorem on the divergence of vortex beams. Physical Review A. 2016;**94**: 023802

[83] Mazilu M, Mourka A, Vettenburg T, Wright EM, Dholakia K. Determination of the azimuthal and radial mode indices for light fields possessing orbital angular momentum. Frontiers in Optics/Laser Science XXVIII, Paper FW4A.4. 2012

[84] Mazilu M, Mourka A, Vettenburg T, Wright EM, Dholakia K. Simultaneous

determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum. Applied Physics Letters. 2012;**100**(23):231115

[85] Yang Y, Chen M, Mazilu M, Mourka A, Liu YD, Dholakia K. Effect of the radial and azimuthal mode indices of a partially coherent vortex field upon a spatial correlation singularity. New Journal of Physics. 2013;**15**:113053

[86] Yang Y, Dong Y, Zhao C, Liu Y, Cai Y, McMorran BJ, et al. Autocorrelation properties of fully coherent beam with and without orbital angular momentum. Optics Express. 2014;**22**:2925

[87] Vanitha P, Manupati B, Reddy SG, Annapureddy V, Prabhakar S, Singh RP. Intensity correlations in perturbed optical vortices: Diagnosis of the topological charge. 2022. DOI: 10.48550/ arxiv.2206.02365

[88] Peatross J, Ware M. Physics of light and optics: A free online textbook. Frontiers in Optics/ Laser Science XXVI, Paper JWA64. 2010

[89] Dogariu A, Acevedo CH. Nonevolving spatial coherence function. Optics Letters. 2018;**43**(23):5761-5764

[90] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. Washington, USA: Academic Press; 2014

[91] Goodman JW. Introduction to Fourier optics. Colorado, USA: Robert and Company Publishers; 1996

[92] Kumar A, Banerji J, Singh RP, Prabhakar S. Revealing the order of a vortex through its intensity record. Optics Letters. 2011;**36**(22):4398-4400 [93] Yue Y, Yan Y, Ahmed N, Yang JY, Zhang L, Ren Y, et al. Mode and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber. IEEE Photonics Journal. 2012; **4**(2):535-543

[94] Ostrovsky AS, Rickenstorff-Parrao C, Arrizón V. Generation of the perfect optical vortex using a liquid-crystal spatial light modulator. Optics Letters. 2013;**38**(4):534-536

[95] Anaya Carvajal N, Acevedo CH, Torres Moreno Y. Generation of perfect optical vortices by using a transmission liquid crystal spatial light modulator. International Journal of Optics. 2017; **2017**:6852019

[96] Yuan W, Xu Y, Xu Y, Zheng K, Fu S, Fu S, et al. Experimental generation of perfect optical vortices through strongly scattering media. Optics Letters. 2021; **46**(17):4156-4159

[97] Kotlyar VV, Kovalev AA, Porfirev AP. Elliptic perfect optical vortices. Optik. 2018;**156**:49-59

[98] Chen M, Mazilu M, Arita Y, Wright EM, Dholakia K. Dynamics of microparticles trapped in a perfect vortex beam. Optics Letters. 2013; **38**(22):4919-4922

[99] Rusch L, Vaity P. Perfect vortex beam: Fourier transformation of a Bessel beam. Optics Letters. 2015;**40**(4): 597-600

[100] Kumar A, Singh RP, Reddy SG, Prabhakar S. Experimental generation of ring- shaped beams with random sources. Optics Letters. 2013;**38**(21): 4441-4444

[101] Gori F, Guattari G, Padovani C. Bessel-Gauss beams. Optics

*Correlations in Scattered Phase Singular Beams DOI: http://dx.doi.org/10.5772/intechopen.106484*

Communications. 1987;**64**(6): 491-495

[102] McLeod JH. The Axicon: A new type of optical element. Journal of the Optical Society of America. 1954;**44**(8): 592-597

[103] Reddy SG, Chithrabhanu P, Vaity P, Aadhi A, Prabhakar S, Singh RP. Non- diffracting speckles of a perfect vortex beam. Journal of Optics. 2016;**18**: 055602

[104] Vanitha P, Rani A, Annapureddy V, Reddy SG, Singh RP. Diffracting and Non-diffracting random fields. 2021. https://doi.org/10.48550/arxiv.2111. 12388.

[105] Vaudenay S. A Classical Introduction to Cryptography. New York: Springer; 2008

[106] Stallings W. Cryptography and Network Security Principles and Practice. New York: Prentice Hall; 2011

[107] Pappu R, Recht B, Taylor J, Gershenfeld N. Physical one-way functions. Science. 2002;**297**(5589): 2026-2030

[108] Bohm C, Hofer M, Physical Unclonable functions in theory and practice. New York: Springer Publishers; 2013

[109] Chen K, Huang F, Wang P, Wan Y, Li D, Yao Y. Fast random number generator based on optical physical unclonable functions. Optics Letters. 2021;**46**(19):4875-4878

[110] Shamsoshoara A, Korenda A, Afghah F, Zeadally S. A survey on physical unclonable function (PUF) based security solutions for Internet of Things. Computer Networks. 2020;**183**: 107593

[111] Lee B-G, Guo C, Muniraj I, Ryle JP, Healy JJ, Sheridan JT, et al. Low photon count based digital holography for quadratic phase cryptography. Optics Letters. 2017;**42**(14):2774-2777

[112] Liu S, Guo C, Sheridan JT. A review of optical image encryption techniques. Optics & Laser Technology. 2014;**57**: 327-342

[113] Muniraj I, Ryle JP, Healy JJ, Sheridan JT, Wan M, Chen N, et al. Orthographic projection images-based photon-counted integral Fourier holography. Applied Optics. 2019; **58**(10):2656-2661

[114] Lam EY, Chen N, Ren Z. Highresolution Fourier hologram synthesis from photographic images through computing the light field. Applied Optics. 2016;**55**(7):1751-1756

[115] Javidi B, Refregier P. Optical image encryption based on input plane and Fourier plane random encoding. Optics Letters. 1995;**20**(7):767-769

[116] Jumarie G. Relative Information. Berlin: Springer-Verlag; 1990

[117] Hayes MH. In: Stark H, editor. Image Recovery: Theory and Application. San Diego, CA: Academic; 1987

[118] Dainty JC, Fienup JR. In: Stark H, editor. Image Recovery: Theory and Application. San Diego, CA: Academic; 1987

## *Edited by Joseph Rosen*

Holography of today is a broad field developed in the meeting between optics and the digital world of computers. A hologram usually contains more or different information on the observed scene than a regular image of the same scene. The development of the field has been accelerated lately due to the improvement of digital cameras, computers, light sources, and spatial light modulators. As a multidisciplinary area, holography connects experts in electro-optical engineering, image processing, and computer algorithms. More experts are needed when holography is utilized in various applications such as microscopy, industrial inspection, biomedicine, and entertainment. This book provides an overview of the world of holography from the aspect of concepts, system architectures, and applications.

Published in London, UK © 2023 IntechOpen © ALIOUI Mohammed Elamine / iStock

Holography - Recent Advances and Applications

Holography

Recent Advances and Applications

*Edited by Joseph Rosen*