Anamorphic fractional Fourier transforms graded index lens designed using transformation optics

Xiao-bo Yang; Jin Hu

doi:10.1364/OE.26.027528

1. Introduction

It is widely accepted that Fourier transform (FT) is one of the cornerstones of signal processing [1–4]. A general form of FT is fractional Fourier transform (FrFT), which can be interpreted as the counterclockwise rotation of the signal in the time (or space)-frequency plane, with the FT corresponding to the case when the rotation angle is $π / 2$ [5,6]. In the context of optical signal processing, two-dimensional (2D) FrFT can be implemented basically by two types of devices: bulk systems and continuous systems. Bulk systems contain one or multiple spherical lenses [7], and continuous systems contain a graded index (GRIN) lens [8–10]. Both of the systems are rotation invariant, i.e., rotating the devices cannot provide more information in the output facets. In contrast, the anamorphic FrFT (AFrFT) systems generally refer to the optical devices that can implement the FrFT in the fractional Fourier domain with the two variables that correspond to the two spatial coordinates in the FrFT output facet are nonsymmetrical, i.e., the devices are rotation variant [11,12]. In addition, some AFrFT devices are nonseparable, i.e., the transform cannot be expressed as a repetition along two directions independently [13]. Of course, the anamorphic FT (AFT) is a special case of AFrFT. Because of the anamorphic and nonsymmetrical characteristics in the transform domain, AFrFT systems can lead to some interesting applications for which the conventional FrFT devices are incapable of achieving or is inconvenient to perform. For example, AFT systems can be used for pseudocoloring [14], proving the angular discrimination in the Fourier plane [11,15], building anamorphic multiple matched filters [16–19], simultaneously detecting several objects [20], and encrypting the optical holographic memories [21]; AFrFT systems are recommended to perform anamorphic fractional correlation, efficient multiplexing, anamorphic chirp filtering [22], and signal restoration for distorted images with particular additive-noise [13]. While normal FrFT (including FT) can be implemented by both bulk and continuous systems, AFrFT (including AFT) can only be structured by bulk systems so far because it relies on the assembly of multiple cylindrical lenses that are active along only one direction [12–23]. Compared with the GRIN lens, the bulk systems have some obvious defects. First, the bulk systems lack the flexibility to adjust the order of the FrFT. In a bulk device, all the spherical or cylindrical lenses should be specifically designed to have particular focal lengths to obtain the given FrFT order. Thus, if the order value must be changed, all the lenses should be replaced by new lenses with new focal lengths, and the lens locations must be adjusted; that is, one must rebuild the whole system. This inflexibility is more serious in the AFrFT system, whose order is not 1, because the number of the lenses is greater [13,20–23]. Second, the anamorphic patterns in the transform domains are limited by the operating mechanism of the bulk systems. Anamorphoses of these systems are obtained by manipulating the cylindrical lenses, which are active only along one direction; this approach is equivalent to a distortion of the transform process by imposing a specific transforming operator in one direction. Using the cylindrical lenses-based anamorphic systems, it is difficult (if not impossible) to constitute more complicated anamorphic patterns, such as deforming the image in a curvilinear manner. In view of this issue, new types of AFrFT devices that improve performance by overcoming the aforementioned shortcomings are in demand. In this study, a new type of AFrFT lens comprised of a GRIN media is proposed with the help of transformation optics (TO). TO is a powerful tool that can be used to obtain the necessary distribution of material parameters to result in the prescribed electromagnetic wave behaviors in a region [24–27]. The most interesting application of TO might be the invisible cloak, which can guide the wave to avoid an object, thus preventing it from being detected via the scattered waves [25]. As the wave behaviors are the physical bases of the optical FrFT, it is then expected that TO may have broad application prospects in the field of new FrFT (including FT) devices design. The proposed lens is an example of such designs. The proposed lens can provide a much more complex anamorphic image in the fractional Fourier domain compared to the images obtained using cylindrical lenses-based systems; moreover, it can offer different FrFT orders in the same lens by merely choosing the detection locations (the detection facets are along the optical axis), making it more flexible than the bulk systems. The new AFrFT lens may lead to many new optical applications. Here, we demonstrate three examples: higher distinguishability in the fractional Fourier domain, more precise recognition matched filter system and AFrFT-based image encryption with stronger security. With the development of metamaterials (including three-dimensional printing) technologies [28–34], it has become more convenient to obtain GRIN media; thus, the proposed lens may have vast application prospects in signal processing or communication. The design also demonstrates the ability and flexibility of TO in Fourier optics.

The paper is arranged as follows: the AFrFT lens design is explained in Section 2, the application samples of the proposed lens are presented in Section 3, and the discussion and conclusions are presented in Section 4.

2. Design of the AFrFT lens

We start the design by considering the conventional quadratic GRIN lens that can implement the normal FrFT [9,35]. The refractive index distribution of the lens is $n^{2} (r) = n_{1}^{2} [1 - {(r / ξ)}^{2}],$ where $r$ is the radial distance from the optical axis, and $(n_{1}, ξ)$ are the GRIN medium parameters. The propagation of the wave field in the lens is an FrFT operator based on Hermite–Gaussian function decomposition [2,9]. The 2D normal FrFT achieved by this lens can be expressed as:

g_{p} (u, v) = F^{p} [f (x, y)] (u, v) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (x, y) K_{p} (x, y; u, v) d x d y,

where

p

is a real number called the order of the FrFT (here, we consider a situation in which the orders for the x-axis and y-axis are equal) and

K_{p} (x, y; u, v) = K_{p x} (x, u) K_{p y} (y, v)

is the kernel [36,37] with:

K_{p x} (x, u) = {\begin{cases} A_{p} \exp (- \frac{2 π}{s^{2}} i x u \csc α) \exp [i \frac{π}{s^{2}} (x^{2} + u^{2}) \cot α], α \neq n π; \\ δ (u - x), α = 2 n π; \\ δ (u + x), α = (2 n + 1) π, \end{cases}

where

A_{p} = 1 / [j s {(2 π \sin α)}^{1 / 2}],

s^{2} = λ ξ / n_{1}

is the scale factor related to the lens parameters and the wavelength

λ,

and

n

is an integer.

K_{p y}

is similar

K_{p x},

with only the variables

(x, u)

replaced by

(y, v) .

The important parameter

α = p (π / 2)

can be interpreted as the counterclockwise rotation angle of the signal in the x-frequency or y-frequency plane. The above kernel shows that the 2D normal FrFT is separable, and the transform can proceed in the x and the y directions independently. The p-order

(0 \leq p \leq 1)

FrFT of an input signal can be found in the facet, for which the distance to the input facet is given by [9]

b = p L

where

L = (π / 2) ξ

is focal length of the lens. A sketch of this detection process is shown in Fig. 1. If

P = 1

or

α = π / 2

then the FrFT lens is reduced to the well-known FT lens.

Fig. 1 The relation of the FrFT order p between the detection position b in the GRIN lens.

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Next, consider the output facet where the normal FrFT of the input signal is displayed. If the wave field here can be distorted as desired, then the AFrFT is achieved. To achieve this task, we employ a particular spatial mapping on the lens region, and according to TO principle, the changed material parameters corresponding to the space mapping will steer the wave field to redistribute as if the field is attached at every point of the space [38–40]. To show the space mapping, we first sketch the conformal mapping in a cross section of the lens, as shown in Fig. 2.

Fig. 2 The sketch of the conformal mapping in a cross section of the lens. (a) The original space; (b) the mapped space, where the center of original circular area is mapped to a given position.

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This mapping is a linear fractional transformation in complex planes [41]:

w = \frac{z + a}{1 + \bar{a} z}

where

a = l + m i

is a given point and

\bar{a}

is its conjugate. The original center of this circular cross section in the

z

plane is mapped to point

a

in the

w

plane, and we call this point the anamorphic center in the new plane. In Cartesian coordinates, this mapping reads:

\begin{array}{l} u = f (u', v') = \frac{(u^{'} + l) (1 + l u^{'} + m v^{'}) - (v^{'} + m) (m u^{'} - l v^{'})}{{(1 + l u^{'} + m v^{'})}^{2} + {(m u^{'} - l v^{'})}^{2}}, \\ v = g (u', v') = \frac{(v^{'} + m) (1 + l u^{'} + m v^{'}) + (u^{'} + l) (m u^{'} - l v^{'})}{{(1 + l u^{'} + m v^{'})}^{2} + {(m u^{'} - l v^{'})}^{2}}, \end{array}

where

u', v'

are the original coordinates and

u, v

are the transformed coordinates. It is shown that the deformation of the input facet is equivalent to changing the input signal [37]; hence, to avoid influencing the input signal, the positions of the anamorphic centers are gradually varied from the input facet to the output facet in the lens, as shown in Fig. 3.

Fig. 3 The positions of the anamorphic centers from the input facet to the output facet in the lens are shown by the red dashed line.

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The link line of the anamorphic centers in different cross sections is a plane quadratic curve

\begin{array}{l} l = l (z) = \frac{l_{0}}{L^{2}} z^{2}, \\ m = m (z) = \frac{m_{0}}{L^{2}} z^{2}, \end{array}

where the optical axis is along the

\hat{z}

axis and (

l_{0}

,

m_{0}

) are the coordinates of the anamorphic center in the focal plane. In the input facet the anamorphic center

a = (0, 0)

and the tangent of the line is perpendicular to the input facet, means there are no distortions in this facet, and the input signal should not be affected by space mapping at that location. In any cross section, according to Eq. (5) and the TO principle, the kernel becomes:

K_{p} (x, y; u, v) = {\begin{cases} A_{p}^{2} \exp {- i x [\frac{ζ (u - l) - η (v - m)}{ζ^{2} + η^{2}}] \csc α} \\ \times \exp [\frac{1}{2} i (x^{2} + {[\frac{ζ (u - l) - η (v - m)}{ζ^{2} + η^{2}}]}^{2}) \cot α], α \neq n π; \\ \times \exp {- i y [\frac{ζ (v - m) + η (u - l)}{ζ^{2} + η^{2}}] \csc α} \\ \times \exp [\frac{1}{2} i (y^{2} + {[\frac{ζ (v - m) + η (u - l)}{ζ^{2} + η^{2}}]}^{2}) \cot α] \\ δ (u - x) δ (v - y), α = 2 n π; \\ δ (u + x) δ (v + y), α = (2 n + 1) π, \end{cases}

where

ζ = 1 - l u - m v,

η = l v - m u .

As expected, the new kernel is obviously asymmetric and inseparable with respect to the two orthometric coordinates

u

and

v

in the output facet (which are also the two variables in the fractional Fourier domain).

Although the space mapping in the cross sections is 2D, one must consider the whole three-dimensional (3D) space mapping to obtain the transformation materials of the lens. The deformation view of TO [42] is utilized for this calculation, with which the transformed material parameters at the mapped space can be expressed in a geometrical manner as:

ε' = ε_{0} diag [\frac{λ_{1}}{λ_{2} λ_{3}}, \frac{λ_{2}}{λ_{1} λ_{3}}, \frac{λ_{3}}{λ_{1} λ_{2}}], μ' = μ_{0} diag [\frac{λ_{1}}{λ_{2} λ_{3}}, \frac{λ_{2}}{λ_{1} λ_{3}}, \frac{λ_{3}}{λ_{1} λ_{2}}],

where

ε_{0}

and

μ_{0}

are the original permittivity and permeability, respectively,

λ_{1},

λ_{2}

and

λ_{3}

are the three principal stretches of a spatial element during the space mapping, and the equations are established in the local principal system of the deformation. As the lens is operating under the paraxial approximation, the length of the link line of the anamorphic centers is approximately equal to the optical axis length, implying that the out-of-plane stretch in a cross section can be ignored, i.e.,

λ_{z} = 1.

The in-plane mapping of a cross section is conformal; thus, the principal stretches there are equal, i.e.,

λ_{x} = λ_{y} \equiv λ,

where

λ

can be calculated from Eq. (4) by

| d w / d z |

or from Eq. (5) by the determinant of the Jacobian matrix. According to Eq. (8), these principal stretches indicated that the transformed material of the new lens should have the following form:

ε' = diag [ε_{0}, ε_{0}, \frac{1}{λ^{2}} ε_{0}], μ' = diag [μ_{0}, μ_{0}, \frac{1}{λ^{2}} μ_{0}],

thus, the transformation lens is of uniaxial anisotropy. Let

n_{c} = 1 / λ n_{0}

and

n_{z} = n_{0}

to denote the in-plane and out-of-plane refractive indices of the cross sections, respectively, and let

n_{0} (r) = \sqrt{ε_{0} (r) μ_{0} (r)}

represent the original refractive index of the quadratic GRIN lens. Fresnel’s equation of wave normals for this lens reduces to [43]:

(\frac{1}{n^{2}} - \frac{1}{n_{c}^{2}}) [(\frac{1}{n^{2}} - \frac{1}{n_{z}^{2}}) \sin^{2} θ + (\frac{1}{n^{2}} - \frac{1}{n_{c}^{2}}) \cos^{2} θ] = 0,

where

n

is the whole anisotropic refractive index of the designed lens to be determined, and

θ

denotes the angle of the wave normal makes with the optical axis. Noting again the paraxial approximation for the perpendicular or quasi-perpendicular incidence signal,

θ

is approximately equal to zero; thus, according to above normal wave equation, the effect of

n_{z}

can be ignored and the solution of the equation is

n \approx n_{c} .

This result is very important because it implies that the anisotropic transformation materials can be implemented approximately by isotropic materials. The isotropy of the proposed lens materials is significant in practice because it has fewer challenges for fabrication and is more likely to be used in broadband and low-loss applications. To verify the designed lens, numerical simulations are utilized by comparing the ideal transformation of a signal with the output of the designed isotropic lens. The parameters are as follows:

ξ = 1.2732,

n_{1} = 1.5,

and the focal length L = 2 mm and the radius of the lens r = 1 mm. The coordinates of the anamorphic centers in the final output facet of the two samples are (0.2 mm, 0.2 mm) and (0.4 mm, 0mm), respectively. Because the input signal is a circle function, its FT (i.e., FrFT with order p = 1) is a besinc function [44]. As shown in Fig. 4 and Table. 1, the output images of the designed lenses are very similar to those ideal images, indicating that the design lens indeed can work well in the paraxial approximation.

Fig. 4 Comparison of the ideal mapped wave fields with the outputs of the AFrFT lens. The lines in the field are the contour lines. (a) The output of a circle function in the normal GRIN lens is obtained by the wave optics module of COMSOL Multiphysics, which is the source image requiring transformation; (b) and (d) the ideal mappings of the source image obtained by directly using Eq. (4) through MATLAB, and the anamorphic centers are (0.2 mm, 0.2 mm) and (0.4 mm, 0 mm), respectively; (c) and (e) the output results of the same input as (a) in the designed AFrFT lens obtained by wave optics module of COMSOL Multiphysics, and the anamorphic centers at the output facet of the lens is designed at (0.2 mm, 0.2 mm) and (0.4 mm, 0 mm), respectively.

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Table 1. Comparison of the ideal mapping with the results obtained by the AFrFT lens

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The designed AFrFT lens is quite different from conventional cylindrical lens-based anamorphic systems. This GRIN media-based lens can adjust the FrFT order without reconstruction of the system and with only changes in the detection positions. These positions changes can be accomplished with the help of tomography technology [45,46]. In addition, the anamorphic pattern offered by the proposed AFrFT lens is much more complex than that of conventional systems. As shown in Fig. 2, the image in the transform domain experiences conformal mapping, with some areas being magnified and others being compressed. The original Cartesian coordinate lines become specific curves, which is difficult to achieve by assembling the cylindrical lenses as done in conventional AFrFT systems. The proposed lens may have vast application prospects. In the following section, we present three examples.

3. Applications samples of the proposed AFrFT lens

3.1 Improving the distinguishability in the fractional Fourier domain

The wave field distributions in the fractional Fourier domains carry important information that can help in recognizing or analyzing the input signal. In the proposed AFrFT lens, the deformation in the fractional Fourier domain (the output facet of the lens) is conformal, i.e., any crossing angle of the lines in the domain can remain, and the magnification effect implies that some regions in the fractional Fourier domain can have enhanced distinguishability. This enhanced distinguishability is particularly useful for systems with insufficient space-bandwidth products. To demonstrate this character, numerical simulation based on COMSOL Multiphysics is processed for the AFrFT lens with an anamorphic center located at (0.3 mm, 0 mm), and the other parameters are as same as those in Sec. 2. The input is a cosine signal $f (x, y) = \cos (5.3 x)$ within a circular aperture, and the input wave length is 0.8 mm. In the fractional Fourier domains, more than one local peak energy area should exist, and the distribution of these areas is the important feature of this input signal [4]. However, as shown in Figs. 5(a), 5(c), 5(e) and 5(g), because of the limit of the space-bandwidth products of the conventional lens in the fractional Fourier domains, the level + 1 and −1 energy peaks are difficult to distinguish from the peak of the level 0, where the FrFT orders are 1.0 and 0.9, respectively. In contrast, under the same conditions, the proposed AFrFT lens can clearly distinguish the level 0 energy peak from others, as shown in Figs. 5(b), 5(d), 5(f), and 5(h), indicating the enhanced distinguishability provided by the lens.

Fig. 5 Comparison of the distinguishability of the conventional GRIN lens with that of the AFrFT lens. (a) and (e) The distribution of the wave fields in the output facets of conventional GRIN lenses with FrFT orders of 1 and 0.9, respectively. (c) and (g) The corresponding amplitude distributions along the lines in (a) and (e), respectively. (b) and (f) The distribution of the wave fields in the output facets of the AFrFT lenses with FrFT orders of 1 and 0.9, respectively. (d) and (h) The corresponding amplitude distributions along the lines in (b) and (f), respectively.

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3.2 Anamorphic matched filter of correlation recognition for images with nuance

Optical image recognition is a popular research topic in the field of optical information processing. The concept of a matched filter, which is a typical application of Fourier optics, plays an important role in the pattern recognition problem [1]. Denoting $S$ as the FT of the input signal $s,$ if the frequency plane filter has an amplitude transmittance proportional to $S^{*}$ (the conjugation of $S$ ), then it is called the matched filter of $s .$ A sketch of the 4f-system-based matched filter correlation recognition is shown in Fig. 6. After the first FT, the field distribution of the input signal $s$ is converted to the wave field distribution proportional to $S,$ which is then incident on the matched filter. Because of the transmittance of the filter, the wave field distribution transmitted there is proportional to $S S^{*},$ which can be considered as the input signal of the second FT. According the autocorrelation theorem [1,4], the FT of $S S^{*}$ is equivalent to performing an autocorrelation for $s$ and will lead the correlation peak on the last output screen. From the optical point of view, the matched filter cancels all the curvature of the incident wave front (noting that the wave field $S S^{*}$ is real), and this plane wave (generally of nonuniform intensity) forms the bright focus of the correlation peak after the second FT. If the first input signal is other than $s,$ then the autocorrelation will be weakened (or the planarity of the input wave in second FT will be disturbed), and the correlation peak will be lower. Therefore, the condition of the correlation peak in the last output can be used to recognize the input signal $s .$

Fig. 6 Sketch of the matched filter correlation recognition based on a 4f system.

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The abovementioned principle indicates that the operation between $S$ and $S^{*}$ is the core of this system. It is known that, in the frequency plane, the detail of an image is contained in the high-frequency region, which is located around the center low-frequency region [47]. This feature implies that the areas far from the center in both $S$ and $S^{*}$ planes are important in recognizing the details of the image. The proposed AFrFT lens can be used to adjust the wave field distribution in the frequency domain, causing more high-frequency areas to be magnified and moved to the center; hence, more such areas can be included in the operation between $S$ and $S^{*},$ resulting in the enhancement of recognition of the details of input image $s .$ It is difficult to use conventional AFrFT or AFT systems based on cylindrical lenses to achieve such deformation.

To verify this minutia recognition ability, numerical simulations based of two types of matched filter correlation recognition systems are performed; the corresponding block diagrams are shown in Fig. 7. The difference between the two systems lies only in the first stage FT and the matched filter. One system (called A) is the conventional one, and the other (called B), for comparison, is based on the proposed AFT (i.e., AFrFT with order = 1), where the first FT is replaced by AFT, and the matched filter is the conjugation of the AFT of the image that requires recognition. The two systems have the same input signals, two images (called I1 and I2) with nuance are shown in Fig. 8, and it is expected that the nuance can be detected and the right image (I1), which is the source of the matched filter, can be recognized.

Fig. 7 The block diagrams of the two types of matched filter correlation recognition systems. (a) The traditional matched filter correlation recognition systems, system A; and (b) the matched filter correlation recognition systems based on the proposed AFrFT, system B.

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Fig. 8 Two images with nuance used as the inputs of both of the systems. (a) Image I1; and (b) image I2.

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The simulation results obtained by MATLAB are shown in Fig. 9, Fig. 10 and Table 2. Clearly, in system A, the two different input images have same value of correlation peaks, i.e., the correct image cannot be recognized; in contrast, in system B, the right image has an obviously higher (more than 10%) correlation peak than the reference, indicating that the nuance of the images takes effect in the anamorphic matched filter and that more precise recognition is achieved.

Fig. 9 The result of matching correlation recognition of system A. (a)(c)€, and (g): input image is I1; (b)(d)(f), and (h): input image is I2. (a) and (b) the spectra of the input image; (c) and (d) the synthesis spectra of the input image and the temple(SS*); (e) and (f) the output wave fields; and (g) and (h) the 3D contour plots of the correlation peaks.

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Fig. 10 The result of matching correlation recognition of system B. (a), (c), (e) and (g), and (g): input image is I1; (b)(d)(f), and (h): input image is I2. (a) and (b) the spectra of the input image; (c) and (d) the synthesis spectra of the input images and the temples; (e) and (f) the output wave fields; and (g) and (h) the 3D contour plots of the correlation peaks.

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Table 2. Comparison of the correlation peak values between systems A and B

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3.3 Image encryption based on AFrFT lens

The optical FrFT has been used in optical image encryption to improve the security and attack resistance of the system [48–53]. Compared with the companionate encryption system based on FT, the additional parameter of the FrFT order provides one more key for the decryption and thus accomplishes the security enhancement. Moreover, if the FrFT is nonseparable, then the encryption strength will be further stronger because the transform cannot be decomposed into two independent one-dimensional (1D) transforms, causing the dimensionality reduction in decryption to become impossible [54,55]. Notably, the proposed AFrFT absorbs the coordinates of the anamorphic center as two independent parameters beside the FrFT order and is nonseparable; it is then expected that the lens can provide stronger security in optical image encryption than conventional FrFT.

To demonstrate this application, the optical image encryption technique based on the proposed AFrFT lens is utilized in the numerical simulation sample. The FrFT encryption systems usually includes random phase encoding schemes [48,52]; however, it is not inevitable [51,53]. Here, without loss of the illustrative effect, the simpler system without random phase masks is adopted. The device diagram of the system is shown in Fig. 11, and the corresponding flow charts, including the encryption and decryption processes, are sketched in Fig. 12, where the conventional FrFT lens is replaced by the proposed AFrFT lens. The decryption is exactly the inverse process of the encryption; more specifically, the encrypted image should first undergo the inverse mapping of Eq. (4) and then -p order FrFT. Therefore, both the FrFT order and the two independent coordinates of the anamorphic center must be known in the decryption.

Fig. 11 The device diagram of the encryption system. (a) The traditional encryption system and (b) the proposed encryption system, where the conventional FrFT is replaced by the designed AFrFT.

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Fig. 12 Flow charts of the encryption and decryption process of the sample.

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In the simulation, the FrFT order is set as p = 0.8 and the anamorphic center located at (0.35 mm, 45°) in the polar coordinate system of the output facet. As shown in Fig. 13, the original images, with size of width = height = 256 pixels, can be encrypted and decrypted with right keys. However, even in the appropriate FrFT order, the tiny errors of the anamorphic center coordinates can lead the encryption unsuccessful. Because the parameter space of the anamorphic center coordinates is huge, the system is very strong against the blind decryption. We further investigate the sensibility of the three parameters in the decryption by estimating their Mean Square Error (MSE) curves; this parameter is the typical means to quantify the difference between two images and is defined as

M S E = \frac{1}{M \times N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} {| h_{2} (i, j) - h_{1} (i, j) |}^{2}

where

h_{1} (i, j)

and

h_{2} (i, j)

represent the gray values at point

(i, j)

of the original and decrypted images, respectively, and M and N are image sizes. Obviously, the greater the MSE, the more difference between the two images. As shown in Fig. 14, as the FrFT order, angular and radial coordinate of the anamorphic center deviate, their corresponding MSE become greater. The MSE is demonstrated to be more sensitive to the coordinates of the anamorphic center than the FrFT order, manifesting the strong security encryption strength brought by the proposed AFrFT.

Fig. 13 Simulation results of the encryption system. (a) Original image; (b) encrypted image; (c) reconstructed image with all correct keys; (d)-(k) reconstructed images with correct FrFT order but with just one correct coordinate of the anamorphic center, where (d)-(g) use the incorrect radial coordinate keys, and (h)-(k) use the incorrect angular coordinate keys.

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Fig. 14 MSE plotted as a function of errors in the decryption keys. (a) MSE with respect to the FrFT order error, (b) MSE with respect to the angular coordinate error of the anamorphic center, and (c) MSE with respect to the radial coordinate error of the anamorphic center.

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4. Discussion and conclusion

The FrFT (including the FT) is a basic and widely used function in optical signal processing and communication. Consequently, designing and discovering new devices that can perform optical FrFT with specific advantages or functionalities in some situations is an important task in both academia and engineering. Among the optical fittings, the GRIN lenses generally have several of advantages over the curved-surface lenses, such as small size, low weight, short focal distances, easy mounting and adjustment, and stable configuration [56,57]. As the TO can steer the electromagnetic waves at will by redistributing the materials parameters in an area, it is then significant to introduce TO as a powerful tool to design new or improve existing GRIN lenses in view of the fundamental role of these devices. Note that the development of metamaterials and 3D printing technologies will definitely promote the fabrication of GRIN materials, and we believe the GRIN lenses will have broad prospects in optical and electromagnetic applications. Some works have shown the potential of TO and metamaterials in construction of functional GRIN media [58–60]. In this paper, the GRIN-based AFrFT lens was proposed for the first time that allows a particular anamorphic pattern can be achieved in any FrFT order; moreover, some properties of the AFrFT were indicated. The design notion and calculation were presented, and the conformal linear fractional mapping in the cross sections of the lens was found to make the transformation material becomes isotropic. The design clearly showed the TO’s ability and flexibility in obtaining GRIN lens for special goals. Compared with the conventional AFrFT or AFT systems, which are based on cylindrical and spherical lenses, the proposed lens can offer some advantages via the characteristics of the GRIN materials. Three sample applications of the AFrFT lens were demonstrated: improving the distinguishability in the fractional Fourier domain, enhancing the recognition precise of the matched filter system and intensifying the security strength of FrFT based optical encryption. Additional functions of this lens are worthy of further exploration. The complexity of the refractive index distribution of the AFrFT lens might cause challenges in the fabrication, for example, the traditional techniques such as chemical vapor deposition (CVD) [61], partial polymerization [62], ion exchange [63] or ion stuffing [64] should require higher precision control in the manufacture. If the wavelengths are in suitable ranges, the nowadays material fabrication technologies such as 3D printing with a high resolution ratio [65] or direct laser writing [66] are of help in the fabrication.

Funding

National Natural Science Foundation of China (NSFC) (61575022, 61421001).

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	Theoretical location of anamorphic center (mm)	Simulation location of anamorphic center (mm)
Sample 1	(0.2,0.2)	(0.195,0.198)
Sample 2	(0.4,0)	(0.4,0.016)

System	Autocorrelation peak (X1): input I1	Cross-correlation peak (X2): input I2	Relative error $\| X_{1} - X_{2} \| / X_{1}$
System A	6.56e5	6.56e5	0
System B	1.98e5	1.75e5	11.88%

	Theoretical location of anamorphic center (mm)	Simulation location of anamorphic center (mm)
Sample 1	(0.2,0.2)	(0.195,0.198)
Sample 2	(0.4,0)	(0.4,0.016)

System	Autocorrelation peak (X1): input I1	Cross-correlation peak (X2): input I2	Relative error $\| X_{1} - X_{2} \| / X_{1}$
System A	6.56e5	6.56e5	0
System B	1.98e5	1.75e5	11.88%

Anamorphic fractional Fourier transforms graded index lens designed using transformation optics

Abstract

1. Introduction

2. Design of the AFrFT lens

3. Applications samples of the proposed AFrFT lens

3.1 Improving the distinguishability in the fractional Fourier domain

3.2 Anamorphic matched filter of correlation recognition for images with nuance

3.3 Image encryption based on AFrFT lens

4. Discussion and conclusion

Funding

References

Cited By

Figures (14)

Tables (2)

Equations (11)

Optics Express