An anamorphic fractional Fourier transform (AFrFT) lens based on graded index (GRIN) materials and designed with the help of transformation optics is proposed. Cross sections of the new lens are mapped from those of a standard quadratic GRIN lens via gradually varied conformal transformations. This lens can afford complicated anamorphic patterns in the fractional Fourier domain for any fractional order, possibly leading to many new applications. Three samples are shown, which offer higher distinguishability in the fractional Fourier domain, a more precisely recognized matched filter, and stronger security of the AFrFT-based optical encryption. With metamaterials development, including three-dimensional printing technologies, GRIN media fabrication has become more convenient; thus, the proposed lens may have vast application prospects in signal processing. The design also demonstrates the ability and flexibility of the transformation optics in exploring new Fourier optics devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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 [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 , 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 . 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 , proving the angular discrimination in the Fourier plane [11,15], building anamorphic multiple matched filters [16–19], simultaneously detecting several objects , and encrypting the optical holographic memories ; AFrFT systems are recommended to perform anamorphic fractional correlation, efficient multiplexing, anamorphic chirp filtering , and signal restoration for distorted images with particular additive-noise . 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 . 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 where is the radial distance from the optical axis, and 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:36,37] with:9]Fig. 1. If or then the FrFT lens is reduced to the well-known FT lens.
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.
This mapping is a linear fractional transformation in complex planes :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.
The link line of the anamorphic centers in different cross sections is a plane quadratic curveEq. (5) and the TO principle, the kernel becomes:
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  is utilized for this calculation, with which the transformed material parameters at the mapped space can be expressed in a geometrical manner as:Eq. (4) by 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:43]: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.
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 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 . 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.
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 . Denoting as the FT of the input signal if the frequency plane filter has an amplitude transmittance proportional to (the conjugation of ), then it is called the matched filter of 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 is converted to the wave field distribution proportional to which is then incident on the matched filter. Because of the transmittance of the filter, the wave field distribution transmitted there is proportional to which can be considered as the input signal of the second FT. According the autocorrelation theorem [1,4], the FT of is equivalent to performing an autocorrelation for 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 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 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
The abovementioned principle indicates that the operation between and 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 . This feature implies that the areas far from the center in both and 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 and resulting in the enhancement of recognition of the details of input image 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.
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.
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.
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 asFig. 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.
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) , partial polymerization , ion exchange  or ion stuffing  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  or direct laser writing  are of help in the fabrication.
National Natural Science Foundation of China (NSFC) (61575022, 61421001).
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