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Abstract
Gradient refractive index (GRIN) lens-based chirp signal chirpiness detection usually relies on the fractional Fourier transform (FRFT) functionality of a quadratic GRIN lens and is limited by paraxial conditions. In this paper, a non-FRFT mechanism-based chirpiness detection GRIN lens is proposed that converts the Luneburg lens’ focus capacity of input plane waves to the designed lens’ focusing of input chirp waves using transformation optics, and the source chirpiness can be obtained by sweeping the illumination wavelength rather than locating the focusing pulse, consequently greatly increasing the upper limit of the chirpiness detection range. The feasibility and robustness of the method are verified through numerical simulations.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Taking a one-dimensional (1D) case as an example, chirp signals refer to signals having the form of $f(x) = \textrm{exp} [ - i(1/2)m{x^2} + {m_0}x + {m_1}]$, where m, m0 and m1 are real values and x can be temporal or spatial coordinates. As the frequency increases or decreases linearly with x, the signal, also known as the linear frequency modulation (LFM) signal, is one of the most representative nonstationary signals in the field of signal processing and is widely used in radar, sonar, communication, biology, geological exploration and other fields due to its autocorrelation and energy focusing characteristics in the original domain and transform domain [1–4]. Chirp signals in the spatial domain are widely used in optical signal processing, such as optical measurement, optical estimation and image processing [1,5]. The parameter m indicates how fast the signal changes its frequency and is the pivotal information of a chirp signal, called the chirpiness or frequency change rate. As this parameter describes the unique characteristic of such a signal, it is usually the target in the context of chirp signal detection.
Most chirpiness detection methods of chirp signals are based on time (or space)-frequency representation, such as short-time Fourier transform (STFT) [6,7], Wigner Ville distribution [8], fractional Fourier transform (FRFT) [9], high-order phase function [10], maximum likelihood estimation [11], minimum absolute deviation estimation [12], and so on. These methods usually need to traverse search every line or column of the measured signal or image, which means that even for a (1D) signal, a two-dimensional (2D) space search is necessary, so it is time-consuming with high computational complexity, and the detection accuracy is limited by the scanning step size.
Based on the FRFT properties of the quadratic gradient refractive index (GRIN) lens [13,14], Chen et al. [15,16] proposed a faster method by using a new type of GRIN lens to locate the pulse position on the optical axis inside the lens corresponding to the chirpiness of the input spatial chirp signal to determine its chirpiness. Later, the exposure lens proposed by Shi et al. [17] can expose and extend the optical axis inside the traditional GRIN lens body on the surface of the media and thus is convenient for direct and more precise measurements; this ability is demonstrated by chirpiness detection. The idea of an exposure lens was extended by He et al. [18] to improve the range of chirpiness that can be detected in the exposure lens.
All these GRIN lenses can be regarded as variants of the traditional quadratic GRIN lens, and their chirpiness detection functionality relies on obtaining the pulse position corresponding to the chirpiness of the input chirp signal, which is the corollary of the governing FRFT mechanism. Although these methods are faster, from a wider and prospective perspective, in some cases, for example, in very narrow work regions, it is not convenient to accomplish the related spatial search or location. Moreover, these GRIN lenses are limited by the paraxial approximation [13,14,19]. When the chirpiness is high, the pulse position will be close to the input surface and make the paraxial conditions poor, resulting in large errors or even failure of the systems [18]. In view of this, more chirpiness detection methods are in demand. In the current work, a new type of GRIN lens is proposed based on deformation of the Luneburg lens instead of the traditional quadratic GRIN lens, which uses illumination wavelength sweeping to detect the chirpiness of the spatial chirp signal and avoid the mentioned problems of the methods based on pulse location in the GRIN lens.
The canonical Luneburg lens is a sphere-shaped medium with a center-symmetric GRIN, which has the function of gathering and capturing plane waves from any direction and can also send or feed plane waves in any direction [20–22]. The device is interesting, but its input and output surfaces are curved, which limits its practical application. Some studies have attempted to overcome this disadvantage by turning the curved surface into a straight surface and use the characteristics of the Luneburg lens to control wave propagation [23,24]. However, the lenses obtained by these methods are usually anisotropic, which is challenging for preparation techniques.
In this paper, the transformation optics (TO) method [25,26] is adopted to deform the curved surface Luneburg lens into a flat GRIN lens that remains isotropic. From a deformation point of view, TO indicates that the physical quantities can be viewed as attached at the spatial element and moved and deformed together with the element during the space transformation [27–32], hence it is a powerful tool to manipulate the electromagnetic wave behavior in a region through the relationship between the material parameter distribution and the space deformation. The proposed lens can focus chirp signals with particular chirpiness at the output facet, and this particular chirpiness is called the eigenchirpiness of this lens, which is related to the input wavelength. For a screen or structure that contains the input spatial chirp signal, the illumination wavelength can modulate the chirpiness inputted into the system. By sweeping wavelengths and obtaining the eigenchirpiness, the source chirpiness can be detected. Compared to locating the pulse position, a wavelength-controlled system has its own unique flexibility in signal processing [33].
The rest of the paper is organized as follows. In Section 2, the design of the flat isotropic GRIN lens within the framework of TO is introduced, and in Section 3, the chirp-focus feature of the proposed lens is analysed. Then, in Section 4, some chirpiness detection examples are presented by numerical simulation to verify the designed lens and its robustness. Finally, Section 5 gives the discussion and conclusion.
2. Design of the flat isotropic GRIN lens
The traditional Luneburg lens is a sphere GRIN lens with a refractive index distributed as ${n_w} = \sqrt {2 - {{(r/R)}^2}} $, where R is the radius of the Luneburg lens and r is the radial distance from a point to the center of the sphere [20]. Its refractive index increases radially from the outer surface to the center, and it has the ability to focus and collimate waves. Parallel light incident in any direction can be converged to a point on the other side of the lens surface [22]. This characteristic is valid under both geometrical optics and wave optics, as shown in Fig. 1. This property actually means that the Luneburg lens can achieve a perfect Fourier transform [20].
For convenience, this paper considers the design of a 2D flat Luneburg lens. The term “flat Luneburg lens” here denotes the flat lens is transformed from Luneburg lens; it does not means the lens has same function as that of the traditional Luneburg lens, and should be distinguished from other flat Luneburg lenses [34–36]. By rotating the 2D lens around its optical axis, a three-dimensional (3D) cylindrical Luneburg lens can be obtained, where the input and output surfaces are both planar, which can detect 2D signals. According to the principle of TO, if the original material is isotropic and the transformation medium needs to be isotropic, the space mapping should be conformal. To do this, the complex variable function is adopted, and the conformal transformation is [37]:
where $W = u + iv$ and $Z = x + iy$ are the points on the original (or virtual) and transformed (or physical) complex planes, respectively. This transformation satisfies the Cauchy-Riemann condition, i.e., it is conformal, and via this space mapping, a circular region is transformed to an infinite strip region, as shown in Fig. 2. According to the deformation view of TO [27], supposing the isotropic principal stretch of the deformation at a point is $\gamma $, the refractive index at that point of the Z plane can be obtained as ${n_z} = {n_w}/\gamma $, where ${n_w}$ is the refractive index of the corresponding point in the W plane, and the principal stretch can be obtained as:
Fig. 2. Diagram of the conformal transformation of the flat Luneburg lens. The left half of the unit circle arc in W-space is mapped to the left boundary $x ={-} \pi /4$ in Z-space, and the right half of the unit circle arc in W-space is mapped to the right boundary $x = \pi /4$ in Z-space.
The original Luneburg lens has a radius of 1 and a center point of $(0,0)$. The range of the transformed zonal region is $ - \pi /4 < x < \pi /4$ and $ - \infty < y < \infty $. The transformation maps the points $W = ({\pm} 1,i0)$ to $Z = ({\pm} \pi /4,i0)$, disassembles the points $W = (0, \pm i)$ and maps them to $Z = (0, \pm i\infty )$; that is, the left half of the unit circle arc is mapped to line $x ={-} \pi /4$ and the right half of the unit circle arc is mapped to line $x = \pi /4$.
After transformation, the refractive index of the Z-plane strip region is axis-symmetrically distributed with respect to $x = 0$ and $y = 0$, respectively. Since the strip region is infinite in the y direction, we take the middle part within $ - 1 \le y \le 1$ as the lens body, so a rectangular lens is obtained. Since the refractive index less than 1 is difficult to achieve in practice, the region with the refractive index less than 1 is truncated to be 1, which has almost no effect on the results. The refractive index distribution before and after transformation is shown in Fig. 3.
Fig. 3. Principal stretch distribution and refractive index distribution of the lens before and after transformation. (a) The isotropic principal stretch distribution in the transformed space; (b) the refractive index distribution in the original Luneburg lens with a radial gradient; (c) the refractive index distribution in the flat Luneburg lens, which is symmetrical with respect to axes $x = 0$ and $y = 0$, respectively, where a refractive index less than 1 is truncated.
3. Chirp signal detection possibility of the proposed lens
Now let us examine some properties of the proposed lens by investigating the corresponding wave pattern variation during the space transformation. The normal incidence plane waves on the proposed lens can only focus at the center, as shown in Fig. 4, because it is equivalent to the radial input waves on the original lens. Without loss of generality, suppose the right half arc of the lens is the input surface and the left half arc is the output surface before the transformation. As shown in Fig. 1, the incident plane wave was parallel to the u axis and focused at the output surface of the Luneburg lens; when it reached the input surface, the phase distribution on the surface was nonuniform because the input surface was arc-shaped. To maintain this focusing phenomenon after transformation, the wave field attached on the input boundary of the transformed space should experience corresponding distortion during the transformation. It is interesting to determine the kind of input signal for this nonuniform phase distribution on the straight boundary of the transformed flat lens.
Fig. 4. The propagation of normal incidence plane waves in the proposed lens. (a) Ray optics; (b) wave optics.
To this end, the wave pattern transformation on the input boundaries is investigated. As shown in Fig. 5, the arc length S on the W plane, starting from $\theta = 0$, is:
where R is the radius of the circle, $\theta $ is the polar angle, and u and v are the horizontal and vertical coordinates, respectively. From the conformal transformation $W = \tan (Z)$, the relationship between u, v and x, y can be obtained:Fig. 5. Geometric schematic diagram for solving the signal on the circular arc of the W plane at the incidence of parallel light.
Substituting Eq. (4) into Eq. (3) and set the radius of the circle $R = 1$, the right semiarc on the W plane becomes line $x = \pi /4$ on the Z plane, so the relationship between the arc length S on the W plane and the vertical coordinate y on the Z plane is obtained:
The horizontal distance $\Delta u$ from a point on the right half arc to (R,0) is:
Now, considering that a plane wave propagates horizontally into the input surface and that the initial phase is ignored, the signal incident on the circular arc is:
This means that if one wants to strictly map the focalization phenomenon of the plane waves in the original Luneburg lens to the flat Luneburg lens, the input signal in the latter should have such an unusual form. To obtain a simpler and applied form of this signal, its Taylor expansion is written:
In other words, the original Luneburg lens can focus parallel light on the output surface, while the flat Luneburg lens can focus chirp signals on the output surface with specific input chirpiness. As the chirpiness term of a chirp signal is customarily expressed as $(1/2)m$, the chirpiness formula can be obtained from Eq. (10):
This is an intrinsic feature of the flat Luneburg lens, and the particular chirpiness of the lens is called its eigenchirpiness, which is only related to wavelength $\lambda$. For a source signal (such as a screen or test specimen) illuminated by coherent light, the phase information of the source can be maintained when the light propagates into the system [38]. If the source is a chirp, the illumination wavelength will determine the chirpiness that is actually transported into the lens. When the modulated input chirpiness results in the best focusing effect in the flat Luneburg lens, i.e., the modulated input chirpiness is equal to the eigenchirpiness, then the source chirpiness can be calculated according to the chirpiness equation in (11). One can set a wavelength range and sweep the illumination wavelength in this range to find the eigenchirpiness as well as the corresponding eigenwavelength and in turn obtain the source chirpiness.
4. Example and numerical simulation verification
Chirp signal chirpiness detection has been widely applied in many fields. For example, Newton's ring is a typical two-dimensional chirp signal, which is often encountered in interferometry [39]. Physical parameters such as the radius of curvature can be estimated during the analysis of Newton's rings [39]. Suppose the screen has the image of a Newton's ring, and any radial line of the ring is a 1D chirp signal [39]. Under coherent illumination, a radial line of the ring is set as the input of the proposed lens, indicated as $g(y)$, which is the complex amplitude distribution on the input surface, whose modulus represents the amplitude at each point, and the argument represents the initial phase at each point [38]. In other words, a chirp signal is obtained on the input surface, and the input chirpiness is modulated by the illumination wavelength, which can be used to obtain the source chirpiness, i.e., the chirpiness of the chirp image in the screen.
Through numerical simulations based on COMSOL Multiphysics, the chirpiness detection of the designed lens is verified. In the simulations, the lens size and the wavelength have the same length unit, such as millimeters (mm) or centimeters (cm), and the choice of length unit will not influence the results because the result depends only on the relative size of the wavelength compared to the device, so we omit the length unit to keep the simulation examples as general as possible. Suppose the 1D source chirp signal is $f(y) = \textrm{exp} [ - i(1/2m{y^2})]$ with $m = 359$ ($m \approx 8\pi /0.07$). Then, the eigenwavelength of the source chirpiness is $\lambda = 8\pi /m \approx 0.07$ according to the chirpiness equation in (11). The chirpiness of the input signal is detected by the abovementioned sweeping wavelength method. The aperture width, which can be used to satisfy the approximation condition of Eq. (10), is 0.4. The wavelength sweeping range is (0.04∼0.1), and the sweeping step size is 0.02. Ideally, when the illumination wavelength is 0.07, the input signal can be well focused on the output surface.
The different wave propagations under different illumination wavelengths are shown in Fig. 6. When the illumination wavelength is 0.04, the wavelength is shorter than the eigenwavelength, so that the modulated chirpiness is higher than the eigenchirpiness, and the focus of the signal is inside the output surface; in contrast, when the illumination wavelength is 0.1, the wavelength is longer than the eigenwavelength, and the modulated chirpiness is lower than the eigenchirpiness, while the focus of the signal is outside the output surface. When the illumination wavelengths are others (0.06 and 0.08), the output signals focused better, that is, around the output surface, agreeing with the analytical model. One can check the amplitude distribution along the output boundary to find this.
Fig. 6. Simulation results of the wave field |Ez| distribution in the lens for a source chirpiness 359 with an eigenwavelength of 0.07 and its amplitude distribution along the output boundary. (a) and (e) Illumination wavelength of 0.04; (b) anf (f) illumination wavelength of 0.06; (c) and (g) illumination wavelength of 0.08; (d) and (h) illumination wavelength of 0.1.
This rough wavelength sweeping with a larger sweep step can find the general scope of the source chirpiness. This procedure can be skipped if the general scope of the source chirpiness is known. To further obtain a more precise measurement, the parameterized wavelength range was reduced to (0.055∼0.085), the step was refined to 0.005, and the results are shown in Fig. 7.
Fig. 7. Simulation results of the wave field |Ez| distribution in the lens for a source chirpiness of 359 with an eigenwavelength of 0.07 and its amplitude distribution along the output boundary. (a) and (g) Illumination wavelength of 0.055; (b) and (h) illumination wavelength of 0.06; (c) and (i) illumination wavelength of 0.065; (d) and (j) illumination wavelength of 0.07; (e) and (k) illumination wavelength of 0.075; (f) and (l) illumination wavelength of 0.08. The red circle indicates the first zero location in each boundary graph.
To subtly distinguish the focus qualities of different illumination wavelengths, the waveform of the output signal is investigated. The original Luneburg lens is a perfect Fourier transform operator, and the incident uniform plane wave should lead to a sinc distribution of the wave field on the output plane, i.e., the Fourier transform of a square wave. As a consequence, the output signal should be the stretching or compression of the sinc function in the y direction of the flat Luneburg lens, and it can be found that these sinc graphs are different with respect to the first zero status, as shown in Fig. 7. In the Rayleigh criterion for evaluating the optical quality of the system, the first zero is an important indicator [38,40], and based on this concept, the first null beam width (FNBW) becomes an important lobe diagram parameter [41]. The indicator is also crucial for the filters, as the first zero should be as shallow as possible to effectively suppress clutter and avoid the loss of weak targets trapped in it [42]. Theoretically, the amplitude of the first zero should be zero, but due to system errors such as the approximation effect in the design derivation, the amplitude of the first zero cannot be as ideal as zero. In this case, the one that is closest to zero is chosen as the best one; that is, the illumination wavelength that leads the focus with the lowest first zero amplitude is considered as the eigenwavelength.
The detailed detection data are shown in Table 1. Based on this rule, the obtained eigenwavelength of this example is 0.065, and according to the chirpiness formula, the source chirpiness is obtained as $m = 8\pi /0.065 \approx 386.6576$, with a relative error of approximately 7.7% compared with the real value of 359. Although the wavelength of 0.07 (related to real source chirpiness) is unrecognized due to system error, the accuracy is better than that of the pulse position-based traditional quadratic GRIN lens and similar to that of improved GRIN lenses [15–18]. It is also noted that the distinguishability of the lowest first zero is quite high because the relative errors shown in the third column in Table 1 are considerable; it is easy to find the minimum first zero amplitude. In addition, the first zero points are located in a very small region, providing convenience of detection.
Table 1. Amplitude of the first zero at each sweeping wavelength where the eigenwavelength is 0.07
To verify the generality of the method, more cases with different source chirpinesses are surveyed. Setting the source chirpiness as $m = 251$ ($m \approx 8\pi /0.1$), the corresponding eigenwavelength is $\lambda = 8\pi /m \approx 0.1$. The parameterized sweeping range of the illumination wavelength is (0.085∼0.115) with a step size of 0.005, where rough wavelength sweeping is skipped. The result is shown in Fig. 8, and the wavelength with the best focusing effect is 0.095. According to the chirpiness formula, the obtained source chirpiness is $m = 8\pi /0.095 \approx 264.5552$, and the relative error is approximately 5.4% compared with the real value of 251. The detailed detection data are shown in Table 2.
Fig. 8. Simulation results of the wave field |Ez| distribution in the lens for a source chirpiness of 251 with an eigenwavelength of 0.1 and its amplitude distribution along the output boundary, where the dotted circles show the first zero points. (a) and (g) Illumination wavelength of 0.085; (b) and (h) illumination wavelength of 0.09; (c) and (i) illumination wavelength of 0.095; (d) and (j) illumination wavelength of 0.1; (e) and (k) illumination wavelength of 0.105; (f) and (l) illumination wavelength of 0.11.
Table 2. Amplitude of the first zero at each sweeping wavelength when the eigenwavelength is 0.1
The source chirpiness is set as $m = 193$ ($m \approx 8\pi /0.13$) with the corresponding eigenwavelength $\lambda = 8\pi /m \approx 0.13$. The parameterized sweeping range is (0.115∼0.145), and the step size is 0.005. As shown in Fig. 9, when the illumination wavelength is 0.125, the focusing effect is the best. According to the chirpiness formula, the obtained source chirpiness is $m = 8\pi /0.125 \approx 201.0619$, with a relative error of approximately 4.2% compared with the real value of 193. The detailed detection data are shown in Table 3.
Fig. 9. Simulation results of the wave field |Ez| distribution in the lens for a source chirpiness of 193 with an eigenwavelength of 0.13 and its amplitude distribution along the output boundary. (a) and (g) Illumination wavelength of 0.115; (b) and (h) illumination wavelength of 0.12; (c) and (i) illumination wavelength of 0.125; (d) and (j) illumination wavelength of 0.13; (e) and (k) illumination wavelength of 0.135; (f) and (l) illumination wavelength of 0.14.
Table 3. Amplitude of the first zero at each sweeping wavelength when the eigenwavelength is 0.13
It is necessary to reveal the interesting capacity of the proposed lens for detecting high chirpiness values. When the illumination wavelength is small, the corresponding source chirpiness is large according to Eq. (11). For example, choosing the length unit of the system as mm and when the eigenillumination wavelength is 0.05 mm, the obtained source chirpiness can be up to $m = 8\pi /0.05 \approx 502.6548$, which is an ultrahigh chirpiness for spatial chirps. GRIN lenses based on pulse position detection are difficult, if not impossible, to detect such high chirpiness because the focus pulse is too close to the input surface to meet the paraxial approximation condition. For an exposure lens [17,18], a sufficiently high chirpiness can also lead to failure of exposure efficacy. The proposed flat Luneburg lens, however, works beyond the FRFT mechanism, and the focus of interest is always located on the output surface, which can still achieve accurate detection of high chirpiness. The limitation of the system might be the sweeping range of the illumination wavelength as well as the fineness of the sweeping step.
The robustness of the flat Luneburg lens against a nonideal work environment is worth examining due to its new working mechanism. As the flat Luneburg lens is no longer circular symmetric, the input location matters. In the above examples, the center of the input aperture (and the signal) is required to be located in the middle of the lens boundary, i.e., $y = 0$ in the input surface. The reason is as follows. For the original Luneburg lens, if the input aperture is shifted upward $\Delta S$ along the arc, that is, the incident plane wave is not along the u axis. Then, after the transformation, the phase at arc length S is equal to the phase at $S - \Delta S$ when incident along the u axis. From Eq. (7) and Eq. (8), the following is obtained:
At this time, on the flat Luneburg lens, the corresponding shifted signal cannot be approximated as a chirp signal. However, in practice, it probably cannot strictly satisfy the symmetrical input condition, so it is necessary to check the sensitivity of the system to the signal offset. In the simulation, the input chirp signals on the input surface of the flat Luneburg lens are shifted upward by 0.1, 0.15, and 0.22, and the aperture width is still 0.4, i.e., the aperture changes to [−0.1, 0.3], [−0.05, 0.35], and [0.02, 0.42], respectively; for example, the input signal changes to $f(y )= \textrm{exp} [ - i(1/2m{(y - 0.1)^2})]$ after an upward offset of 0.1 for source chirpiness $m = 359$ ($m \approx 8\pi /0.07$) with the corresponding eigenwavelength $\lambda = 8\pi /m \approx 0.07$. The sweeping wavelength range and step length remain the same as those of the corresponding example with symmetrical input. The results are shown in Fig. 10, where the best focusing effects of different offsets are displayed. The detailed data are shown in Tables 4–6. As the input is asymmetrical, the amplitude of the left and right first zeros can be unequal, and if so, the criterion used in the method is the average amplitude of the two. Even when the center offset is up to 0.1, as large as 50% of the half aperture width, the same eigenillumination wavelength of 0.065 as its counterpart with symmetrical input is obtained; thus, the detection precision is not disturbed by the offset error. This shows that the designed lens has some tolerability to the signal center offset error. When the offset distance is too large, as in this example, 0.15 and 0.22, the obtained eigenwavelengths are 0.06 and 0.055, respectively, and the detection precision is lower than that of the corresponding symmetrical input case.
Fig. 10. Simulation results of the wave field |Ez| distribution in the lens and its amplitude distribution along the output plane when the signal is best focused after an upward offset for a source chirpiness of 359 with an eigenwavelength of 0.07. (a) and (d) Upward offset of 0.1 and illumination wavelength of 0.065; (b) and (e) upward offset of 0.15 and illumination wavelength of 0.06; (c) and (f) upward offset of 0.22 and illumination wavelength of 0.055.
Table 4. Eigenwavelength of 0.07 and upward offset of 0.1
Table 5. Eigenwavelength of 0.07 and upward offset of 0.15
Table 6. Eigen wavelength of 0.07 and upward offset of 0.22
Another consideration of nonideal work circumstances is noise, and the anti-noise performance of the system should be determined. Under noisy conditions, the input signal becomes ${f_n}(y )= \textrm{exp} [ - i(1/2m{y^2})] + n(y )$, where n(y) is noise. In the simulation, source chirpiness $m = 359$ ($m \approx 8\pi /0.07$) with the corresponding eigenwavelength $\lambda = 8\pi /m \approx 0.07$; random noise with different signal-to-noise ratios (SNRs) is added to the source signal, where SNR = 0 dB, −3.52 dB, and −6.02 dB, respectively, as shown in Fig. 11. After noise is introduced, the amplitude of the left and right first zeros could be unequal, so the criterion of amplitude of the first zeros is taken as the average of the two. The sweeping illumination wavelength range and step remain the same as that of the corresponding example with noiseless input. The corresponding numerical simulation results of the wave field with the best focusing effect and its amplitude distribution along the output surface are obtained, as shown in Fig. 12, and more detailed data are listed in Tables 7–9. Interestingly, for the three noises with the given SNR, the detection results are uninfluenced, and the obtained eigenwavelength is still 0.065, which is the same as its noise-free counterpart; this means that the system has a strong anti-noise property. Nevertheless, a sensible guess is that as the noise becomes stronger, the imbalance of the two first zeros and the distortion of the output signal become more serious, which could disturb normal detection.
Fig. 11. Signal with random noise. The blue line is the input signal, and the red line is the random noise. (a) Noise with SNR = 0 dB; (b) noise with SNR = −3.52 dB; (c) noise with SNR = −6.02 dB.
Fig. 12. Simulation results of the wave field |Ez| distribution in the lens and its amplitude distribution along the output plane for the best focusing effect after adding random noise to the source signal for a source chirpiness of 359 with an eigenwavelength of 0.07. (a) and (d) SNR = 0 dB and illumination wavelength of 0.065; (b) and (e) SNR= −3.52 dB and illumination wavelength of 0.065; (c) and (f) SNR= −6.02 dB and illumination wavelength of 0.065.
Table 7. Eigenwavelength of 0.07 and SNR= 0 dB
Table 8. Eigenwavelength of 0.07 and SNR= −3.52 dB
Table 9. Eigenwavelength of 0.07 and SNR= −6.02 dB
5. Discussion and conclusion
Detecting the chirpiness of a chirp signal is an important topic in many information fields. Compared to the search algorithms based on time (or space)-frequency representations, the GRIN lens-based method can quickly obtain the input chirpiness by determining the corresponding impulse position in the lenses. These lenses require locating the pulse position and are limited by the paraxial conditions because the FRFT functionality of the GRIN lenses works in a paraxial approximation; thus, high chirpiness cannot be detected in these systems. In this paper, a canonical circular Luneburg lens is deformed to a flat Luneburg lens using TO, and the new GRIN lens is found to focus the input chirp signal in the case of the source chirpiness and illumination wavelength satisfying a given relationship. This feature is derived from the plane wave focusing of the circular Luneburg lens and thus is a new, non-FRFT mechanism. This makes it possible to use illumination wavelength sweeping instead of spatial position scanning to detect the source chirpiness via the flat Luneburg lens, and it can achieve high chirpiness detection that cannot be realized by other GRIN lenses. The numerical simulations show that the detection precision is acceptable and that the lens is robust against center offset error and noise.
This lens is isotropic and has advantages in low loss and broadband applications, with a reasonable refractive index range, and its preparation is possible [43]. Currently, 3D printing technology with submicron spatial resolution and multiple materials can manufacture structures directly from digital models without the need for molds, making it an ideal choice for manufacturing efficient media with high geometric or exponential complexity [44,45]. When the length unit of the system is cm or mm, a possible experimental scheme of the lens fabrication is to build dielectric plate with holes of different spatial sizes, with the help of effective medium theory [46] and 3D printing, and the effective indices can be controlled well in this way [47].The proposed lens is expected to provide a flexible and powerful tool for a wide range of chirpiness detections and has broad application prospects.
Funding
National Natural Science Foundation of China (61975015); Beijing Natural Science Foundation (L191004).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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