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Abstract
Speckle with non-Rayleigh amplitude distribution has significant research value in imaging and measurement using structured illumination. However, existing speckle customizing schemes have been limited in generation speed due to the refresh rate of spatial light modulators (SLMs). In this work, we proposed a method to rapidly generate non-Rayleigh distributed speckle fields using a digital micro-mirror device (DMD). In contrast to SLMs that allow for gray-scale phase modulation, DMD is limited to binary amplitude control. To solve this limitation, we design a Gerchberg-Saxton-like algorithm based on super-pixel method, this algorithm enables the customization of non-Rayleigh speckle with arbitrary intensity probability density function. Statistical analyses of experimental results have demonstrated that the customized speckles exhibit excellent stability in their lateral statistical properties, while also maintaining consistent propagation characteristics with Rayleigh speckle in the longitudinal direction. This method provides a new approach for high-speed and arbitrary intensity speckle customization, holding potential applications in imaging, measurement, and encryption fields.
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1. Introduction
Speckle is a well-known optical phenomenon that occurs when coherent light scatters off a rough surface or through a random medium. Initially considered to be the great obstacle to coherent optical imaging and display, the particles in speckle would obscure the high-frequency information in the image [1,2]. With further research, researchers have discovered that some unique characteristics of speckle hold significant application value in various fields, including imaging [3–9], metrology [10,11], and encryption [12,13]. Typically, naturally generated speckles mostly follow the Rayleigh distribution, whose amplitudes follow the Rayleigh distribution, resulting in a negative exponential intensity distribution. This characteristic is dependent on that the interference of a large number of independent scattering waves with random phases uniformly distributed over a range of $0$ to $2\pi$ [14].
Recently, customizing speckles with non-Rayleigh distribution has drawn garnered attention due to their numerous practical applications. It has been demonstrated that super-Rayleigh and other types of customized speckles can improve image resolution, visibility, and signal to noise ratio [15–17]. Considering the interference generation of speckle field, a common approach to tailoring non-Rayleigh speckles is designing phase-only patterns to create the desired statistical distributions in its Fourier plane [18–23]. This procedure is very similar to computational Fourier holography. Initially, an intensity pattern is numerically generated with designed intensity PDF, then the inverse Fourier transform is computed, the resulting phase of inverse Fourier transform is applied to a phase-only liquid crystal on silicon spatial light modulator (Lcos-SLM). Specifically, iterative constraint using the Gerchberg-Saxton (G-S) algorithm often yields better results than direct Fourier transform [24]. Employing these methods, researchers have successfully achieved speckle customization with arbitrary statistical distributions in both longitudinal and horizontal directions [25]. While various methods have been proposed to generate customized speckle patterns, they often face limitations in terms of modulation speed and complexity. These limitations are mainly attributed to the SLM, as the speed of speckle generation is restricted by the refresh rate of the liquid crystal SLM, which typically reaches a maximum of a few hundred Hertz.
Unlike Lcos-SLM, which has limitations in refresh rate, digital micro-mirror device (DMD) offers a much higher refresh rate up to 22.7 kHz, and has found applications in various fields for high-speed light field modulation [26–28]. Based on this advantage, we present a fast approach to generate non-Rayleigh speckle with arbitrary intensity PDF by using DMD. However, DMD is typically limited by the property of binary mirror tilt, which restricts the ability for the continuous complex amplitude modulation. In this work, we design a Gerchberg-Saxton-like algorithm based on super-pixel method to achieve continuous complex amplitude speckle generation [29]. Different with computational Fourier holography, the target speckle is generated in the conjugate plane of DMD. It means that we can control every pixel of target field pixel independently, rather than a global transformation like Fourier holography method. By applying numerical intensity transformation to a Rayleigh speckle, we can construct the target speckle with arbitrary intensity PDFs. Specifically, due to the bandwidth limitation of super-pixel method, the high-frequency distribution of target speckle field will be obstructed, which will cause the divergence from the initial intensity PDF. Therefore, the Gerchberg–Saxton-like iteration filtering is proposed in this intensity transformation procedure to ensure that the spatial frequency of generated speckle satisfy the bandwidth limitation of super-pixel method. To assess the feasibility of our method, we perform a series of statistical analyses on the customized speckle patterns. The results have demonstrated the stationary characteristics of the customized non-Rayleigh speckles in the lateral direction. However, we also observe the rapid evolution of customized non-Rayleigh speckles in the propagation direction, eventually developing into similar Rayleigh distributions. This similar behavior is attributed to the preservation of the phase distribution during the intensity transformation process, which also inspires us to consider the possibility of tailoring the longitudinal characteristics of the speckle field by simultaneously designing its phase distribution. we will explore its feasibility in future work.
2. Method
2.1 Generating non-Rayleigh speckle
The key to generating speckle with a non-Rayleigh distribution lies in customizing its intensity distribution while preserving the common properties of speckles. To address this, we employ a numerical intensity transformation to construct an arbitrary intensity probability density function (PDF) from a Rayleigh speckle. Throughout this transformation process, the phase distribution of the Rayleigh speckle remains unchanged, ensuring that fundamental properties of the generated speckle, such as particle size, stationary, and spatial correlation length, align with those of a fully developed Rayleigh speckle.
Rayleigh speckle typically emerges in the far field after a fully randomized modulation or reflection for a coherent light. We achieve this process numerically by applying a phase-only modulation with a uniform distribution ranging from $0$ to 2$\pi$. The modulated field is then approximately transformed to the far field using the Fourier transform, and a low-pass filter is applied to simulate the bandwidth limitation in practical experimental system. The intensity PDF of Rayleigh speckle can be expressed by a negative exponential distribution $\rho (I)=\frac {1}{<I>}e^{\frac {-I}{<I>}}$, where $<I>$ is the average intensity of Rayleigh speckle pattern. By knowing the characteristics of the Rayleigh speckle pattern, the intensity distribution of target Non-Rayleigh speckle can be mathematically transformed from the intensity PDF of Rayleigh speckle and expressed as
Nevertheless, due to the spatial bandwidth limitations of practical experimental system, the high-frequency information of generated speckle field will be blocked and leading to deviations from the expected PDF distribution. To eliminate these deviations, we introduce a low-pass constraint in the frequency domain, together with the PDF constraint in spatial domain, to ensure the generated speckles can satisfy both the PDF condition and the bandwidth restriction. The detailed transformation process involves a Gerchberg-Saxton-like iterative method and illustrated in Fig. 1.
Fig. 1. The diagram of intensity transformation steps. The intensity transform is started from a Rayleigh speckle field $E_0$, and outputs a new speckle field $E_t$ with target PDF distribution. Subsequently, a low-pass filter is applied to the target complex field $E_t$, aims to simulate the bandwidth limit of the system. The size of filter window is consistent with the iris that used in experiment. The iterative process ends when the MSE between $E_t$ and $E_{i+1}$ falls below a predetermined threshold, indicating that the target speckle field complies with the bandwidth limitation of practical experimental setup.
We start with a Rayleigh speckle field ($E_0$), and generate a new speckle field ($E_{t}$) through the transformation procedure as depicted in Eq. (1), ensuring its intensity conforms to the desired target PDF. In this process, only the intensity of the initial speckle field is transformed, while the phase remains unchanged. Subsequently, the low-pass filter, designed according to the numerical aperture of experimental system, is imposed to the transformed speckle as
where $S$ denotes the aperture function which performs as a low-pass filter, $\mathcal {F}$ represents the Fourier transform, and $\cdot$ denotes the multiplication operator. This filtering step may cause slight deviations in the intensity PDF from the target one. To guarantee both the PDF condition and the bandwidth restriction are met simultaneously, we repeat the aforementioned procedure by replacing the initial speckle field with the updated field ($E_i = E_{i+1}$) until a stabilized output speckle is achieved. Here, the mean square error (MSE) criterion is adopted to ascertain whether the iteration converges. The iteration will terminate when the MSE between the updated field and target field falls below a predetermined threshold value, which we have set as $0.001$. Typically, the iteration rapidly converges after several tens of cycles, yielding the speckle field with the desired intensity PDF as the output.2.2 High-speed display of target speckle field
In pursuit of an arbitrarily intensity PDF for the speckle field with a high refresh rate, we chose DMD over Locs-SLM due to its ultra-high modulation speed. To achieve gray-scale modulation with the binary modulation of DMD, we introduce super-pixel method to modulate both the phase and amplitude of the generated speckle field.
The schematic diagram of experimental set-up is depicted in Fig. 2. A He-Ne laser operating at $632.8$ nm is expanded by a beam expander and uniformly illuminates on the DMD chip (Vialux V-7000). The resolution of DMD is $1024 \times 768$, and its pixel pitch is 13.68 $\mathrm{\mu}$m. An off-axis 4$f$-configuration is used to transform the field at the DMD plane to the target plane. The focal lengths of $4f$-configuration are $f_1$ = 250 mm and $f_2$ = 200 mm, respectively. An iris is positioned at the frequency-plane of the 4$f$-configuration, which is off-axis respect to the first lens, but coaxial with the second lens. A camera (Tucsen BSI-400) with the pixel pitch of 6.5 $\mathrm{\mu}$m is placed at the back focal plane to capture the intensity of target speckle field.
Fig. 2. Schematic diagram of experimental setup. A He-Ne laser is expanded by a beam expander and uniformly illuminates on the DMD chip. An off-axis 4$f$-configuration ($f_1$ and $f_2$) is used to transform the field at the DMD plane to the target plane, The iris located at the frequency-plane provides a stationary phase difference and low-pass filtering to make the superposition of the fields reflected from the individual micro-mirrors within each super-pixel.
The off-axis configuration of the iris introduces a stationary phase difference between the field at the DMD plane and the camera plane. Through controlling the off-axis position of the iris, the DMD pixels are effectively divided into a set of super-pixels, each consisting of $n \times n$ micro-mirrors, corresponding to a total phase difference of $2\pi$. Furthermore, the iris blocks the high spatial frequencies information generated by the modulation of DMD, such that the individual DMD pixels are no longer resolved on the camera plane, but become a coherent superposition with the adjacent pixels. By controlling the size of the iris, the output field can be regarded as the superposition of the fields reflected from the individual micro-mirrors within each super-pixel.
With aforementioned discussions, the position of iris is determined by size of super-pixels, the maximum phase difference in each super-pixel should be set as $2\pi$. Therefore, the position of iris can be determined at $(x, y)\ = (\frac {-\lambda f_1}{n^2d},\frac {\lambda f_1}{nd})$, which corresponds to $(u, v) = (\frac {2\pi }{n^2d},\frac {2\pi }{nd})$ in the frequency domain. Here the origin of coordinates is set at the optical center of $L_1$, $\lambda$ is the wavelength of the light, $f_1$ is the focal length of $L_1$, and $d$ represents the pixel pitch of DMD. After filtering, the output field behind the iris can be denoted as
where $circ(\sqrt {u^2+v^2}) = \begin {cases} 1 & \sqrt {u^2+v^2}<r\\ 0 & else \end {cases}$ is the function of spatial filter, and $r$ is the radius of the filter iris. $M(u,v)$ represents modulation field in frequency domain, due to the off-axis filtering, an extra phase offset is added as $M(u-\frac {2\pi }{n^2d},v+\frac {2\pi }{nd})$. Afterwards, an inverse Fourier transform is imposed by the second lens $L_2$ to obtain the target field in the camera plane, the output field can be expressed as where $\frac {J_1(\rho )}{\rho }$ is the inverse Fourier transform of $circ(\sqrt {u^2+v^2})$, $J_1(\rho )$ is the one order Bessel function, and $\rho = 2rd/\lambda f_1 \sqrt {x^2+y^2}$.3. Results and discussions
3.1 Statistical distribution of customized speckle patterns
To verify the feasibility of the proposed method, we customized four different speckle distributions with an MSE less than 0.001 on a computer equipped with an AMD Ryzen Threadripper 3990X 64-core processor running at a frequency of 2.90 GHz and with 256 GB of memory, each following uniform, sinusoidal, linear, and negative exponential intensity PDFs. The speckle with negative exponential intensity distribution does not need intensity transformation, and the average intensity transformation time of other three distributions is about 15.97 s, 16.08 s and 61.21 s respectivly. Compared to the other two types of speckles, the difference between the PDF of the linear distribution speckle and the original Rayleigh speckle is greater. The intensity distribution of Rayleigh speckle is mostly concentrated below the mean value of intensity (we have normalized the mean intensity of generated speckle to 1), while the intensity probability peaks of linear distribution speckle is concentrated above 1. This distribution difference makes its convergence slower than other two types of speckles.
The first row of Fig. 3 shows the captured speckle distribution of these four different speckle fields, the size of the captured pictures in Fig. 3 is $400 \times 400$ pixels. The topology structure of the customized speckle exhibits marked differences from that of Rayleigh speckle. Specifically, the dark channel in uniformly and linearly distributed speckle is significantly reduced compared to Rayleigh speckle, and the all the distributions appear more discontinuous than that of Rayleigh speckle. The second row of Fig. 3 illustrates the intensity PDF curves. The black line in intensity PDF curves are generated from experiment while the red line is the theoretical distribution. The experimental results exhibit excellent conformity with the theoretical curves. Nevertheless, some deviations between the experimental and theoretical curves can be observed, particularly in regions where the intensity curves undergo abrupt changes. In the bottom row of Fig. 3, the amplitude PDF curves are also presented. The results exhibit a similar trend to the intensity curves, with deviations occurring in areas where the curves experience abrupt changes. The reason is that, we use 16 mirror elements to form a superpixel. Due to the limited number of mirror elements, only 6531 complex-valued modulations can be achieved currently in our experiment. Moreover, these modulations must comply with the filter aperture bandwidth limit. This bandwith limit restrict the complex value of target speckle field. When certain required complex values for a speckle field cannot be obtained, we have to choose approximate tunable substitutions, which introduce some degree of error.
Fig. 3. Customized speckle patterns and the corresponding intensity and amplitude PDF. The four images in the top row are spatial distributions of speckle patterns with (a) uniform, (b) sinusoidal, (c) linear, and (d) Rayleigh speckle. The middle row is the PDF curve of their intensity, and the bottom is the PDF curve of their amplitude. The red curve is the simulation result and the black curve is the experimental result.
In addition to the PDF, some other statistical properties, such as independence and particle size, are still essential in the application of speckles. The independence between different speckle patterns can be quantitatively described using cross correlation. Figure 4(a) illustrates the normalized cross correlation (NCC) for 35000 different speckle patterns [30]. All these patterns follow uniform intensity PDF distribution but are transformed from different Rayleigh speckles. The NCC between the patterns $U(x,y)$ and $V(x,y)$ can be calculated by
The NCC coefficient falls within the range of $[-1,1]$, where $c = -1$ or $c = 1$ represents extremely strong correlation, and $c = 0$ indicates completely uncorrelation. The NCC results shown in Fig. 4(a) are all distributed around 0, which indicates that the generated speckles have hardly any correlation. This property holds significant potential for applications in the field of structured illumination, where the correlation between the illumination speckles directly influences the validity of the corresponding samplings. Moreover, we also calculate the statistical distribution of these 35000 patterns and subsequently randomly plotted 263 of them in Fig. 4(b). The intensity PDF of these 263 patterns exhibits excellent stationary which means the proposed method for customizing speckle pattern is absolutely reliable.
Fig. 4. The analysis for the independence and stability of generated speckle. (a) NCC for 35000 different speckle patterns, all of these patterns follow uniform intensity PDF distribution but are transformed from different Rayleigh speckles. (b) The intensity PDF of 263 uniform speckles.
We also analyse the speckle size by autocorrelation of target speckle patterns, the autocorrelation is a critical indicator for the average particle size of the speckle, which can be determined by the Full Width at Half Maximum (FWHM) of its autocorrelation. Figure 5 plots the 2D section of the speckle autocorrelation of the four different speckles discussed in Fig. 3. The FWHM is approximately 10 pixels for all cases, and since each pixel on the CMOS camera is 6.5 $\mathrm{\mu}$m, the average particle size of the speckle is calculated to be 65 $\mathrm{\mu}$m. After the intensity transformation from Rayleigh distribution, the particle size of the generated speckle remains almost unchanged.
Fig. 5. The autocorrelation of different speckles. The dotted red line marks the position of the FWHM. (a) Uniformly distributed speckle, (b) sinusoidal distributed speckle, (c) linearly distributed speckle, (d) Rayleigh distributed speckle.
To further investigate this phenomenon, we calculate the structural similarity index (SSIM) between any two speckles [31]. As shown in Table 1, the generated speckles still have some similarity in spatial structure, especially between the speckles with sinusoidal and uniform distributions. The intensity transformation only changes the bright-dark relation, but most of the spatial structures are still unchanged. This is the reason why the speckle particle size remains constant.
Table 1. SSIMs between the speckles with different intensity PDFs
The contrast of speckle is a vital parameter in speckle measurement applications. It describes the intensity variation of the speckle pattern relative to its mean intensity and can be defined as follows:
where $\left \langle I \right \rangle$ denotes spatial average operation. Table 2 shows the contrast of generated speckles and its higher-order intensity distributions. The Rayleigh speckle has the largest contrast value because its intensity distribution has a higher probability of deviating from its mean intensity. In contrast, the PDF curves of the other three customized speckles show their maximum points to be distributed around the average intensity, resulting in lower contrast values compared to the Rayleigh distribution. Typically, the higher-order intensity distributions have the increasing contrast as $\left \langle I^n\right \rangle$, which means they perform with larger contrast as $n$ increases. Consequently, one can directly enhance the contrast of the target speckle by replacing its intensity distribution with its higher-order intensity moments, rather than meticulously designing its intensity PDF distribution.Table 2. The contrast of generated speckles and its higher-order intensity distributions
To analyze the correlation between the real part and imaging part of the non-Rayleigh distributed speckle field, we present the joint complex-amplitude probability density function (PDF) between the real part and imaging part of the speckle field in Fig. 6. In contrast to a Rayleigh distributed speckle, whose joint complex-amplitude PDF is distributed as a circular Gaussian statistical distribution, the joint complex-amplitude PDF of a non-Rayleigh distributed speckle presents a different statistical distribution, even though the phase distribution of the customized speckle remains the same.
Fig. 6. Joint probability density function of four different complex speckle fields. (a) Uniformly distributed speckle, (b) sinusoidal distributed speckle, (c) linearly distributed speckle, (d) Rayleigh distributed speckle.
3.2 Propagation properties of the customized speckle patterns
In the above discussions, we focus on customizing the intensity distribution of the speckle field while keeping its random phase distribution unchanged. However, it is important to note that the propagation of the light field is primarily determined by its phase distribution. As a result, their intensity distribution will not remain stationary along the propagation, leading to further development of intensity structures and the PDF. Figure 7 exhibits the intensity evolution procedure with the propagation of a uniformly distributed speckle. During the propagation process, the uniformly distributed speckle gradually develops into a Rayleigh distribution. The similar phenomena also manifest in other customized speckles, and indicate that the speckle patterns generated by proposed method are exclusively formed at the image plane.
Fig. 7. Propagation of uniformly distributed speckle patterns. Speckle pattern and PDF from (a) $0$ cm, (b) $1$ cm, (c) $5$ cm, and (d) $10$ cm behind the image plane.
Furthermore, Fig. 8 displays the intensity cross-correlation between the propagating speckle and origin customized speckle. In all cases, the NCC coefficient progressively decreases with propagation distance and stabilizes at approximately $0.1$ after a propagation distance of 5 cm. Therefore, it can be inferred that all customized speckle patterns exhibit the same spatial correlation length.
Fig. 8. Normalized cross correlation between the intensity of customized speckle and the speckle after a distance of $d$ cm.
More interestingly, when the speckles propagate the same distance, the intensity distribution of non-Rayleigh speckles exhibits very high similarity to that of the Rayleigh speckles. Table 3 presents the SSIMs calculated from the speckle patterns captured at a distance of 5 cm behind the image plane. We attribute this phenomenon to the fact that all customized speckles share the hardly same phase distributions. In the transmission process, the phase information plays a major role in the distribution of the light field, resulting in the speckles with different intensity PDFs developing into similar Rayleigh distributions after propagating some distance.
Table 3. SSIMs of the speckle patterns captured at a distance of 5 cm behind the image plane
4. Conclusions and discussions
In this work, we present a new scheme to generate non-Rayleigh distributed speckle field by using of DMD. Although the generation of speckle through iteration may require a certain amount of time, our primary concern lies in optimizing the display speed of speckle in most scenarios. Compared with Lcos-SLM, DMD has the advantage of high refresh rate, but is limited to binary intensity modulation. To overcome this limitation, we introduce the super-pixel method to realize both amplitude and phase modulation. We posit that this approach can expedite the production of speckle illumination with diverse characteristics in experiments such as correlated imaging, thereby, facilitating rapid experimentation and comprehensive analysis of the relationship between speckle properties and imaging outcomes. A series of statistical analyses demonstrate that the speckle generated by the proposed method is sufficiently stationary in the lateral direction. However, we also observe that the customized non-Rayleigh speckles rapidly evolve in the propagation direction, and eventually develop into a Rayleigh distribution. This phenomenon is determined by the phase distribution of the customized non-Rayleigh speckle fields, as their random phase distribution is preserved during the intensity transformation procedure. This observation also inspires us that one can manipulate the lateral and longitudinal characteristics of the speckle fields by controlling the amplitude and phase simultaneously, and we will explore its feasibility in future work.
Funding
National Key Research and Development Program of China (2022YFC2807702); National Natural Science Foundation of China (12274262); Natural Science Foundation of Shandong Province (ZR2023QF069); Shandong University Inter-discipline Research Grant; Chinesisch-Deutsche Zentrum für Wissenschaftsförderung (M-0044); State Key Laboratory of Precision Measurement Technology and Instruments (pilab2205).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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