A method for eliminating computer holographic speckle by using phase probability regulation optimization algorithm

Abstract

The present application relates to a kind of methods for eliminating the speckle of computer holography using phase probability control optimization algorithm, the key parameters of computer holographic image, the key parameters of computer holographic image include root mean square error of computer holographic image, standard deviation and optical efficiency, update cost function;Update mask phase, based on mask phase, use Rayleigh-Sommerfeld diffraction integral to calculate the diffraction from input surface to target surface, determine whether the holographic image of output surface contains speckle;If it contains speckle, the next iteration is carried out, and finally the holographic image without speckle is obtained.The present application can effectively eliminate the inherent speckle existing in computer holographic imaging, design and process the phase device for eliminating speckle using the proposed holographic optimization algorithm, and use it for holographic lithography, expand the practical application scenarios of computer holography technology.

Description

TECHNICAL FIELD

[0001] The application relates to a method for eliminating speckle in computer holography by using a phase probability regulation optimization algorithm, and belongs to the technical field of computer holography and holographic lithography. BACKGROUND

[0002] In computer holography, a mask composed of randomly distributed phase or amplitude units can reconstruct a pre-designed holographic image under the illumination of a high-coherence light source. The intensity distribution of the light field at each position on the image plane can be regarded as the coherent superposition of the diffraction fields of all pixel units in the holographic mask. However, the uncertain phase or amplitude distribution in the mask will cause random coherent constructive or destructive interference of the light source, thereby generating random bright and dark spots on the imaging plane. Such intensity fluctuation of bright and dark on the imaging plane is called holographic speckle, which introduces background noise and destroys the uniformity of the holographic image, thereby limiting the application of holography in optical imaging and lithography. Therefore, speckle elimination is the key to the practical application of optical holography.

[0003] In order to reconstruct an arbitrary irregular holographic image, it is necessary to introduce a random phase or amplitude distribution in the mask. At present, almost all the methods reported for eliminating holographic speckle have a common starting point: average processing of holographic speckle in different scopes. For example, in the spatial domain, a degenerate laser with multiple orthogonal modes or a rotating diffuser is used to reduce the spatial coherence of the light source, thereby suppressing speckle; in the frequency domain, the bandwidth of the laser (or light-emitting diode) is expanded (or reduced) to realize the average processing of speckle. In addition, some people also propose average processing in the time domain: collecting holographic speckle at different times in a time sequence for analysis.

[0004] Although the above-mentioned average processing techniques can effectively suppress speckle, they also cause the edges of the holographic image to become blurred and the image details to be lost, and at the same time, a tedious and time-consuming processing process is required. SUMMARY

[0005] The application solves the problem of overcoming the shortcomings of the prior art and providing a method for eliminating speckle in computer holography by using a phase probability regulation optimization algorithm, which can effectively eliminate the inherent speckle in computer holography, design and process a phase-type device for eliminating speckle by using the proposed holographic optimization algorithm, and use it in holographic lithography, thereby expanding the practical application scenarios of computer holography technology.

[0006] The technical solution of the application is a method for eliminating speckle in computer holography by using a phase probability regulation optimization algorithm, and the implementation steps are as follows:

[0007] Step 1. Calculate the key parameters of the holographic image, wherein the key parameters of the holographic image include the root mean square error (RMSE) of the holographic image i, standard deviation SD i and optical efficiency η i ;

[0008] The calculation formula is as follows: optical efficiency η i = P image / P incident , wherein I n is the intensity at the nth pixel on the entire target surface, and there are N pixels in total; is the average intensity of the region with an amplitude of 1 in the holographic image, I m is the intensity at the mth pixel in the region of the holographic image, and there are M pixels in total; I ideal is the intensity of the ideal holographic image; P image is the total energy of the region with an amplitude of 1 in the holographic image, P incident is the total energy of the input surface;

[0009] Step 2. Dynamically adjusting the key parameters of the holographic image obtained in step 1 to update the cost function CF i , wherein the cost function CF i = w1(i)·RMSE i + w2(i)·SD i + w3(i)·(1-η i ), i is the iteration number, w1(i), w2(i) and w3(i) are weight factors of RMSE i , SD i and η i respectively. Among them, the weight factors dynamically change with the iteration number i, and the change of the weight factors adopts the following formula:

[0010] w1(i) = w RMSE {tanh[(i-i0) / D0]+1} / 2,

[0011] w2(i) = w SD {tanh[-(i-i0) / D0]+1} / 2,

[0012] w3(i) = w Eff {tanh[-(i-i0) / D0]+1} / 2.

[0013] , w RMSE , w SD and w Eff are the change ranges of the weight factors, i0 (i0 = 175) is the point of mutation of the weight factors, D0 (D0 = 40) is the length of the mutation duration, and tanh is the hyperbolic tangent function.

[0014] Step 3. According to the cost function CF i, update the mask phase, set w RMSE = 1, in the ith iteration, the n-th pixel position (x n , y n ) on the target surface (x n and y n are its horizontal and vertical coordinates respectively) of the CF i The gradient of the phase is defined as The mask phase

[0015] Step 4. Based on the mask phase Use the Rayleigh-Sommerfeld diffraction integral to calculate the diffraction from the input plane to the target plane to determine whether the holographic image of the output plane contains speckle; if it contains speckle, proceed to the next iteration, and finally obtain a speckle-free holographic image.

[0016] In this step 4, after obtaining the mask phase , any phase-type device among spatial light modulators (SLM), digital micromirror devices (DMD), metasurface devices (Metasurface), and diffractive optical devices (DOE) is loaded with the mask phase, and a speckle-free holographic image is diffracted and generated.

[0017] Compared with the prior art, the present application has the following advantages:

[0018] (1) An optimization algorithm capable of dynamically adjusting the weights of key parameters (root mean square error, standard deviation, and efficiency) is proposed. The phase probability control (PPM) algorithm can control the phase distribution of the mask, thereby eliminating speckle in the holographic image and improving image uniformity.

[0019] (2) The present application can use any phase-type device, such as a spatial light modulator (SLM), a digital micromirror device (DMD), a metasurface (Metasurface), and a diffractive optical device (DOE), to generate a holographic image with high uniformity, high edge sharpness, and no speckle.

[0020] (3) The speckle-free holographic image based on the phase-type device of the present application can be used to realize high-quality holographic lithography, promoting the practicality and industrialization of holographic imaging. BRIEF DESCRIPTION OF DRAWINGS

[0021] Figure 1 is the implementation flowchart of the method of the present application;

[0022] Figure 2Figures for eliminating speckle by phase probability control; wherein a is a schematic diagram of diffraction propagation of a random phase distribution mask, the mask pixel is 512*512, the period is 0.25 μm, the propagation distance z = 250 μm, and the working wavelength is 0.633 μm; b is the Gaussian phase distribution of the mask under different σ; c is the intensity probability density of the target surface under different σ.

[0023] Figure 3 Figures for calculating holographic imaging and dynamic value of weight factor; wherein a is a schematic diagram for calculating holographic imaging; b is w1(i), w2(i) and w3(i) dynamically changing with the iteration number i;

[0024] Figure 4 Figures for holographic speckle and phase probability density under different w SD and w Eff ; wherein a is the influence of different w RMSE and w SD on holographic speckle when w Eff = 1; b is the holographic image obtained by optimizing A point (w SD = 1, w Eff = 0.1) and B point (w SD = 0.1, w Eff = 1); c is the phase probability density of the mask of A point (w SD = 1, w Eff = 0.1) and B point (w SD = 0.1, w Eff = 1);

[0025] Figure 5 Figures for generating holographic image by phase probability control optimization algorithm and GS optimization algorithm; wherein a is the mask phase obtained by phase probability control optimization algorithm (key parameters w RMSE = 1, w SD = 0, w Eff = 0); b is the mask phase obtained by GS optimization algorithm; c is the phase probability density of the mask phase obtained by the two algorithms; d and e are the simulated holographic images of the two algorithms, respectively, the scale is 20 μm; f and g are scanning electron micrographs of the two silicon-based metasurfaces, respectively, the scale is 300 nm; h and i are experimental holographic images obtained by diffraction of the metasurfaces, respectively, the scale is 20 μm.

[0026] Figure 6The images show holographic lithography experiments on silicon nitride metasurfaces. In this image, a is a schematic diagram of ultraviolet lithography based on the silicon nitride metasurface at a wavelength of 405 nm, with an inset showing a SEM image of the silicon nitride metasurface; b is a measured holographic image of a QR code; c is a measured holographic image of a fork-shaped grating; d is a microscopic image of the QR code after exposure and development; and e is a microscopic image of the fork-shaped grating after exposure and development.

[0027] Figure 7 The vortex beam generated by the fork grating under laser irradiation at different wavelengths (410nm-590nm, step size 20nm). Detailed Implementation

[0028] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0029] like Figure 1 As shown, this invention provides a method for eliminating computational holographic speckle using a phase probability modulation optimization algorithm. This method can utilize any phase-type device, such as a spatial light modulator (SLM), digital micromirror device (DMD), metasurface, or diffractive optics (DOE), to generate holographic images with high uniformity, high edge sharpness, and no speckle. Here, we take a commonly used subwavelength planar device, a metasurface, as an example to describe the specific implementation process and its application in holographic lithography.

[0030] First, based on the random phase distribution of the mask, the statistical properties of holographic speckle need to be derived. Figure 2 (a) If the phase at the nth pixel in the mask Satisfies Gaussian distribution (where σ determines the phase probability distribution), then the electric field at position (x,y,z) can be expressed as E=u+iv=Ae iθ In the formula, u, v, A, and θ represent the real part, imaginary part, amplitude, and phase of the electric field, respectively. Since Since u and v follow a Gaussian distribution, they are also Gaussian random variables. According to the central limit theorem, the intensity probability density function P at position (x, y, z) can be obtained. I (x,y,z,I) (where I=A) 2 ). For P I Integrating (x,y,z,I) yields the intensity probability density of the speckle pattern within the region ∑ on the z-plane:

[0031]

[0032] In the formula, S is the area of ​​region ∑, and the mean of variable X is... Based on the Gaussian phase distribution of masks with different σ values ​​( Figure 2In step b), the intensity distribution of the target surface can be calculated using the Rayleigh-Sommerfeld diffraction integral formula, and its intensity probability density is as follows: Figure 2 As shown in c in the figure. The calculated probability density is consistent with the theoretical intensity probability density derived from equation (1). When the value of σ is large (σ = 0.6π or 0.7π), the intensity probability density of the target surface is more dispersed, indicating that the intensity fluctuation of the holographic image is large, that is, there are many speckles. To eliminate holographic speckles, the phase distribution of the mask needs to be considered while optimizing the holographic image, so that it is as concentrated as possible (small σ value).

[0033] Figure 3 Figure 'a' is a schematic diagram of holographic imaging. The gradient descent-based iterative algorithm can be divided into the following four steps:

[0034] 1. Diffraction propagation process. The diffraction from the input surface to the target surface is calculated using a rigorous Rayleigh-Sommerfeld diffraction integral. In the i-th iteration, the root mean square error is calculated using the simulated electric field intensity of the target surface. Standard deviation and optical efficiency η i =P image / P incident Key parameters, such as I. n Let N be the intensity at the nth pixel on the target surface (there are N pixels in total). I represents the average intensity of the holographic image region. m P represents the intensity at the m-th pixel in the holographic image region (out of a total of M pixels). image P represents the total energy of the region with amplitude 1 in the holographic image. incident This represents the total energy of the input surface.

[0035] 2. Update key parameters. RMSE i It can evaluate the similarity between the generated holographic image and the preset image; SD i It is crucial for eliminating holographic speckle and enhancing the uniformity of holographic images; η i The use of these parameters can improve the optical efficiency of holographic images. Therefore, the above key parameters need to be updated in each iteration.

[0036] 3. Construct the cost function (CF) i Cost function CF i =w1(i)·RMSE i +w2(i)·SD i +w3(i)·(1-η i ), where i is the iteration number, and w1(i), w2(i), and w3(i) are the RMSE values ​​respectively. i SD i and η iThe weighting factor is determined by the number of iterations, i, where the weighting factor changes dynamically. Here, we use:

[0037] w1(i)=w RMSE {tanh[(i-i0) / D0]+1} / 2,

[0038] w2(i)=w SD {tanh[-(i-i0) / D0]+1} / 2,

[0039] w3(i)=w Eff {tanh[-(i-i0) / D0]+1} / 2.

[0040] In the formula, w RMSE w SD and w Eff The range of weighting factors is specified: i0 = 175, D0 = 40. Figure 3 b). Therefore, the standard deviation SD i and efficiency η i It only dominates at the beginning of the iteration, while the root mean square error (RMSE) i In the next iteration of CF i The only key parameter.

[0041] 4. Optimize the phase using the Adam gradient descent algorithm. The Adam algorithm can provide an independent increment for each phase, allowing for dynamic adjustment of the cross-sectional area (CF). i To form the desired phase probability density. In the i-th iteration, the position (x) n ,y n CF at ) i phase The gradient is defined as To update the phase at all locations in a single iteration, N gradient calculations (where N is the total number of pixels) are required. Therefore, when N is large, the time and computational costs are relatively high.

[0042] In the actual optimization process, the root mean square error (RMSE) i The quality of a holographic image is critically important, therefore setting w is crucial. RMSE =1. Different w SD and w Eff The effect of the value of on holographic speckle is as follows Figure 4 As shown in Figure 'a', the range of values ​​for the absence of speckle is significantly larger than that for the presence of speckle, indicating that the proposed optimization algorithm has strong adaptability. Specifically, point A (w) in the figure... SD =1, w Eff =0.1) and point B (w SD =0.1, w Eff=1) The optimized holographic image is as follows Figure 4 As shown in b, the holographic image corresponding to point B contains a large number of speckles, which reduces the uniformity of the image. Figure 4 In this context, 'c' represents the phase probability density of the masks corresponding to points A and B. The phase probability density of the mask corresponding to point B is more dispersed within the range of 2π (with a larger value for σ), which aligns with conclusions drawn from the statistical properties of phase. Therefore, as long as we take... Figure 4 Any group (w) corresponding to the speckle-free range of a in the middle. SD w Eff This allows for the optimization to obtain a speckle-free holographic image.

[0043] Figure 5 a and Figure 5 In the equation, 'b' represents the optimization algorithm using the phase probability modulation (PPM) algorithm (key parameter w). RMSE =1,w SD =0, w Eff =0, the point is located Figure 4 The mask phase is obtained from the range of speckle-free values ​​of 'a' in the formula and the commonly used Gerchberg-Saxton (GS) optimization algorithm. Figure 5 In this context, 'c' represents the probability density of the phase of the two masks mentioned above. The phase distribution obtained by the GS optimization algorithm is significantly more dispersed (a superposition of uniform and Gaussian distributions), which also means that the intensity fluctuation of the optimized holographic image is larger and there are more speckles. Figure 5 (e in the original text). Conversely, the holographic image optimized by the phase probability modulation algorithm exhibits better uniformity (e in the original text). Figure 5 (d) in the middle. Based on Figure 5 a and Figure 5 The mask phase in b and the scanning electron microscopy (SEM) images of the fabricated geometric phase metasurface are shown below. Figure 5 f and Figure 5 As shown in g in the figure. Figure 5 h and Figure 5 The experimental holographic image shown in i matches the simulated holographic image, proving that the phase probability modulation algorithm can indeed eliminate holographic speckle and improve image uniformity.

[0044] To verify the feasibility of holographic lithography, a silicon nitride metasurface was designed at a wavelength of 405 nm using a phase probability modulation algorithm. Figure 6 (a) Using different mask phases, this silicon nitride metasurface can generate uniform QR codes at a 405 nm ultraviolet wavelength. Figure 6 b) and fork grating ( Figure 6c) holographic image. The generated holographic image was magnified by the objective lens group and projected onto the surface of the photoresist (Microposit S1813, Shipley) for 10 seconds. The results after development are shown in d and e of FIG. 6, respectively. The pattern after exposure and development has rich details and no obvious speckle, indicating that the intensity of the holographic image is very uniform. Figure 6 c) holographic image. The generated holographic image was magnified by the objective lens group and projected onto the surface of the photoresist (Microposit S1813, Shipley) for 10 seconds. The results after development are shown in d and e of FIG. 6, respectively. The pattern after exposure and development has rich details and no obvious speckle, indicating that the intensity of the holographic image is very uniform. Figure 6 c) holographic image. The generated holographic image was magnified by the objective lens group and projected onto the surface of the photoresist (Microposit S1813, Shipley) for 10 seconds. The results after development are shown in d and e of FIG. 6, respectively. The pattern after exposure and development has rich details and no obvious speckle, indicating that the intensity of the holographic image is very uniform.

[0045] In addition, the exposure-generated cross grating can also be used to verify the uniformity of holographic lithography. The vortex beam generated by the cross grating under illumination of laser beams at different wavelengths (410 nm-590 nm, step 20 nm) is shown in FIG. 7. Figure 7 The beam is scattered to the left and right of the zero-order spot, and the intensity zero point in the middle of the positive and negative first-order spots confirms the expected vector vortex phase. This fully demonstrates that the uniformity of holographic lithography using the method of the present application has reached a practical level.

[0046] Here, the present application uses a phase-controlled metasurface to prove that the phase probability control algorithm can indeed eliminate computational holographic speckle. Further, the high uniformity, high sharpness, and speckle-free holographic image can also be used for holographic lithography, expanding the application scenarios of computational holographic technology and promoting the practicality and industrialization of holographic imaging.

[0047] The above embodiments are provided only for the purpose of describing the present application, and are not intended to limit the scope of the present application. The scope of the present application is defined by the appended claims. Various equivalent substitutions and modifications made without departing from the spirit and principles of the present application should be encompassed within the scope of the present application.

Claims

1. A method for eliminating computational holographic speckles using a phase probability regulation optimization algorithm, characterized in that, The implementation steps are as follows: Step 1. Compute hologram image key parameters, including root mean square error RMSE of the hologram image, standard deviation SD, and optical efficiency η in the i-th iteration i i i ;​​ Step 2. Dynamically adjust the key parameters of the holographic image obtained in step 1, update the cost function CF i ; The cost function CF i = w1(i) · RMSE i + w2(i) · SD i + w3(i) · (1 - η i ), i is the iteration number, w1(i), w2(i) and w3(i) are weight factors of RMSE i , SD i and η i , respectively, wherein the weight factors dynamically change with the iteration number i, and the weight factor change adopts the following formula: w1(i) = w RMSE {tanh[(i-i0) / D0]+1} / 2, w2(i) = w SD {tanh[-(i-i0) / D0]+1} / 2, w3(i) = w Eff {tanh[-(i-i0) / D0]+1} / 2. where w RMSE , w SD , and w Eff are the ranges of variation of the weight factors, i0 is the point of mutation of the weight factors, D0 is the length of the mutation persistence, and tanh is the hyperbolic tangent function. Step 3. Update the mask phase according to the cost function CF i , Step 4. Updating the mask phase based on the updated mask phase The diffraction from the input plane to the target plane is calculated using the Rayleigh-Sommerfeld diffraction integral to determine whether the holographic image of the output plane contains speckle; if it contains speckle, the next iteration is performed, and a speckle-free holographic image is finally obtained.

2. The method for eliminating the speckle of computer hologram by using phase probability regulation optimization algorithm according to claim 1, characterized in that: In this step 1, the root mean square error RMSE i , the standard deviation SD i and the optical efficiency η i of the holographic image The calculation formula is as follows: Optical efficiency η i = P image / P incident , wherein I n is the intensity at the nth pixel on the entire target surface, and there are N pixels; I is the average intensity of the region with an amplitude of 1 in the holographic image, I m is the intensity at the mth pixel in the region of the holographic image, and there are M pixels; I ideal is the intensity of the ideal holographic image; P image is the total energy of the region with an amplitude of 1 in the holographic image, P incident is the total energy of the input surface.

3. The method of claim 1, wherein the phase probability is controlled by an optimization algorithm to eliminate the speckle in the computer-generated hologram. The mask phase is updated in step 3, which requires that w RMSE = 1, and the CF n at the nth pixel position (x n , y i ) on the target surface in the ith iteration is defined as The gradient of the phase Mask phase x n and y n are the horizontal and vertical coordinates of the nth pixel position, respectively.

4. The method of claim 1, wherein the phase probability is controlled by an optimization algorithm to eliminate the speckle in the computer-generated hologram. In this step 4, any phase type or amplitude type device in a liquid crystal spatial light modulator SLM, a digital micro-mirror device DMD, a super surface device, and a diffractive optical device is loaded with the obtained mask phase, that is, a speckle-free holographic image is diffractively generated.

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