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Gradient descent based algorithm of generating phase-only holograms of 3D images

Shujian Liu and Yasuhiro Takaki

Open Access Open Access

Abstract

Fraunhofer diffraction based computer generated holograms (CGH) adopts a Fourier transform lens that reconstructs the image on the Fourier plane. Fresnel diffraction based CGH directly reconstruct the image on the near field, however, the reconstructed image is much farther, which brings difficulty of application. In this paper, a Fresnel transform with the utilization of a Fourier transform lens and a gradient descent based algorithm is proposed to generate holograms of 3D images.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Hologram, first invented by Dennis Gabor [1,2] is a technique that records the amplitude and phase of the wavefront through interfering with a reference beam. By recording the interference pattern using a film, the wavefront is recorded and can be reconstructed through light on the film with the same reference beam by diffraction on the holographic film.

With the invention of spatial light modulator (SLM), controlling the wavefront of the beam by computer become possible. SLM can modulate amplitude, phase, or polarization of the light. SLMs are widely utilized in many fields such as laser shaping [3,4], lithographic [5,6], holographic 3D display [7,8], etc. We consider the utilization of phase only SLM and phase only Computer-Generated Holograms (CGH) for holographic 3D display in this research.

There are two main approximation methods in physics for diffraction. Fraunhofer diffraction, also known as far field diffraction, which reconstruct the image on the observation plane far enough from the hologram. With the utilization of a Fourier transform lens, the reconstructed image is located at the Fourier plane (focal plane). This enables the objective plane to locate at a near position from the hologram. Many algorithms produce the holograms in Fourier field, such as simulated annealing [9], genetic algorithms [10], machine learning algorithms [11–13], Gerchberg-Saxton (GS) algorithm [14], gradient descent algorithm [15], etc. However, Fourier transform based CGH produces only the hologram of 2D image.

Fresnel diffraction, known as near field diffraction, which reconstructs the image on the observation plane directly. CGHs of Fresnel field are able to produce holograms that can reconstruct 3D images. Various research focus on developing phase only CGH algorithms in Fresnel field [16,17]. However, due to the pixel pitch of the SLMs at present are too big comparing with the wavelength of the lasers, the near field of Fresnel transform is still too long in practice. Thus, a new transform is desired for CGHs for regenerating 3D images with the utilization of the Fourier transform lens for a nearer imaging position and a larger field of view. Such transform would realize portable designs of 3D display systems.

Gradient Descent is an optimization method that reduces the error by changing the variant opposite the gradient each iteration. Gradient Descent algorithms are widely used in deep learning in recent years [18,19]. Gradient Descent based algorithm for phase only CGH was proposed in previous research [15], which showed a higher efficiency and precision. In this research, we propose a Gradient Descent based algorithm for phase only CGH of 3D images with the utilization of a Fresnel transform lens.

2. Theory

2.1 Proposal of transform for 3D image and hologram

When wavefront generated from a three-dimensional (3D) object having a finite depth distribution is calculated, the Fresnel or Fourier transforms should be performed repeatedly in the depth direction. The brightness of 3D images can be increased by use of phase-only holograms because the amplitude of light is not modulated. For the optimization of phase-only holograms, the Fourier iterative techniques [11–13] have been used to improve the reconstructed images. Because the Fourier transform provides a two-dimensional (2D) complex-amplitude distribution on a diffraction plane located at a fixed distance from the hologram plane, the phase-only hologram optimized by the Fourier iterative techniques can generate 2D images on the diffraction plane, not 3D images with depth. Although the phase optimization techniques which support multiple diffraction planes have been developed [20], these techniques require to perform multiple Fourier transforms, resulting in the increase of the calculation time. In this study, we propose an approximated calculation technique to obtain the wavefront generated from a 3D object when a distribution of a 3D object is represented by $E(x, y, z) = E(x, y, z(x, y))$, i.e., z position of each point of the 3D object is given by the function with the variables of $x$ and $y$.

First, the conventional diffraction calculations are reviewed. For the Fraunhofer diffraction, when a Fourier transform lens with a focal length of $f$ is used and the object plane and the diffraction plane are located on the two focal planes of the Fourier transform lens as shown in Fig. 1, the light diffraction is given by the Fourier transform as below:

$$E'(x', y') = \frac{e^{i\frac{2\pi f}{\lambda}}}{i\lambda f} \iint^{+\infty}_{-\infty} E(x, y)e^{{-}i\frac{2\pi}{\lambda f}(x'x + y'y)} dxdy$$
where $\lambda$ is the wavelength of light, the object plane is denoted by $(x, y)$, and the diffraction plane is denoted by $(x', y')$. The distribution of a 2D object located on the object plane is denoted by $E(x, y)$, and the distribution on the diffraction plane is denoted by $E'(x', y')$. A point at $(x, y)$ on the object plane generates a plane wave $e^{-i\frac {2\pi }{\lambda f}\left (xx'+yy'\right )}$ on the diffraction plane. The inclination of the plane wave depends on the position of the point $(x, y)$ on the object plane and is given by $(\frac {2\pi x}{\lambda f}, \frac {2\pi y}{\lambda f})$.

 figure: Fig. 1.

Fig. 1. Hologram Generation by Fourier transform

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For the Fresnel diffraction, when the distance between the object plane and the diffraction plane is denoted by $z$ as shown in Fig. 2, the light diffraction is given by the following Fresnel transform:

$$E'(x', y') = \iint^{+\infty}_{-\infty} E(x,y,0)\frac{e^{i\frac{2\pi z}{\lambda}}}{i\lambda z}e^{i\frac{\pi}{\lambda z}\left[(x' - x)^2 + (y' - y)^2 \right]}$$

 figure: Fig. 2.

Fig. 2. Hologram Generation by Fresnel transform

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Here, we propose an approximated transform for an object having a 3D distribution. The proposed transform was intuitively derived from the Fourier and Fresnel transforms described above. This study assumes the optical setup including a Fourier transform lens illustrated in Fig. 3. The 3D object is located in the vicinity of one focal plane of the Fourier transform lens and the diffraction plane is located on the other focal plane. The 3D object is represented by $E(x,y,z(x,y))$. The virtual image of the object is represented by $E_0\left (X,Y,Z\right )$. The origin of the coordinates for the 3D object is set at the left focal point of the Fourier transform lens.

 figure: Fig. 3.

Fig. 3. Optical system of the proposed transform

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Where $u$ is the image distance and $Z$ is the distance of the image from the diffraction plane. $X, Y, Z$ is the coordinates of the image of the object.

The Fourier transform lens produces the virtual image of the 3D object far from the Fourier transform lens. Considering the Fresnel transform represented by Eq. (2), the electric field of each point on the virtual image is $E_0\left (X,Y,Z\right )$. The convert of the electric field of the object and the virtual image is explained as following.

The image distance $u$ can be calculated through the convex lens imaging formula.

$$u = \frac{f}{z}(f - z)$$
The convert relation of the coordinate $X, Y$ and $x, y$ is shown as below:
$$\begin{aligned} \frac{X}{u} &= \frac{x}{f - z},\hspace{10pt} \frac{Y}{u}=\frac{y}{f - z}. \\ X &= \frac{f}{z}x, \hspace{20pt} Y=\frac{f}{z}. \end{aligned}$$
The convert relation of the depth distance $Z$ and $z$ is shown as below:
$$Z = u + f = \frac{f^2}{z}$$
The convert relation of the electric field of the object $E\left (x,y,z\right )$ and virtual image $E_0\left (X,Y,Z\right )$ is divided by amplitude convert and phase convert. The electric field can be expressed as:
$$\begin{aligned} E(x,y,z) &= A(x,y,z)e^{i\theta(x,y,z)} \\ E_0(X,Y,Z) &= A_0(X,Y,Z)e^{i\theta_0 (X,Y,Z)} \end{aligned}$$
where $A\left (x,y,z\right )$ and $A_0\left (X,Y,Z\right )$ are the amplitude of the object and the virtual image. $\theta \left (x,y,z\right )$ and $\theta _0\left (X,Y,Z\right )$ are the phase of the object and the virtual image.

We first consider the amplitude relation of the object and the virtual image. The virtual image and the object show a same light intensity on the center of the Fourier transform lens.

$$\frac{A(x,y,z)dxdy}{f-z} = \frac{A_0(X,Y,Z)dXdY}{u}$$
The differential relation can be inferred from Eq. (4)
$$dX = \frac{f}{z}dx, dY = \frac{f}{z}dy$$
Substrate the Eqs. (3) and (8) into Eq. (7), the amplitude convert can be expressed as following:
$$A_0(X,Y,Z)dXdY = \frac{f}{z} A(x,y,z)dxdy$$
The phase convert can be calculated by the distance of the object and the virtual image.
$$\theta(x,y,z) - \theta_0(X,Y,Z) = \frac{2\pi}{\lambda}\sqrt{[u - (f - z)]^2 + (X - x)^2 + (Y - y)^2}$$
Substrate the Eqs. (3) and (4) into Eq. (10), the first-order approximation is
$$\theta(x,y,z) - \theta_0(X,Y,Z) = \frac{2\pi}{\lambda z}(f - z)^2 + \frac{\lambda}{\pi z}(x^2 + y^2)$$
The Fresnel diffraction of the virtual image can be expressed as below
$$E'(x',y') = \iint^{+\infty}_{-\infty}E_0(X,Y,Z) \frac{e^{i\frac{2\pi}{\lambda}Z}}{i\lambda Z} e^{i\frac{\pi}{\lambda Z}\left[ (X - x')^2 + (Y - y')^2 \right]} dXdY$$
Change the expression of the Eq. (12) into amplitude and phase:
$$E'(x', y') = \iint^{+\infty}_{-\infty} A_0(X,Y,Z)e^{i\theta_0(X,Y,Z)}\frac{e^{i\frac{2\pi}{\lambda}Z}}{i\lambda Z} e^{i\frac{\pi}{\lambda Z}\left[ (X - x')^2 + (Y - y')^2 \right]} dXdY$$
Substrate $X,Y,Z,A_0\left (X,Y,Z\right )dXdY,\theta _0\left (X,Y,Z\right )$ into function Eq. (13)
$$E'(x', y') = \iint^{+\infty}_{-\infty} A(x,y,z) e^{i\theta(x,y,z)}\frac{e^{i\frac{2\pi}{\lambda}(2f-z)}}{i\lambda f} e^{{-}i\frac{2\pi}{\lambda f}(xx' + yy')} e^{i\frac{\pi z}{\lambda f^2}(x'^2 + y'^2)} dxdy$$
Thus, the Fresnel transform with the utilization of Fourier lens can be expressed as below:
$$E'(x', y') = \iint^{+\infty}_{-\infty} E(x,y,z) \frac{e^{i\frac{2\pi}{\lambda}(2f-z)}}{i\lambda f} e^{{-}i\frac{2\pi}{\lambda f}(xx' + yy')} e^{i\frac{\pi z}{\lambda f^2}(x'^2 + y'^2)} dxdy$$
The transform above possesses a counter part of the Fourier transform, $e^{-i\frac {2\pi }{\lambda f}\left (xx\prime +yy\prime \right )}$, which is phase distribution of a plane wave. The main difference of the transform above and the Fourier transform is $e^{i\frac {\pi z}{\lambda f^2}\left (x^{\prime 2}+y^{\prime 2}\right )}$, which is a phase distribution of a centralized spherical wave. When the object point is nearer to the lens, the depth coordinate $z\left (x,y\right )$ is positive, where the spherical wave is divergence (Fig. 4). When the object point is farther to the lens, the depth coordinate $z\left (x,y\right )$ become negative, the phase diffraction of the spherical wave is converging (Fig. 5).

 figure: Fig. 4.

Fig. 4. Object on nearer plane

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 figure: Fig. 5.

Fig. 5. Object on farther plane

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Similar to the inverse Fresnel transform, the inverse transform of the transform above is expressed as below:

$$E(x,y,z) = \frac{1}{2\pi} \iint^{+\infty}_{-\infty} E'(x', y') i\lambda f e^{{-}i\frac{2\pi}{\lambda}(2f-z)} e^{i\frac{2\pi}{\lambda f}(xx' + yy')} e^{{-}i\frac{\pi z}{\lambda f^2}(x'^2 + y'^2)} dx'dy'$$

2.2 Computer simulations

The computer simulations were performed to verify the inversibility of the proposed transforms and the accuracy of the diffraction calculation.

The discrete form of the proposed transform and inverse transform can be proposed through a convert relation. The SLM is located at the diffraction plane. Thus, the coordinates of the diffraction side can be expressed as:

$$\begin{aligned} x' &= p_0 p\left( -\frac{P}{2} \leq p \leq \frac{P}{2} - 1 \right) \\ y' &= p_0 q\left( -\frac{Q}{2} \leq p \leq \frac{Q}{2} - 1 \right) \end{aligned}$$
where $p$ and $q$ are the row and column coordinates of the SLM, and $P$ and $Q$ are the numbers of rows and columns, and $p_0$ is the pixel pitch of the SLM.

Consider the object side, the size of the object was limited in the effective range. The effective row and column ranges are determined by the pixel pitch.

$$l_0 = \frac{f\lambda}{p_0}$$
where $l_0$ is the effective row and column length of the object.

We set the same row, column, and depth ranges. Thus, the convert relation of the object coordinate is:

$$\begin{aligned} x &= l_0 \frac m M = \frac{f\lambda}{p_0}\frac m M \hspace{10pt} \left( -\frac M 2 \leq m \leq \frac M 2 - 1 \right) \\ y &= l_0 \frac n N = \frac{f\lambda}{p_0}\frac n N \hspace{10pt} \left( -\frac N 2 \leq n \leq \frac N 2 - 1 \right) \\ z &= \frac{l_0}{2} z_{mn} = \frac{f\lambda}{2 p_0} z_{mn} \hspace{10pt} ({-}1 \leq z_{mn} \leq 1) \end{aligned}$$
where $m$ and $n$ are row and column coordinates of the object, and $M$ and $N$ are the numbers of the rows and columns, and $z_{mn}$ is the relative depth position of the object.

Substrate the coordinates into the proposed transform. The discrete form of the proposed transform can be expressed as below.

$$A'_{pq} e^{i\theta'_{pq}} = \sum_{m ={-}\frac{M}{2}}^{\frac{M}{2} - 1} \sum_{n ={-}\frac{N}{2}}^{\frac{N}{2} - 1} A_{mn} e^{i\theta_{mn}} e^{i\frac{2\pi}{\lambda}\left( 2f - \frac{f\lambda}{2p_0}z_{mn} \right)} e^{{-}i2\pi \left( \frac{mp}{M} + \frac{nq}{N} \right)} e^{i\frac{\pi p_0}{2f}z_{mn}(p^2 + q^2)}$$
Similar to the inverse Fresnel transform, the inverse transform is shown as below:
$$A_{mn} e^{i\theta_{mn}} = \frac{1}{PQ} \sum_{p ={-}\frac{P}{2}}^{\frac{P}{2} - 1} \sum_{q ={-}\frac{Q}{2}}^{\frac{Q}{2} - 1} A'_{pq} e^{i\theta'_{pq}} e^{{-}i\frac{2\pi}{\lambda}\left( 2f - \frac{f\lambda}{2p_0}z_{mn} \right)} e^{i2\pi \left( \frac{mp}{M} + \frac{nq}{N} \right)} e^{{-}i\frac{\pi p_0}{2f}z_{mn}(p^2 + q^2)}$$
Equation (21) can also be expressed as:
$$A_{mn} e^{i\theta_{mn}}e^{i\frac{2\pi}{\lambda}\left( 2f - \frac{f\lambda}{2p_0}z_{mn} \right)} = \frac{1}{PQ} \sum_{p ={-}\frac{P}{2}}^{\frac{P}{2} - 1} \sum_{q ={-}\frac{Q}{2}}^{\frac{Q}{2} - 1} A'_{pq} e^{i\theta'_{pq}} e^{i2\pi \left( \frac{mp}{M} + \frac{nq}{N} \right)} e^{{-}i\frac{\pi p_0}{2f}z_{mn}(p^2 + q^2)}$$
There is a common item in Eqs. (20) and (27), $e^{i\theta _{mn}}e^{i\frac {2\pi }{\lambda }\left (2f-\frac {f\lambda }{2p_0}z_{mn}\right )}$, which is the phase of the object and a phase compensation determined by the depth coordination. We can substitute this item with $e^{i\varphi _{mn}}$:
$$\begin{aligned} \varphi_{mn} &= \theta_{mn} + \frac{2\pi}{\lambda}\left( 2f - \frac{f\lambda}{2p_0}z_{mn} \right) \\ e^{i\varphi_{mn}} &= e^{i\theta_{mn} + \frac{2\pi}{\lambda}\left( 2f - \frac{f\lambda}{2p_0}z_{mn} \right)} \end{aligned}$$
The discrete transform and inverse transform can be thus expressed as following:
$$A'_{pq} e^{i\theta'_{pq}} = \mathcal{T}(A_{mn}e^{i\theta{mn}}, z_{mn}) = \sum_{m ={-}\frac{M}{2}}^{\frac{M}{2} - 1} \sum_{n ={-}\frac{N}{2}}^{\frac{N}{2} - 1} A_{mn} e^{i\varphi_{mn}} e^{{-}i2\pi \left( \frac{mp}{M} + \frac{nq}{N} \right)} e^{i\frac{\pi p_0}{2f}z_{mn}(p^2 + q^2)}$$
$$A_{mn} e^{i\varphi_{mn}} = \mathcal{T}^{{-}1}(A'_{pq}e^{i\theta'_{pq}}, z_{mn}) = \frac{1}{PQ} \sum_{p ={-}\frac{P}{2}}^{\frac{P}{2} - 1} \sum_{q ={-}\frac{Q}{2}}^{\frac{Q}{2} - 1} A'_{pq} e^{i\theta'_{pq}} e^{i2\pi \left( \frac{mp}{M} + \frac{nq}{N} \right)} e^{{-}i\frac{\pi p_0}{2f}z_{mn}(p^2 + q^2)}$$
The proposed transform and inverse transform cannot be applied as fast Fourier transform (FFT) because the proposed transform is equivalent as Fresnel transform for 3D object, which cannot be applied the fast transform.

The conventional algorithms based on Fourier or Fresnel transforms are point-based methods that each point stand for a pixel. However, most of the phase only CGHs are based on Fourier or Fresnel transforms. The main advantage over the layered Fourier transform is the optimization algorithm. The layered Fourier transform based phase only CGH algorithm so far does not able to optimize the whole image on different layers simultaneously [16], optimization of a certain layer may bring extra error to other layers. Thus, more layers bring bigger error. The proposed transform in this paper enables the optimization algorithm optimizes the image on a certain 3D surface, this enables the advantage over the layered Fourier transform that image parts on all the layers could be optimized simultaneously.

The adopted SLM in experiments is PLUTO-2.1 LCOS Spatial Light Modulator Phase Only (Reflective) (HOLOEYE Photonics AG). The pixel pitch is 8 $\mu$m. The resolution is 1,080 $\times$ 1,920. The focal distance of the Fourier transform lens is 150 mm. The adopted laser is a He-Ne laser with the wavelength of 632.8nm.

We did the simulation to test the proposed transform. We adopted two images with different depth option (Figs. 6 and 7), which the resolution was 300$\times$300. In the "$\mathcal {LR}$" image (Fig. 6), the letter "$\mathcal {L}$" is located on the preset minimum range to the Fourier transform lens; the letter "$\mathcal {R}$" is in the preset maximum range to the Fourier transform lens. In the "grid" image (Fig. 7), the depth option is an inclined screen.

 figure: Fig. 6.

Fig. 6. "$\mathcal {LR}$" image, depth option and image of depth option

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 figure: Fig. 7.

Fig. 7. "grid" image, depth option and image of depth option

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To test the validity of the proposed algorithm, such simulation was executed that do the inverse transform of the transform of the image. The desired result is that the inverse transform reconstructs the target image on the depth position from the transform of the target image and all the pixels are focused. Figure 8 shows the transform of the "$\mathcal {LR}$" image with a random phase which the resolution was changed to 1,920$\times$1,080 to fit the SLM adopted in the experiment. Figure 9 shows the reconstructed image by making the inverse transform of the Fig. 8. Figure 10 shows the transform of the "grid" image with a random phase, and Fig. 11 shows the reconstructed image.

 figure: Fig. 8.

Fig. 8. Transform of the "$\mathcal {LR}$" image. Left: Intensity; Right: phase (1920$\times$1080)

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 figure: Fig. 9.

Fig. 9. Reconstructed "$\mathcal {LR}$" image (300$\times$300)

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 figure: Fig. 10.

Fig. 10. Transform of the "grid" image. Left: Intensity; Right: phase (1920$\times$1080)

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 figure: Fig. 11.

Fig. 11. Reconstructed "grid" image (300$\times$300)

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The validity of the transform was tested through this simulation. The reconstructed image succeeded in reconstructing the target image with a theoretical noise. Because the transform cannot be applied as fast transform like fast Fourier transform (FFT), the proposed transform is time consuming, same with the raw Fresnel transform. The time of transform takes about 70 seconds and the inverse transform takes about 68 seconds.

3. Phase optimization using proposed transforms

The proposed transform can generate the diffraction of 3D objects or reconstruct the 3D object from the diffraction, however, to generate phase only holograms, we need phase retrieval optimization algorithms.

3.1 Modified Gerchberg-Saxton algorithm

A modified GS algorithm is designed as the Fig. 12 shows. This algorithm simply replaces the Fourier transform with the proposed transform, same with the method in [16].

 figure: Fig. 12.

Fig. 12. Modified GS Algorithm

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The beginning hologram of the iteration is the proposed transform of the target image with a random phase which is equivalent to the random phase adopted in the former simulation. The hologram refreshes each iteration. The holograms of 1, 2, 5, 10, 50, 100 iterations are shown as Fig. 13 ("$\mathcal {LR}$" image).

 figure: Fig. 13.

Fig. 13. Holograms of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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The reconstructed images are shown as Fig. 14.

 figure: Fig. 14.

Fig. 14. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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The grid image was also set as the target image in simulation. The hologram and the reconstructed image is shown as Fig. 15 and Fig. 16.

 figure: Fig. 15.

Fig. 15. Holograms of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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 figure: Fig. 16.

Fig. 16. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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The error estimation function is set as following:

$$E = \frac{1}{MN} \sum_{m ={-}\frac{M}{2}}^{\frac{M}{2} - 1} \sum_{n ={-}\frac{N}{2}}^{\frac{N}{2} - 1} \left( \frac{I_{mn}}{\overline{I}} - \frac{I_{0-mn}}{\overline{I_0}} \right)^2$$
where $M$ and $N$ are the pixel range of the rows and columns of the target image, $I_{mn}$ is the regenerated intensity of the pixel $m$ and $n$, $\overline {I}$ is the average intensity of the regenerated image, $I_{0-mn}$ is the intensity of the target image, $\overline {I_0}$ is the average intensity of the target image.

The MSE-iteration graphs are shown as Fig. 17.

 figure: Fig. 17.

Fig. 17. MSE-iteration graphs

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The graphs show that the modified GS algorithm does not reach the convergence after 100 iterations. The optimization effectiveness reduces through iterations as the "$\mathcal {LR}$" image was set as target. In the state of grid image, the MSE even increases in part of the iterations.

3.2 Gradient descent based algorithm

To calculate the hologram of a 3D image, the method of gradient descent can be utilized. Similar to the former research, there are two essential elements in gradient descent method, which are error estimation function and the gradient of the error and the retrieval variant.

In this research, we set the error estimation function as the Mean Square Error of the intensity of the regenerated image and the target image. The retrieval variant is the phase distribution of the hologram.

The gradient of the error and the retrieval variant can be calculated through chain rule (27).

$$\frac{\partial E}{\partial \theta'_{pq}} = \sum_{m ={-}\frac{M}{2}}^{\frac{M}{2} - 1} \sum_{n ={-}\frac{N}{2}}^{\frac{N}{2} - 1} \frac{\partial E}{\partial I_{mn}} \frac{\partial I_{mn}}{\partial \theta'_{pq}}$$
where $\theta '_{pq}$ is the phase of the pixel $p$ and $q$ on the hologram.

The Eq. (27) can be calculated separately. calculation of $\frac {\partial E}{\partial I_{mn}}$:

$$\frac{\partial E}{\partial I_{mn}}=\frac{2}{\bar{I}MN}\left(\frac{I_{mn}}{\bar{I}}-\frac{I_{0-mn}}{\bar{I_0}}\right)$$
calculation of $\frac {\partial I_{mn}}{\partial \theta '_{pq}}$:

The intensity of the regenerated image can be calculated through the phase-only hologram:

$$\begin{aligned} I_{mn} &= \frac{1}{PQ}\left\{\sum_{p'={-}\frac{P}{2}}^{\frac{P}{2}-1}\sum_{q'={-}\frac{Q}{2}}^{\frac{Q}{2}-1}\sin{\left[\theta_{p' q'}'+2\pi\left(\frac{mp'}{M}+\frac{nq'}{N}\right)-\frac{\pi p_0}{2f}z_{mn}\left(p^{\prime2}+q^{\prime2}\right)\right]}\right\}^2 \\ &+ \frac{1}{PQ}\left\{\sum_{p'={-}\frac{P}{2}}^{\frac{P}{2}-1}\sum_{q'={-}\frac{Q}{2}}^{\frac{Q}{2}-1}\cos{\left[\theta_{p' q'}'+2\pi\left(\frac{mp'}{M}+\frac{nq'}{N}\right)-\frac{\pi p_0}{2f}z_{mn}\left(p^{\prime2}+q^{\prime2}\right)\right]}\right\}^2 \end{aligned}$$
The gradient of the intensity and the hologram is:
$$\frac{\partial I_{mn}}{\partial\theta_{pq}'}=\frac{2}{PQ}A_{mn}\sin{\left[\varphi_{mn}-\theta_{pq}'-2\pi\left(\frac{mp}{M}+\frac{nq}{N}\right)+\frac{\pi p_0}{2f}z_{mn}\left(p^2+q^2\right)\right]}$$
Now the gradient can be calculated:
$$\frac{\partial E}{\partial\theta_{pq}'}=\frac{4}{\overline{I}MNPQ}\mathrm{Im}\left\{e^{{-}i\theta_{pq}'}\mathcal{T}\left[\left(\frac{I_{mn}}{\overline{I}}-\frac{I_{0-mn}}{\overline{I_0}}\right)A_{mn}e^{i\varphi_{mn}},z_{mn}\right]\right\}$$
According to the principle of the gradient descent method, the phase change of each iteration can be expressed as (32).
$$\Delta\theta_{t-pq}'={-}\gamma\frac{\partial E}{\partial\theta_{pq}'}$$
where $\gamma$ is the optimization rate of each iteration.

The phase of the next iteration is shown as following.

$$\theta_{t+1-pq}'=\theta_{t-pq}'+\Delta\theta_{t-pq}'$$
The optimization rate can be also optimized through some method. Similar to the research before, the optimization rate can be calculated through the estimated error of the next iteration and the gradient of this iteration.
$$E'=\frac{1}{MN}\sum_{m={-}\frac{M}{2}}^{\frac{M}{2}-1}\sum_{n={-}\frac{N}{2}}^{\frac{N}{2}-1}\left(\frac{I_{mn}+\Delta I_{mn}}{\overline{I}}-\frac{I_{0-mn}}{\overline{I_0}}\right)^2$$
where the $\mathrm {\Delta }I_{mn}$ is the estimated change of the intensity of the present iteration, which can be calculated through chain rule (35).
$$\Delta I_{mn}\approx\sum_{p={-}\frac{P}{2}}^{\frac{P}{2}-1}{\sum_{q={-}\frac{Q}{2}}^{\frac{Q}{2}-1}\frac{\partial I_{mn}}{\partial\theta_{pq}'}\Delta\theta_{pq}'}$$
Substitute (30) into (35).
$$\Delta I_{mn}=2A_{mn}\mathrm{Im}e^{i\varphi_{mn}}\left[ \mathcal{T}\left( \Delta \theta'_{pq}e^{i\theta'_{pq}}, z_{mn} \right) \right]^*$$
In order to optimize the optimization rate, $\gamma$ should minimize the estimated error of the next iteration.
$$\frac{\partial E'}{\partial\gamma_0}=0$$
The calculation of the optimization rate is shown as (38).
$$\gamma_0={-}\frac{\sum_{m={-}\frac{M}{2}}^{\frac{M}{2}-1}\sum_{n={-}\frac{N}{2}}^{\frac{N}{2}-1}I_{mn}\cdot2A_{mn}\mathrm{Im} e^{i\varphi_{mn}}\left\{\mathcal{T}^{{-}1}\left[-\frac{\partial E}{\partial\theta_{pq}'}e^{i\theta_{pq}'},z_{mn}\right]\right\}^\ast}{\sum_{m={-}\frac{M}{2}}^{\frac{M}{2}-1}\sum_{n={-}\frac{N}{2}}^{\frac{N}{2}-1}\left\{2A_{mn}\mathrm{Im} e^{i\varphi_{mn}}\left\{\mathcal{T}^{{-}1}\left[-\frac{\partial E}{\partial\theta_{pq}'}e^{i\theta_{pq}'},z_{mn}\right]\right\}^\ast\right\}^2}$$
The optimized rate calculated by the consider only the first order differential as the approximation, which the higher order differential act on the practical is ignored. To reduce the error caused by the higher order differential, we can reduce the optimization by a constant rate $\gamma _r$ to reduce the error caused by higher order differential.
$$\gamma=\gamma_0\gamma_r$$
In the simulations below, contrast rate $\gamma _r$ is set as 0.5.

Based on the calculation of the previous, the proposed algorithm based on Gradient Descent method is designed as the Fig. 18.

 figure: Fig. 18.

Fig. 18. Flow chart of the proposed algorithm

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The effectiveness of the Gradient Descent based phase only holograms generation algorithm was tested through simulation.

We do simulations same with the modified GS algorithm. The holograms of 1, 2, 5, 10, 50, 100 iterations are shown as Fig. 19 ("$\mathcal {LR}$" image).

 figure: Fig. 19.

Fig. 19. Holograms of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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The reconstructed images are shown as Fig. 20.

 figure: Fig. 20.

Fig. 20. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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We reconstruct from the hologram of 100 iterations with the depth option of minimum range and maximum range to show the reconstructed images of different planes as Fig. 21.

 figure: Fig. 21.

Fig. 21. Reconstructed in minimum and maximum range. Left: minimum; Right: maximum

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The grid image was also set as the target image in simulation. The holograms and the reconstructed images are shown as Fig. 22 and Fig. 23.

 figure: Fig. 22.

Fig. 22. Holograms of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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 figure: Fig. 23.

Fig. 23. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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The MSE-iteration graph is shown as Fig. 24.

 figure: Fig. 24.

Fig. 24. MSE-iteration graphs

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We know that the MSE convergences after about 15 iterations. The MSE reaches the local minimum and no longer changes after convergence.

The method in [21] successfully in generating the diffraction on tilted planes, however, when the tilt angle up to 45 degrees, the reconstructed image would appear a significant deformation. The topic of reconstructing the "grid" image is a special simple example. The proposed transform is not designed only for planes or tilted planes.

3.3 Comparison of the gradient descent based algorithm and modified GS algorithm

We can know the difference of the effectiveness of the algorithms through the MSE-iteration graphs shown in Fig. 25.

 figure: Fig. 25.

Fig. 25. MSE-time graphs of two algorithms

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In the simulation of "$\mathcal {LR}$" image, the Gradient Descent based algorithm reaches the convergence in 10 iterations and no longer changes in the rest iterations. The modified GS algorithm does not reach the convergence in 100 iterations. The convergence MSE is smaller than the 100th MSE of the modified GS algorithm.

In the simulation of grid image, the Gradient Descent based algorithm also reaches the convergence in 15 iterations. The modified GS algorithm does not show the feature of monotonical decreasing as the naive GS algorithm shows.

The Gradient Descent based algorithm shows a better stability and a higher effectiveness than the modified GS algorithm.

3.4 Special simulation with an irregular depth option

We did a special simulation using "$\mathcal {LR}$" image with an irregular depth option. To test the robustness of the proposed transform and gradient descent based optimization algorithm, we adopt a marble texture image as the depth option, shown as Fig. 26(a).

 figure: Fig. 26.

Fig. 26. A simulation with an irregular depth option

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The hologram generated by the Gradient Descent based optimization algorithm is shown as Fig. 26(b), and reconstructed image is shown as Fig. 26(c). The result shows that the proposed algorithm can generate proper holograms under irregular depth options.

4. Experimental results

We did experiments to verify the correctness of the proposed algorithm. The optical system was set as the Fig. 27 shows.

 figure: Fig. 27.

Fig. 27. Optical system

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In the experiments of the "$\mathcal {LR}$" image, the screen was set in different depth position from the Fourier transform lens. As the Fig. 6 shows, the letter "$\mathcal {L}$" was set on the plane in the preset minimum range, and the letter "$\mathcal {R}$" was set on the plane in the preset maximum range. To take the clear letter "$\mathcal {L}$", we first set the screen at the preset minimum range. The result of the reconstructed images by the holograms produced by Gradient Descent based algorithm and modified GS algorithm are shown as Fig. 28.

 figure: Fig. 28.

Fig. 28. Reconstructed letter "$\mathcal {L}$". Left: Gradient Descent; Right: modified GS

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As we take the clear letter "$\mathcal {L}$", the letter "$\mathcal {R}$" blurred because the letter "$\mathcal {R}$" is in another plane. To take the clear letter "$\mathcal {R}$", the screen was set at the preset maximum range. The result of the letter "$\mathcal {R}$" is shown as Fig. 29.

 figure: Fig. 29.

Fig. 29. Reconstructed letter "$\mathcal {R}$". Left: Gradient Descent; Right: modified GS

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The results of "$\mathcal {LR}$" image shows that both Gradient Descent based algorithm and modified GS algorithm can produce holograms that able to reconstruct images clear enough.

In the experiments of the grid image, we set a tilted screen as the Fig. 7 shows. The results of "grid" image produced by two algorithms are shown as Fig. 30.

 figure: Fig. 30.

Fig. 30. Reconstructed "grid" images. Left: Gradient Descent; Right: modified GS

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Both algorithms show a correctness on reconstructing the grid image in an inclined screen. Gradient Descent based algorithm shows a higher precision on the white zone of the image, and the modified GS algorithm shows a higher precision on the black zone of the image. This is because the GS algorithm is equivalent as the Gradient Descent algorithm that minimize the error evaluated through the amplitude of the reconstructed image, however, the error evaluation of the proposed Gradient Descent based algorithm is the intensity of the reconstructed image. This difference resulting in the proposed algorithm shows a higher accuracy in white zone, and the modified GS algorithm show a higher accuracy in black zone.

5. Conclusion

We proposed a Fresnel transform with the utilization of a Fourier transform lens. The transform succeeded in generating holograms of 3D images. Moreover, based on the transform, we proposed a Gradient Descent based algorithm for phase-only computer-generated holograms that shows a high effectiveness than modified GS algorithm.

Funding

Japan Society for the Promotion of Science (19H02189).

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.

References

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4. S. Li, Y. Wang, Z. Lu, L. Ding, C. Cui, Y. Chen, D. Pengyuan, D. Ba, Z. Zheng, and H. Yuan, “Spatial beam shaping for high-power frequency tripling lasers based on a liquid crystal spatial light modulator,” Opt. Commun. 367, 181–185 (2016). [CrossRef]  

5. J. Lutkenhaus, D. George, M. Moazzezi, U. Philipose, and Y. Lin, “Digitally tunable holographic lithography using a spatial light modulator as a programmable phase mask,” Opt. Express 21(22), 26227–26235 (2013). [CrossRef]  

6. S. Behera, M. Kumar, and J. Joseph, “Submicrometer photonic structure fabrication by phase spatial-light-modulator-based interference lithography,” Opt. Lett. 41(8), 1893–1896 (2016). [CrossRef]  

7. J. Geng, “Three-dimensional display technologies,” Adv. Opt. Photonics 5(4), 456–535 (2013). [CrossRef]  

8. H. Yu, K. Lee, J. Park, and Y. Park, “Ultrahigh-definition dynamic 3d holographic display by active control of volume speckle fields,” Nat. Photonics 11(3), 186–192 (2017). [CrossRef]  

9. A. G. Kirk and T. J. Hall, “Design of binary computer generated holograms by simulated annealing: coding density and reconstruction error,” Opt. Commun. 94(6), 491–496 (1992). [CrossRef]  

10. M. Wen, J. Yao, W. K. Wong, and G. C. Chen, “Holographic diffuser design using a modified genetic algorithm,” Opt. Eng. 44(8), 085801 (2005). [CrossRef]  

11. P. Cheremkhin, N. Evtikhiev, V. Krasnov, V. Rodin, D. Rymov, and R. Starikov, “Machine learning methods for digital holography and diffractive optics,” Procedia Comput. Sci. 169, 440–444 (2020). [CrossRef]  

12. R. Horisaki, R. Takagi, and J. Tanida, “Deep-learning-generated holography,” Appl. Opt. 57(14), 3859–3863 (2018). [CrossRef]  

13. M. H. Eybposh, N. W. Caira, M. Atisa, P. Chakravarthula, and N. C. Pégard, “Deepcgh: 3d computer-generated holography using deep learning,” Opt. Express 28(18), 26636–26650 (2020). [CrossRef]  

14. R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

15. S. Liu and Y. Takaki, “Optimization of phase-only computer-generated holograms based on the gradient descent method,” Appl. Sci. 10(12), 4283 (2020). [CrossRef]  

16. H.-E. Hwang, H. T. Chang, and W.-N. Lie, “Fast double-phase retrieval in fresnel domain using modified gerchberg-saxton algorithm for lensless optical security systems,” Opt. Express 17(16), 13700–13710 (2009). [CrossRef]  

17. D. Leseberg, “Sizable fresnel-type hologram generated by computer,” J. Opt. Soc. Am. A 6(2), 229–233 (1989). [CrossRef]  

18. S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv preprint arXiv:1609.04747 (2016).

19. H. Pan, X. Niu, R. Li, Y. Dou, and H. Jiang, “Annealed gradient descent for deep learning,” Neurocomputing 380, 201–211 (2020). [CrossRef]  

20. H. Zhang, Y. Zhao, L. Cao, and G. Jin, “Layered holographic stereogram based on inverse fresnel diffraction,” Appl. Opt. 55(3), A154–A159 (2016). [CrossRef]  

21. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20(9), 1755–1762 (2003). [CrossRef]  

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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Figures (30)
Fig. 1.
Fig. 1. Hologram Generation by Fourier transform

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Fig. 2.
Fig. 2. Hologram Generation by Fresnel transform

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Fig. 3.
Fig. 3. Optical system of the proposed transform

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Fig. 4.
Fig. 4. Object on nearer plane

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Fig. 5.
Fig. 5. Object on farther plane

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Fig. 6.
Fig. 6. "$\mathcal {LR}$" image, depth option and image of depth option

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Fig. 7.
Fig. 7. "grid" image, depth option and image of depth option

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Fig. 8.
Fig. 8. Transform of the "$\mathcal {LR}$" image. Left: Intensity; Right: phase (1920$\times$1080)

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Fig. 9.
Fig. 9. Reconstructed "$\mathcal {LR}$" image (300$\times$300)

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Fig. 10.
Fig. 10. Transform of the "grid" image. Left: Intensity; Right: phase (1920$\times$1080)

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Fig. 11.
Fig. 11. Reconstructed "grid" image (300$\times$300)

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Fig. 12.
Fig. 12. Modified GS Algorithm

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Fig. 13.
Fig. 13. Holograms of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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Fig. 14.
Fig. 14. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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Fig. 15.
Fig. 15. Holograms of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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Fig. 16.
Fig. 16. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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Fig. 17.
Fig. 17. MSE-iteration graphs

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Fig. 18.
Fig. 18. Flow chart of the proposed algorithm

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Fig. 19.
Fig. 19. Holograms of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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Fig. 20.
Fig. 20. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("$\mathcal {LR}$" image)

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Fig. 21.
Fig. 21. Reconstructed in minimum and maximum range. Left: minimum; Right: maximum

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Fig. 22.
Fig. 22. Holograms of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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Fig. 23.
Fig. 23. Reconstructed images of 1, 2, 5, 10, 50, 100 iterations ("grid" image)

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Fig. 24.
Fig. 24. MSE-iteration graphs

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Fig. 25.
Fig. 25. MSE-time graphs of two algorithms

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Fig. 26.
Fig. 26. A simulation with an irregular depth option

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Fig. 27.
Fig. 27. Optical system

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Fig. 28.
Fig. 28. Reconstructed letter "$\mathcal {L}$". Left: Gradient Descent; Right: modified GS

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Fig. 29.
Fig. 29. Reconstructed letter "$\mathcal {R}$". Left: Gradient Descent; Right: modified GS

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Fig. 30.
Fig. 30. Reconstructed "grid" images. Left: Gradient Descent; Right: modified GS

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Equations (39)

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E ′ ( x ′ , y ′ ) = e i 2 π f λ i λ f ∬ − ∞ + ∞ E ( x , y ) e − i 2 π λ f ( x ′ x + y ′ y ) d x d y
E ′ ( x ′ , y ′ ) = ∬ − ∞ + ∞ E ( x , y , 0 ) e i 2 π z λ i λ z e i π λ z [ ( x ′ − x ) 2 + ( y ′ − y ) 2 ]
u = f z ( f − z )
X u = x f − z , Y u = y f − z . X = f z x , Y = f z .
Z = u + f = f 2 z
E ( x , y , z ) = A ( x , y , z ) e i θ ( x , y , z ) E 0 ( X , Y , Z ) = A 0 ( X , Y , Z ) e i θ 0 ( X , Y , Z )
A ( x , y , z ) d x d y f − z = A 0 ( X , Y , Z ) d X d Y u
d X = f z d x , d Y = f z d y
A 0 ( X , Y , Z ) d X d Y = f z A ( x , y , z ) d x d y
θ ( x , y , z ) − θ 0 ( X , Y , Z ) = 2 π λ [ u − ( f − z ) ] 2 + ( X − x ) 2 + ( Y − y ) 2
θ ( x , y , z ) − θ 0 ( X , Y , Z ) = 2 π λ z ( f − z ) 2 + λ π z ( x 2 + y 2 )
E ′ ( x ′ , y ′ ) = ∬ − ∞ + ∞ E 0 ( X , Y , Z ) e i 2 π λ Z i λ Z e i π λ Z [ ( X − x ′ ) 2 + ( Y − y ′ ) 2 ] d X d Y
E ′ ( x ′ , y ′ ) = ∬ − ∞ + ∞ A 0 ( X , Y , Z ) e i θ 0 ( X , Y , Z ) e i 2 π λ Z i λ Z e i π λ Z [ ( X − x ′ ) 2 + ( Y − y ′ ) 2 ] d X d Y
E ′ ( x ′ , y ′ ) = ∬ − ∞ + ∞ A ( x , y , z ) e i θ ( x , y , z ) e i 2 π λ ( 2 f − z ) i λ f e − i 2 π λ f ( x x ′ + y y ′ ) e i π z λ f 2 ( x ′ 2 + y ′ 2 ) d x d y
E ′ ( x ′ , y ′ ) = ∬ − ∞ + ∞ E ( x , y , z ) e i 2 π λ ( 2 f − z ) i λ f e − i 2 π λ f ( x x ′ + y y ′ ) e i π z λ f 2 ( x ′ 2 + y ′ 2 ) d x d y
E ( x , y , z ) = 1 2 π ∬ − ∞ + ∞ E ′ ( x ′ , y ′ ) i λ f e − i 2 π λ ( 2 f − z ) e i 2 π λ f ( x x ′ + y y ′ ) e − i π z λ f 2 ( x ′ 2 + y ′ 2 ) d x ′ d y ′
x ′ = p 0 p ( − P 2 ≤ p ≤ P 2 − 1 ) y ′ = p 0 q ( − Q 2 ≤ p ≤ Q 2 − 1 )
l 0 = f λ p 0
x = l 0 m M = f λ p 0 m M ( − M 2 ≤ m ≤ M 2 − 1 ) y = l 0 n N = f λ p 0 n N ( − N 2 ≤ n ≤ N 2 − 1 ) z = l 0 2 z m n = f λ 2 p 0 z m n ( − 1 ≤ z m n ≤ 1 )
A p q ′ e i θ p q ′ = ∑ m = − M 2 M 2 − 1 ∑ n = − N 2 N 2 − 1 A m n e i θ m n e i 2 π λ ( 2 f − f λ 2 p 0 z m n ) e − i 2 π ( m p M + n q N ) e i π p 0 2 f z m n ( p 2 + q 2 )
A m n e i θ m n = 1 P Q ∑ p = − P 2 P 2 − 1 ∑ q = − Q 2 Q 2 − 1 A p q ′ e i θ p q ′ e − i 2 π λ ( 2 f − f λ 2 p 0 z m n ) e i 2 π ( m p M + n q N ) e − i π p 0 2 f z m n ( p 2 + q 2 )
A m n e i θ m n e i 2 π λ ( 2 f − f λ 2 p 0 z m n ) = 1 P Q ∑ p = − P 2 P 2 − 1 ∑ q = − Q 2 Q 2 − 1 A p q ′ e i θ p q ′ e i 2 π ( m p M + n q N ) e − i π p 0 2 f z m n ( p 2 + q 2 )
φ m n = θ m n + 2 π λ ( 2 f − f λ 2 p 0 z m n ) e i φ m n = e i θ m n + 2 π λ ( 2 f − f λ 2 p 0 z m n )
A p q ′ e i θ p q ′ = T ( A m n e i θ m n , z m n ) = ∑ m = − M 2 M 2 − 1 ∑ n = − N 2 N 2 − 1 A m n e i φ m n e − i 2 π ( m p M + n q N ) e i π p 0 2 f z m n ( p 2 + q 2 )
A m n e i φ m n = T − 1 ( A p q ′ e i θ p q ′ , z m n ) = 1 P Q ∑ p = − P 2 P 2 − 1 ∑ q = − Q 2 Q 2 − 1 A p q ′ e i θ p q ′ e i 2 π ( m p M + n q N ) e − i π p 0 2 f z m n ( p 2 + q 2 )
E = 1 M N ∑ m = − M 2 M 2 − 1 ∑ n = − N 2 N 2 − 1 ( I m n I ¯ − I 0 − m n I 0 ¯ ) 2
∂ E ∂ θ p q ′ = ∑ m = − M 2 M 2 − 1 ∑ n = − N 2 N 2 − 1 ∂ E ∂ I m n ∂ I m n ∂ θ p q ′
∂ E ∂ I m n = 2 I ¯ M N ( I m n I ¯ − I 0 − m n I 0 ¯ )
I m n = 1 P Q { ∑ p ′ = − P 2 P 2 − 1 ∑ q ′ = − Q 2 Q 2 − 1 sin ⁡ [ θ p ′ q ′ ′ + 2 π ( m p ′ M + n q ′ N ) − π p 0 2 f z m n ( p ′ 2 + q ′ 2 ) ] } 2 + 1 P Q { ∑ p ′ = − P 2 P 2 − 1 ∑ q ′ = − Q 2 Q 2 − 1 cos ⁡ [ θ p ′ q ′ ′ + 2 π ( m p ′ M + n q ′ N ) − π p 0 2 f z m n ( p ′ 2 + q ′ 2 ) ] } 2
∂ I m n ∂ θ p q ′ = 2 P Q A m n sin ⁡ [ φ m n − θ p q ′ − 2 π ( m p M + n q N ) + π p 0 2 f z m n ( p 2 + q 2 ) ]
∂ E ∂ θ p q ′ = 4 I ¯ M N P Q I m { e − i θ p q ′ T [ ( I m n I ¯ − I 0 − m n I 0 ¯ ) A m n e i φ m n , z m n ] }
Δ θ t − p q ′ = − γ ∂ E ∂ θ p q ′
θ t + 1 − p q ′ = θ t − p q ′ + Δ θ t − p q ′
E ′ = 1 M N ∑ m = − M 2 M 2 − 1 ∑ n = − N 2 N 2 − 1 ( I m n + Δ I m n I ¯ − I 0 − m n I 0 ¯ ) 2
Δ I m n ≈ ∑ p = − P 2 P 2 − 1 ∑ q = − Q 2 Q 2 − 1 ∂ I m n ∂ θ p q ′ Δ θ p q ′
Δ I m n = 2 A m n I m e i φ m n [ T ( Δ θ p q ′ e i θ p q ′ , z m n ) ] ∗
∂ E ′ ∂ γ 0 = 0
γ 0 = − ∑ m = − M 2 M 2 − 1 ∑ n = − N 2 N 2 − 1 I m n ⋅ 2 A m n I m e i φ m n { T − 1 [ − ∂ E ∂ θ p q ′ e i θ p q ′ , z m n ] } ∗ ∑ m = − M 2 M 2 − 1 ∑ n = − N 2 N 2 − 1 { 2 A m n I m e i φ m n { T − 1 [ − ∂ E ∂ θ p q ′ e i θ p q ′ , z m n ] } ∗ } 2
γ = γ 0 γ r
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