Computational Theory and Experimental Research on Coherent Optical Imaging

Abstract

Since the advent of lasers, coherent optical imaging has found significant applications in optical precision measurement. Among these, digital holography is a key research area. In detection studies, accurately obtaining the amplitude and phase of the object being measured through digital holographic image reconstruction is a critical research task. However, current coherent optical imaging formulas can only calculate the amplitude distribution of the image light field when the object size is smaller than one-quarter of the diameter of the optical system’s entrance pupil, making it difficult to meet the needs of applied research. This paper derives formulas that are not restricted by the size of the incident pupil and can calculate the amplitude and phase distributions of the image light field. Based on the mathematical analysis of the formulas, this paper introduces a technique for obtaining the amplitude and phase distribution of detected objects and provides experimental evidence.

1. Introduction

In modern optical precision measurement, microscopic digital holography is an important technology [1]. In recent years, to improve measurement accuracy, researchers have employed illumination light with different phase structures to illuminate the object plane, effectively enhancing the amplitude resolution of the image light field [2,3,4,5,6,7,8]. However, in detection studies, accurately obtaining the amplitude and phase of the object under coherent light illumination is equally important research work [9,10,11]. When using illumination light with different phase structures to illuminate the object plane, the phase of the reconstructed hologram image is inevitably altered. To eliminate phase interference from the illumination light and accurately obtain the complex amplitude of the object being measured, it is necessary to establish a quantitative computational relationship between the complex amplitude of the light wavefield on the object plane and the complex amplitude of the light wavefield on the image plane. However, the imaging theory currently widely adopted is the formula for the amplitude distribution of the image light field derived from the optical classic Introduction to Fourier Optics [12]. Since the theoretical derivation of this formula approximates the impulse response of the imaging system, it can only be used when the object size is smaller than one-quarter of the diameter of the optical system’s entrance pupil. Therefore, it is difficult to obtain the amplitude and phase of the object based on this formula. Additionally, according to this formula, a coherent optical imaging system is a linear space-invariant system, and the amplitude resolution on the image plane is the same everywhere, which does not align with experimental measurements. Therefore, in 2016, Horstmeyer et al. [13] cited the imaging theory in [12] and suggested that when reporting research results on improving the reconstruction image resolution of microscopic digital holography, one should provide amplitude images of a radiation pattern grating (Siemens star) at different positions on the image plane. They also recommended that scientists provide mathematical methods for obtaining the amplitude and phase of the object light field based on digital holograms. However, this issue remains unresolved to this day.

After years of effort, we derived the formula for calculating the complex amplitude of the image field in an imaging system described by 2 × 2 matrix elements [14]. Further research showed that, following the research method described by Professor Goodman in Introduction to Fourier Optics, but without approximating the impulse response of the imaging system, it is also possible to derive an expression that can accurately calculate the complex amplitude of the image field.

This paper is based on the impulse response expression derived by Professor Goodman [12] and presents two different approximations for the impulse response. It introduces the amplitude distribution calculation formula provided in Introduction to Fourier Optics and the formula for approximating the complex amplitude of the image light field. Subsequently, through rigorous mathematical analysis, an accurate formula for calculating the complex amplitude of the image light field is derived, the new physical significance of the transfer function is introduced, and theoretical calculations and comparisons with digital holographic experimental image reconstruction are performed for the three formulas. For practical application, experimental and digital processing techniques for obtaining the complex amplitude of the detected object are introduced [15]. To prove that coherent optical imaging systems are not linear space-invariant systems, the image light amplitude resolution of the Siemens star at different positions on the image plane is calculated using the formula. Finally, this research is discussed and conclusions are drawn.

2. Three Formulas for Calculating the Imaging Light Field of a Lens Imaging System

Following Professor Goodman’s research method [12], we define the spatial rectangular coordinate system O-xyz, with the z-axis coinciding with the optical axis. Figure 1 shows a schematic diagram of a single-lens imaging system illuminated by coherent light along the optical axis. Let the distance from the object plane to the first principal plane of the lens be z₁ and the distance from the image plane to the second principal plane of the lens be z₂.

The optical system is regarded as a linear system, and the object plane coordinates are $\left(\xi ,\eta \right)$, the optical wave field is $U_0\left(\xi ,\eta \right)$, the image plane coordinates are $\left(u,v\right)$, and the image light field $U_i\left(u,v\right)$ can be expressed as the lower integral (the formula of reference [12] (6–28)).

$$U_i\left(u,v\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{U}_0\left(\xi ,\eta \right)h\left(u,v;\xi ,\eta \right)\mathrm{d}\xi \mathrm{d}\eta }$$

(1)

where $h\left(u,v;\xi ,\eta \right)$ is the image made in the image plane by a light wave emitted from a point source with coordinates $\left(\xi ,\eta \right)$ in the object plane, called the impulse response. The focal length of the imaging lens is f, and the pupil function is P(x,y), and their expressions are as follows (the formula of reference [12] (6–33)):

$$\begin{array}{l}h\left(u,v;\xi ,\eta \right)={\displaystyle \frac{1}{{\lambda }^2{z}_1{z}_2}}\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]\exp \left[j{\displaystyle \frac{k}{2z_1}}\left({\xi }^2+{\eta }^2\right)\right]\\ \hspace{1em}\times {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}P\left(x,y\right)}\exp \left\{\left[j{\displaystyle \frac{k}{2}}\left({\displaystyle \frac{1}{z_1}}+{\displaystyle \frac{1}{z_2}}-{\displaystyle \frac{1}{f}}\right)\left(x^2+y^2\right)\right]\right\}\\ \hspace{1em}\times \exp \left\{-jk\left[\left({\displaystyle \frac{\xi }{z_1}}+{\displaystyle \frac{u}{z_2}}\right)x+\left({\displaystyle \frac{\eta }{z_1}}+{\displaystyle \frac{v}{z_2}}\right)y\right]\right\}\mathrm{d}x\mathrm{d}y\end{array}$$

(2)

where $j=\sqrt{-1}$, $k=2\pi /\lambda ,\hspace{0.33em}\lambda$ is the wavelength of light.

When substituting Equation (2) into Equation (1) to calculate the light wave field at the image plane, the resulting expression remains highly complex. Therefore, the literature [12] states the following: “Equations (6–28) and (6–33) now provide a formal solution specifying the relationship between the object U₀ and the image U_i. However, it is difficult to determine the conditions under which U_i can reasonably be considered an image of U₀ unless further simplifications are made.”

Through a series of simplifications [12], we finally get (formula of reference [12] (6–42))

$$U_i\left(u,v\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}\tilde{h}\left(u-\xi ,v-\eta \right)\left[\frac{1}{\left|M\right|}{U}_0\left(\frac{\xi }{M},\frac{\eta }{M}\right)\right]}\mathrm{d}\xi \mathrm{d}\eta$$

(3)

where $M=-z_2/z_1$ is the transverse magnification of the image,

$$\tilde{h}\left(u,v\right)={\displaystyle \frac{1}{{\lambda }^2{z}_2^2}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}P\left(\lambda z_2{f}_{\xi },\lambda z_2{f}_{\eta }\right)\exp \left[-j2\pi \left(u{f}_{\xi }+vy\right)\right]}\mathrm{d}{f}_{\xi }\mathrm{d}{f}_{\eta }$$

(4)

where $f_{\xi },f_{\eta }$ is the spectral coordinate. According to the study of chapter 7 in reference [8], the Fourier transform and inverse transform symbol are introduced, and Equation (3) can be rewritten as

$$U_i\left(u,v\right)=F^{-1}\left\{F\left\{{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\right\}P\left(-\lambda z_2{f}_{\xi },-\lambda z_2{f}_{\eta }\right)\right\}$$

(5)

where $F\left\{{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\right\}$ is the spectrum of the ideal image, and $P\left(-\lambda z_2{f}_{\xi },-\lambda z_2{f}_{\eta }\right)$ is called the amplitude transfer function [12]. For a circularly symmetric imaging system, its physical meaning is a low-pass filter for the spectrum of the ideal image.

If the phase factor $\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]$ in Equation (2) is retained, because the term does not participate in the integration operation of Equation (1), the approximate calculation formula of the complex amplitude of the imaging light field based on Equation (5) can be written as

$$U_i\left(u,v\right)=\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]F^{-1}\left\{F\left\{{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\right\}P\left(-\lambda z_2{f}_{\xi },-\lambda z_2{f}_{\eta }\right)\right\}$$

(6)

Since the image plane $\left({\displaystyle \frac{1}{z_1}}+{\displaystyle \frac{1}{z_2}}-{\displaystyle \frac{1}{f}}\right)=0$, the quadratic phase factor within the formula integral can be omitted, and using $M=-z_2/z_1$, Equation (2) can be rewritten using Equation (4) as

$$\begin{array}{l}h\left(u,v;\xi ,\eta \right)=-M\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]\\ \hspace{1em}\hspace{1em}\times \exp \left[j{\displaystyle \frac{k}{2z_1}}\left({\xi }^2+{\eta }^2\right)\right]\tilde{h}\left(u-M\xi ,v-M\eta \right)\end{array}$$

(7)

put Equation (7) into Equation (1)

$$\begin{array}{l}{U}_i\left(u,v\right)=-M\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]\\ \hspace{1em}\times {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{U}_0\left(\xi ,\eta \right)\exp \left[j\frac{k}{2z_1}\left({\xi }^2+{\eta }^2\right)\right]\tilde{h}\left(u-M\xi ,v-M\eta \right)\mathrm{d}\xi \mathrm{d}\eta }\end{array}$$

(8)

Let ${\xi }^{\prime }=M\xi$ and ${\eta }^{\prime }=M\eta$, which results in

$$\begin{array}{l}{U}_i\left(u,v\right)=\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]\\ \times {\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}-\frac{1}{M}{U}_0\left(\frac{{\xi }^{\prime }}{M},\frac{{\eta }^{\prime }}{M}\right)\exp \left[j\frac{k}{2M^2{z}_1}\left({{\xi }^{\prime }}^2+{{\eta }^{\prime }}^2\right)\right]\tilde{h}\left(u-{\xi }^{\prime },v-{\eta }^{\prime }\right)\mathrm{d}{\xi }^{\prime }\mathrm{d}{\eta }^{\prime }}\end{array}$$

(9)

For ease of comparison with Equations (5) and (6), Equation (9) can be written as a calculation formula expressed by Fourier transform and inverse transform

$$\begin{array}{l}{U}_i\left(u,v\right)=\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]\times \\ F^{-1}\left\{F\left\{{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\exp \left[j{\displaystyle \frac{k}{2M^2{z}_1}}\left(u^2+v^2\right)\right]\right\}P\left(-\lambda z_2{f}_{\xi },-\lambda z_2{f}_{\eta }\right)\right\}\end{array}$$

(10)

here, three different forms of the image light field calculation formula are derived.

The following study on the transmission of physical meaning will show [16] that Equation (10) can be applied to the calculation of the complex amplitude of the image light field for any size of the exit pupil.

3. Physical Meaning of Transfer Function

Expand the Fourier transform term in Equation (10)

$$\begin{array}{l}F\left\{{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\exp \left[j{\displaystyle \frac{k}{2M^2{z}_1}}\left(u^2+v^2\right)\right]\right\}\\ ={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}}{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\exp \left[j{\displaystyle \frac{k}{2M^2{z}_1}}\left(u^2+v^2\right)\right]\exp \left[-j2\pi \left(f_{\xi }u+f_{\eta }v\right)\right]\mathrm{d}u\mathrm{d}v\end{array}$$

(11)

Let $f_{\xi }={\displaystyle \frac{x}{\lambda M^2{z}_1}},\hspace{0.33em}{f}_{\eta }={\displaystyle \frac{y}{\lambda M^2{z}_1}}$. Since $\left(x,y\right)$ and $\left(u,v\right)$ have the same spatial scale units, the calculation results of the above equation are similar to the diffraction field of the ideal image ${\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)$ on the spatial plane $\left(x,y\right)$ after Fresnel diffraction at a distance $M^2{z}_1$ under plane wave illumination. Based on this result, when examining Equation (10), the transfer function $P\left(-\lambda z_2{f}_{\xi },-\lambda z_2{f}_{\eta }\right)$ acts as a spatial filter for the Fresnel diffraction field of the ideal image in geometric optics during frequency-domain filtering via the transfer function. Since the transfer function window is finite, if the diffraction field of the ideal image projected along the optical axis is partially intercepted by the window portion of the transfer function, the inverse Fourier transform calculation will not yield a complete image.

4. Theoretical Simulation and Experimental Proof

To validate the findings of this paper, we conducted a digital holographic experiment. The optical path of the experimental system is shown in Figure 2. In the figure, a HeNe laser (200 mw) with a wavelength of $\lambda =632.8\hspace{0.33em}\mathrm{nm}$ passes through a spatial filter (Daheng Optics, Beijing, China) and beam expander system, SF, to become a plane wave, which is then directed toward the beam splitter, BS1 (Daheng Optics, Beijing, China). The light waves passing through BS1 are reflected by the plane mirror, M1 (Daheng Optics, Beijing, China), and illuminate the object plane, OBJ. The object plane is formed by a transparent aperture in the shape of the Chinese character for “light” (光). The light waves passing through the aperture are imaged on a CCD (Point Grey, Wilsonville, OR, USA) detector controlled by a microcomputer PC via the imaging lens, L, and the beam splitter, BS2. The focal length of the convex lens L is $f=300\hspace{0.33em}\mathrm{mm}$, and the aperture is 75 mm. To verify whether the derived formulas are unaffected by the size of the incident light pupil, a circular aperture diaphragm, P, with a diameter $D=8\hspace{0.33em}\mathrm{mm}$ is placed at a distance $d=45\hspace{0.33em}\mathrm{mm}$ in front of the lens. The reference light for the digital holographic system comes from the reflective surface of beam splitter BS1. This light beam is reflected by mirror M2 and beam splitter BS2 before reaching the CCD. (The light waves passing through lateral beam splitter BS2, which are irrelevant to the theoretical study, are omitted in the figure.) In the experimental study, the object distance $Z_1=900\hspace{0.33em}\mathrm{mm}$ and the image distance $Z_2=450\hspace{0.33em}\mathrm{mm}$, i.e., the magnification ratio $M=-0.5$. Two off-axis digital holograms were recorded by the CCD with and without the diaphragm, P.

The pixel width of the experimental CCD is 0.0052 mm, with a pixel count of 1280 × 1024. When processing the experimental data, zero padding is used to transform the hologram into a square hologram with a width of 1280 pixels and a width of L_i = 6.656 mm. After processing the hologram to eliminate zero-order diffraction interference [1], the spectrum image is obtained by the Fast Fourier Transform (FFT), as shown in Figure 3a. After extracting the spectrum from the lower right corner, the reconstructed image of the amplitude of the image light field is obtained by the Inverse Fast Fourier Transform (IFFT), as shown in Figure 3b, and Figure 3c shows the phase distribution image of the image light field.

In order to use FFT to simulate the three formulas derived above, the diameter of the filter window with diaphragm P is first calculated. According to Figure 2, the exit pupil is the image formed by diaphragm P through lens L, located 52.94 mm below the lens, and the diameter of the exit pupil is D_p = 9.41 mm. Thus, the distance between the exit pupil and the image plane is d_pi = 502.94 mm, and the diameter of the filter window formed by the spectral plane $D_f=D_p/\lambda d_{pi}=35.17\hspace{0.33em}{\mathrm{mm}}^{-1}$. Because the spectral plane width is $N=1280$ pixels, the filter window is $N_f=N{D}_f/L_i\approx 234$ diameter in pixels.

A theoretical simulation image of Equation (10) with the diaphragm is shown in Figure 4. Figure 4a is a spectral amplitude image of the ideal image with a secondary phase factor in Equation (10), where the ring is a filter window formed by the transfer function. The imaging light field amplitude obtained by inverse Fourier transformation based on the spectrum within the filter window is shown in Figure 4b, and its phase image is shown in Figure 4c. It can be seen that the spectral amplitude image of the ideal image with a secondary phase factor is similar to the Fresnel diffraction image with a distance of $M^2{z}_1=112.5\hspace{0.33em}\mathrm{mm}$. Because the “spatial filtering” effect of the transfer function window makes the reconstruction look like an incomplete image, the physical significance of the transfer function discussed above is a good proof. A comparison with Figure 3 shows that the theoretical simulations yield excellent agreement with the experimental measurements.

According to the experiment with diaphragm P, the use conditions of Equation (5) are not met. Figure 5 shows a simulated calculation image using Equation (6).

An analysis of Figure 5 shows that the amplitude image of the simulated image does not match the experimental measurements. Since the exit pupil in the calculation process of Equation (6) is a low-pass filter of the ideal image, the image obtained after Fourier transformation is a complete image that has lost some high-frequency components. Although Equation (6) can calculate the phase distribution of the image light field, it cannot accurately obtain the same phase as the experimental measurements (Figure 3c). A quantitative analysis will be performed later.

Figure 6 shows the reconstructed image of the digital hologram without diaphragm P. Specifically; the spectral images obtained by using FFT are shown in Figure 6a. After removing the lower right frequency spectrum, the image of IFFT is obtained, as shown in Figure 6b, and Figure 6c is the phase distribution image of the image light field.

When there is no diaphragm P, the 75 mm diameter lens pupil essentially meets the usage conditions of Equation (5), allowing the image light field to be simulated using three separate formulas. Since the exit pupil is the same as the lens pupil at this point, the distance from the exit pupil to the image plane is dpi = 450 mm, and the filter window diameter is 1754 pixels, which is greater than the calculated plane width. Therefore, all numerical values from the spectral plane can be directly used for calculations in all three formulas. Figure 7 shows a simulated calculation image using Equation (10). It is evident that the amplitude and phase of the reconstructed image are in excellent agreement with the experimental image in Figure 6.

Figure 8 shows theoretical simulation images using Equations (5) and (6), where (Figure 8a) and (Figure 8b) are the spectrum of the ideal image and the amplitude of the reconstructed image, respectively, and Figure 8c is the phase image simulated by Equation (6). Compared with Figure 6, it can be seen that the three formulas can simulate the amplitude distribution of the imaging light field quite well.

To quantitatively assess the differences between the phase distributions simulated by Equations (6) and (10) and the experimentally measured values, we examined the phase within the horizontal line range of the transmitted light hole image. We counted the number of stripes showing phase changes in Figure 6c, Figure 7c and Figure 8c, yielding the following results: 60 stripes in Figure 6c, 58 stripes in Figure 7c, and 26 stripes in Figure 8c. Since the phase change value between adjacent lines is 2π, compared to Equation (6), Equation (10) yields phase values that better align with experimental measurements. Therefore, Equation (10) can accurately calculate the amplitude and phase of the image light field.

5. Experimental Techniques for Accurately Obtaining the Complex Amplitude of the Object Light Field

In digital holographic detection research, accurately obtaining the amplitude and phase of the object plane light wave field through digital holograms is equally important. Over the past decade, significant progress has been made in optical frequency shifting technology, which uses illumination light with different phase structures to illuminate objects in order to improve the amplitude resolution of reconstructed images [17,18]. However, it is not difficult to see from Equation (10) that if the illumination light is represented by function $E_0\left(u,v\right)$, Equation (10) can be rewritten as

$$\begin{array}{l}{U}_i^{\prime }\left(u,v\right)=\exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]\times \\ F^{-1}\left\{F\left\{{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)E_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\exp \left[j{\displaystyle \frac{k}{2M^2{z}_1}}\left(u^2+v^2\right)\right]\right\}P\left(-\lambda z_2{f}_{\xi },-\lambda z_2{f}_{\eta }\right)\right\}\end{array}$$

(12)

This means that digital holographic reconstruction images will introduce amplitude and phase interference from the illumination light, which is inconvenient for phase-sensitive optical detection. Here, we introduce the technique we proposed in reference [9] to eliminate this interference.

If the exit pupil of the optical system is sufficiently large, transfer function $P\left(-\lambda z_2{f}_{\xi },-\lambda z_2{f}_{\eta }\right)\approx 1$ in Equation (12) can be simplified to

$$U_i^{\prime }\left(u,v\right)\approx \exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]{\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)E_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\exp \left[j{\displaystyle \frac{k}{2M^2{z}_1}}\left(u^2+v^2\right)\right]$$

(13)

If there are no objects on the object plane, then the image formed by the illumination light on the image plane is simplified to

$$U_{ie}\left(u,v\right)\approx \exp \left[j{\displaystyle \frac{k}{2z_2}}\left(u^2+v^2\right)\right]{\displaystyle \frac{1}{\left|M\right|}}{E}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\exp \left[j{\displaystyle \frac{k}{2M^2{z}_1}}\left(u^2+v^2\right)\right]$$

(14)

from the above two equations, we obtain

$${\displaystyle \frac{1}{\left|M\right|}}{U}_0\left({\displaystyle \frac{u}{M}},{\displaystyle \frac{v}{M}}\right)\approx {\displaystyle \frac{1}{\left|M\right|}}{U^{\prime }}_i\left(u,v\right)/U_{ie}\left(u,v\right)$$

(15)

This means that when performing digital holographic inspection using structured light illumination, an “Null-object digital hologram” can first be recorded without the inspection object. Using this hologram, image $U_{ie}\left(u,v\right)$ of the illumination light can be obtained. Subsequently, the inspection object is placed, and after obtaining ${U^{\prime }}_i\left(u,v\right)$ using the hologram, the image light field can be obtained using Equation (15), facilitating the accurate determination of the amplitude and phase of the object light field. Since Equation (15) has been experimentally validated [9], for simplicity, Figure 9 shows a theoretical simulation example of the optical system in Figure 2 without diaphragm P.

Figure 9a,b show the amplitude and phase distribution images of the image light field obtained using a hologram. Since the object plane is formed by the “光” character, the transmission holes should have an equal phase distribution. However, Figure 9b clearly shows a spherical wave phase distribution influenced by the illumination light and the imaging components of the imaging system, which does not match the actual situation. However, after obtaining the illumination light image light field without an object through numerical simulation, the accurate phase image in Figure 9c can be obtained using Equation (15).

6. A Study on the Amplitude Resolution of Light Fields in Different Regions of a Plane Image

Based on the theoretical analysis of Equation (10), coherent optical imaging systems are no longer linear space-invariant systems. To clearly verify this conclusion, we follow the suggestion in reference [9] and replace the object plane in the experimental system shown in Figure 2 with a 40-petal Siemens star while retaining diaphragm P. We then perform theoretical calculations on the image light field of the Siemens star’s center, both on the optical axis and off the optical axis. The theoretical simulations indicate that the amplitude resolution varies across different regions of the image plane, with the resolution increasing as the distance from the optical axis increases. As an example, Figure 10 presents theoretical study images of resolution across different regions of the image plane. Figure 10a,b show the amplitude distribution of the image field normalized to a grayscale range of 0–255 when the center of the Siemens star coincides with the optical axis and when the center coordinates of the Siemens star are (u = 1 mm, v = 1 mm). The lower left corner of each figure displays a magnified image of the observed region. To study changes in amplitude resolution, sampling rings with a radius of 0.26 mm were drawn on each image with the center of the Siemens star as the center. Figure 10c shows the amplitude curve of the ring sampled counterclockwise along the right side of the ring in Figure 10a, and Figure 10d shows the amplitude curve of the ring sampled counterclockwise along the right side of the ring in Figure 10b.

The improvement in image resolution deviating from the optical axis can be explained by the physical meaning of the coherent transfer function. For this experimental system, the circular hole-shaped transfer function is similar to the ideal image passing through a spatial filter window with a distance $M^2{z}_1$ of Fresnel diffraction. If the image plane is regarded as a combination of several small pixels, the small pixels in the edge area have higher spectral values than those in the central area during imaging. In addition, due to the “Fresnel diffraction” of small pixels in the boundary area at a distance of $M^2{z}_1$, the low-frequency components of their spectrum are gradually blocked to a greater extent, and finally, the amplitude value of the image formed in the boundary area gradually decreases.

7. Discussion

In the theoretical study of coherent optical imaging [12], given that the impulse response expression of the imaging system is a double integral of a relatively complex complex function, it is difficult to obtain a mathematical expression for the image light field that is convenient for calculation through superposition integration. In order to obtain a simpler calculation formula, Professor Goodman simplified the impulse response expression and derived a formula that can only calculate the amplitude distribution of the image light field. The work presented in this paper can be regarded as solving a challenging mathematical problem proposed by Professor Goodman over 50 years ago.

Since Professor Goodman’s impulse response is derived from the Fresnel diffraction integral, we recently directly tracked the spatial propagation of light waves through the imaging system using the Fresnel diffraction integral [19] and derived the same calculation formula as in Equation (10).

In laser application research, formulas capable of calculating the complex amplitude of an image light field are not only important theoretical research tools for microscopic digital holography but also hold significant importance in application studies related to coherent optical imaging. As far back as the 1980s, when we used the formula derived from reference [12] to study an infrared strong laser transformation system with a wavelength of 10.6 μm, we had to artificially introduce the phase of the image light field to approximately describe the interference issues between image light fields in experimental studies. Now, using Equation (10) derived in this paper, we can successfully resolve this challenge encountered 30 years ago [20].

We sincerely hope that the work presented in this paper will serve as a useful supplement to research on coherent optical imaging and its applications.

8. Conclusions

Based on the research methods for coherent optical imaging described in the optical classic Introduction to Fourier Optics, this paper derives theoretical formulas for calculating the amplitude and phase distribution of the image light field without approximating the impulse response of the imaging system. The feasibility of these formulas is experimentally verified using digital holography. To facilitate the practical application of digital holographic detection, this paper introduces experimental techniques that have been experimentally proven to accurately obtain the complex amplitude of the object light field.