Investigation of the Space-Variance Effect of Imaging Systems with Digital Holography

Abstract

In classical Fourier optics, an optical imaging system is regarded as a linear space-invariant system, which is only an approximation. Especially in digital holography, the space-variance effect has a great impact on the image quality and cannot be ignored. Therefore, it is comprehensively investigated in this article. Theoretical analyses indicate that the space-variance effect is caused by linear frequency modulation and ideal low-pass filtering, and it can be divided into three states: the approximate space-invariance state, the high-frequency distortion state, and the boundary-diffraction state. Classical Fourier optics analysis of optical imaging systems only considers the first. Regarding the high-frequency distortion state, the closer the image field is to the edge, the more severe the distortion of high-frequency information is. As for the boundary-diffraction state, in addition to the distortion of high-frequency information in the margin, a prominent boundary-diffraction phenomenon is observed. If the space-variance effect of the imaging lens is ignored, we predict that no space-variance effect in image holography will occur when the hologram is recorded at the back focal plane of the imaging lens. Simulation and experimental results are presented to validate our theoretical prediction.

1. Introduction

In signal and systems theory, a system whose properties do not change with its spatial location is called a space-invariant system. In particular, the system response only depends on the input signal and the system characteristics and is independent of the spatial location where the input signal is imposed. Due to the linear space invariance of optical imaging systems, classical Fourier optics describes the optical imaging system in terms of the frequency response from the perspective of signals and systems [1].

However, the space invariance of optical imaging systems is only a simplification. Related studies have pointed out that the space variance lowers the image quality of the part of the image field that is farther away from the optical axis. Accordingly, this simplification may be inappropriate for some cases. Yan et al. [2] discussed the axial measurement error caused by the space-variance effect in digital holography by performing numerous simulations of point and line spread functions. Lohmann and Paris [3] defined the cross-correlation of two-line spread functions in an optical system as the evaluation index of the space-variance effect of the system. Moreover, Brainis [4] investigated the space-variance effect in aperture and lens imaging by analyzing the point spread function.

The aforementioned articles researched optical systems by analyzing the point spread function, which is classical in Fourier optics. The responses of any point or edge in a space-invariant system are the same, and every point or line is representative. A linear space-invariant system satisfies the convolution theorem, and its transfer function can be expressed by the Fourier transform of its point spread function. Therefore, point and edge spread functions can accurately describe the properties of space-invariant systems. However, no transfer function is present in the space-variant system, and the point spread function of a certain point can only describe the local response of the system to the point, which is unrepresentative. Therefore, it cannot holistically or comprehensively describe the system.

This study focuses on the causes of the space-variance effect, which have been studied by some scholars. Tichenor and Goodman [5] identified the quadratic phase factor as breaking the space-invariance condition of the single-lens imaging system. However, they only studied the conditions under which the quadratic phase factor can be ignored. Thus, although the optical imaging system can be simplified as a space-invariant system, the properties of the optical imaging system under space-variance effects cannot be analyzed. Pan et al. [6] showed that the spectrum broadening attributed to the quadratic phase factor was an important contribution to the space-variance effect, but their discussion on the space-variance effect was qualitative and not comprehensive. Herein, the role of the quadratic phase factor in the space-variance effect and its propagation law in digital holographic imaging systems are elucidated through a rigorous mathematical derivation.

Digital holography, an important three-dimensional measurement technology, can simultaneously record the intensity and phase information of the measured object. It is widely used in cell observation [7,8,9], particle and flow field measurement [10,11,12], and topography [13,14,15] and tomography [8,16,17] measurement, among others. According to the space-invariance approximation criterion proposed by Tichenor and Goodman [5], the width of the object field of view (FOV) should be less than 1/4 of the width of the aperture. This is easy to achieve for lens imaging. However, due to the limitation of the resolution and magnification in digital holography, the transverse size of the object under test or the virtual image of the front imaging system is usually comparable to the size of the CCD/CMOS chip, so the space-variance effect is very common in digital holography, especially digital Fresnel holography [18,19,20,21].

This paper analyzes the mathematical model of the Fresnel holographic imaging system and determines that the image field is not the result of the ideal low-pass filtering of the object field, as described in classical Fourier optics, but the result of the object field first modulated by the linear frequency modulation (LFM) signal and then filtered by the ideal low-pass filter. Thereafter, according to the ratio of the space–bandwidth product between the LFM signal and the aperture, the space-variance effect of the holographic imaging system is categorized into approximate space-invariance, high-frequency distortion, and boundary-diffraction states. Notably, the classical Fourier optical analysis of optical imaging systems only considers the approximate space-invariance state, which can be reduced to a space-invariant system. In the high-frequency distortion state, the closer the image field is to the FOV edge, the more severe the high-frequency information distortion is. In the boundary-diffraction state, in addition to the distortion of high-frequency information in the FOV margin, boundary-diffraction fringes are prominent. Specifically, the Fresnel diffraction pattern of the aperture stop can be observed in the image field. To validate our theory, we predict that no space-variance effect occurs in image holography [22,23] when the hologram is recorded in the back focal plane of the imaging lens if the space-variance effect of the imaging lens is ignored. Simulations and experiments were conducted to confirm this prediction. For simplicity, the theoretical derivation in this paper is based on the one-dimensional imaging case, but it can be easily extended to the two-dimensional case.

The rest of this article is organized as follows. In Section 2, the mathematical model of Fresnel holography considering the space-variance effect is established. In Section 3, the space-variance effect is divided into three states. In Section 4, the prediction is proposed and confirmed by the results of simulations and experiments.

2. Materials and Methods

2.1. Mathematical Model of Space-Variant Fresnel Holographic Imaging Systems

Diffraction in free space is linear space-invariant. According to the Huygens–Fresnel principle, “every unobstructed point of a wavefront, at a given instant, serves as a source of spherical quadratic wavelets. The amplitude of the optical field at any point beyond is the superposition of all these wavelets [24].”. The Huygens–Fresnel principle comprises two elements: the spherical wavelets hypothesis and the combination mode of spherical wavelets—interference superposition. If free-space diffraction is considered, the above points, respectively, correspond to the space-invariant and linear characteristics of free-space diffraction.

However, although the diffraction-limited optical system is linear, it is not space-invariant [3,25]. Without loss of generality, only the one-dimensional imaging process is considered. Taking holography as an example, Figure 1 shows the recording and reconstruction process of Fresnel holography, where R is the reference wave. For simplicity, the back-propagation is represented to the right. The planes $x_0$, $x_1$, and $x_2$ represent the object plane, the CMOS chip (or hologram plane), and the observation plane, respectively.

From the angular spectrum point of view, diffraction in free space occurs from the object plane to the CMOS chip, but due to the finite aperture, the CMOS will only selectively receive the frequency components of each object point. In Figure 1, the frequency components of A and B recorded by the CMOS are different, and the response on the observation plane markedly differs. In other words, the wavelet emitted by different positions of the object plane received by the CMOS is different. Therefore, diffraction-limited optical systems are space-variant systems.

Considering the object field $u_o\left(x_o\right)$, diffraction from the object plane to the hologram plane is expressed using the Fresnel diffraction formula as follows:

$$u_1\left(x_1\right)=\frac{exp\left(jkd\right)}{j\lambda d}\underset{-\infty }{\overset{\infty }{\int }}{u}_0\left(x_0\right)w_0\left(x_0\right)exp\left[\frac{jk}{2d}{\left(x_1-x_0\right)}^2\right]d{x}_0$$

(1)

where $d$ is the recording distance, $-d$ is the reconstruction distance, and $k=2\pi /\lambda$. If the object field has a rectangular boundary and the side length is $L_0$, then $w_0\left(x_0\right)=rect\left(\frac{x_0}{L_0}\right)$, where $rect\left(·\right)$ represents the rectangular window function. Let the reference wave be

$$R\left(x_1\right)=a_{r}exp\left[j\psi \left(x_1\right)\right]$$

(2)

where $a_r=C_1$, $\psi \left(x_1\right)=C_2{x}_1$, and $C=constant$. The hologram can then be obtained from the interference between the object wave and the reference wave:

$$I_1\left(x_1\right)={\left|u_1\left(x_1\right)+R\left(x_1\right)\right|}^2={\left|u_1\right|}^2+{\left|R\right|}^2+R^{*}{u}_1+R{u}_1^{*}$$

(3)

By phase shifting or applying the off-axis technique, the object wave $u_1$ recorded by the hologram can be extracted. Through back-propagation, the reconstructed wave front $u_2\left(x_2\right)$ can be written as

$$u_2\left(x_2\right)=\frac{{a_r}^2}{{\lambda }^2{d}^2}\underset{-\infty }{\overset{\infty }{\int }}{w}_1\left(x_1\right)\left\{\underset{-\infty }{\overset{\infty }{\int }}{u}_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)exp\left[\frac{jk}{2d}{\left(x_1-x_0\right)}^2\right]d{x}_0\right\}\times exp\left[{\frac{-jk}{2d}\left(x_2-x_1\right)}^2\right]d{x}_1$$

(4)

where $w_1\left(x_1\right)=rect\left(\frac{x_1}{L_1}\right)$ is the rectangular window function determined by the size of the CMOS chip, which can be regarded as the aperture stop of the Fresnel holographic imaging system. Ignoring the constant term in Equation (7), we can derive the reconstructed wave front $u_2\left(x_2\right)$ as

$$u_2\left(x_2\right)=exp\left(\frac{-jk}{2d}{x_2}^2\right)\times {{FFT}^{-1}\left\{rect\left(\frac{f_x}{L_1/\lambda d}\right){FFT\left[u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)exp\left(\frac{jk}{2d}{x_0}^2\right)\right]}_{f_{x0}=\frac{x_1}{\lambda d}}\right\}}_{t_{fx0}=x_0=x_2}$$

(5)

where $L_1$ is the physical size of the CMOS chip; $FFT$ and ${FFT}^{-1}$ represent the Fourier transform and inverse Fourier transform, respectively; $f_{x0}$ is the frequency-domain coordinates after the Fourier transform; and $t_{fx0}$ is the spatial-domain coordinates after the inverse Fourier transform.

The system represented by Equation (5) is shown in Figure 2 as a block diagram. According to communication theory, the input signal, namely, the object field $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)$, is the modulation signal, and its spectrum distribution in phase space is shown in Figure 3a [26,27]. $exp\left(\frac{jk}{2d}{x_0}^2\right)$ is the carrier signal, which is also called the LFM signal or chirp signal. $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)exp\left(\frac{jk}{2d}{x_0}^2\right)$ is the modulated signal. The Fresnel holographic imaging system consists of the following three parts:

(1)

Modulation of input signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)$ by carrier signal $exp\left(\frac{jk}{2d}{x_0}^2\right)$. If $\phi =\frac{k}{2d}{x_0}^2$, the frequency shift introduced by the carrier signal $exp\left(\frac{jk}{2d}{x_0}^2\right)$ when applying the frequency shift theorem of the Fourier transform is

$${∆f}_x=\frac{1}{2\pi }·\frac{\partial \phi }{\partial x_0}=\frac{x_0}{d\lambda }$$

(6)

Equation (6) shows that the frequency shift is dependent on the spatial coordinates, which indicates that this process is space-variant. The distribution of the modulated signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)exp\left(\frac{jk}{2d}{x_0}^2\right)$ in phase space is shown in Figure 3b.

(2)

Ideal low-pass filtering of modulated signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)exp\left(\frac{jk}{2d}{x_0}^2\right)$. $rect\left(\frac{f_x}{L_1/\lambda d}\right)$ represents a rectangular ideal low-pass filter, as shown in Figure 4, whose passband is determined by the angular aperture of the CMOS target plane, namely, $-\frac{L_1}{2\lambda d}\le f_x\le \frac{L_1}{2\lambda d}$. As shown in Figure 3b,c, after the frequency shift of the modulation signal, the original low-frequency information becomes high-frequency. In this case, low-pass filtering will block the original low-frequency information of the modulation signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)$. The farther away it is from the optical axis, the more low-frequency information of the modulation signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)$ that will be lost. Therefore, for the input signal of the system–object field $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)$, the aperture is not reflected as an ideal low-pass filter due to the frequency shift effect of Equation (6) but as a frequency-selective filter, with the constant passband width and center frequency changing with the position in the object plane. Such a filter presents different frequency responses with different spatial locations of the modulation signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)$; that is, the filtering process is space-variant for the modulation signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)$.

(3)

Modulation of ideal low-pass-filtered signal by carrier signal $exp\left(-\frac{jk}{2d}{x_0}^2\right)$. When the reconstruction distance is equal to the recording distance, the two carrier signals before and after filtering are conjugate and cancel each other, and the Fresnel holography reconstruction automatically completes the demodulation process. Therefore, the problem of quadratic phase aberration is not encountered in Fresnel holography. The distribution of the reconstructed wave field in phase space is shown in Figure 3d.

The above analyses reveal that LFM and ideal low-pass filtering are critical to the space-variance effect. Without LFM, the ideal low-pass filtering is strict with respect to the object field, and the system will be space-invariant. Moreover, without ideal low-pass filtering, the two carriers will cancel out, and the reconstructed wave front will not differ from the object wave field.

Both LFM and ideal low-pass filtering are important for space-variance effects, but different optical systems require different considerations. For example, the space-variance effect can be effectively suppressed by using larger optical elements to increase the passband width of the low-pass filter. Usually, this requirement is easy to fulfill, such as by using a larger lens. However, in some cases, such as the demand for a more compact design or when some optical elements are very small, one can only rely on the control of LFM to suppress the space-variance effect. For example, in digital holography, the size of CMOS chips is usually very small, about a few millimeters.

2.2. Three Stages of the Space-Variance Effect

The solution to Equation (5) is derived to further analyze the space-variance effect. From the perspective of signal processing, the chirp rate of the LFM signal $exp\left(\frac{jk}{2d}{x_0}^2\right)$ is $K_0=1/d\lambda$, its bandwidth is $B_0=K_0{L}_0$, and its space–bandwidth product is ${SBP}_0=B_0{L}_0=K_0{L_0}^2$. The object wave field $u_0\left(x_0\right)$ is assumed to be a slowly varying function compared with the LFM signal $exp\left(\frac{jk}{2d}{x_0}^2\right)$; therefore, it can be ignored. This approximation is reasonable for phase-only samples with slow phase changes, such as cells. The modulated signal can be reduced to $rect\left(\frac{x_0}{L_0}\right)exp\left(j\pi K_0{x_0}^2\right)$, and its Fourier transform is

$$U\left(f_{x_0}\right)=\underset{-L_0/2}{\overset{L_0/2}{\int }}exp\left[j\pi \left(K_0{x_0}^2-2f_{x_0}{x}_0\right)\right]d{x}_0=exp\left(-j\pi \frac{{f_{x_0}}^2}{K_0}\right)\underset{-L_0/2}{\overset{L_0/2}{\int }}exp\left[j{\pi \left(\sqrt{K_0}{x}_0-\frac{f_{x_0}}{\sqrt{K_0}}\right)}^2\right]d{x}_0$$

(7)

Assuming that ${\left(\sqrt{K_f}f-\frac{x}{\sqrt{K_f}}\right)}^2=v^2/2$, $v=\sqrt{2K_0}{x}_0-\frac{\sqrt{2}{f}_{x_0}}{\sqrt{K_0}}$, we have

$$U\left(f_{x_0}\right)=\frac{1}{\sqrt{2K_0}}exp\left(-j\frac{{{\pi f}_{x_0}}^2}{K_0}\right)\underset{-V_1}{\overset{V_2}{\int }}exp\left(j\frac{\pi v^2}{2}\right)dv$$

(8)

The integral term of Equation (8) is the Fresnel integral, where

$$\begin{array}{c}{V}_1=\frac{\sqrt{2K_0{L_0}^2}}{2}+\sqrt{2/K_0}{f}_{x_0}\\ V_2=\frac{\sqrt{2K_0{L_0}^2}}{2}-\sqrt{2/K_0}{f}_{x_0}\end{array}$$

(9)

The spectrum $U\left(f_{x_0}\right)$ of the signal $rect\left(\frac{x_0}{L_0}\right)exp\left(j\pi K_0{x_0}^2\right)$ is

$$U\left(f_{x_0}\right)=\frac{1}{\sqrt{2K_0}}exp\left(-j\pi \frac{{f_{x_0}}^2}{K_0}\right)\left[C\left(V_1\right)+jS\left(V_1\right)+C\left(V_2\right)+jS\left(V_2\right)\right]$$

(10)

where

$$\begin{array}{c}C\left(V\right)=\underset{0}{\overset{V}{\int }}cos\left(\frac{\pi v^2}{2}\right)dv\\ S\left(V\right)=\underset{0}{\overset{V}{\int }}sin\left(\frac{\pi v^2}{2}\right)dv\end{array}$$

(11)

As shown in Figure 4, the red line is the magnitude spectrum of $U\left(f_{x_0}\right)$ with a rectangular envelope, and its main energy is relatively evenly distributed between $-\frac{B_0}{2}$ and $\frac{B_0}{2}$, rather than concentrated near the zero frequency such as in the classical distribution of the magnitude spectrum [28]. This phenomenon is called central frequency spreading [29,30]. Equation (10) offers a good approximation of the spectrum of the modulated signal $u_0\left(x_0\right)rect\left(\frac{x_0}{L_0}\right)exp\left(\frac{jk}{2d}{x_0}^2\right)$. Notably, the spectrum $U\left(f_{x_0}\right)$ is only meaningful in the system represented in Equation (5) or Figure 2 and is independent of the object wave $u_1\left(x_1\right)$ recorded by the hologram.

The low-pass-filtering process can be divided into three states.

(1)

Approximate space-invariance state. In this state, the filter passband width $B_{L1}=L_1/\lambda d$ is much larger than the bandwidth $B_0=L_0/d\lambda$ of the LFM signal; that is, the width of the aperture $L_1$ is much larger than the width of the object field $L_0$. The blue box in Figure 4 represents the passband of the low-pass filter in this case. Tichenor and Goodman [5] pointed out that when $L_1>4L_0$, the space-variance effect is negligible. The classical Fourier optical analysis of optical imaging systems rests on the assumption of approximate space invariance; in other words, only the $L_1>4L_0$ case is considered.

(2)

High-frequency distortion state. The passband width of the filter in this state satisfies $4B_0>B_{L1}>B_0$. The purple box in Figure 4 represents the passband of the low-pass filter in this case. The energy blocked by the filter is mainly from the high-frequency information of the object field far from the optical axis. The farther away the object wave field is from the optical axis, the higher the frequency modulated by the LFM signal is. Therefore, more information is lost in these areas after low-pass filtering, and the image quality is worse, which is mainly reflected in the distortion of the abrupt phase in the margin of the image wave field.

(3)

Boundary-diffraction state. The green box in Figure 4 represents the passband of the low-pass filter when $B_{L1}<B_0$. In this case, the high-frequency information and a mass of low-frequency information in the margin of the object wave field are filtered out. Because the main energy of the optical field is concentrated in the low-frequency information, high energy loss occurs in areas where low-frequency information is filtered out, which leads to a decrease in the signal-to-noise ratio, resulting in phase distortion.

When the space–bandwidth product $TBP=K_0{L_0}^2$ of the LFM signal $exp\left(j\pi K_0{x_0}^2\right)$ is sufficiently large, the $U\left(f_{x_0}\right)$ spectrum can be approximated as

$$U\left(f_{x_0}\right)=\frac{1}{\sqrt{K_0}}exp\left(-j\pi \frac{{f_{x_0}}^2}{K_0}+\frac{\pi }{4}\right), -\frac{B_0}{2}<f_{x_0}<\frac{B_0}{2}$$

(12)

Equation (12) demonstrates that the spectrum of the band-limited LFM signal can still be regarded as a band-limited LFM signal. Curlander and McDonough [31] pointed out that when the spatial bandwidth product of the LFM signal is greater than 100, Equation (12) is sufficient for obtaining the exact spectrum. Therefore, when the filter passband width $B_{L1}=L_1/\lambda d$ is less than the bandwidth $B_0=L_0/d\lambda$ of the LFM signal, that is, when the aperture width $L_1$ is less than the object wave field width $L_0$, ignoring the constant phase, the spectrum after ideal low-pass filtering can be approximated as

$$U\left(f_{x_0}\right)=\frac{1}{\sqrt{K_0}}exp\left(-j\pi \frac{{f_{x_0}}^2}{K_0}\right), -\frac{B_{L1}}{2}<f_{x_0}<\frac{B_{L1}}{2}$$

(13)

The signal presented in Equation (13) is considered a new LFM signal. Let $K_1=1/K_0=d\lambda$; its chirp rate is $-K_1$. The bandwidth $B_1=K_1{B}_{L1}=L_1$ is the physical width of the aperture. The space–bandwidth product is ${SBP}_1=K_1{B_{L1}}^2=\frac{{L_1}^2}{\lambda d}$, which is the same form as the Fresnel number.

The inverse Fourier transform of $U\left(f_{x_0}\right)$ is

$${\tilde{u}}_2\left(x_2\right)=\frac{1}{\sqrt{K_0}}\underset{-B_{L1}/2}{\overset{B_{L1}/2}{\int }}exp\left[j\pi \left(-K_1{f_{x_0}}^2+2f_{x_0}{x}_2\right)\right]d{f}_{x_0}$$

(14)

From the derivation in Equations (7)–(10), we can obtain

$${\tilde{u}}_2\left(x_2\right)=\frac{1}{\sqrt{2}}exp\left(j\frac{\pi {x_2}^2}{K_1}\right)\left[C\left(X_1\right)-jS\left(X_1\right)+C\left(X_2\right)-jS\left(X_2\right)\right]$$

(15)

where

$$\begin{array}{c}{X}_1=\frac{\sqrt{2K_1{B_{L1}}^2}}{2}+\sqrt{2/K_1}{x}_2=\sqrt{2N_{F1}}\left(1+2x_2/L_1\right)\\ X_2=\frac{\sqrt{2K_1{B_{L1}}^2}}{2}-\sqrt{2/K_1}{x}_2=\sqrt{2N_{F1}}\left(1-2x_2/L_1\right)\end{array}$$

(16)

where $N_{F1}=\frac{{L_1}^2}{4d\lambda }$. Substituting Equation (15) into Equation (5), the quadratic phase factor in Equation (15) cancels out. The reconstructed wave front is

$$u_2\left(x_2\right)=\frac{1}{\sqrt{2}}\left[C\left(X_1\right)-jS\left(X_1\right)+C\left(X_2\right)-jS\left(X_2\right)\right]$$

(17)

which proves that no quadratic phase aberration is present in the Fresnel-holography-reconstructed wave front. The above derivation assumes that the object field changes slowly, so the solution to Equation (17) can also be regarded as the wavefront error, which is caused by the space-variance effect on the low frequency of the image field.

Equation (17) is obtained from the back-propagation of $R^{*}{u}_1$. The corresponding wavefront error $u_2\left(x_2\right)$ obtained from the forward propagation of $R^{*}{u}_1$ is

$$u_2\left(x_2\right)=\frac{1}{\sqrt{2}}\left[C\left(X_1\right)+jS\left(X_1\right)+C\left(X_2\right)+jS\left(X_2\right)\right]$$

(18)

Equation (18) is exactly the same as the Fresnel diffraction pattern of the square aperture. The square aperture here is the CMOS chip. Therefore, $B_{L1}<B_0$ is an instance of the boundary-diffraction state. Moreover, Equations (17) and (18) reveal that the boundary diffraction will disturb the entire image wave field, including the paraxial state.

The experimental results of the three stages of the space-variance effect are shown in Figure 5. The sample is a laser-etched “XJTU” quartz plate with a maximum width of about 3 mm. The size of the CMOS chip used in the experiment is 12.8 × 12.8 ${\mathrm{m}\mathrm{m}}^2$ (resolution 5120 × 5120, pixel size 2.5 × 2.5 ${\mathsf{\mu }\mathrm{m}}^2$). Therefore, if the object wave field of the measured sample is exactly located in the center of the CMOS chip, the approximate space-invariance state $L_1>4L_0$ is satisfied, as shown in Figure 5a. In order to compare the measurement results of the same location of the same sample under different conditions, the sample is moved so that the relative position of the CMOS chip and the object wave field changes, which is equivalent to using a sample with a larger size. If part of the main energy of the object wave field irradiates outside the CMOS chip, the boundary-diffraction state $B_{L1}<B_0$ or $L_1<L_0$ is satisfied, as shown in Figure 5c. The condition between the cases shown in Figure 5a,c corresponds to the high-frequency distortion state, as shown in Figure 5b.

From the perspective of signal processing, this boundary-diffraction disturbance can be considered a ringing artifact, namely, Gibb’s phenomenon [6]. Cuche et al. [32] utilized the apodization method to suppress the boundary-diffraction disturbance in digital holographic imaging and achieved good results in the application of holographic measurements [33]. In general, in addition to consuming more time and computational resources and having a reduced FOV, the apodization method can satisfactorily suppress the boundary-diffraction perturbation of the intensity map. However, the apodization method does not fundamentally change the space-variance effect of the optical imaging system, so it cannot suppress the high-frequency information distortion in the marginal area of the reconstructed image. In addition, because the apodization method reduces the intensity of the hologram marginal area, the signal-to-noise ratio of the reconstructed image in the corresponding area decreases, which leads to more severe distortion of the phase map.

2.3. Eliminating the Space-Variance Effect by Recording Holograms at the Back Focal Plane of the Imaging Lens

Due to the size of a CMOS chip, the numerical aperture of a Fresnel holographic imaging system is usually much smaller than that of a lens imaging system. Moreover, increasing the numerical aperture by increasing the CMOS chip size is expensive and inefficient. Image holography [22,23] is a hologram-recording method that combines holography with a lens imaging system. We deductively predicted that hologram recording at the back focal plane of the imaging lens would not produce a space-variance effect, and this prediction is proved by simulations and experiments. As shown in Figure 6, if the constant term is ignored, the expression of single-lens imaging is

$$u_i(x_i)=exp\left(\frac{jk}{2d_2}{x_i}^2\right)\times {{FFT}^{-1}\left\{{FFT\left[u_o(x_o)exp\left(\frac{jk}{2d_1}{x_o}^2\right)\right]}_{f_{xo}=\frac{x_l}{{\lambda d}_1}}\times P\left(f_{xo}\right)\right\}}_{t_{fxo}=x_o=\frac{{-d}_1}{d_2}{x}_i}$$

(19)

where $P\left(·\right)$ is the pupil function, and

$$\frac{1}{d_1}+\frac{1}{d_2}=\frac{1}{f}$$

(20)

where $f$ is the focal length of the lens.

In the image holography schematic shown in Figure 7, the image holography consists of single-lens imaging and Fresnel holography when the diffraction between each plane is located in the Fresnel diffraction region. The image formed by the lens can be regarded as the virtual object in Fresnel holography. The recording distance is ${-d}_3$, and the reconstruction distance is $d_3$, with their signs being opposite to those in ordinary Fresnel holography. In this case, the formula for Fresnel holographic imaging is as follows:

$$u_{ci}\left(x_{ci}\right)=exp\left(\frac{jk}{2d_3}{x_{ci}}^2\right)\times {FFT}^{-1}{\left\{FFT{\left[u_i\left(x_i\right)exp\left(\frac{-jk}{2d_3}{x_i}^2\right)\right]}_{f_{xi}=\frac{-x_c}{\lambda d_3}}\times rect\left(\frac{f_{xi}}{L_x/\lambda d_3}\right)\right\}}_{t_{fxi}=x_i=-x_{ci}}$$

(21)

Substituting Equation (19) into Equation (21) yields a cumbersome expression.

$$\begin{array}{c}{u}_{ci}\left(x_{ci}\right)=exp\left(\frac{jk}{2d_3}{x_{ci}}^2\right) \\ \begin{array}{l}\times {FFT}^{-1}\{FFT\{exp\left(\frac{jk}{2d_2}{x_i}^2\right)\\ \times {{FFT}^{-1}\left\{{FFT\left[u_o\left(x_o\right)exp\left(\frac{jk}{2d_1}{x_o}^2\right)\right]}_{f_{xo}=\frac{x_l}{{\lambda d}_1}}\times P\left(f_{xo}\right)\right\}}_{t_{fxo}=x_o=\frac{{-d}_1}{d_2}{x}_i}\\ {{\times exp\left(\frac{-jk}{2d_3}{x_i}^2\right)\}}_{f_{xi}=\frac{-x_c}{\lambda d_3}}\times rect\left(\frac{f_{xi}}{L_x/\lambda d_3}\right)\}}_{t_{fxi}=x_i=-x_{ci}}\end{array}\end{array}$$

(22)

As shown in Figure 8, the optical imaging system presented in Equation (22) can be represented by a block diagram, and the former is analyzed in the phase space, as shown in Figure 9. The modulation of the LFM signal in phase space is manifested as the shearing of the original signal, and the low-pass filtering is manifested as the cutting of the spectrum along the frequency axis. Because shearing occurs before cutting, different states retain different frequency components after cutting, as shown in Figure 9b, which will lead to the space-variance effect. In Figure 7, the virtual object field of holography is the image of the previous lens, which results in the spectrum undergoing reverse shearing and makes it possible to “correct” the spectrum before passing through the next low-pass filter, the CMOS, as shown in Figure 9d. In this case, the frequency response of the CMOS low-pass filter to each spatial position is the same, and no new space-variance effect will be produced.

When

$$\frac{jk}{2}\left(\frac{{x_o}^2}{d_1}+\frac{{x_i}^2}{d_2}-\frac{{x_i}^2}{d_3}\right)=0$$

(23)

is satisfied, where $x_o=\frac{{-d}_1}{d_2}{x}_i$, that is,

$$d_3=\frac{{d_2}^2}{d_1+d_2}$$

(24)

the distance between the lens and the CMOS target plane is

$$d_2-d_3=d_2-\frac{{d_2}^2}{d_1+d_2}=\frac{{d_{1}d}_2}{d_1+d_2}$$

(25)

According to Equation (20), we have

$$f=\frac{{d_{1}d}_2}{d_1+d_2}$$

(26)

and the following formula:

$$d_4=d_2-d_3=f$$

(27)

That is, when the distance between the lens and the CMOS chip is the focal length of the lens, the holographic imaging process will not produce a new space-variance effect. In addition, it is necessary to exclude the influence of the lens as much as possible since our focus is on the space-variance effect in digital holography. A large aperture lens can be used to ensure this point. For example, in our experiment, a lens with a diameter of 50.8 mm was used as the image lens, thus satisfying the approximate space-invariance state. When the space-variance effect of the imaging lens can be ignored, the whole holographic imaging system remains space-invariant. From the perspective of Fourier optics, the back focal plane of the imaging lens is the Fourier plane of the object field; thus, the aperture stop located in the back focal plane is equivalent to the ideal low-pass filtering of the object field, so the imaging process is space-invariant. In addition, for different object distances of the same imaging lens, its Fourier plane remains the same, so the recording position of focal plane holography also remains the same.

3. Results

The prediction was verified by simulations and experiments involving microlens array topography measurements. As shown in Figure 10, an ordinary Mach–Zehnder interferometer was used for the experiment. A laser beam with a wavelength of 523.5 nm was used in our experiment; the focal length of the imaging lens was 200 mm, and the sample was placed 400 mm in front of the imaging lens to maintain a vertical-axis magnification of 1×. The tested object was an lbtek MLAS10-F15-P300-AB microlens array with a period of 300 μm and window size of 9 mm × 9 mm. A CMOS camera with a resolution of 5120 × 5120 and pixel size of 2.5 μm was used for recording. To better compare the image quality of the three hologram-recording methods, the holograms were cropped to a resolution of 3800 × 3800 with a physical size of 9.5 mm.

For comparison, the following three sets of simulations and experiments were performed: Fresnel holography, image holography with the recording of holograms at the back focal plane (IHWF), and image holography without recording holograms at the back focal plane of the imaging lens (IHWOF). To maintain the same numerical aperture as that in IHWF, the CMOS was placed 200 mm behind the sample in Fresnel holography and 200 mm behind the image plane of the imaging lens in IHWOF. Calculations show that the recording distance satisfies the Fresnel approximation condition and the spectrum separation condition of off-axis holography.

The simulation processes of Fresnel holography and image holography are based on the block diagrams shown in Figure 2 and Figure 8, and the simulation results under the above conditions are shown in Figure 11 and Figure 12, respectively.

Recording the hologram directly at the back focal plane of the imaging lens in the IHWF experiment is difficult because the energy of the zero-frequency component is too concentrated, and the overexposure of the low-frequency information is severe. Therefore, the CMOS target plane can slightly deviate from the focal plane to record the hologram and still maintain the approximate space invariance.

The experimental results are shown in Figure 13 and Figure 14. In Figure 14c, because of the space-variance effect, the high-frequency information at the edge of the microlens array is lost, while the phase in Figure 14d is in order. In addition, there is a phenomenon of a “phase increase” at the edge in Figure 14, which may result from phase aberration. The experimental results in [34,35,36,37,38,39,40] indicate that in digital holographic phase measurement, it is difficult to perfectly control or eliminate the phase aberration in the whole field of view. However, the “phase increase” in Figure 14 does not affect any of the analyses and conclusions of this article.

The simulation and experimental results reveal that the imaging quality of IHWF is significantly higher than that of Fresnel holography and IHWOF, and the imaging quality of IHWOF is worse than that of Fresnel holography. The latter can easily be explained by Figure 8 and Figure 9. For example, the bandwidth of the LFM signal in IHWOF is twice that of the LFM signal in Fresnel holography, so only the area in the center, which occupies approximately 1/4 of the total area in Figure 11k and Figure 13k, has relatively high imaging quality. These results indicate that IHWF has the potential to suppress or even eliminate the space-variance effect and further indicate the correctness of the proposed theory.

4. Discussion

In this article, Fresnel holography and holography with a lens are studied. It is precisely because the object of study is very simple that it is possible to obtain theoretical solutions. However, modern optical systems are often complex and contain a large number of optical elements. Furthermore, different optical elements modulate the beam differently, and the superposition effects are difficult to strictly derive.

It is meaningful to analyze the space-variance effect of complex optical systems to improve the imaging capability of these systems. Unfortunately, these cases are beyond the scope of this article. More research is needed on how the spatial effect will affect the system and what kind of experimental phenomena occur in a system with a large number of optical elements, such as grating and spatial light modulators.

One of the significant contributions of this manuscript is that it provides a good framework: that is, through Fourier analysis and phase space analysis for each step of the diffraction process, the space-variance effect of the whole system is finally obtained. This lays a foundation for the future study of the space-variance effects of complex systems.

5. Conclusions

In this study, the digital holography imaging system is regarded as a space-variant system. In digital Fresnel holography, the space-variance effect severely affects the measurement results due to the small size of the CMOS chip. Therefore, it is inappropriate to treat it as a space-invariant system. The discussion of space-variance effects in Section 2 and Section 3 of this paper is based on Fresnel holography, but it is also applicable to ordinary optical imaging systems. The presented analysis reveals that the space-variance effect is caused by LFM and ideal low-pass filtering. If one of these two conditions is destroyed, the space-variance effect can be eliminated. Furthermore, three states of the space-variance effect are pointed out: approximate space-invariance state, high-frequency distortion state, and boundary-diffraction state. In this study, the two physical phenomena of high-frequency distortion and boundary diffraction are added to the classical Fourier optical analysis of an optical imaging system, which only considers the approximate space-invariance state, making it more consistent with the fact.

Based on the theory proposed in this study, we have proved through theoretical analysis, simulation, and experiments that the space-variance effect can be controlled by adjusting the distance between the optical elements in the imaging system, especially when recording the hologram in the back focal plane of the imaging lens, in which case the holographic imaging process will not produce a new space-variance effect. In other words, when the space-variance effect of the imaging lens can be ignored, there will be no space-variance effect in the imaging system. Thus, this article provides a comprehensive understanding of digital holographic imaging systems with the space-variance effect.