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Adaptive phase-shifting interferometry based on a phase-shifting digital holography algorithm

Xianxin Han, Yuheng Wang, Zhengyang Bu, Xiaoya Bu, Hongyun Li, Shengde Liu, Liyun Zhong, and Xiaoxu Lu

Open Access Open Access

Abstract

Phase-shifting interferometry (PSI) requires accurate phase shifts between interferograms for realizing high-accuracy phase retrieval. This paper presents an adaptive PSI through synchronously capturing phase shifts measurement interferograms and phase measurement interferograms, in which the former is a series of spatial carrier frequency phase-shifting interferograms generated by an additional assembly and the phase shifts are calculated with the single-spectrum phase shifts measurement algorithm (SS-PSMA), the latter is employed for phase retrieval with an adaptive phase-shifting digital holography algorithm (PSDHA) based on complex amplitude recovery. In addition to exhibiting excellent reliability, high-accuracy phase retrieval (0.02 rad), and short calculation time (<25 ms), the proposed adaptive PSDHA is suitable for various interferograms with different fringe shapes and numbers. Importantly, both simulation analysis and experimental result demonstrate that this adaptive PSI based on PSDHA can effectively eliminate phase-shifting errors caused by phase shifter and external disturbance, ensuring high-accuracy phase shifts measurement and phase retrieval, meanwhile significantly reducing phase-shifting interferograms acquisition time and phase retrieval calculation time.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Phase-shifting interferometry (PSI) [1], a high-accuracy phase retrieval method, has been found wide-ranging applications in various measurement fields, such as three-dimensional surface profiling [2], refractive index measurement [3], small displacement and deformation measurement [4], biological cell morphology [5], and precise mechanical measurement [6]. In general, PSI requires at least 3-frame phase-shifting interferograms with equal and known phase shifts for realizing phase retrieval [7]. Since the proposal of the least-square algorithm (LSA) based on signal error minimization [8,9], it realizes phase retrieval using multiple interferograms without the constraint of equal phase shifts. Although the LSA improves the signal-to-noise ratio, it still requires the prior knowledge of phase shifts. Several typical approaches such as 4-step [10,11], 5-step [12], and N-step [9,13] phase-shifting algorithms can be attributed to LSA. These typical algorithms are performed with equal phase shifts in an integer period, significantly reducing calculation complexity while ensuring the accuracy of phase retrieval. Nonetheless, in cases where the phase shifts are unknown or deviate from the nominal phase shifts, phase retrieval accuracy will be degraded.

Due to the lack of phase shifts measurement technology, various iterative and non-iterative phase retrieval algorithms have been proposed for unknown phase shifts situation, such as the advanced iterative algorithm (AIA) [14], a cross-iterative optimization solution based on temporal and spatial least squares algorithm. Furthermore, the principal component analysis (PCA) [15] is proposed by extracting two mutually orthogonal eigenvalues from interferograms to obtain phase shifts. Another approach is the normalization and orthogonalization phase-shifting algorithm (NOPSA) [16], it eliminates the influence of phase shifts deviation (PSD) by the orthogonal properties of complex cosine functions. Among these algorithms, AIA is the best due to its high-accuracy phase shifts measurement and phase retrieval. Note that these algorithms all utilize the orthogonality of cosine functions, so the accuracy of phase retrieval is closely related to the fringe shape and number in the interferogram. If the fringe number in the interferogram is less than one, the error of phase retrieval is large and unstable, moreover, these algorithms are relatively complex and time-consuming.

To solve the problem of PSD, Peng proposed a phase shifts calibration method based on dual-interference optical path configuration [17,18], in which a special three-step phase-shifting algorithm is used for calculating the amplitude and phase of the sample [19]. Additionally, Zhou proposed a solution by synchronously measuring phase shifts and performing differential phase retrieval for differential interference contrast (DIC) microscopy using the LSA [20]. Although these methods can effectively eliminate the influence caused by the fringe shape and number, the former is still affected by external disturbance and the latter requires to be improved.

In this paper, we first introduce an adaptive phase retrieval solution based on phase-shifting digital holography algorithm (PSDHA). In addition to revealing the advantages of simplicity, high-accuracy, and rapid-speed, this PSDHA is suitable for various interferograms with any fringe shape and number. Meanwhile, we propose an adaptive PSI system through synchronously capturing a series of phase shifts measurement interferograms and phase measurement interferograms, in which the former is the spatial carrier frequency phase-shifting interferograms generated by an additional assembly and the phase shifts are calculated with the single-spectrum phase shifts measurement algorithm (SS-PSMA), so it can accurately and rapidly realize phase shifts measurement online, avoiding the influence caused by external disturbance and other factors. After that, using the adaptive phase-shifting digital holography algorithm (PSDHA), we realize high-accuracy phase retrieval by the captured phase measurement interferograms.

2. Phase-shifting digital holography algorithm (PSDHA)

In PSI, the phase-shifting interferograms captured by digital camera can be expressed as

$${{I_n}(x,y) = a(x,y) + b(x,y)\cos [{\varphi (x,y) + {\delta_n}} ]},$$
where $n = 1,2,3 \cdot{\cdot} \cdot N$ represents the order of the phase-shifting interferograms; x and y denote the pixel coordinates of the interferograms, here $1 \le x \le X$ and, $1 \le y \le Y$, in which X and Y represent the maximum number of pixels in the x and y directions, respectively; $a(x,y)$ and $b(x,y)$ are the background and modulation amplitude; $\varphi (x,y)$ represents the measured phase; and ${\delta _n}$ is the actual phase shifts of the nth-frame phase-shifting interferogram.

By rearranging the intensity matrix corresponding to the nth-frame phase-shifting interferogram (Y rows and X columns) into an M × 1 column matrix, we have

$${I_n} = \left[ {\begin{array}{c} {{I_{n,1}}}\\ {{I_{n,2}}}\\ \vdots \\ {{I_{n,m}}}\\ \vdots \\ {{I_{n,M}}} \end{array}} \right] = \left[ {\begin{array}{c} {{a_1} + {b_1}\cos ({{\varphi_1} - {\delta_n}} )}\\ {{a_2} + {b_2}\cos ({{\varphi_2} - {\delta_n}} )}\\ \vdots \\ {{a_m} + {b_m}\cos ({{\varphi_m} - {\delta_n}} )}\\ \vdots \\ {{a_M} + {b_M}\cos ({{\varphi_M} - {\delta_n}} )} \end{array}} \right],$$
where $m\textrm{ = }(y - 1)X + x$ represents the order of pixels in the interferogram, and $M\textrm{ = }X \times Y$; am, bm, and φm are the background, modulation amplitude and phase at pixel m, respectively.

A series of N-frame phase-shifting interferograms can be represented as a matrix of M × N:

$$H = \left[ {\begin{array}{cccccc} {{I_1}}&{{I_2}}& \cdots &{{I_n}}& \cdots &{{I_N}} \end{array}} \right].$$

To retrieve the phase $\varphi (x,y)$, we reconstruct the object field by phase-shifting digital holography method. Firstly, we construct a complex exponential function matrix E using ideal phase shifts $\delta _n^0 = 2\pi (n - 1)/N$ in an integer period as following

$$E = {\left[ {\begin{array}{*{20}{c}} {{e^{j\delta_1^0}}}&{{e^{j\delta_2^0}}}& \cdots &{{e^{j\delta_n^0}}}& \cdots &{{e^{j\delta_N^0}}} \end{array}} \right]^T},$$
where ${[\;]^T}$ is transpose operator.

Then, the complex interference term ${I_c}$ can be obtained through matrix operation

$${I_c} = HE = \left[ {\begin{array}{c} {\sum\nolimits_{n = 1}^N {{b_1}\cos ({{\varphi_1} - {\delta_n}} ){e^{j\delta_n^0}}} }\\ {\sum\nolimits_{n = 1}^N {{b_2}\cos ({{\varphi_2} - {\delta_n}} ){e^{j\delta_n^0}}} }\\ \vdots \\ {\sum\nolimits_{n = 1}^N {{b_m}\cos ({{\varphi_m} - {\delta_n}} ){e^{j\delta_n^0}}} }\\ \vdots \\ {\sum\nolimits_{n = 1}^N {{b_M}\cos ({{\varphi_M} - {\delta_n}} ){e^{j\delta_n^0}}} } \end{array}} \right].$$

If the actual phase shifts and the ideal phase shifts are the same, both the background and conjugate terms in the interferogram will be eliminated due to the orthogonality of cosine functions, so the reconstruction in Eq. (5) contains only the object optical field. However, if there is deviation between the actual phase shifts and the ideal phase shifts, the background term can be eliminated while the conjugate term still exists, thus Eq. (5) can be expressed as real and imaginary components

$${I_{cRe }} = \left[ {\begin{array}{c} {\sum\nolimits_{n = 1}^N {{b_1}\cos ({{\varphi_1} - {\delta_n}} )\cos ({\delta_n^0} )} }\\ {\sum\nolimits_{n = 1}^N {{b_2}\cos ({{\varphi_2} - {\delta_n}} )\cos ({\delta_n^0} )} }\\ \vdots \\ {\sum\nolimits_{n = 1}^N {{b_m}\cos ({{\varphi_m} - {\delta_n}} )\cos ({\delta_n^0} )} }\\ \vdots \\ {\sum\nolimits_{n = 1}^N {{b_M}\cos ({{\varphi_M} - {\delta_n}} )\cos ({\delta_n^0} )} } \end{array}} \right],{\kern 1pt} \,and\,{I_{c{\mathop{\rm Im}\nolimits} }} = \left[ {\begin{array}{*{20}{c}} {\sum\nolimits_{n = 1}^N {{b_1}\cos ({{\varphi_1} - {\delta_n}} )\sin ({\delta_n^0} )} }\\ {\sum\nolimits_{n = 1}^N {{b_2}\cos ({{\varphi_2} - {\delta_n}} )\sin ({\delta_n^0} )} }\\ \vdots \\ {\sum\nolimits_{n = 1}^N {{b_m}\cos ({{\varphi_m} - {\delta_n}} )\sin ({\delta_n^0} )} }\\ \vdots \\ {\sum\nolimits_{n = 1}^N {{b_M}\cos ({{\varphi_M} - {\delta_n}} )\sin ({\delta_n^0} )} } \end{array}} \right].$$

Subsequently, we construct a coefficient matrix using the actual phase shifts ${\delta _n}$ and the ideal phase shifts $\delta _n^0$:

$$\left[ {\begin{array}{cc} {{A_{11}}}&{{A_{12}}}\\ {{A_{21}}}&{{A_{22}}} \end{array}} \right] = \left[ {\begin{array}{cccccc} {\cos {\delta_1}}&{\cos {\delta_2}}& \cdots &{\cos {\delta_n}}& \cdots &{\cos {\delta_N}}\\ {\sin {\delta_1}}&{\sin {\delta_2}}& \cdots &{\sin {\delta_n}}& \cdots &{\sin {\delta_N}} \end{array}} \right]\left[ {\begin{array}{cc} {\cos \delta_1^0}&{\sin \delta_1^0}\\ {\cos \delta_2^0}&{\sin \delta_2^0}\\ \vdots & \vdots \\ {\cos \delta_n^0}&{\sin \delta_n^0}\\ \vdots & \vdots \\ {\cos \delta_N^0}&{\sin \delta_N^0} \end{array}} \right].$$

The real and imaginary components of ${I_c}$ can be rewritten as

$${I_{cRe }} = {\alpha _{Re }}\left[ {\begin{array}{*{20}{c}} {{b_1}\cos ({{\varphi_1} + {\Delta _{Re }}} )}\\ {{b_2}\cos ({{\varphi_2} + {\Delta _{Re }}} )}\\ \vdots \\ {{b_m}\cos ({{\varphi_m} + {\Delta _{Re }}} )}\\ \vdots \\ {{b_M}\cos ({{\varphi_M} + {\Delta _{Re }}} )} \end{array}} \right]\textrm{,}\;\textrm{and}\;{I_{c{\mathop{\rm Im}\nolimits} }} = {\alpha _{{\mathop{\rm Im}\nolimits} }}\left[ {\begin{array}{*{20}{c}} {{b_1}\cos ({{\varphi_1} + {\Delta _{{\mathop{\rm Im}\nolimits} }}} )}\\ {{b_2}\cos ({{\varphi_2} + {\Delta _{{\mathop{\rm Im}\nolimits} }}} )}\\ \vdots \\ {{b_m}\cos ({{\varphi_m} + {\Delta _{{\mathop{\rm Im}\nolimits} }}} )}\\ \vdots \\ {{b_M}\cos ({{\varphi_M} + {\Delta _{{\mathop{\rm Im}\nolimits} }}} )} \end{array}} \right],$$
where
$${\alpha _{Re }} = {({A_{11}^2 + A_{21}^2} )^{\frac{1}{2}}},$$
$${\alpha _{{\mathop{\rm Im}\nolimits} }} = {({A_{12}^2 + A_{22}^2} )^{\frac{1}{2}}},$$
$${\Delta _{Re }} = \arctan ({{A_{21}}/{A_{11}}} ),$$
$${\Delta _{{\mathop{\rm Im}\nolimits} }} = \arctan ({{A_{22}}/{A_{12}}} ).$$

When the actual phase shifts ${\delta _n}$ are known, ${\alpha _{Re }}$, ${\alpha _{{\mathop{\rm Im}\nolimits} }}$, ${\Delta _{Re }}$, and ${\Delta _{{\mathop{\rm Im}\nolimits} }}$ are all constants. To retrieve the phase $\varphi (x,y)$, we calculate the sum of $\begin{array}{*{20}{l}} {{I_{cRe }}/{\alpha _{Re }}} \end{array}$ and $\begin{array}{*{20}{l}} {{I_{c{\mathop{\rm Im}\nolimits} }}/{\alpha _{{\mathop{\rm Im}\nolimits} }}} \end{array}$, and their difference, respectively.

$$\begin{aligned} S &= \begin{array}{*{20}{l}} {{I_{cRe }}/{\alpha _{Re }}} \end{array} + \begin{array}{*{20}{l}} {{I_{c{\mathop{\rm Im}\nolimits} }}/{\alpha _{{\mathop{\rm Im}\nolimits} }}} \end{array}\\ &= 2\cos [({\Delta _{{\mathop{\rm Im}\nolimits} }} - {\Delta _{Re }})/2]\left[ {\begin{array}{*{20}{c}} {{b_1}\cos ({{\varphi_1} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )}\\ {{b_2}\cos ({{\varphi_2} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )}\\ \vdots \\ {{b_m}\cos ({{\varphi_m} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )}\\ \vdots \\ {{b_M}\cos ({{\varphi_M} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{s_{11}}}\\ {{s_{21}}}\\ \vdots \\ {{s_{m1}}}\\ \vdots \\ {{s_{M1}}} \end{array}} \right], \end{aligned}$$
$$\begin{aligned} D &= \begin{array}{*{20}{l}} {{I_{cRe }}/{\alpha _{Re }}} \end{array} - \begin{array}{*{20}{l}} {{I_{c{\mathop{\rm Im}\nolimits} }}/{\alpha _{{\mathop{\rm Im}\nolimits} }}} \end{array}\\ &= 2\sin [({\Delta _{{\mathop{\rm Im}\nolimits} }} - {\Delta _{Re }})/2]\left[ {\begin{array}{*{20}{c}} {{b_1}\sin ({{\varphi_1} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )}\\ {{b_2}\sin ({{\varphi_2} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )}\\ \vdots \\ {{b_m}\sin ({{\varphi_m} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )}\\ \vdots \\ {{b_M}\sin ({{\varphi_M} + ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2} )} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{d_{11}}}\\ {{d_{21}}}\\ \vdots \\ {{d_{m1}}}\\ \vdots \\ {{d_{M1}}} \end{array}} \right]. \end{aligned}$$

Based on the above, we can calculate the phase at order m in the phase matrix ${\varphi _{M \times 1}}$ by

$${\varphi _m} = \arctan \{ [{d_{m1}}/{s_{m1}}]/\tan [({\Delta _{{\mathop{\rm Im}\nolimits} }} - {\Delta _{Re }})/2]\} - ({\Delta _{{\mathop{\rm Im}\nolimits} }} + {\Delta _{Re }})/2.$$

So that ${\varphi _{Y \times X}}$ can be obtained by rearranging the phase matrix ${\varphi _{M \times 1}}$.

Evidently, using this PSDHA based on complex amplitude recovery, we can conveniently realize phase retrieval, but it requires accurate phase shifts, which is still a challenge in PSI.

3. Experimental system and phase shifts measurement

To obtain accurate phase shifts and verify the proposed method, as shown in Fig. 1, we build an experimental system consisting of phase measurement section and phase shifts measurement section, in which the phase-shifting interferograms acquisition and phase shifts measurement can be synchronously realized. In the phase measurement section, the laser emitted from a short-coherence laser source (central wavelength of 639.67 nm, bandwidth of 0.836 nm) passes through the polarizer P1 and the e-light is phase-shifted by the liquid crystal phase shifter LC, generating orthogonal polarization light. The e-light, whose polarization plane is perpendicular to the paper surface, is reflected by the polarizing beam splitter PBS1. It then goes through the optical path compensator consisting of a quarter-wave plate QWP1 and a mirror M1, generating the reference beam with its polarization plane parallel to the paper surface. The o-light, with its polarization plane parallel to the paper surface, passes through PBS1 and PBS2. It is modulated by another optical path compensator QWP2 and M2, generating the object beam with its polarization plane perpendicular to the paper surface. The reference beam is converged by the microscope objective Obj2 and then collimated by the tube lens TL2. A part of the reference beam is reflected by the beam splitter BS onto the target surface of phase measurement camera Cam1 (BFS-U3-16S2M-CS, 1440 × 1080 pixels, pixel size of 3.45μm). The object beam passes through Obj1 and TL1, a part of it passes through BS and is imaged onto the target surface of Cam1. The phase shifted reference beam and the object beam pass through the 1st surface of BS and polarizer P2 in coaxial and common path, and then interfere on the Cam1, generating phase-shifting interferograms.

 figure: Fig. 1.

Fig. 1. Sketch of the proposed adaptive PSI system. P1-3: polarizer; LC: liquid crystal phase shifter; QWP1,2: quarter wave plate; PBS1,2: polarizing beam splitter; BS: beam splitter; M1-3: mirror; S: sample; Obj1,2: objective (NA = 0.4, 20×); TL1,2: tube lens; WP: Wollaston prism; L: lens; Cam1,2: CMOS camera.

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Another section in Fig. 1 is a synchronous phase shifts measurement assembly (SPSMA) consisting of a Wollaston prism WP, a converging lens L, a polarizer P3, and a camera Cam2 (BFS-U3-16S2M-CS, sensor format: 1440 × 1080 pixels, pixel size of 3.45μm). In the 2nd surface of BS, the mutually orthogonal polarization object beam and reference beam pass through the SPSMA, their angle is adjusted as 1° by the WP. After being converged by L and passing through P3, they interfere on the Cam2, then the spatial carrier-frequency phase-shifting interferograms are generated and used for the phase shifts calculation through the spectral analysis at the carrier frequency position.

Since the interference optical path for phase measurement and phase shifts measurement is identical, after passing through BS, the phase measurement and phase shifts measurement are transmitted in the same path. Even if there is external disturbance in the interference optical path, as long as the acquisition of Cam1 and Cam2 is synchronous, the phase shifts measured by the SPSMA are completely the same and synchronized with the phase shifts between phase measurement interferograms.

To extract the phase shifts from the carrier frequency phase-shifting interferograms, we perform the following analysis. The nth-frame carrier frequency phase-shifting interferograms captured by Cam2 can be expressed as

$$\begin{aligned} {I_{PSn}}(x^{\prime},y^{\prime})&={a_{PS}}(x^{\prime},y^{\prime}) + {b_{PS}}(x^{\prime},y^{\prime})\cos [{2\pi {f_{x0}}x^{\prime} + 2\pi {f_{y0}}y^{\prime} + {\varphi_{PS}}(x^{\prime},y^{\prime}) + {\delta_n}} ]\\ &= {a_{PS}}(x^{\prime},y^{\prime}) + 0.5{b_{PS}}(x^{\prime},y^{\prime})\exp \{{j[{2\pi {f_{x0}}x^{\prime} + 2\pi {f_{y0}}y^{\prime} + {\varphi_{PS}}(x^{\prime},y^{\prime}) + {\delta_n}} ]} \}\\ &\quad + 0.5{b_{PS}}(x^{\prime},y^{\prime})\exp \{{ - j[{2\pi {f_{x0}}x^{\prime} + 2\pi {f_{y0}}y^{\prime} + {\varphi_{PS}}(x^{\prime},y^{\prime}) + {\delta_n}} ]} \}, \end{aligned}$$
here, $n = 1,2,3 \cdot{\cdot} \cdot N$ represents the order of the carrier frequency phase-shifting interferograms; $x^{\prime}$ and $y^{\prime}$ denote the pixel coordinates; ${a_{PS}}(x^{\prime},y^{\prime})$ and ${b_{PS}}(x^{\prime},y^{\prime})$ are the background and modulation; ${f_{x0}}$ and ${f_{y0}}$ are the spatial frequency; ${\varphi _{PS}}(x^{\prime},y^{\prime})$ denotes the phase of interferogram; and ${\delta _n}$ is the phase shifts. By performing the Fourier transform on Eq. (16), we can obtain the spectrum of ${I_{PSn}}(x^{\prime},y^{\prime})$ as
$$\begin{aligned} {\mathbf{\mathscr{I}}_{PSn}}({f_x},{f_y}) &={\mathbf{\mathscr{I}}_{PS}}({f_x},{f_y}) + {\mathbf{\mathscr{C}}_{PS}}({f_x},{f_y})\exp (j{\delta _n}) \ast \delta ({f_x} - {f_{x0}},{f_y} - {f_{y0}}) + \\ &\quad conj[{{\mathbf{\mathscr{C}}_{PS}}({f_x},{f_y})\exp (j{\delta_n})} ]\ast \delta ({f_x} + {f_{x0}},{f_y} + {f_{y0}}), \end{aligned}$$
where ${\mathbf{\mathscr{C}}_{PS}}({f_x},{f_y})$ represents the Fourier transform of $0.5{b_{PS}}(x^{\prime},y^{\prime})\exp [j{\varphi _{PS}}(x^{\prime},y^{\prime})]$, * represents the convolution, and conj is the conjugate. When ${f_{x0}}$ and ${f_{y0}}$ are large enough, the three terms in Eq. (17) can be separated. By extracting the first-order term and simplifying it, we have
$$\mathbf{\mathscr{I}}_{PSn}^{ + 1}({f_x},{f_y})\textrm{ = }{\mathbf{\mathscr{C}}_{PS}}({f_x} - {f_{x0}},{f_y} - {f_{y0}})\exp (j{\delta _n}).$$

For each phase-shifting carrier frequency interferogram, ${\mathbf{\mathscr{C}}_{PS}}({f_x} - {f_{x0}},{f_y} - {f_{y0}})$ is the same, so the phase shifts can be calculated at any single frequency. When ${f_x} = {f_{x0}}$ and ${f_y} = {f_{y0}}$, the spectral amplitude reaches the maximum, we can rapidly obtain the most accurate result with highest signal-to-noise ratio. Therefore, by using the phase shifts of the 1st-frame carrier frequency phase-shifting interferogram as a reference, the phase shifts of the nth-frame interferogram can be calculated by

$${\delta _n} = \arg [\mathbf{\mathscr{I}}_{PSn}^{ + 1}({f_{x0}},{f_{y0}})/\mathbf{\mathscr{I}}_{PS1}^{ + 1}({f_{x0}},{f_{y0}})].$$

It is referred to the single-spectrum phase shifts measurement algorithm (SS-PSMA).

Subsequently, we perform the simulation analysis and experimental research of the proposed method. All calculations are conducted in MATLAB R2022b on a PC equipped with an AMD Ryzen 7 3800X 8-Core Processor (3.89 GHz) and 64 GB of RAM.

4. Analysis and discussion

To illustrate the characteristics of the proposed method, we first perform the simulation analysis of PSDHA on its accuracy, calculation time, tolerance to PSD, and then compare the results obtained with different algorithms (LSA, AIA, and the 4-step phase-shifting algorithm (4-PSA)). Meanwhile, the characteristics of SS-PSMA is also analyzed.

In our simulation, the reference phase $\varphi (x,y)$ is based on the actual measured phase of neural progenitor cells, as shown in Fig. 2(a), with total number of pixels of 1440 × 1080 and pixel size of $\Delta x = \Delta y = 3.45\mu m$, which matches the camera used in our experiment. The center of the interferogram is defined as the coordinate origin, with x and y representing pixel coordinates, where $- 2.484mm \le y \le 2.484mm$ and $- 1.863mm \le y \le 1.863mm$. The object beam is set as

$$O(x,y) = (11/\sqrt 2 )\exp [ - 0.05({x^2} + {y^2})/2]\exp [j\varphi (x,y)],$$
while the reference beam on Cam1 is set as
$${R_n}(x,y) = (11/\sqrt 2 )\exp [ - 0.05({x^2} + {y^2})/2]\exp \{{j[{n_f}2\pi ({x^2} + {y^2})/9.641 + {\delta_n}]} \},$$
where $x_{\max }^2 + y_{\max }^2\textrm{ = }9.641$, so that the phase measurement interferograms on Cam1 can be expressed as
$${I_n}(x,y) = 121\exp [ - 0.05({x^2} + {y^2})]\{ 1 + \cos [\varphi (x,y) - {n_f}2\pi ({x^2} + {y^2})/9.641 - {\delta _n}]\} ,$$
where ${n_f}$ represents the circular fringe number in the interferogram. The reference beam on phase shifts measurement Cam2 is set as
$${R_{PSn}}(x,y) = (11/\sqrt 2 )\exp [ - 0.05({x^2} + {y^2})/2]\exp \{ j[2\pi {n_f}({x^2} + {y^2})/9.641 + 2\pi {f_x}x + 2\pi {f_y}y + {\delta _n}]\} .$$

 figure: Fig. 2.

Fig. 2. Comparison of phase retrieval accuracy with different algorithms (PSDHA, LSA, AIA and 4-PSA). (a) Reference phase; (b) a phase-shifting interferogram for phase measurement when the fringe number nf =5; (c) the carrier-frequency phase-shifting interferogram for phase shifts measurement of (b); (d) the spatial spectrum of the white dashed rectangle in (c); (e) a phase-shifting interferogram for phase measurement when the fringe number nf <1; (f) the carrier-frequency phase-shifting interferogram for phase shifts measurement of (e); (g) the spatial spectrum of the white dashed rectangle in (f); (h)(i) comparison of phase retrieval accuracy with different algorithms when the PSD are Δδn = 0, −0.7485, −0.2319, −0.6267 rad for nf =5 and nf <1, respectively.

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For simplicity, a tilted reference beam is utilized to introduce the spatial carrier frequency. ${f_y} = 16.034m{m^{ - 1}}$ and ${f_y} = 17.713m{m^{ - 1}}$ represent the carrier frequency along x and y direction, respectively, which correspond to the actual experiment. Consequently, the carrier frequency phase-shifting interferogram for phase shifts measurement on Cam2 can be expressed as

$${I_{PSn}}(x,y) = 121\exp [ - 0.05({x^2} + {y^2})]\{ 1 + \cos [\varphi (x,y) - 2\pi {n_f}({x^2} + {y^2})/9.641 - 2\pi ({f_x}x + {f_y}y) - {\delta _n}]\} .$$

When the fringe number ${n_f} = 5$, Figs. 2(b) and 2(c) show one of the phase measurement interferograms and the corresponding phase shifts measurement interferograms, respectively. Figures 2(e) and 2(f) are the corresponding interferograms when the fringe number ${n_f} < 1$. In our study, the white dashed rectangle (200 × 200 pixels) in Figs. 2(c) and 2(f) are chosen to calculate the phase shifts as the size of these areas is sufficient to meet the accuracy requirements of phase shifts measurement. Figures 2(d) and 2(g) are the spatial spectra of the white dashed rectangle in Figs. 2(c) and 2(f), respectively. And the phase shifts are calculated by Eq. (19) using the positive first-order spectrum in Figs. 2(d) and 2(g). Based on the actual experience, a random intensity noise with maximum gray value of 5 is added to the interferograms.

Taking 4-frame carrier frequency phase-shifting interferograms as the example, the accuracy and calculation time with the proposed PSDHA are analyzed. The ideal phase shifts are set as $\delta _n^0 = 0,\pi /2,\pi ,3\pi /2\;rad$, and the deviation between the actual shifts and the ideal phase shifts randomly is varied between 0 and ±0.8 radians (rad). When the phase shifts of 4-frame phase-shifting interferograms are ${\delta _n} = 0,\,0.8223,\,2.9097,\,4.0848\;rad$, the corresponding PSD are $\Delta {\delta _n} = 0,\, - 0.7485,\, - 0.2319,\, - 0.6267\;rad$. The phases are calculated with the PSDHA, LSA, AIA, and 4-PSA when ${n_f} = 5$ and ${n_f} < 1$, respectively. In our calculation, the PSDHA and LSA use the actual measured phase shifts, AIA does not require the known phase shifts, and the 4-PSA use the ideal phase shifts. Table 1 presents a comparison of the accuracy and calculation time of phase retrieval with different algorithms under varying the PSD, ranging randomly from 0 to ±0.8 rad. Note that Table 1 are the average results from 100 repeated calculations. RMSE and PVE represent the root mean square error and peak to valley error, respectively.

Tables Icon

Table 1. Comparison of the accuracy and calculation time of phase retrieval with different algorithms

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As shown in Fig. 2(h), we can see that when ${n_f} = 5$, meaning large PSD (about 0.8 rad), the accuracy with the PSDHA, LSA, and AIA is the same while the 4-PSA has lower accuracy, indicating that the PSD seriously affects the accuracy of traditional 4-PSA algorithm due to the requirement of equal phase shifts in integer period. From Fig. 2(i), it is found that when ${n_f} < 1$, the accuracy of the AIA is lower, which is consistent with the results in Table 1. The reason behind this can be attributed to the phase-shifting interferograms not satisfying the spatial summation characteristics when ${n_f} < 1$, which leads to inaccurate calculation of phase shifts. The phase shifts are global calculation parameter in the temporal and spatial iteration process of AIA. Therefore, the error values differ at various phase distributions.

As for the calculation time, the proposed PSDHA not only guarantees high-accuracy, but also takes little calculation time (22 ms), much less than the LSA and AIA. This is because LSA needs to construct the component matrices and invert it, and AIA requires iteration, while PSDHA is a simple and direct phase recovery algorithm. Note that the 4-PSA takes the shortest calculation time (15 ms), but it comes at the cost of sacrificing accuracy. Once the phase shifts are obtained, it is not needed to use the AIA.

Subsequently, we analyze the influence of PSD on the accuracy of phase retrieval with different algorithms, as shown in Figs. 3(a) and 3(b). It can be observed that in the case of large PSD, the accuracy of phase retrieval with the 4-PSA is lower than those with the PSDHA, LSA, and AIA, and both the PSDHA and LSA maintain high-accuracy in any fringe shape. When ${n_f} = 5$, the RMSE of phase retrieval with the PSDHA, LSA, and AIA are less than 0.02 rad. In contrast, when the PSD is less than 0.05 rad, the RMSE with the 4-PSA is close to those with other three methods, but as the PSD increases, the RMSE rapidly increases. When the PSD reaches 0.8 rad, the RMSE is larger than 0.15 rad. From Fig. 3(b), we can see that when ${n_f} < 1$, the RMSE with PSDHA and LSA is the same as ${n_f} = 5$, but both the error and the stability of the AIA are significantly reduced. Therefore, AIA is not suitable for phase recovery of phase-shifting interferograms with fringe number less than 1 due to its inherent characteristics, which limits its applications.

 figure: Fig. 3.

Fig. 3. Influence of PSD on the accuracy of phase retrieval with different algorithms (a) nf =5; (b) nf < 1.

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In a word, regardless of whether there is a PSD or fringe number in the interferogram, the PSDHA and LSA algorithms obtain high-accuracy phase, and the former significantly reduce calculation time.

Also, we analyze the accuracy and calculation time of phase shifts measurement in practical experiment. By introducing a certain spatial carrier frequency into the reference beam, we can obtain the phase measurement interferograms with spatial frequency ${f_x} = 6.039m{m^{ - 1}}$ and ${f_y} = 5.368m{m^{ - 1}}$, and the phase shifts measurement interferograms with spatial frequency ${f_x} = 17.512m{m^{ - 1}}$ and ${f_y} = 24.423m{m^{ - 1}}$, the phase shifts during liquid crystal phase-shifting are calculated by Eq. (19). The obtained results indicate that the RMSE of phase shifts measurement is less than 0.007 rad, meeting the requirement of high-accuracy phase measurement. Specially, the calculation time for all phase shifts of 4-frame phase-shifting interferograms is only 4 ms, it has almost no influence on the calculation of phase retrieval.

5. Experiment

To verify the effectiveness of the proposed method in practical application, we perform the quantitative phase imaging of neural progenitor cells (NPCs) differentiated from induced pluripotent stem cells (iPSCs) using the experimental setup (Fig. 1), in which a calibrated liquid crystal (LC) device is employed to realize phase shifts of $\delta _n^0 = 0,\pi /2,\pi ,3\pi /2\;rad$. The curvature radius of reference beam is adjusted to generate circular interference fringe on Cam1, when the fringe number ${n_f} = 5$ and ${n_f} < 1$, we perform the phase shifts measurement and phase measurement, respectively.

When the fringe number ${n_f} = 5$, Figs. 4(a) and 4(b) show the phase measurement interferogram on Cam1 and the corresponding phase shifts measurement interferogram (${f_{x0}} = 16.304m{m^{ - 1}}$ and ${f_{y0}} = 17.713m{m^{ - 1}}$) on Cam2, respectively. The white dashed rectangle (200 × 200 pixels) in Fig. 4(b) is chosen to calculate the phase shifts, and its spatial spectrum is shown in Fig. 4(c). The results of phase shifts measurement are ${\delta _n} = 0,\;1.535,\;3.943,\;5.270\;rad$. The phases retrieved by PSDHA, LSA, and AIA are presented in Figs. 4(d), 4(e) and 4(f), respectively. The difference between the phases obtained by PSDHA and LSA is shown in Fig. 4(g), and the difference between PSDHA and AIA is shown in Fig. 4(h).

 figure: Fig. 4.

Fig. 4. (a) A phase measurement interferogram of NPCs when fringe number nf =5; (b) the corresponding phase shifts measurement interferogram of (a); (c) the spatial spectrum of the white dashed rectangle in (b); the phases retrieved with different algorithms (d) PSDHA; (e) LSA; (f) AIA; the differences of phase retrieval between (g) PSDHA and LSA; (h) PSDHA and AIA.

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As shown in Figs. 4 (d)-4(h), in the case of a large fringe number nf =5, the consistency between the three algorithms (PSDHA, LSA and AIA) is very good, and the RMSE between the two is less than 0.01 rad, indicating that the accuracy of phase retrieval with the proposed method is very high.

Similarly, when the fringe number ${n_f} < 1$, Figs. 5(a) and 5(b) show the phase measurement interferogram on Cam1 and the corresponding phase shifts measurement interferogram (${f_{x0}} = 17.915m{m^{ - 1}}$ and ${f_{y0}} = 16.908m{m^{ - 1}}$) on Cam2, respectively. The white dashed rectangle (200 × 200 pixels) in Fig. 5(b) is chosen to calculate the phase shifts, and its spatial spectrum is shown in Fig. 5(c). And the results of phase shifts measurement are ${\delta _n} = 0,\;1.571,\;4.015,\;5.416\;rad$. The phases retrieved by PSDHA, LSA, and AIA are presented in Figs. 5(d), 5(e) and 5(f), respectively. The difference between the phases obtained by PSDHA and LSA is shown in Fig. 5(g), and the difference between PSDHA and AIA is shown in Fig. 5(h).

 figure: Fig. 5.

Fig. 5. (a) A phase measurement interferogram of NPCs when fringe number nf <1; (b) the corresponding phase shifts measurement interferogram of (a); (c) the spatial spectrum of the white dashed rectangle in (b); the phases retrieved with different algorithms (d) PSDHA; (e) LSA; (f) AIA; the differences of phase retrieval between (g) PSDHA and LSA; (h) PSDHA and AIA.

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Clearly, when nf <1, the phases obtained with the PSDHA and LSA are very close, and the RMSE between the two is less than 0.005 rad, this is consistent with the results with large fringe number in the interferogram. However, the phases obtained with the AIA not only have a significant error (RMSE = 0.0522 rad), but are also extremely unstable. The error is associated with the distribution of phases, which results in a significant measurement error of the sample phase.

In our experiment, both Cam1 and Cam2 capture interferograms at a frame rate of 60/s with a shutter time of 50 μs. The synchronization time difference between the two cameras is less than 10 μs. The time interval for each π/2 phase-shifting in LC is less than 10 ms, and the maximum phase-shifting speed during the hysteresis period of the phase shifter is about 20 rad/s. This guarantees that the variation of phase shifts during each interferogram acquisition is less than 0.001 rad, meeting the requirement of phase shifts accuracy. Consequently, the acquisition time for 4-frame phase-shifting interferograms is less than 66 ms, and the corresponding calculation time of phase retrieval are the same as those in Table 1. The calculation time of phase retrieval with the PSDHA, including the calculation time of phase shifts measurement is only 21.7 ms while the LSA takes 86.8 ms. The calculation time with the PSDHA is only 25% of that of LSA, indicating a large improvement of PSDHA in calculation efficiency. Obviously, this is very beneficial for real-time quantitative phase imaging of embryonic stem cells, and it has significant significance for monitoring and studying live biological samples.

Importantly, the above experimental results demonstrate that even in the presence of external disturbance, phase shifter hysteresis, or other factors that may cause inaccurate phase shifts measurement, the proposed adaptive PSI method based on PSDHA can obtain actual phase shifts, ensuring high-accuracy phase retrieval. In all situation, the PSDHA is the least time-consuming high-precision algorithm.

6. Conclusion

In summary, this paper presents an adaptive PSI through synchronously capturing phase shifts measurement interferograms and phase measurement interferograms, in which the former is a series of spatial carrier frequency phase-shifting interferograms generated by an additional assembly and the phase shifts are calculated with a high-accuracy single-spectrum phase shifts measurement algorithm (SS-PSMA), the latter is employed for phase retrieval with an adaptive phase-shifting digital holography algorithm (PSDHA) based on complex amplitude recovery. This adaptive PSI method based on PSDHA offers several benefits of notable capacity to reduce the influence of external disturbance and inaccurate phase shifts measurement, consequently guaranteeing and enhancing the accuracy and efficiency of phase retrieval. Specially, it can obtain accurate phase shifts during the phase-shifter hysteresis, significantly reducing the acquisition time of phase-shifting interferograms. In addition to taking short calculation time, the proposed method is suitable for various interferograms with different fringe shapes and numbers. Importantly, the proposed adaptive PSI method based on PSDHA addresses long-standing problems in PSI, such as external disturbance, phase shifter inaccuracy, long-time acquisition for phase-shifting interferograms, and certain limitations in existing phase-shifting algorithms. It not only guarantees and enhances the accuracy of phase measurement, but also presents promising prospect for its application in dynamic phase measurement.

Funding

National Natural Science Foundation of China (62275083, 62175041, 61727814, 62335002).

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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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 (5)
Fig. 1.
Fig. 1. Sketch of the proposed adaptive PSI system. P1-3: polarizer; LC: liquid crystal phase shifter; QWP1,2: quarter wave plate; PBS1,2: polarizing beam splitter; BS: beam splitter; M1-3: mirror; S: sample; Obj1,2: objective (NA = 0.4, 20×); TL1,2: tube lens; WP: Wollaston prism; L: lens; Cam1,2: CMOS camera.

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Fig. 2.
Fig. 2. Comparison of phase retrieval accuracy with different algorithms (PSDHA, LSA, AIA and 4-PSA). (a) Reference phase; (b) a phase-shifting interferogram for phase measurement when the fringe number nf =5; (c) the carrier-frequency phase-shifting interferogram for phase shifts measurement of (b); (d) the spatial spectrum of the white dashed rectangle in (c); (e) a phase-shifting interferogram for phase measurement when the fringe number nf <1; (f) the carrier-frequency phase-shifting interferogram for phase shifts measurement of (e); (g) the spatial spectrum of the white dashed rectangle in (f); (h)(i) comparison of phase retrieval accuracy with different algorithms when the PSD are Δδn = 0, −0.7485, −0.2319, −0.6267 rad for nf =5 and nf <1, respectively.

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Fig. 3.
Fig. 3. Influence of PSD on the accuracy of phase retrieval with different algorithms (a) nf =5; (b) nf < 1.

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Fig. 4.
Fig. 4. (a) A phase measurement interferogram of NPCs when fringe number nf =5; (b) the corresponding phase shifts measurement interferogram of (a); (c) the spatial spectrum of the white dashed rectangle in (b); the phases retrieved with different algorithms (d) PSDHA; (e) LSA; (f) AIA; the differences of phase retrieval between (g) PSDHA and LSA; (h) PSDHA and AIA.

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Fig. 5.
Fig. 5. (a) A phase measurement interferogram of NPCs when fringe number nf <1; (b) the corresponding phase shifts measurement interferogram of (a); (c) the spatial spectrum of the white dashed rectangle in (b); the phases retrieved with different algorithms (d) PSDHA; (e) LSA; (f) AIA; the differences of phase retrieval between (g) PSDHA and LSA; (h) PSDHA and AIA.

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Tables (1)

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Table 1. Comparison of the accuracy and calculation time of phase retrieval with different algorithms

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Equations (24)

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I n ( x , y ) = a ( x , y ) + b ( x , y ) cos ⁡ [ φ ( x , y ) + δ n ] ,
I n = [ I n , 1 I n , 2 ⋮ I n , m ⋮ I n , M ] = [ a 1 + b 1 cos ⁡ ( φ 1 − δ n ) a 2 + b 2 cos ⁡ ( φ 2 − δ n ) ⋮ a m + b m cos ⁡ ( φ m − δ n ) ⋮ a M + b M cos ⁡ ( φ M − δ n ) ] ,
H = [ I 1 I 2 ⋯ I n ⋯ I N ] .
E = [ e j δ 1 0 e j δ 2 0 ⋯ e j δ n 0 ⋯ e j δ N 0 ] T ,
I c = H E = [ ∑ n = 1 N b 1 cos ⁡ ( φ 1 − δ n ) e j δ n 0 ∑ n = 1 N b 2 cos ⁡ ( φ 2 − δ n ) e j δ n 0 ⋮ ∑ n = 1 N b m cos ⁡ ( φ m − δ n ) e j δ n 0 ⋮ ∑ n = 1 N b M cos ⁡ ( φ M − δ n ) e j δ n 0 ] .
I c R e = [ ∑ n = 1 N b 1 cos ⁡ ( φ 1 − δ n ) cos ⁡ ( δ n 0 ) ∑ n = 1 N b 2 cos ⁡ ( φ 2 − δ n ) cos ⁡ ( δ n 0 ) ⋮ ∑ n = 1 N b m cos ⁡ ( φ m − δ n ) cos ⁡ ( δ n 0 ) ⋮ ∑ n = 1 N b M cos ⁡ ( φ M − δ n ) cos ⁡ ( δ n 0 ) ] , a n d I c Im = [ ∑ n = 1 N b 1 cos ⁡ ( φ 1 − δ n ) sin ⁡ ( δ n 0 ) ∑ n = 1 N b 2 cos ⁡ ( φ 2 − δ n ) sin ⁡ ( δ n 0 ) ⋮ ∑ n = 1 N b m cos ⁡ ( φ m − δ n ) sin ⁡ ( δ n 0 ) ⋮ ∑ n = 1 N b M cos ⁡ ( φ M − δ n ) sin ⁡ ( δ n 0 ) ] .
[ A 11 A 12 A 21 A 22 ] = [ cos ⁡ δ 1 cos ⁡ δ 2 ⋯ cos ⁡ δ n ⋯ cos ⁡ δ N sin ⁡ δ 1 sin ⁡ δ 2 ⋯ sin ⁡ δ n ⋯ sin ⁡ δ N ] [ cos ⁡ δ 1 0 sin ⁡ δ 1 0 cos ⁡ δ 2 0 sin ⁡ δ 2 0 ⋮ ⋮ cos ⁡ δ n 0 sin ⁡ δ n 0 ⋮ ⋮ cos ⁡ δ N 0 sin ⁡ δ N 0 ] .
I c R e = α R e [ b 1 cos ⁡ ( φ 1 + Δ R e ) b 2 cos ⁡ ( φ 2 + Δ R e ) ⋮ b m cos ⁡ ( φ m + Δ R e ) ⋮ b M cos ⁡ ( φ M + Δ R e ) ] , and I c Im = α Im [ b 1 cos ⁡ ( φ 1 + Δ Im ) b 2 cos ⁡ ( φ 2 + Δ Im ) ⋮ b m cos ⁡ ( φ m + Δ Im ) ⋮ b M cos ⁡ ( φ M + Δ Im ) ] ,
α R e = ( A 11 2 + A 21 2 ) 1 2 ,
α Im = ( A 12 2 + A 22 2 ) 1 2 ,
Δ R e = arctan ⁡ ( A 21 / A 11 ) ,
Δ Im = arctan ⁡ ( A 22 / A 12 ) .
S = I c R e / α R e + I c Im / α Im = 2 cos ⁡ [ ( Δ Im − Δ R e ) / 2 ] [ b 1 cos ⁡ ( φ 1 + ( Δ Im + Δ R e ) / 2 ) b 2 cos ⁡ ( φ 2 + ( Δ Im + Δ R e ) / 2 ) ⋮ b m cos ⁡ ( φ m + ( Δ Im + Δ R e ) / 2 ) ⋮ b M cos ⁡ ( φ M + ( Δ Im + Δ R e ) / 2 ) ] = [ s 11 s 21 ⋮ s m 1 ⋮ s M 1 ] ,
D = I c R e / α R e − I c Im / α Im = 2 sin ⁡ [ ( Δ Im − Δ R e ) / 2 ] [ b 1 sin ⁡ ( φ 1 + ( Δ Im + Δ R e ) / 2 ) b 2 sin ⁡ ( φ 2 + ( Δ Im + Δ R e ) / 2 ) ⋮ b m sin ⁡ ( φ m + ( Δ Im + Δ R e ) / 2 ) ⋮ b M sin ⁡ ( φ M + ( Δ Im + Δ R e ) / 2 ) ] = [ d 11 d 21 ⋮ d m 1 ⋮ d M 1 ] .
φ m = arctan ⁡ { [ d m 1 / s m 1 ] / tan ⁡ [ ( Δ Im − Δ R e ) / 2 ] } − ( Δ Im + Δ R e ) / 2.
I P S n ( x ′ , y ′ ) = a P S ( x ′ , y ′ ) + b P S ( x ′ , y ′ ) cos ⁡ [ 2 π f x 0 x ′ + 2 π f y 0 y ′ + φ P S ( x ′ , y ′ ) + δ n ] = a P S ( x ′ , y ′ ) + 0.5 b P S ( x ′ , y ′ ) exp ⁡ { j [ 2 π f x 0 x ′ + 2 π f y 0 y ′ + φ P S ( x ′ , y ′ ) + δ n ] } + 0.5 b P S ( x ′ , y ′ ) exp ⁡ { − j [ 2 π f x 0 x ′ + 2 π f y 0 y ′ + φ P S ( x ′ , y ′ ) + δ n ] } ,
I P S n ( f x , f y ) = I P S ( f x , f y ) + C P S ( f x , f y ) exp ⁡ ( j δ n ) ∗ δ ( f x − f x 0 , f y − f y 0 ) + c o n j [ C P S ( f x , f y ) exp ⁡ ( j δ n ) ] ∗ δ ( f x + f x 0 , f y + f y 0 ) ,
I P S n + 1 ( f x , f y )  =  C P S ( f x − f x 0 , f y − f y 0 ) exp ⁡ ( j δ n ) .
δ n = arg ⁡ [ I P S n + 1 ( f x 0 , f y 0 ) / I P S 1 + 1 ( f x 0 , f y 0 ) ] .
O ( x , y ) = ( 11 / 2 ) exp ⁡ [ − 0.05 ( x 2 + y 2 ) / 2 ] exp ⁡ [ j φ ( x , y ) ] ,
R n ( x , y ) = ( 11 / 2 ) exp ⁡ [ − 0.05 ( x 2 + y 2 ) / 2 ] exp ⁡ { j [ n f 2 π ( x 2 + y 2 ) / 9.641 + δ n ] } ,
I n ( x , y ) = 121 exp ⁡ [ − 0.05 ( x 2 + y 2 ) ] { 1 + cos ⁡ [ φ ( x , y ) − n f 2 π ( x 2 + y 2 ) / 9.641 − δ n ] } ,
R P S n ( x , y ) = ( 11 / 2 ) exp ⁡ [ − 0.05 ( x 2 + y 2 ) / 2 ] exp ⁡ { j [ 2 π n f ( x 2 + y 2 ) / 9.641 + 2 π f x x + 2 π f y y + δ n ] } .
I P S n ( x , y ) = 121 exp ⁡ [ − 0.05 ( x 2 + y 2 ) ] { 1 + cos ⁡ [ φ ( x , y ) − 2 π n f ( x 2 + y 2 ) / 9.641 − 2 π ( f x x + f y y ) − δ n ] } .
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