Single-shot higher-order transport-of-intensity quantitative phase imaging based on computer-generated holography

Naru Yoneda; Naru Yoneda; Naru Yoneda; Aoi Onishi; Aoi Onishi; Yusuke Saita; Koshi Komuro; Takanori Nomura

doi:10.1364/OE.415598

1. Introduction

In the biomedical field, there is a demand for visualizing transparent objects such as living cells with minimal invasion. Although fluorescent imaging is widely used, chemical staining causes photobreaching and phototoxicity. Zernike phase [1] and differential interference contrast microscopies [2] have been proposed for visualizing phase distributions and are applicable to label-free imaging of phase objects. However, these methods can only qualitatively detect phase distributions. Quantitative phase imaging (QPI) is required to measure the phase distributions of transparent objects in detail [3]. The quantitative phase distributions of target samples have valuable information and can be used to extract various physical quantities, such as the refractive index, thickness [4], dry mass [5], disorder strength [6], shear stress [7], aberrations [8], molecular vibration [9], and orbital angular momentum [10]. QPI can be performed with the interferometric or non-interferometric approach. Digital holography (DH) is an interferometric method that can quantitatively detect phase information [11–13]. In general, DH-based QPI requires a coherent light source and interferometer. However, a coherent light source causes speckle noise and the interferometer makes the optical setup complicated [14]. In contrast, non-interferometric methods employ a partially coherent light source, which can mitigate the speckle problem. Additionally, non-interferometric methods have a simple and compact optical setup than interferometric methods. Although a Shack–Hartmann wavefront sensor can measure a wavefront or phase information in real time without an interferometer [15–17], the spatial resolution of the detected information depends on the lens pitch of the lenslet array. Transport of intensity equation (TIE) based QPI is also one of non-interferometric techniques [18–20]. In TIE-based QPI, the phase distribution is obtained from the intensity derivative along the optical axis. The accuracy of the phase measurement can be improved using a higher-order derivative for the approximation with multiple defocused intensity distributions [21,22]. Because of the flexibility of TIE-based QPI, it has been applied to the fields of tomography [23,24], fluorescent imaging [25], single-pixel imaging [26,27], phase unwrapping [28,29], autofocusing [30,31], and optical memory [32,33].

For TIE-based QPI, the finite difference approximation with two or more defocused intensity distributions is often used to approximate the intensity derivative along the optical axis. Detecting multiple defocused intensity distributions generally requires axially scanning an image sensor or object, which demands precise alignment of a setup. Multiple defocused intensity distributions can be obtained without axial scanning using chromatic aberration [34], variable focus lens [35], or geometric phase lens [36]. However, these approaches require physically replacing the band pass filter, changing the focal length, or a polarized direction. Refractive index variation makes it unnecessary to axially scan an image sensor or change the parameters of optical elements [37,38]. However, the sequential detection process still limits the imaging frame rate. There are other works combining digital holography with the TIE to achieve single-shot phase retrieval [39–41]. However, the digital holography based techniques require a coherent light source and an interferometer. Alternatively, single-shot techniques have been proposed, such as using a volume hologram [42], chromatic aberrations and an RGB camera [43], tilted mirror and spatial light modulator (SLM) [44], two cameras [45], flipping imaging module [46], or distorted grating [47,48]. While these methods can simultaneously detect multiple intensity distributions, the number of intensity distributions and the defocus distance are fixed. Because the optimal defocus distance depends on the phase distributions of objects, a system that can flexibly vary the defocus distance is preferable.

In this paper, we introduce a computer-generated hologram (CGH) technique to TIE-based QPI. By inserting a CGH displayed on an SLM to the Fourier plane of measurement targets, multiple defocused intensity distributions can be simultaneously detected for a higher-order approximation. The proposed method allows the defocus distance to be flexibly changed owing to a CGH generated from arbitrary defocus point spread functions (PSFs). We term the proposed technique as single-shot higher-order transport-of-intensity (SHOT) QPI. In similar methods, a distorted grating is displayed on an SLM [49,50] to realize multiplane intensity imaging [51,52]. In contrast, SHOT-QPI uses the TIE to detect not only multiplane images but also a quantitative phase distribution. Moreover, SHOT-QPI has a potential for multimodal imaging because general TIE-based QPI can reconstruct not only phase images but also phase contrast and difference interference images [53] after the phase measurement process.

2. Principle of single-shot higher-order transport-of-intensity quantitative phase imaging

2.1 Overview of phase imaging using the transport-of-intensity equation with higher-order approximation

The complex amplitude distribution of an object at $z=z_{0}$ is described as follows:

(1)$$u(x,y;z_{0})=\sqrt{I(x,y;z_{0})} \textrm{exp} \left\{ i \phi(x,y;z_{0}) \right\},$$

where $\phi (x,y;z_{0})$ and $I(x,y;z_{0})$ denote the phase and intensity distributions, respectively, of an object at $z=z_{0}$. The TIE relates the phase distribution of an object to its intensity derivative along the optical axis [18]:

(2)$$\nabla_{{\perp}}\cdot\{I(x,y;z_{0})\nabla_{{\perp}}\phi(x,y;z_{0})\}={-}k\frac{\partial I(x,y;z_{0}) }{\partial z},$$

where $\nabla _{\perp }$ and $k$ are the gradient operator in the lateral dimensions $(x,y)$ and the wavenumber, respectively. When an object intensity distribution $I(x,y;z_{0})$ is uniform, (i.e., a pure-phase object), Eq. (2) is regarded as a two-dimensional Poisson equation and can be rewritten as follows:

(3)$$\nabla^{2}_{{\perp}}\phi(x,y;z_{0})={-}\frac{k}{I_{0}}\frac{\partial I(x,y;z_{0}) }{\partial z},$$

where $I_{0}$ has a constant value. In the general case where the object intensity distribution is not uniform, the phase distribution can be obtained by solving two Poisson equations with an auxiliary function [54]. Equation (3) is solved for $\phi (x,y;z_{0})$ through a Fourier-transform-based method [54,55]:

(4)$$\phi(x,y;z_{0})=\frac{k}{I_{0}}\textrm{IFT}\Biggl[\frac{1}{4\pi^{2}(\mu^{2}+\nu^{2})+\alpha}\textrm{FT}\Biggl[\frac{\partial I(x,y;z_{0}) }{\partial z}\Biggr]\Biggr],$$

where $\textrm {FT}[\cdots ]$, $\textrm {IFT}[\cdots ]$, and $(\mu ,\nu )$ are the Fourier transform operator, and inverse Fourier transform operator, spatial frequency components along the $x$ and $y$ directions, respectively. $\alpha$ is a regularization parameter to avoid division by zero [56]. The intensity derivative in Eq. (4) is estimated using the Taylor expansion and then approximated as follows:

(5)$$\frac{\partial I(x,y;z_{0}) }{\partial z}\approx \sum_{j={-}n}^n\frac{a_{j}I(x,y;j\Delta z)}{\Delta z},$$

where $j=-n,-(n-1),\ldots 0,\ldots ,n-1,n$ and $2n$ is the number of defocused intensity distributions. $a_{j}$ is the coefficient of $I(x,y;j\Delta z)$. $\Delta z$ denotes a defocus interval. Various approximation methods have been proposed to improve the phase measurement accuracy [22,57,58]. In this paper, we employ basic higher-order polynomial fitting [21] to calculate the coefficients. Other approximation methods are also applicable to SHOT-QPI.

2.2 Multi-blurred image detection based on a computer-generated hologram

For SHOT-QPI, a CGH is used to simultaneously detect multiple defocused intensity distributions on the image sensor plane. A Fourier type CGH is inserted in the Fourier plane of an object as shown in Fig. 1. A CGH is an interference fringe pattern between the signal and reference beams. In this case, a defocused PSF based on the angular spectrum method [59] is the signal beam and is described as follows:

(6)$$\begin{aligned} p(x,y) &= \textrm{FT}[P(\mu,\nu)]\\ &=\textrm{FT}\left[\textrm{exp}\Biggl\{i2\pi \Delta z\sqrt{\frac{1}{\lambda^{2}}-\mu^{2}-\nu^{2}}\Biggr\}\right], \end{aligned}$$

where $\lambda$ is the wavelength. In the proposed method, PSFs are located away from the origin on the $(x,y)$ plane so that multiple different defocused intensity distributions can be simultaneously obtained. A laterally shifted PSF $p_{j}(x,y)$ as shown in Fig. 2(a) is described as follows:

(7)$$\begin{aligned} p_{j}(x,y) = & p(x,y) \ast \delta(x-x_{j},y-y_{j})\\ = &\textrm{FT}[P_{j}(\mu,\nu)]\\ =&\textrm{FT}\Biggl[\textrm{exp}\Biggl\{i2\pi j\Delta z\sqrt{\frac{1}{\lambda^{2}}-\mu^{2}-\nu^{2}}\Biggr\}\textrm{exp}\left\{i2\pi\left(x_{j}\mu + y_{j}\nu\right)\right\}\Biggr], \end{aligned}$$

where $\delta (x,y)$, $x_{j}$ and $y_{j}$ are the Dirac’s delta function and center position of the PSF in the $x$ and $y$ directions, respectively. The interference fringe pattern between the Fourier spectrum of a defocused PSF and the plane wave can be described as follows:

(8)$$\begin{aligned} H'(\mu,\nu)=&\ |P_{j}(\mu,\nu)+1|^{2}\\ =&\ |P_{j}(\mu,\nu)|^{2}+1+P_{j}(\mu,\nu)+P_{j}^{{\ast}}(\mu,\nu), \end{aligned}$$

where $^*$ indicates a complex conjugate. In Eq. (8), because the first and second terms are unnecessary, a CGH is generated by extracting only the interference term and substituting the minimum value $I_{\textrm {b}}$ for this term. The CGH is then described as follows:

(9)$$H^{\prime\prime}(\mu,\nu)=\ P_{j}(\mu,\nu)+P_{j}^{{\ast}}(\mu,\nu)-I_{\textrm{b}}.$$

When a CGH is generated from multiple different PSFs, it is described as follows:

(10)$$H(\mu,\nu)=\ \sum_{j=1}^n \left\{P_{j}(\mu,\nu)+P_{j}^{{\ast}}(\mu,\nu)\right\}-I'_{\textrm{b}},$$

where $I'_{\textrm {b}}$ is the minimum value of the first term in Eq. (10). The CGH illuminated by a plane wave is used to reconstruct the PSFs on the Fourier plane as shown in Fig. 2(c).

Fig. 1. Schematic diagram for SHOT-QPI. (a) Optical configuration, (b) an intensity distribution at an image sensor plane, and (c) reconstruction process of a phase distribution through higher-order TIE.

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Fig. 2. Schematic diagram for the generation and reconstruction processes of a CGH: (a) generation process of a PSF distributed away from the origin, (b) generation process of the CGH, and (c) reconstruction process when the plane wave is illuminated to the CGH.

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In SHOT-QPI, the CGH is displayed on an SLM and is inserted to the back focal plane of lens 1, as shown in Fig. 1(a). The complex amplitude distribution at the plane is described as follows:

(11)$$S(\mu,\nu)=U(\mu,\nu)H(\mu,\nu),$$

where $U(\mu ,\nu )$ is the Fourier transform of the object beam $u(x,y,z_{0})$. From Eq. (11), the complex amplitude distribution $s(x,y)$ on the back focal plane of lens 2 in Fig. 1(a) is described as follows:

(12)$$s(x,y)=u(x,y) \ast h(x,y),$$

where $h(x,y)$ and $\ast$ are the inverse Fourier transform of $H(\mu ,\nu )$ and the convolution operator, respectively. The intensity distribution on the image sensor plane is described as follows:

(13)$$\begin{aligned} I'(x,y)=&|u(x,y) \ast h(x,y)|^{2}\\ =&\Biggl|\ \sum_{j=1}^n \Biggl\{u(x,y) \ast p_{j}(x,y) +u(x,y) \ast p_{j}^{{\ast}}(x,y)\Biggr\} -I'_{\textrm{b}}u(x,y) \ast \delta(x, y)\Biggr|^{2}. \end{aligned}$$

When the field of view for an object is appropriately designed, the individual terms in Eq. (13) are spatially separated as shown in Fig. 1(b). Therefore, these terms do not interfere with each other and the cross terms can be neglected. Equation (13) can be rewritten as follows:

(14)$$\begin{aligned} I'(x,y) =&\sum_{j=1}^n \Biggl\{|u(x,y) \ast p_{j}(x,y)|^{2}+|u(x,y) \ast p_{j}^{{\ast}}(x,y)|^{2}\Biggr\}+|-I'_{\textrm{b}}u(x,y) \ast \delta(x, y)|^{2}\\ =& \sum_{j=1}^n \Biggl\{ I(x,y;j\Delta z) + I(x,y;-j\Delta z)\Biggr\} + I'_{\textrm{b}}I(x,y;0). \end{aligned}$$

On the right-hand side of Eq. (14), the first term indicates multiple defocused intensity distributions and the second term is the in-focus intensity distribution multiplied by $I'_{\textrm {b}}$. Equation (14) shows that multiple defocused intensity distributions can be simultaneously obtained.

3. Numerical simulation

3.1 Simulation conditions

The proposed method was numerically evaluated with the optical setup shown in Fig. 1(a). The wavelength of the light source was 532 nm. The Fourier transforms by lens 1 and 2 were calculated using the fast Fourier transform algorithm. The magnification factor of the 4-$f$ configuration was unity. The pixel pitch of the image sensor was 12.5 $\mu$m. The measurement target comprised 160 $\times$ 160 pixels. The defocus interval $\Delta z$ was set to 1 mm. Eight defocused intensity distributions were obtained. Pseudorandom Gaussian noise with a mean value of 0 and standard deviation of 2% were added to the defocused intensity distributions, which represented hard noise conditions [60]. The regularization parameter $\alpha$ in Eq. (4) was set to $10^{-7}$. The phase measurement accuracy was evaluated using the root mean squared error (RMSE) as expressed below:

(15)$$\textrm{RMSE} = \sqrt{\dfrac{1}{XY}\sum_{x=1}^{X}\sum_{y=1}^{Y}\left\{\varphi_m(x,y)-\varphi_t(x,y)\right\}^2},$$

where $X$ and $Y$ are the number of pixels along the $x$ and $y$ directions, respectively. $\varphi _m(x,y)$ and $\varphi _t(x,y)$ are the measured and true phase distributions, respectively.

3.2 Simulation results

The targets are shown in Fig. 3(a)-(c). Three different objects were used to evaluate whether there is a correlation between quality and approximation methods. Two conventional approximation methods were used for comparison: general finite difference approximation with two defocused intensity distributions and first-order polynomial fitting with eight defocused intensity distributions. For the conventional methods, the defocused intensity distributions were detected by scanning an image sensor with the angular spectrum method. For the proposed method, first-order polynomial fitting was used. The reconstructed phase distributions with these conventional methods and the proposed method are shown with their RMSEs in Fig. 3(d)-(i) and 3(j)-(o), respectively. Although the proposed method had a lower RMSE than the conventional method with the finite difference approximation, it had a higher RMSE than the conventional method with the higher-order approximation. The degradation of the RMSE may have been caused by the CGH including sampling errors for the PSFs. The errors can be calibrated by subtracting a background phase; the results are shown in Fig. 3(m)-(o). The calibration is explained in detail in the following subsection.

3.3 Calibration of point spread functions

The residual errors between the theoretical PSF and reconstructed PSFs from a CGH were investigated. To evaluate the dependence on the diffracted position, a CGH was generated from four defocused PSFs placed at different transverse positions, as shown in Fig. 4(a). The four defocused PSFs from the CGH had the same defocus interval of 1 mm. The theoretical PSF is shown in Fig. 4(b). The error distributions between the theoretical and reconstructed PSFs are shown with their RMSEs in Fig. 4(c)-(f). The distributions of each reconstructed PSF slightly differed from that of the theoretical PSF. The results indicated that the error distribution depends on the lateral positions diffracted from the CGH. The difference would be caused by the effect of the discrete sampling. Notably, the error distribution at position (d) had a different shape compared with the other error distributions because position (d) was affected by the higher-order diffracted components from the reconstructed PSFs at positions (c) and (e).

Fig. 3. Phase distributions (a)-(c) measurement targets, (d)-(f) conventional method with two defocused intensity distributions, (g)-(i) conventional method with eight defocused intensity distributions, (j)-(l) proposed method with a background phase, and (m)-(o) proposed method without a background phase.

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Fig. 4. Difference between the theoretical PSF and reconstructed PSFs from a CGH: (a) diffracted position of PSFs, (b) theoretical PSF, and (c)-(f) reconstructed PSFs and error map compared to positions in (a).

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Because the shapes of the reconstructed PSFs from the CGH depended on the positions, the measurement accuracy was degraded as shown in Fig. 3(j)-(l). Firstly, to assess the effect of the difference of the theoretical and actual reconstructed PSFs, the measument accuracy is evaluated with Fig. 3(b) under the condition without noise. Figure 5(a) and (b) show the numerical simulation results without noise in conventional and SHOT-QPI, respectively. For the results, SHOT-QPI has some large error, which indicates SHOT-QPI has the trade-off between the simplicity and the measurement accuracy.

Fig. 5. Simulation results without noise. (a) conventional method with eight defocused intensity distributions, (b) SHOT-QPI with a background phase, (c) a background phase, and (d) SHOT-QPI subtracting a background phase.

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The cause for the accuracy degradation is considered to be an intrinsic property of the optical system even if the measured object is changed. With this prediction, the phase distribution without the object is measured in advance. In this paper, the pre-measured phase distribution is called as the background phase. To mitigate the degradation due to the differences in the reconstructed PSFs, a background phase distribution was obtained first and then subtracted from the reconstructed phase distributions because the background phase preserved errors from the reconstructed PSFs. We predicted that subtracting the background phase distribution as a degradation factor would improve the measurement accuracy.

As the example, a background phase without noise is shown in Fig. 5(c), and the measured phase subtracting the background phase is shown in Fig. 5(d). In addition to the case without noise, we evaluate the effect of subtracting the background phase under the case with noise. Figure 3(m)-(o) shows that the measurement accuracy could be improved by subtracting the background phase. Note that the proposed method without the background phase still had a lower RMSE than the conventional method with the higher-order approximation. This indicates that subtracting the background phase distribution does not fully improve the accuracy. The measurement accuracy can be further improved using a CGH optimization method [61,62]. The additional optimization method such as the iterative method could be helpful to improve the measurement accuracy for SHOT-QPI.

4. Optical experiment

4.1 Experimental conditions

An optical experiment was demonstrated to verify the feasibility of the proposed method. The experimental setup is shown in Fig. 6(a). A green light-emitting diode (LED) (Ushio Inc. SugarCUBE) with a center wavelength of 523 nm and the full width at half maximum of 39 nm was used as a light source to suppress coherent speckle noises. A band pass filter was used to increase the temporal coherence and it had a center wavelength and width of 532 and 3 nm, respectively. A pinhole with the diameter of 400 $\mu$m was placed just after the LED to increase the spatial coherence. Note that the TIE under the coherent condition (i.e., the original equation in Eq. (4)) was used in this experiment, although we used a partially coherent light source. Therefore, introducing the TIE to a partially coherent light source should improve the phase measurement accuracy [63–65]. A phase-only SLM 1 (Hamamatsu Photonics K.K. X13138-01) was used as the measurement target for the experiment, which is the same as the object used for the numerical simulation described in Section 3. The number of pixels and the pixel pitch of SLM 1 were 1272 $\times$ 1024 and 12.5 $\mu$m, respectively. The object beam was projected to an aperture plane with a 4-$f$ setup. An aperture with dimensions of 2.6 mm $\times$ 2.6 mm was used to limit the region of interest of the objects. The 4-$f$ setup with a magnification factor of 2 consisted of lens 2 and 3 with focal lengths of 300 and 150 mm, respectively. The object beam was Fourier transformed by lens 4 and, then, modulated by SLM 2 (Holoeye GAEA) displaying a CGH. The number of pixels and the pixel pitch of SLM 2 were 3140 $\times$ 2160 and 3.74 $\mu$m, respectively. The diffraction angles of the reconstructed PSFs from the CGH with respect to the $x$ and $y$ axes were set to 1.34 degree to separate each defocused intensity distribution. The diffracted intensity distributions were detected with a complementary metal-oxide-semiconductor (CMOS) camera (Hamamatsu Photonics K.K. C11440-52U30) with a resolution of 2048 $\times$ 2048 pixels and a pixel pitch of 6.5 $\mu$m. The 4-$f$ setup with a magnification factor of 2 consisted of lens 4 and 5 with focal lengths of 300 and 150 mm, respectively.

Fig. 6. Experimental conditions: (a) optical setup, (b) amplitude modulation property of SLM 2, (c) the CGH, and (d) the background phase. HWP, half-wave plate; BS, beam splitter; SF, spatial filter; L, lens; A, aperture; P, polarizer; BF, band pass filter.

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Here, we describe how we modulated the amplitude distribution according to the CGH with SLM 2. A transparent SLM is widely used to display an amplitude CGH [32,66,67]. However, the SLM has the disadvantage of the small fill factor, which may affect the measurement accuracy. In order to prevent the affect by the small fill factor, we used the reflective SLM as an amplitude-only SLM. Since a liquid crystal on silicon (LCoS) SLM has a polarization dependence, this property can be used to modulate the amplitude distribution by combining the half-wave plate (HWP) and polarizer [68]. When the horizontal polarization component can be modulated by the SLM, the vertical component are not modulated. These polarization components are aligned by the polarizer and interfere with each other. The interference generates the constructive and destructive points, which leads to the amplitude modulation. First, the polarization direction of the object beam was diagonally rotated with a half-wave plate (HWP). In a prior experiment, we evaluated the amplitude modulation property of SLM 2, as shown in Fig. 6(b). In the prior experiment, the patterns at SLM 2 was projected to the image sensor with a 4-$f$ setup. By displaying the blank pattern with phase values of 0 to 2$\pi$ sequentially, the modulation property of SLM2 was obtained. As shown in Fig. 6(b), the modulation property has slight fluctuations and depends on the positron at the SLM plane. Because SLM 2 is placed at the Fourier plane of the object, the center position of SLM 2 modulates most of incident beams. Therefore, the modulation property at the center position was used for the fitting. The amplitude distribution of the CGH was fitted to the modulation property, as shown in Fig. 6(c).

4.2 Experimental results

One of the detected intensity distributions is shown in Fig. 7(a). The defocus interval of these defocused intensity distributions was 0.5 mm. These defocused intensity distributions are extracted according to the diffracted positions which are arbitrarily determined in the generation process of the CGH. If there is no prior information, correlation-based techniques [69,70] could be useful to obtain the position information. Each defocused intensity distribution had a ring-shaped artifact. These artifacts were caused by the specific distortion of SLM 1. Although these artifacts affected the defocused intensity distributions, a phase distribution could clearly be reconstructed because the artifacts could be reduced by subtracting the background phase distribution.In addition to the ring-shaped artifact caused by SLM 1, the background phase distribution shown in Fig. 6(d) contains the aberrations of the optical system. Therfore, the aberration can be also compensated by subtracting the background phase.

Fig. 7. Experimental results: (a) one of the detected intensity distributions, (b)-(d) results for the conventional finite difference approximation with two defocused intensity distributions, and (e)-(g) results with the first-order polynomial fitting.

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To compare the imaging quality, phase distributions were reconstructed through the conventional finite difference approximation with two defocused intensity distributions and the first-order polynomial approximation with eight defocused intensity distributions. The results are shown in Figs. 7(b)-(d) and (e)-(g), respectively. The first-order polynomial fitting with eight defocused intensity distributions reconstructed the phase distribution with higher quality than the finite difference approximation with two defocused intensity distributions.

4.3 Comparison of approximation methods

For the conventional TIE-based QPI, the finite difference approximation with two defocused intensity distributions improves the contrast of the reconstructed phase distribution using a longer defocus interval but the higher frequency components vanish [60]. Therefore, we also evaluated the relationship between the defocus interval and approximation method. The defocus interval was set to 0.25, 0.5, 1.0, and 2.0 mm. For the approximation methods, the finite difference approximation with two defocused intensity distributions, first-order polynomial approximation, and eighth-order polynomial approximation with eight defocused intensity distributions were considered to investigate trends for the difference in approximations and polynomial order. The reconstructed phase distributions are shown in Fig. 8. For the first-order polynomial approximation, reducing the defocus interval increased the sharpness of the reconstructed phase distribution. This is because the coefficients of the defocused intensity distributions increased with the defocus distance. However, for the eighth-order polynomial approximation, reconstructed phase distributions with a shorter defocus interval could not be distinguished as the measurement target. The degradation was caused by overfitting. These results indicate that lower-order approximation is preferable with a shorter defocus interval and vice versa. In contrast, for the finite difference approximation with two defocused intensity distributions, artifacts always overlapped at lower spatial frequencies. These results indicate the effectiveness of the higher-order approximation.

Fig. 8. Experimental evaluation of the relationship between the defocus interval and approximation method.

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4.4 Effectiveness of SHOT-QPI for various phase differences

We also evaluated the effectiveness of SHOT-QPI with various phase differences. The measurement target was set to the distribution shown in Fig. 3(a). The phase differences were set to $\pi /2$, $\pi$, and $2\pi$. The reconstructed phase distributions with maximum phase differences of $\pi /2$, $\pi$, and $2\pi$ are shown in Fig. 8(a)-(c), respectively. The cross sectional profiles corresponding to Fig. 9(a)-(c) are shown in Fig. 9(d). The defocus interval was set to 0.5 mm, and first-order polynomial fitting was used. The results showed that SHOT-QPI can measure various phase differences. Although the steeper phase gradient of the measurement target in Fig. 9(c) caused a greater error than the other phase differences, the error can be mitigated using other fitting methods [22,57,58].

Fig. 9. Experimental results with different phase differences: (a)-(c) measurement target with a maximum phase difference of $\pi /2$, $\pi$, and $2\pi$, respectively; (d)-(f) reconstructed phase distributions of (a)-(c); and (g) sectional profiles of (a)-(f).

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5. Discussion

5.1 Regularization parameter and defocus distance

In this subsection, we evaluate the effect of the regularization parameter and defocused distance to overcome the low frequency noise (known as the cloudy noise) with the parameters in Section 3. The regularization parameter was changed every $10^{8}$ from $10^{-7}$ to $10^{9}$ and the defocus distance was changed by a factor of 10 from 0.1 to 10 mm as shown in Fig. 10. Actually, this effect has been already evaluated in [60]. From the result, when $\Delta z = 10$ mm, the RMSE was lower than the result in SHOT-QPI. However, as shown in Fig. 10, the high frequency components are lost. This results indicate that the conventional TIE with two defocused intensity distribution does not work well under the highly noise condition. The ambiguity of the regularization parameter would be overcame by solving the nonlinear optimization problems [23,71].

Fig. 10. Simulation results from two defocused intensity distributions in different regularization parameter and defocus interval. The phase distribution surrounded by the red square is the same as Fig. 3(e).

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5.2 Phase measurement accuracy and spatial bandwidth product

In SHOT-QPI, multiple defocused intensity distributions are simultaneously detected at an image sensor. Hence, the field of view for the phase distribution becomes smaller as the number of defocused intensity distributions increases. In other words, there is trade-off between the number of defocused intensity distributions and the spatial resolution. This relationship can be also considered as the trade-off between the phase measurement accuracy and spatial bandwidth product (SBP) [72] because the phase measurement accuracy becomes better as the number of defocused intensity distributions increases [21].

Jingshan and co-authors have proposed one of higher-order TIE-based QPI with Gaussian process regression (GP-TIE) to improve the phase measurement accuracy and to reduce the through-focus intensity distributions. GP-TIE could mitigate the trade-off between the phase measurement accuracy and the SBP in SHOT-QPI because of the ability. In the next subsection, we assess the applicability of GP-TIE in SHOT-QPI.

5.3 Exponentially spaced and equally spaced transport of intensity equation

To mitigate the trade-off between the SBP and the measurement accuracy, GP-TIE is introduced to SHOT-QPI. GP-TIE can reconstruct the phase distribution with lower MSEs under the low signal-to-noise ratio condition compared with other higher-order TIE approaches. GP-TIE can reconstruct the phase distribution with less defocused intensity distributions with exponential defocus interval [73]. Because SHOT-QPI can flexibly reconstruct the defocused intensity distributions with various defocus distances, SHOT-QPI is compatible to GP-TIE.

The measurement accuracy was numerically evaluated. The exponential defocus intervals of 1, 2, 4, and 8 mm were employed for GP-TIE-based SHOT-QPI. For comparison, the result of the first-order polynomial fitting is shown in Fig. 11(a) and this result is the same as Fig. 3(n). The result of GP-TIE-based SHOT-QPI is shown in Fig. 11(b). For the results of the RMSEs, the measurement accuracy can be improved by applying GP-TIE.

Fig. 11. The comparison results. (a) and (b) are the schematic and the reconstructed phase distributions by the first-order polynomial fitting and GP-TIE, respectively.

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5.4 Paraxial and non-paraxial approximation

In this subsection, we discuss the generation process for the CGH. In other multi-plane imaging [51,52], the paraxial approximation model is employed to generate the defocus PSFs. Under the paraxial approximation, Eq. (6) can be rewritten as

(16)$$p(x,y) =\textrm{FT}\left[\textrm{exp}\Biggl\{{-}i\pi\lambda \Delta z\left(\mu^{2}+\nu^{2}\right)\Biggr\}\right].$$

As the examples, the phase distributions when $\Delta z=10$ mm under the non-paraxial and paraxial approximation conditions are shown in Fig. 12(a). From Fig. 12(a), these phase distribution are the almost same. In addition, we evaluate the difference under the condition of Section 3. The evaluation results are shown in Fig. 12(b). The mean values of the differences in Fig. 12(a) and (b) are $1.9\times 10^{-3}$ and $-5.9\times 10^{-6}$, respectively. The standard deviations of the differences in Fig. 12(a) and (b) are $2.0\times 10^{-3}$ and $3.1\times 10^{-5}$, respectively. The results indicate that the same phase distribution can be obtained whether under the paraxial approximation or not.

Fig. 12. Difference between the defocus phase distributions under the non-paraxial and paraxial approximation conditions.

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5.5 Sampling requirement

In SHOT-QPI, a CGH is discrete sampled. Therefore, the modulatable defocused distance is limited by the sampling requirement of an SLM. The sampling condition of the phase distribution is often evaluated in some holographic application [74–76]. In this paper, we adopted the evaluation method in Ref. [74] because the sampling condition is determined by the spherical phase and grating distributions.

When the spherical phase distribution described in Eq. (6) is displayed on an SLM, the sampling condition is described as [74]

(17)$$\Delta p \left|\frac{\partial\phi_{s}(\mu,0)}{\partial \mu} \right|_{max} \leq \pi\ \ \textrm{and}\ \ \Delta p \left|\frac{\partial\phi_{s}(0,\nu)}{\partial \nu} \right|_{max} \leq \pi,$$

where $\Delta p$ and $\phi _{s}$ are the pixel pitch of the SLM and a defocus phase distribution at the SLM plane, where $\phi _{s}$ is described as

(18)$$\phi_{s}(\mu,\nu)=2\pi \Delta z\sqrt{\frac{1}{\lambda^{2}}-\mu^{2}-\nu^{2}}.$$

Therefore, Eq. (17) can be solved for $\Delta z$:

(19)$$\Delta z \leq \frac{\sqrt{\frac{1}{\lambda^{2}}-\left|\mu_{max}\right|^{2}}}{2\left|\mu_{max}\right|\Delta p} \ \ \textrm{and}\ \ \Delta z \leq \frac{\sqrt{\frac{1}{\lambda^{2}}-\left|\nu_{max}\right|^{2}}}{2\left|\nu_{max}\right|\Delta p},$$

where $N$ is the number of pixels of the CGH. In the case of the numerical simulation in Section 3, the maximum defocused distance is 300.7 mm. In addition to the spherical phase distribution, the CGH includes the linear phase distribution. The sampling condition for the linear phase is described as

(20)$$\Delta p \left|\frac{\partial\phi_{l}(\mu,0)}{\partial \mu} \right|_{max} \leq \pi\ \ \textrm{and}\ \ \Delta p \left|\frac{\partial\phi_{l}(0,\nu)}{\partial \nu} \right|_{max} \leq \pi,$$

where $\phi _{l}$ is the linear phase distribution at the SLM plane. $\phi _{l}$ is described as

(21)$$\phi_{l}(\mu,\nu)={-}2\pi\left(x_{j}\mu + y_{j}\nu\right).$$

From Eq. (7), SHOT-QPI has to satisfy the conditions of Eqs. (17) and (20), and the sampling condition in SHOT-QPI is described as

(22)$$\Delta p \left|\frac{\partial\left\{\phi_{s}(\mu,0)+\phi_{l}(\mu,0)\right\}}{\partial \mu} \right|_{max} \leq \pi\ \ \textrm{and}\ \ \Delta p \left|\frac{\partial\left\{\phi_{s}(0,\nu)+\phi_{l}(0,\nu)\right\}}{\partial \nu} \right|_{max} \leq \pi.$$

Eq. (22) can be solved for $\Delta z$:

(23)$$\Delta z \leq \frac{\sqrt{\frac{1}{\lambda^{2}}-\left|\mu_{max}\right|^{2}}}{2\left|\mu_{max}\right|}\left(\frac{1}{\Delta p}-2\left|x_{j}\right| \right) \ \ \textrm{and}\ \ \Delta z \leq \frac{\sqrt{\frac{1}{\lambda^{2}}-\left|\nu_{max}\right|^{2}}}{2\left|\nu_{max}\right|}\left(\frac{1}{\Delta p}-2\left|y_{j}\right| \right),$$

The maxmum diffraction angle is also calculated by solving Eq. (23) for $x_{j}$ and $y_{j}$ with a constant $\Delta z$. In the case of the parameters in Section 3, the maximum defocus distance is 150.3 mm. From the result, the CGH in the proposed method satisfies the sampling condition.

5.6 Alignment of computer-generated hologram

In this subsection, we discuss the alignment of a CGH. The relationship between the CGH miss aligment and the PSFs displacement is shown in Fig. 13. When the CGH is rotated along $\mu$ or $\nu$ axis, the multiple defocused intensity distributions are shifted along $x$ or $y$ axis, respectively because the rotation along $\mu$ or $\nu$ axis is regarded as adding the phase shift of $\textrm {exp}(i2\pi \nu \Delta x)$ or $\textrm {exp}(i2\pi \nu \Delta y)$ to the CGH. When the CGH is rotated along $z$ axis, the PSFs are rotated along $z$ axis because the grating vector of the CGH is changed. Therefore, the multiple defocused intensity distributions are convoluted to the rotated PSFs. If the displacement position is preliminary obtained, the rotation error of the CGH can be neglected. On the other hand, when the CGH is shifted along $x$ or $y$ axis, the phase shift of $\textrm {exp}(i2\pi \Delta \nu x)$ or $\textrm {exp}(i2\pi \Delta \nu y)$ is added to the PSFs. Although this phase shift effect can be useful for interferometric applications [77,78], the effect can be neglected in SHOT-QPI because the phase shift is lost in the intensity detection process. When the CGH is shifted along $z$ axis, the phase shift of $\textrm {exp}(i2\pi \Delta z w)$ is added to the PSFs, where $w$ is the spherical distribution. In addition to the phase shift, the magnification of each defocused intensity distribution is changed, which brings that the intensity derivative cannot be obtained.

In summary, precise alignment of the CGH along $z$ axis is the most important in SHOT-QPI. However, this requirement is only the construction process of the optical setup, not the measurement process. Therefore, SHOT-QPI does not require the precise alignment in the measurement process although the conventional TIE-based QPI requires it for scanning an image sensor or object.

Fig. 13. Schematic of the relationship between the CGH alignment and the PSFs displacement.

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6. Conclusions

In this paper, we proposed SHOT-QPI which applies a CGH technique to a higher-order-TIE-based QPI system. To set up SHOT-QPI, a CGH generated from multiple defocused PSFs distributed away from the origin is inserted in the Fourier plane of a measured object. In the image sensor plane, convolutions between defocused PSFs and the object are reconstructed with the CGH. Therefore, SHOT-QPI can simultaneously detect multiple defocused intensity distributions at the image sensor plane simultaneously. In a numerical simulation, we compared the differences in image quality of the conventional methods and proposed method. The simulation results indicated that the quality of the phase distributions measured by SHOT-QPI was slightly inferior to that of the conventional higher-order-TIE-based QPI. Comparable quality could be obtained by SHOT-QPI with additional optimization of the CGH. Additionally, the phase measurement quality could be improved by subtracting the background phase distribution. An optical experiment was performed to confirm the feasibility of SHOT-QPI. The experimental results indicated that SHOT-QPI is applicable to various fields because the defocus interval and approximation method can be flexibly changed and it can accommodate various phase differences.

In this paper, we did not demonstrate the comparison between the measurement accuracy of SHOT-QPI and that of other single-shot techniques because it is difficult to coordinate the experimental condition. We leave the comparison as future works.

Funding

Japan Society for the Promotion of Science (20J10441).

Disclosures

The authors declare no conflicts of interest.

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Single-shot higher-order transport-of-intensity quantitative phase imaging based on computer-generated holography

Abstract

1. Introduction

2. Principle of single-shot higher-order transport-of-intensity quantitative phase imaging

2.1 Overview of phase imaging using the transport-of-intensity equation with higher-order approximation

2.2 Multi-blurred image detection based on a computer-generated hologram

3. Numerical simulation

3.1 Simulation conditions

3.2 Simulation results

3.3 Calibration of point spread functions

4. Optical experiment

4.1 Experimental conditions

4.2 Experimental results

4.3 Comparison of approximation methods

4.4 Effectiveness of SHOT-QPI for various phase differences

5. Discussion

5.1 Regularization parameter and defocus distance

5.2 Phase measurement accuracy and spatial bandwidth product

5.3 Exponentially spaced and equally spaced transport of intensity equation

5.4 Paraxial and non-paraxial approximation

5.5 Sampling requirement

5.6 Alignment of computer-generated hologram

6. Conclusions

Funding

Disclosures

References

Cited By

Figures (13)

Equations (23)

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