1. Introduction

In optical microscopy, the phase of the light from the object contains important information about the morphology, such as structures in cells and small defects on the surface of semiconductor wafers [1]. For visualizing the phase information, phase microscopy involves the interference between the direct light and the scattered light with an additional π/2 phase shift in phase contrast microscopy (PCM) [2], or interfering with another known reference field in digital holographic microscopy (DHM) [3], or interference between two sheared beams from the object in differential interference contrast (DIC) microscopy [4]. These methods were phase sensitive by converting the phase information to intensity contrast, where their magnitudes are dependent on the phase difference between scattered light and the direct light/the reference light in the PCM and the DHM, or between two sheared lights both from the object in DIC [5]. By introducing the initial phase shift on the reference light or two sheared lights to modulate the phase difference, the contrast and therefore the phase sensitivity can be largely improved. However, due to the nonlinear relation between the phase and the intensity contrast, the multiple exposures with the shifted initial phases are required [6,7] to quantitatively acquire the phase distributions, which increase the measuring period. The off-axis DHM can recover phase from single exposure by involving the titled reference wave to separate the scattered light and its twin image in Fourier space, in which the cost is the decreasing spatial resolution [8]. The recently proposed single-exposure DIC uses polarization and spatial multiplexing technique to capture images with three various biases simultaneously, resulting in a decrease in field of view (FOV) [9].

Phase retrieval, based on alternating projection algorithms, provides another solution for quantitative phase recovery from a single exposure of the object, without decreasing the spatial resolution or FOV. As a successful single exposure phase retrieval method, coherent diffraction imaging (CDI) [10] recorded the scattering intensity in Fourier space, then recovered phase by using the iterative Fourier transform on the Fourier intensity and the object-space constraint. However, successful recovery from a single Fourier intensity relies on the uniqueness of the solution. Even though nontrivial ambiguities can be removed by oversampling and finite size constraint, it still suffers from trivial ambiguities, causing algorithm stagnation [11]. To remove the trivial ambiguities, the asymmetrical ‘tight support’ of object [12] was used in CDI. However, it is difficult to obtain the accurate ‘tight support’ for transparent objects. Further, diversity measurements by multiple overlapped illuminations with redundant information were introduced in ptychography [13] and Fourier ptychographic microscopy (FPM) [14], removing the ambiguities by adjacent estimation. However, multi-exposures inevitably increase the measuring period. Recently, single-exposure phase microscopy based on transport-of-intensity equation (TIE) [15] has been developed, which simultaneously captures intensities at several out-of-focus planes using a beam splitter, however, to recover the phase, the weak phase assumption of objects was used. Color-multiplexed differential phase contrast (cDPC) [16] and FPM [17], which simultaneously records the color and angle-varied illuminated images using color image sensor, can also achieve single-exposure phase measurement with the assumption that the object has the same absorption for different wavelengths. In previous work, we proposed single-exposure phase microscopy that simultaneously records the band-limited image and its Fourier intensity (called BIFT microscopy) [18] and applied the intrinsic constraints of microscopy (finite field of view and band-limit) to improve the uniqueness and the convergence [11]. These methods achieved phase retrieval with unique solution from single exposure. However, the image contrast, the scattered intensity in Fourier space and therefore the phase sensitivity only depends on the object.

Efforts to further improve the phase sensitivity in phase retrieval were focused on increasing the scattering intensity in Fourier space or increasing image contrast. When recording in Fourier space, as in CDI, the direct approach to increase the scattering intensity was to increase the illumination intensity, and then to prevent the direct light overloading detector by a beam stop [19]. However, the loss of low-frequency intensity due to the beam stop makes the phase recovery difficult. Further, a reference wave was involved to utilize interference with the scattered light, where the reference wave is from another scattering object placed nearby the object in CDI [20], or simply from part of the known illumination in in-line holography [21]. In these methods, by recording the scattering light with object’s phase into the interfered intensity instead of directly recording the scattering intensity, the scattered light was amplified and therefore the phase sensitivity was improved [22]. When recording in image space, theoretical phase retrieval method based on DIC improved the image contrast the same as in conventional DIC [23]. However, successful recovery with a unique solution still required object constraints to eliminate trivial ambiguities, such as a positive constraint on the object’s absorption normalized by background [24] in in-line holography, or positive absorption and bandlimit in DIC [23]. Thus, achieving single-exposure quantitative phase recovery method with high phase sensitivity is still a challenge.

In this work, we develop a single-exposure phase microscope combining conventional DIC microscopy and BIFT method, which shares the benefits of both techniques, called BIFT-DIC microscopy. By adding a Fourier lens to a conventional DIC microscope, we simultaneously record the DIC image and its Fourier intensity. The phase information was converted into both the image contrast and the scattering intensity in Fourier space. By introducing the initial phase shift between the two sheared lights with the balance between improving the image contrast and increasing the scattering intensity in Fourier space, this method exhibits high phase sensitivity. Further, the phase distribution can be quantitatively recovered from single exposure using phase retrieval, avoiding the multiple exposures with shifted phases in conventional DIC. By adopting the algorithm and constraints of the BIFT, the uniqueness of solution and rapid convergence can be achieved. Owing to the single-exposure and the high phase sensitivity, this work provides an alternative for real-time phase imaging.

2. Method and theory

2.1 Optical schematics and image formation of BIFT-DIC

The BIFT-DIC microscope includes an add-on optical system for the BIFT method in a DIC microscope, as shown in the reflection configuration in Fig. 1. The add-on system for the BIFT method includes a beam splitter, a lens (L1) and an array detector. A coherent laser with wavelength λ as illumination source goes through a linear polarizer with the polarization direction at 45° to the x-axis. Then, the linear polarized light is angularly split by a Nomarski prism into two beams with polarization along x-axis (o-light) and with polarization along y-axis (e-light), respectively. The o-light and e-light passed through an objective and then illuminate the object parallelly with shear distance along x-axis, which is smaller than the spatial resolution of the objective. The reflected o-light and e-light from the object are collected by the same objective, recombined by the same Nomarski prism, and then imaged through the tubelens. The recombined o-light and e-light are transmitted through the analyzer, with the polarization direction at -45° to the x-axis. These two lights, now polarized in the same direction, are able to interfere. A phase shifter was added before the analyzer to adjust the initial phase shift (bias) between the o-light and e-light. To combine the BIFT method, the interference light is split into two beams by the beam splitter. One beam is recorded in the imaging plane as the DIC image. The other beam is Fourier transformed by lens L1 and the resulting Fourier-space image is recorded at the focal plane of L1.

 

Fig. 1. Optical scheme of BIFT-DIC microscope.

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Firstly, we consider the sheared o-light or e-light reflected from the object individually until they interfered. Because they illuminate the object with spatial displacement 2Δx₀ and symmetrically relative to the optical axis of the microscope, their reflected lights from object are given by $t({{x_0} + \Delta {x_0},{y_0}} )$ and $t({{x_0} - \Delta {x_0},{y_0}} )$ [25], where the $t({{x_0},{y_0}} )$ is the response function of object with reflectivity $a({{x_0},{y_0}} )$ and phase $\phi ({{x_0},{y_0}} )$. The o-light or e-light is then projected to the image space (x,y) by the objective and tubelens. After passing through the beam splitter, one half of o-light or e-light to the imaging plane is given by

2.2 Contrast and phase sensitivity

In this section, we will discuss the phase sensitivity limit of BIFT-DIC microscopy, that is the minimum phase variation of the object that can be detected. When testing the phase sensitivity, we can consider an object with uniform reflectivity and weak phase, where t(x₀,y₀) can be approximated as $t({x_0,y_0} )\approx 1 + i\phi ({x_0,y_0} )$[26]. When projected into the image space and the Fourier space, their responses can be approximated as $t^{\prime}({x,y} )\approx a^{\prime}({x,y} )+ i\phi ^{\prime}({x,y} )$ and $T^{\prime}({f_x,f_y} )\approx \overline {T^{\prime}} ({f_x,f_y} )+ i\Phi ^{\prime}({f_x,f_y} )$, respectively, where $a^{\prime}(x,y) = \int\!\!\!\int_{S1} {h(x - x_0,y - y_0)dx_0dy_0}$, $\phi ^{\prime}(x,y) = \int\!\!\!\int_{S1} {\phi (x_0,y_0)h(x - x_0,y - y_0)dx_0dy_0}$, $\overline {T^{\prime}} ({f_x,f_y} )= \int\!\!\!\int {a^{\prime}({x,y} )\textrm{exp} [ - 2\pi i(xf_x + yf_y)]} dxdy$ and $\Phi ^{\prime}({f_x,f_y} )= \int\!\!\!\int {\phi ^{\prime}({x,y} )\textrm{exp} [ - 2\pi i(xf_x + yf_y)]} dxdy$. Substituting these results into the Eqs. (5) and (6), the recorded image and Fourier intensity in BIFT-DIC microscopy can be given as

For the image space, when the contrast C_r are smaller than the standard deviation of the noise σ_r, the phase difference of object cannot be distinguished [28]. The phase sensitivity limit p_{r,sensitivity} can be given by the maximum phase difference of object p_r at where $C_{r|pr} = \sigma_r$

It is shown in Eq. (11) that the smaller bias improves the phase sensitivity with the larger modulation cotθ₀, thus, a small bias is required for weak phase object [5]. When the small bias decreased to zero where the illumination of o-light and e-light interfere destructively which leads ${\overline I _r} = 0$, the phase sensitivity becomes extremely large, however, it is usually not adopted due to the indistinguishable increased intensity of ascending or descending edge of object’s phase. This sensitivity limit matches the Cramer–Rao bound (CRB, representing the theoretical limit of mean squared error in the presence of noise corruption) of single DIC image, by using the normalized intensity into the derivation of Ref. [29]. In comparison, the sensitivity limit of four-step phase-shifting DIC (PS-DIC), which records images with bias of 0, π/2, π, 3π/2, can be given according to its CRB [29] as

For the coherent diffraction imaging and BIFT method, the successful recovery of phase largely relies on the successful detection of scattering intensity at each angular frequency, i.e., it depends on the angular spectrum of object. For the BIFT-DIC, the scattering intensity is further modulated by the interference, as shown in Eqs. (8) and (10). The phase sensitivity limit of BIFT-DIC in Fourier space p_{F,sensitivity} can be given by p_F at each angular frequency where $C_{F|pF} = \sigma_F$

As seen in Eq. (13), for object with given normalized scattering intensity ${\Phi ^{\prime}_{_{01}}}^2(f_x,f_y)/{|{\overline {T^{\prime}} ({0,0} )} |^2}$, the phase sensitivity is improved with the larger modulation $H_{m,norm}^2(f_x,f_y)\textrm{ } = {\sin ^2}(2\pi M\Delta x0\textrm{ }f_x - \theta_0)/{\sin ^2}(\theta_0)$, which increases the normalized scattering intensity at the angular frequencies primarily determining the phase information. This matches the CRB of BIFT [11,30], in which the mean squared phase error for weak phase object is also inversely proportional to the normalized scattering intensity. Based on these, the phase sensitivity of the BIFT method can be given by removing the modulation of scattering intensity from that of BIFT-DIC,

Thus, the key to increase the phase sensitivity of BIFT-DIC is to adjust the modulation of $H_{m,norm}^2(f_x,f_y)\textrm{ }$. The maximum and minimum values of modulation $H_{m,norm}^2(f_x,f_y)\textrm{ }$ can be tuned by the shear distance and the bias, as shown in Fig. 2. With the fixed bias, shown in each row of Fig. 2, the increasing shear distance (i.e., $2\pi M\Delta x_0\textrm{ }f_0$ from π/8, π/4 to π/2) increases the maximum modulation (e.g., from 3.41, 5.82 to 6.82 at θ₀=π/8). However, it has to mention that when increased to π/2, the minimum values decrease to zero, the scattered information is lost, which should be avoided. On the other hand, with the fixed shear distance $2\pi M\Delta x_0\textrm{ }f_0$ smaller than π/2, shown in each column of Fig. 2, the decreasing bias (i.e., θ₀ from π/2, π/4 to π/8) increases the maximum modulation (e.g., from, 1, 1.70 to 3.41 at shear distance $2\pi M\Delta x_0\textrm{ }f_0 = \pi /8$). Meanwhile, it reduces the minimum modulation, which leads the zero points moving to low frequency. To avoid this, a small shear distance $2\Delta x_0 < 1/2Mfc^{\prime}$ is needed to make the interval of the two zero points $\pi /(2\pi M\Delta x_0)$ larger than $2fc^{\prime}$, then a larger bias $2{\theta _0} > 2 \times 2\pi M\Delta {x_0}fc^{\prime}$ can remove the zero point outside the cutoff frequency.

 

Fig. 2. The amplitude of the modulation $H_{m,n\textrm{orm}}^2(fx,fy) = H_m^2(fx,fy)/H_m^2(0,0)$ in Fourier space versus normalized angular frequencies f_x/f₀ and f_y/f₀ (f₀= 1/λ is the cutoff frequency at NA = 1) with various biases θ₀=π/8 (Row 1), π/4 (Row 2), π/2 (Row 3), and with shear distances 2πf₀·MΔx₀=π/8 (Column 1), π/4 (Column 2), π/2 (Column 3). The dash circle: the frequencies containing most scattered information of object, and it is the same in each figure. Color bar: The amplitude. The cross-sections along the dash lines are also given at the right of each figure, and the frequencies along the dash line are the same in each figure.

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It is shown from Eqs. (11) and (13) that, in BIFT-DIC microscopy, the phase sensitivities from the image and Fourier contrasts are differently modulated by the bias. The optimal bias can be adaptively searched with the balance of image and Fourier contrast. In comparison, in BIFT method, without modulation, the phase sensitivity is largely determined by the normalized scattering intensity of object in Fourier space. In conventional DIC microscope, the phase sensitivity of a single image contrast can be improved with small bias. However, its high-sensitive phase information cannot be solved quantitatively, which still needs multiple measurements with various biases [29]. Therefore, the BIFT-DIC methods exhibits phase sensitive as well as single-exposure recovery.

2.3 Recovery algorithm

To quantitatively recover the phase information from the object, an algorithm has been developed based on the BIFT algorithm and accommodating the interference characteristics of DIC microscopy. The algorithm has two steps:

Step 1: Retrieve the amplitude and the phase of the optical fields $E_r(x,y)$ in image space and $E_F(f_x,f_y)$ in Fourier space from the recorded intensities of them I_r(x,y) and I_F(f_x,f_y) using the BIFT algorithm [18]. The BIFT algorithm has been modified based on the HIO algorithm for two intensities measurement [31] by applying the intrinsic bandlimit constraint of microscope to remove the nontrivial ambiguities and combining the recorded intensities constraint to remove the trivial ambiguities [11]. The specific modifications in the jth iteration are: (1) The estimated $E_{r,j}(x,y)$ is imposed to the recorded image intensity that

The similar feedback can also be applied to the image space using the constraints of limited field of view. The “feedback” strategies are according to the phase sensitivity of two intensities and impose the convergence to the plane with better phase sensitivity. For example, the feedback can be applied to Fourier space for strong scattering object, or to image space for weak scattering object. Using this BIFT algorithm, the optical fields $E_r(x,y)$ and $E_F(f_x,f_y)$ can be retrieve with the improved and fast convergence.

Step 2: Extract the amplitude and phase information of object from the retrieved optical field $E_F(f_x,f_y)$ using inverse filter. As the retrieved optical fields $E_r(x,y)$ and $E_F(f_x,f_y)$ are the responses of the ‘differential’ of object from the sheared lights, the direct approaches to recover the amplitude and phase of object is to integrate from $E_r(x,y)$ in image space. However, it will also integrate the error in each pixel, therefore cause a large integrated error and bring the streaking artifacts [32,33]. Alternatively, we can also apply the inverse filter in Fourier space from $E_F(f_x,f_y)$ according to the Fourier integral theorem. Additionally, considering that the response of object in Fourier space $T^{\prime}(f_x,f_y)$ is bandlimited, according to Eq. (4), the final inverse filter can be given by

3. Simulations

3.1 Contrast trend and phase sensitivity limit

Simulations for studying contrast and the therefore phase sensitivity are carried out in this section, using the experiment parameters with the illumination laser wavelength 532 nm, 100× objective, 200 mm tubelens and a 100 mm Fourier lens. The image and Fourier contrasts are calculated by Eqs. (9) and (10), respectively, and further verified with simulations without weak phase approximation (WPA). Because of the bandlimited responses of objective, when object has broad angular spectrum but is constrained by the objective, i.e., $fc^{\prime} \ge fc$, the recorded phase information $\Delta \phi ^{\prime}_{01(x0,y0)}$ in image space is the convolution of $\Delta \phi ^{\prime}_{01(x_0,y_0)}$ and PSF, and the recorded angular spectrum $\Phi ^{\prime}(f_x,f_y)$ in Fourier space has been cut off. To study this bandlimited effect, two samples with different finite angular spectra were chosen for simulation, in which the maximum angular frequency is mainly related to the edge width (i.e., sharpness): (S1) a grating object with sharp edge width 145 nm (Line width L = 2.94 µm, Height h = 10 nm, Fig. 3(a1)); and (S2) a gradient object with smoother edge width 870 nm (L = 1.6 µm, h = 10 nm, Fig. 3(a2)), as their angular spectra shown in Fig. 3(b). The image contrast of S1 and S2 via objective’s NA (f_c= NA/λ) are shown in Fig. 3(c) and (d). Here for the image contrast, the results with and without WPA matches well. For the sharp object S1, the image contrast always increased with the increase of NA, i.e., the image contrast is reduced due to the cut off of the broad angular spectrum by the objective. For the smoother object S2, the image contrast increased with the increase of NA until NA = 0.5, within this aperture the most scattered information has been collected, which covers its angular spectrum in Fig. 3(b). Thus, we can consider that the S2 object’s maximum frequency $fc^{\prime}$ is equal to $fc$ at NA = 0.5.

 

Fig. 3. (a) Shape of samples. (a1) Sharp object S1 and (a2) Smooth object S2. (b) The angular spectra of objects S1 and S2. (c) and (d) Simulated image contrast with various numerical aperture of objective (NA) for (c) Sharp object S1 and (d) Smooth object S2. Both the results with and without weak phase approximation are shown.

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The influence of the bias and shear distance on the image and Fourier contrasts are studied by using the objective with the fixed NA = 0.7. As shown in Fig. 4(a) and (b), the image contrasts reduced largely from 0.57 to 0.01 with the increase of the bias from 0.1π to 0.9π for S1, and the trend of S2 is the same as S1 reduced from 0.23 to 0.006. For the Fourier contrasts, first of all the shear distance should satisfy 2Δx₀<λ/1.4 for S1 and 2Δx₀<λ for S2 to move the zero points out of cutoff frequency $f_c = 0.7/\lambda$, here we choose a shear distance 2Δx₀= 266 nm to be satisfied for both S1 and S2. And the bias should be satisfied as 2θ₀> 0.7π for S1 and 2θ₀> 0.5π for S2. The Fourier contrast at cutoff frequency (the frequency with lowest contrast among all frequencies in this case) with various bias (0.7π to π for S1, and 0.5π to π for S2) are shown in Fig. 4(c) and (d). With the increase of bias, the Fourier contrast strengthened for both S1 and S2. It is shown that both the image and Fourier contrast can be largely modulated by the bias, however, their trends are opposite, as shown in Fig. 4(a, b) and Fig. 4(c, d). Thus, the chosen of bias need a compromise between the increase of image contrast and Fourier contrast. Based on the given contrasts, the phase sensitivity limit can be calculated with various bias according to Eqs. (11) and (12). Regarding the noise, we consider that the noise standard deviation is signal-independent and follows Gaussian distribution, and can be set as the standard deviation of noise in image and Fourier space σ_r=σ_F= 1/2¹⁶ (quantization noise of the 16-bit detector). Figure 4(e) and (f) shows the phase sensitivity limit of S1 and S2 from their image contrast and Fourier contrast at cutoff frequency with various bias at fixed NA = 0.7, 2Δx₀= 266 nm. With increased bias, due to their reduced contrast and strengthened scattered intensity at cutoff frequency, the sensitivity limit in image space increased (i.e., from 0.065 rad to 0.2 for S1 rad and 0.05 rad to 0.33 rad for S2), however, that in Fourier space decreased (i.e., from 0.78 rad to 0.7 rad for S1 rad and from 2.5 rad to 0.23 rad for S2). Both the sensitivity limit in image and Fourier intensity can be improved by an optimal bias.

 

Fig. 4. (a) and (b) Calculated and simulated image contrast with various bias for (a) Sharp object S1 and (b) Smooth object S2. (c) and (d) Simulated Fourier contrast at cutoff frequency with various bias for (c) Sharp object S1 and (d) Smooth object S2. (e) and (f) Calculated phase sensitivity from image and Fourier contrasts with various bias for (e) Sharp object S1 and (f) Smooth object S2. Arrow: relating the real and Fourier space sensitivity limit plots to their respective axes.

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3.2 Recovery and comparison with the BIFT and the phase-shifting DIC methods

In this section, we use the developed algorithm to simulate the recovery process. The image intensity and Fourier intensity were calculated by Eqs. (5) and (6), where the response of objects were obtained by inputting the object response function of S1 and S2 into Eqs. (1) and (2). A root mean square error (RMSE) function $RMSE = \sum {|{\varphi_r(x_0,y_0) - \varphi_0(x_0,y_0)} |} /N$ is given to evaluate the reconstructed phase accuracy, where φ_r(x₀,y₀) and φ₀(x₀,y₀) are the recovered and actual phase of the object respectively, and N is the pixel number. Using an illumination of coherent Gaussian beam and the parameters same as the pervious section, we simulated the image and Fourier intensity of S1 and S2, with the initial bias 2θ₀= 0.7π for S1 and 0.5π for S2 according to the aforementioned discussion, which are shown in Fig. 5(a) and (b). The noise is added with standard deviation mentioned in previous section. The reflectivity and phase of S1 and S2 were recovered using the developed algorithm, as shown in Fig. 5(c) and (d), with their RMSE of recovered phase 0.0055 rad (i.e., 0.47 nm), and 0.0139 rad (i.e., 1.18 nm), respectively. Both S1 and S2 were recovered successfully, and the S1 recovered with smaller error than S1, consistent with calculation of sensitivity limits where S1 better than S2.

 

Fig. 5. (a) and (b) The simulated image and Fourier intensity of (a) S1 and (b) S2, respectively. (c) and (d) Recovered phase distribution of (c) S1 and (d) S2, respectively, with their central sectional height distributions. Color bar: the reflectivity with the non-uniform illumination removed.

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To compare the phase sensitivity limit with BIFT method and four-step phase-shifting DIC method, the recovery simulations of S1 were carried out by all three methods. The simulated image and Fourier intensity of BIFT method uses the model in Ref. [11], as shown in Fig. 6(a), and the BIFT algorithm is used for recovery. The simulated four DIC images of the phase-shifting DIC method is according to Eq. (5) with bias of 0, 0.5π, π and 1.5π, as shown in Fig. 6(b). In these images, the decreasing image contrast at biases of 0, 0.5π, and π has been observed, as discussed in the previous section, and then the algorithm combined four-step phase-shifting and integration is used for recovery [34]. The recovery results are shown in Fig. 6(c-e), with the noise σ=10/2¹⁶ adding into both images and Fourier intensities for three methods. The recovery of BIFT-DIC method has higher image contrast to noise (CNR) 0.044 than 0.003 in BIFT, as shown in Fig. 5(a) and Fig. 6(a), and their RMSE of height are 1.39 nm for BIFT-DIC and 1.74 nm for BIFT. The four-step phase-shifting DIC has high sensitivity with RMSE of height 1.27 nm mainly from the integrated error. This recovery errors can be compared with the theoretical limit of three methods given by the Eqs. (11-14). The sensitivity limits are calculated and shown in Fig. 6(g), where the sampled Fourier intensity is at a diffraction peak of f_x/ f_{cutoff, NA = 1}= 0.3 (as seen in Fig. 3(b)). For the image intensity, the BIFT-DIC method has smaller sensitivity limit than that of four-step phase-shifting DIC at small bias. However, in this case, we can choose the large bias to remove the zero-points outside, resulting in the larger sensitivity limit than four-step phase-shifting DIC. For the Fourier intensity, the BIFT-DIC method has approaching sensitivity limit as that of BIFT method at the sampled angular frequency. As the BIFT-DIC recovers phase information using the image and Fourier intensity simultaneously, the final limit is mainly decided by the sensitivity limit of the space with feedback applied. As the Fig. 6(h) shows the recovery error of BIFT-DIC with feedback applied in image and Fourier space, respectively, compared with their sensitivity limits. When using the image feedback, the recovery error decreased toward the image space limit. Overall, in this case, the recovery results mainly approach to the limit of Fourier contrast in BIFT, of image contrast in BIFT-DIC, and the limit from the four image intensities in phase-shifting DIC. The BIFT-DIC has smaller recovered error than BIFT only due to its acquisition of phase sensitive image intensity. The phase-shifting DIC recovers better, however, it needs four acquisitions of images. Thus, the BIFT-DIC method provides a phase sensitive recovery from noise and it does not need multiple acquisitions.

 

Fig. 6. Simulated reconstructions of S1 by three methods. (a) Simulated image and Fourier intensity by BIFT method. (b) Simulated images by phase-shifting DIC with bias of 0, 0.5π, π and 1.5π. (c)-(e) The recovered height of S1 by (c) BIFT, (d) BIFT-DIC, (e) Phase-shifting DIC with streaking artifacts. (f) The cross-sectional height in (c), (d) and (e). (g) The calculated sensitivity limit of three methods. Arrow: relating the real and Fourier space sensitivity limit plots to their respective axes. (h) The recovery error of BIFT-DIC with convergence to the image and Fourier space.

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4. Results of experiments

The experiments will be carried out in this section. The laser with wavelength 532 nm, Nomarski prism (shear angle 40 µrad), two sCMOS cameras (PCO.panda; 16-bit, 6.45-µm pixel size), objective (Olympus; magnification 20×, 0.4 NA; 50×, 0.5 NA, 50×, 0.75 NA), 200 mm tubelens, and a 100 mm Fourier lens were used. The phase shifter to adjust the bias was achieved by the combination of λ/4 wave plater and analyzer, setting both their initial axis -45° to the shear direction. By rotating the angle of analyzer’s axis, a series set of measurements with various bias can be obtained. A standard grating sample (NT-MDT TGZ1) and an electrode sample were imaged with different angular spectra: (1) The standard grating sample has smooth edge with width 1 µm (larger than the spatial resolution 665 nm for 0.4 NA) and depth 21.7 nm. It is shown that most scattered intensities are concentrated at the angular frequency equivalent to NA = 0.18. (2) The electrode sample has a sharp edge with width 392 nm (smaller than the spatial resolution for 0.4 NA) and depth 25 nm, as measured with DHM and the cross-section height profile measured with AFM, shown in Fig. 7(a).

In the following, the influences of bias are investigated using the 20× objective (shear distance 360 nm). We firstly recorded the image and Fourier intensity with various bias, ensuring they were larger than 0.25π for the grating and 0.55π for the electrode to move the zero-valued modulation outside. The image contrast and Fourier contrast versus bias are shown in Fig. 7 (b) and (c) for electrode sample and shown in Fig. 8 (a) and (b) for grating sample, with the insets indicating the squared area and frequencies used to measure the contrast. It is noted that the contrast in the electrode sample is contributed by both its non-uniform reflectivity and the phase distribution. Furthermore, we measured the noise of the background near the area of interest in each image, i.e., σ_r= 15/2¹⁶ and σ_F= 9/2¹⁶. The phase sensitivity of the electrode sample and the grating sample were calculated from their image and Fourier intensities respectively, shown in Fig. 7(d) and Fig. 8(c). The trend of contrasts and phase sensitivities of two samples versus bias matches the simulation in Section 3.1.

 

Fig. 7. (a) Phase distribution of the electrode sample. Arrowhead: shear direction. (b) Image contrast with various bias. (c) Fourier contrast with various bias. (d) Calculated sensitivity limits based on the measured contrast, converting to height. (e) Image contrasts versus with different NA. Inset in (b, c and e): the corresponding recorded intensity at bias = 0.8π, shown in log scale. The square and the dash line show the area and the frequency used to measure the contrast.

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Fig. 8. Measured contrast of the standard grating sample. (a) Image contrast with various bias. (b) Fourier contrast with various bias. Inset in (a) and (b): the corresponding recorded intensity at bias = 0.5π. The square and the dash square show the area and the frequency used to measure the contrast. Arrowhead: shear direction. (c) Calculated sensitivity limit based on the measured contrast, converting to height.

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We also verify the influence of NA on the image contrast by the electrode sample using two 50× objectives (shear distance 144 nm) with 0.5 NA (the spatial resolution 532 nm) and 0.75 NA (the spatial resolution 354 nm), respectively. The recorded image contrasts versus bias are shown in Fig. 7(e). With same shear distance and bias, the improved contrast by 0.75 NA objective is due to its larger aperture collecting most angular frequencies of the object.

The result of reconstruction of the standard grating sample from the image and the Fourier intensity was shown in Fig. 9(a) and the cross-section of height profile was shown in Fig. 9(b). The result of reconstruction of the electrode sample recovered by the BIFT-DIC method was shown in Fig. 9(c) and the cross-section of height profile shown in Fig. 9(d). The differences between the phase recovery by BIFT-DIC and AFM are calculated and plotted in both Fig. 9(b) and (d). They demonstrated the recovered height from BIFT-DIC matched well with that from the AFM, with the flatness matching the calculated sensitivity limits. The wider edge with larger error is due to the spatial resolution constrained by the objective in BIFT-DIC. For comparison, the electrode sample was also measured by a commercial digital holographic microscope (DHM; Lyncee tec-R1000, 50×,10 frames), with both its height distribution and the difference with AFM shown in Fig. 9(d). The recovery of DHM also suffers from the limited spatial resolution and has the similar error range as the BIFT-DIC. The BIFT-DIC method recovered both samples successfully from only single exposure of the samples.

 

Fig. 9. Reconstructions of the samples. (a) Reconstruction of standard grating sample. Color bar: the reflectivity. (b)The central sectional height distributions of (a), compared with the AFM. Red line: the difference between two curves. (c) Reconstructions of electrode sample. Color bar: the reflectivity. (d) The central sectional height distributions of (c), compared with the AFM and DHM. Red line: the between BIFT-DIC and AFM. Orange line: the difference between DHM and AFM.

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5. Conclusion

We proposed a single-exposure quantitative DIC microscopy based on BIFT phase retrieval method. The bandlimited DIC image and its Fourier image have been recorded simultaneously to uniquely phase retrieval with rapid convergence. It is shown that, with both the simulation and experiment, the image and Fourier contrast can be largely improved by the biases, however, their trends are opposite. By using the optimized bias with the compromise of the contrasts in image and Fourier space, the phase sensitivity of BIFT-DIC can be big improved comparing to BIFT method only. In experiments, we successfully recovered the sample of 25 nm height from a single exposure. Owing to the single-exposure and the high phase sensitivity, our work provides an alternative for real-time phase imaging.