1. Introduction

Optical aberration limits our ability to observe structures at diffraction-limited resolutions and causes inconsistencies between images obtained in different setups. Despite advances in modern microscopy, aberration-free imaging remains difficult to achieve [1]. The main challenge is that aberration is not constant for an imaging system, but changes depending on the sample [2,3] and sample holders [4] because of the mismatch in refractive index and the surface roughness. Thus, it is important to calibrate the aberration in the exact environment where a sample of interest is imaged.

Quantitative phase imaging (QPI) is a powerful label-free technique used for imaging transparent cells and tissues [5–8]. By measuring the complex amplitude of light, QPI provides quantitative phase images and refractive index (RI) tomograms of specimens. However, its sensitivity to subtle changes in the optical field makes QPI susceptible to optical aberrations. Several techniques have addressed phase aberration or wavefront curvature in image planes [9–16]. In contrast, it remains difficult to calibrate pupil aberration because transparent samples and plane-wave illumination make QPI incompatible with adaptive optical techniques based on focused illumination [17–19], diffuse reflection [20] and guide stars [21]. Different approaches have been proposed to address this issue. For example, a pinhole [22,23] and weak diffuser [24] can be introduced to measure pupil aberration from holograms of a point spread function (PSF) and speckle intensity images, respectively. However, these methods must replace a sample with calibration objects. Iterative approaches can recover both pupil aberration and sample images but are limited to thin samples [25–27] or require an aberration model based on prior knowledge [28]. Alternatively, the transmission matrix of an imaging system can be used; however, this requires the aberration in the illumination path to be negligible and system-induced [29].

Here, we propose a method for aberration-free QPI, where pupil aberration is calibrated using a sample of interest. We utilized the cross-spectral density between optical fields at different incident angles to extract pupil information, from which we measured pupil aberration. Unlike previous approaches, the proposed method is compatible with both thin and weakly scattering 3D samples and addresses the aberration induced by the sample due to the roughness of a substrate and the mismatch in refractive index, and by the imaging system. We experimentally demonstrated the method by calibrating pupil aberration using various samples, including a resolution target, breast tissue, and polystyrene beads. By correcting the calibrated pupil aberration, we obtain aberration-free quantitative phase images and RI tomograms under different imaging conditions.

2. Principle

The proposed method recovers the pupil aberration from optical fields scattered by an unknown sample at different incident angles [Fig. 1(a)]. For simplicity, we assume a thin sample (see Supplement 1 for a 3D sample). In QPI, the scattered field is directly measured using interferometry. For a plane incident wave, the measured field can be expressed as [Fig. 1(b)]

 

Fig. 1. (a) Schematic of the experiment where an unknown sample is illuminated by a plane wave. Optical fields at different incident angles are measured by a holographic microscope. (b) Simulated Fourier spectrums of the fields measured at different incident angles. (c) Simulated phase difference between fields at different incident angles and the field at the normal incident angle around the unscattered light [indicated with dashed circles in (b)]. (d) Reconstructed pupil aberration after compensating the relative phase between (c) and removing the ambiguity at the normal incident angle. (e) Experimental Setup. BS: beam splitter; M: mirror; L: lens; P: linear polarizer

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where k is the transverse wave vector ${\mathbf k} = \left( {k_x,k_y} \right)$, ${{\mathbf k}_{{\mathbf {inc}}}}$ is the wave vector of the incident plane wave, $\widetilde S$is the Fourier transform of the sample transmittance, and P is the pupil function that describes the aberration induced by the sample and the imaging system. Equation (1) shows that the complex field contains the pupil aberration arg(P) information where the Fourier spectrum of the sample is non-zero. This indicates that the pupil aberration can be expressed by compensating $\widetilde{S}\left( {{\mathbf k}-{\mathbf k}_{{\mathbf {inc}}}} \right)$ in Eq. (1). Our strategy is to use the modified cross-spectral density W between Eq. (1) and the field at a normal incident angle shifted by k_inc

Thus far, our discussion assumes that the measurement noise is negligible. In practice, the signal-to noise ratio varies depending on the sample spectrum [Eq. (1)]. For this reason, we set B(k) as a small circle because Fourier spectra of transparent samples have large values near the origin. In our experiments, the radius of the circular mask was set to ${{\pi \textrm{NA}} / {({10\lambda } )}}$, where NA is the numerical aperture of the imaging system (see Supplement 1 for discussion about the size of the mask). In addition, we numerically corrected the global phase factor in Eq. (1) induced by the instability of an imaging system and by pupil aberration in the illumination path (see Methods). Throughout the experiments, complex fields at 1009 incident angles were measured using an off-axis holographic microscope equipped with a digital micromirror device (DMD) [Fig. 1(e), see Methods] and used to calibrate pupil aberration (see Supplement 1for the discussion about using a different number of illuminations).

3. Results

3.1 Aberration-free QPI using thin samples

To demonstrate the proposed method, we measured the pupil aberration using a USAF resolution target (#59–153, Edmund Optics, New Jersey, USA) and a human breast tissue slice (see Fig. 2). Figure 2(a) displays the pupil aberration calibrated by imaging the USAF target using an objective lens (UPlanSAPO 20×, NA = 0.75, Olympus). Because the target is imaged without a coverslip, the main cause of the pupil aberration is the objective lens, which is designed to correct the spherical aberration induced by a coverslip. The result exhibits good agreement with the expected pupil aberration induced by a 170-μm-thick coverslip and 110 μm defocus (see Supplement 1, Fig. S1), and the minor difference can be explained by aberrations caused by other optical components. Figure 2(b) depicts an image at a normal incident angle. The magnified image clearly shows that the patterns are degraded by the aberration. After aberration correction, the image shows sharp edges in the pattern [Fig. 2(c)]. The effect of the aberration is more severe at an oblique incident angle than at a normal incident angle [Fig. 2(d)]. Nevertheless, a clear image of the target was obtained after aberration correction [Fig. 2(e)]. To show the effect of the aberration under different illumination conditions simultaneously, we conducted synthetic aperture holography [30] using seven complex fields with an illumination NA of 0.54. The synthetic aperture image is normalized in the Fourier space depending on the number of the Fourier spectrum being measured. Figure 2(f) shows a synthetic aperture image of the target without aberration correction. The result is greatly distorted because the synthesis of complex amplitude images is corrupted by the pupil aberration. However, by correcting the aberration in each image prior to the aperture synthesis, we were able to obtain a clear synthetic aperture image [Fig. 2(g)].

 

Fig. 2. Experimental results for thin samples. (a) Pupil aberration measured by imaging a USAF resolution target using an objective lens up to NA = 0.62. The white circle indicates the incident wave vector of oblique illumination. (b)–(e) Amplitude images of the target before (b) and (d) and after (c) and (e) the aberration correction at normal (b) and (c) and oblique illumination (d) and (e). (f)–(g) Amplitude images of synthetic aperture holography of the target before (f) and after (g) the aberration correction. (h) Pupil aberration by imaging breast tissue using an aspheric lens up to NA = 0.48. The white circle indicates the incident wave vector of oblique illumination. (i)–(l) Quantitative phase image of the tissue before (i) and (k) and after (j) and (l) the aberration correction at normal (i) and (j) and oblique illumination (k) and (l). Scale bars are 20 μm.

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Figure 2(h) shows the pupil aberration measured by imaging the human breast tissue with an aspheric lens (A240TM-A, NA = 0.5, Thorlabs Inc.). The aberration can be explained by the imperfections of the aspheric lens, the coverslip that holds the tissue, and the defocus aberration. Figures 2(i) and 2(j) show the quantitative phase images at the normal incident angle before and after aberration correction. Fibrous structures become evident only when the aberration is corrected [indicated by arrows in Figs. 2(i) and 2(j)]. As before, the image is greatly distorted at the oblique incident angle [Fig. 2(k)] and restored after aberration correction [Fig. 2(l)]. Figures 2(m) and 2(n) are synthetic aperture images of the tissue using 10 complex fields with an illumination NA of 0.43, before and after aberration correction. These results illustrate that the pupil aberration is corrected for complex fields at different incident angles.

To further validate the proposed method, we measured the change in pupil aberration induced by defocusing. We calibrated the pupil aberration by imaging the breast tissue using an objective lens (UPlanSAPO 20×, NA = 0.75, Olympus). Then, we repeated the calibration process after the objective lens was axially displaced by approximately 20 μm. Figure 3(a) shows the aberration measured before the defocus. From the quantitative phase image at the normal incident angle, we can see that the complex fields are focused [Fig. 3(b)]. The aberration after the defocus exhibited a concentric pattern [Fig. 3(c)]. The effect of the defocus is also observed in the image of the tissue at a normal incident angle [Fig. 3(d)]. Figure 3(e) shows the change in pupil aberration before and after defocusing. The result shows good agreement with the propagation kernel for 18.51 μm with a cross-correlation value of 0.9983. Figure 3(f) shows the difference between the change in the pupil aberration and the propagation kernel, which confirms that the proposed method accurately reflects the change in pupil aberration.

 

Fig. 3. Change in pupil aberration due to optical defocus (a) Pupil aberration measured with a focused tissue slice up to NA = 0.62. (b) Quantitative phase image of the focused tissue at normal incident angle. (c) Pupil aberration measured with a defocused tissue slice. (d) Quantitative phase image of the defocused tissue at the normal incident angle. (e) The change in the pupil aberration before (a) and after (c) sample defocus. (f) Difference between (e) and propagation kernel for 18.51 μm. Scale bar is 20 μm.

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3.2 Aberration-free QPI using 3D sample

In this section, we demonstrate that the proposed method is compatible with a three-dimensional sample. In general, the field scattered by a three-dimensional sample cannot be described by Eq. (1). Nevertheless, our method works if the scattering event is described by the Born or Rytov approximation and the size of the binary mask is small enough (see Supplement 1). The pupil aberration was measured using a 7 μm polystyrene bead (n = 1.6119) and a long-working distance objective lens (LUCPLFLN60X, NA = 0.7, Olympus). The bead was immersed in oil (n = 1.5823 at 457 nm, Cargille Laboratory) and sandwiched between two coverslips. Under these imaging conditions, the proposed method works as long as the sample thickness is less than 30 μm (see Supplement 1). Figure 4(a) shows the results. The phase aberration is asymmetric, mainly because of the objective lens. To investigate the effect of aberration in three-dimensional QPI, we conducted diffraction tomography [31] of the bead. We used 91 incident angles with an illumination NA of 0.5. Figure 4(b) shows the resultant RI tomogram based on the Rytov approximation. Without aberration correction, the shape of the bead is distorted [Fig. 4(b), top], which shows that diffraction tomography is vulnerable to pupil aberration. The symmetric shape of the bead is recovered when the aberration in the complex fields is corrected [Fig. 4(b), middle]. To validate aberration correction, we generated an ideal tomogram by creating the scattering potential of a spherical object and by selecting a subset of the scattering potential in the wave-vector domain. The selected subset corresponds to what was measured by diffraction tomography in our experimental condition [31]. Then the ideal tomogram was obtained by expressing the scattering potential in terms of refractive index n(x, y, z). The result after the aberration correction shows good agreement with the ideal tomogram of a spherical object [Fig. 4(b), bottom]. Line profiles of the xy-cross-sections also show that the corrected tomogram exhibit the accurate size and refractive index value [Fig. 4(c)].

 

Fig. 4. Experimental results for a three-dimensional sample. (a) Pupil aberration measured by imaging a 7 μm polystyrene bead using a long working distance objective lens up to NA = 0.62. (b) RI tomogram of the bead reconstructed using raw complex amplitude images (top), RI tomogram after correcting the aberration shown in (a), and an ideal tomogram of a spherical object (bottom). (c) Line profiles of the tomograms along white dashed lines in (b).

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4. Discussion

Our method assumes linear shift invariance and negligible phase aberration in the sample beam. Nevertheless, these assumptions do not restrict the applicability of the method because they can be satisfied by dividing the field of view into small areas [32]. In addition, the phase aberration can be addressed by carefully aligning the condenser lens. The pupil aberration in the illumination path did not affect the calibration result. For example, the calibrations shown in Figs. 2 and 3 were successful despite the glass substrate of the resolution target and slide glass of the tissue in the illumination path. Our current approach can only recover the phase part of the pupil function. In the future, we plan to retrieve a complex pupil function, which could further improve the quality of QPI images.

In our current setup, each interferogram is measured every 25 ms and the total measurement time for 1009 interferogram is 25.2 s. For the USAF resolution target (851 × 851 pixels) shown in Fig. 2, the calibration time of the pupil aberration [Fig. 2(a)] was 13 s using a desktop (Intel i5-4690) after the complex amplitude images are retrieved. We expect that the measurement time can be greatly reduced by illuminating a sample with multiple plane waves as the information about pupil aberration can be captured simultaneously as long as scattered fields are separated in the Fourier space. This multiplexing scheme is effective for transparent samples, although it requires a careful choice of the incident angles. Alternatively, computational algorithms, such as regularization and compressive sensing [33], can be adopted because realistic pupil aberration is smooth and the corresponding PSF is sparse. By exploiting these properties, we can potentially calibrate the aberration with fewer measurements.

In summary, we have demonstrated a method for calibrating pupil aberrations using a sample of interest for QPI. Our method directly retrieves pupil aberration by utilizing the cross-spectral density between optical fields scattered by an unknown sample at different incident angles. Its principle is deterministic and compatible with both thins and weakly scattering three-dimensional samples, differentiated with existing methods [25–27] based on iterative algorithms and limited to two-dimensional samples. We experimentally measured pupil aberration using a resolution target, breast tissue, and polystyrene beads. We achieved aberration-free 2D and 3D QPIs by correcting complex fields in the pupil plane. We believe that this approach broadens the application of QPI in challenging imaging conditions and facilitates technological advances by bringing various optical components, such as metasurfaces [34–36], into practical use.

5. Methods

5.1 Numerical compensation of global phase

Complex-field images measured in the experiments have global phase factors ϕ caused by the instability of the setup and the pupil aberration in the illumination path, which is expressed as

The global phase affects the cross-spectral density in Eq. (2), and in turn, the results of Eq. (3). To compensate for the global phase, we exploit the fact that Eq. (2) is approximated to $\arg \left( P \right)$ around ${\mathbf k} = {{\mathbf k}_{{\mathbf {inc}}}}$ if $\arg \left( P \right)$ varies slowly within the binary mask $BP\sim |{BP} |$. Then the global phase can be estimated by comparing Eq. (2) of slightly different incident angles. The global phase factor ϕ_n for the nth incident angle ${\mathbf k}_{{\mathbf {inc}}}^{(n)}$ is expressed as

5.2 Experimental setup

The experimental setup is an off-axis Mach–Zehnder interferometer combined with an optical microscope [Fig. 1(e)]. The light source is a coherent laser (λ = 457 nm, Cobolt Twist^TM HÜBNER Photonics). Focal lengths of the lenses (L1, L2, L3, and L4) are 250, 300, 250, and 500 mm respectively. The incident angle is controlled using a DMD (DLPLCR9000EVM, Texas Instruments). To ensure plane wave illumination, we adopt a 4-bit time-multiplexing method [35] when displaying a grating pattern to the DMD and filter the first-order diffraction term using an iris [37]. The curvature of the DMD is calibrated using a separate interferometer and compensated by displaying a correction pattern. To select 1009 incident angles, we created a 37 × 37 grid pattern with side length equal to the diameter of a pupil function and then selected 1009 positions that lie within the circular pupil function. The condenser lens (LUCPLFLN40X, NA = 0.6, Olympus) is fixed throughout the experiments and different imaging lenses (UPlanSAPO 20×, NA = 0.75, Olympus for the resolution target and the tissue in Fig. 3, A240TM-A, NA = 0.5, Thorlabs Inc. for the tissue shown in Fig. 2, and LUCPLFLN60X, NA = 0.7, Olympus for the polystyrene beads) are used. The reference beam is tilted 2.3 degree. Interferograms are captured by a camera (Lt425, Lumenera). Complex-field images are measured using conventional off-axis holography [38].