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Curr. Opt. Photon. 2024; 8(5): 502-507

Published online October 25, 2024 https://doi.org/10.3807/COPP.2024.8.5.502

Copyright © Optical Society of Korea.

Single-shot Transport-of-intensity Equation Using a Wollaston Prism for Biological Samples

Joseph Vermont Bunyi Bandoy, Cuong Manh Nguyen, An Nazmus Sakib, Suhyeon Kim, Hyuk-Sang Kwon

Department of Biomedical Science and Engineering, Gwangju Institute of Science and Technology, Gwangju 61005, Korea

Corresponding author: *hyuksang@gist.ac.kr, ORCID 0000-0002-3387-210X

Received: June 26, 2024; Revised: August 18, 2024; Accepted: August 26, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Wollaston prism (WP) has shown promise in enabling single-shot transport-of-intensity equation (ssTIE) measurements, facilitating efficient phase retrieval in microscopy. The 1-degree prism which produces the minor beam-separation angle is used to prevent distortions. An optical-glass plate is employed in the duplicated beam path to introduce defocusing. This configuration is also advantageous when aligning the beams laterally caused by the refraction of the optical-glass plate, thus allowing another method for single-shot measurements. We applied the proposed method to image the red blood cells (RBCs), demonstrating that the proposed method could be useful in various biological and medical applications.

Keywords: Microscopy, Transport of intensity equation, Wollaston prism

OCIS codes: (120.5050) Phase measurement; (170.0170) Medical optics and biotechnology; (180.0180) Microscopy

Quantitative phase imaging (QPI) is a technique with various applications in microscopy, including label-free noninvasive capabilities in the study of cells and tissues [1]. Light’s interaction with a transparent sample causes a phase shift, and this phase shift provides essential information, as it can measure the index of refraction, morphology, thickness, and optical properties [2]. In addition, with the advancements in deep learning, QPI has been integrated with microscopy to produce images with improved resolution, and enhance the overall performance of microscopes [3].

QPI can be implemented using both interferometric and non-interferometric methods, such as digital holography (DH) [4] or transport-of-intensity equation (TIE) [5]. TIE is a non-interferometric QPI technique. In comparison, DH relies on the coherence of the light source, requires another beam path for fringe analysis, and requires phase unwrapping. This is because the phase retrieved from the hologram is bounded in the range of (−π, π), which necessitating a phase-unwrapping algorithm to obtain the true phase [6]. TIE does not require another beam which obviates the phase-unwrapping step, making reconstruction simpler.

TIE was first derived by Teague [7], where the phase retrieval was solved using the Green’s function. It was first applied in optical phase microscopy in 1998 [8]. The conventional method relies on intensity measurements at multiple distances along the propagation direction of the wave [5, 914], and recording is required for every defocus. Single-shot, on the other hand, aims to achieve phase retrieval using a single intensity image. Achieving single-shot TIE remains an area of ongoing investigation, since it eliminates the need for defocus, making it more efficient. The reconstruction of phase information from a single intensity measurement can be achieved utilizing a volume holographic microscope [15]. Another method is based on a Michelson-interferometer configuration, including two mirrors [16] or one mirror with a spatial light modulator (SLM) [17] to generate two beams for phase imaging. Using an SLM enables accurate phase retrieval by providing 2π phase modulation with linear electro-optic characteristics [18]. Furthermore, employing the Greek-ladder sieves [19] create two beams that are focused at a distance apart. In addition, the Wollaston prism (WP), a type of prism that separates orthogonal beams, can be employed in the setup. The WP is commonly rotated to adjust the intensities of the two rays. In 2021, Omar and El-Bakary [20] suggested an optical setup for the duplicated-image TIE that can accurately determine the optical anisotropy in fibers. Their method is mainly applied to determine the refractive indices of anisotropic fibers at different polarization directions, using only a single set of micrographs. However, the setup still obtains multiple images of the same object at different axial positions. To solve this, Picazo-Bueno and Micó [21] implemented a cost-effective, simple, and robust method for single-shot QPI-based TIE using an add-on optical module that can be assembled on any part of the microscope. The optical module is composed of a Stokes lens (SL), a beam-splitter cube (BSC) and an optical-glass (OG) plate, which enables intensity recording without mechanical translation. The initial results of this technique produce astigmatic images caused by the BSC, requiring the SL. Though providing good field of view, the duplicated image is flipped and has different background intensity, requiring another step for normalization.

The conventional methods for TIE typically involve capturing three images and relying on mechanical translation to achieve defocus. This approach is prone to misalignment with the sample, which can affect the accuracy of the results. On the other hand, previous single-shot methods have addressed this issue by capturing three or more images, extending the processing time and the possibility of observing image-quality variations, or increasing calibration efforts.

To address these limitations, we report a method for single-shot TIE. The system uses a Wollaston prism to produce a duplicated image, which eliminates the need for mechanical translation and only requires two images, to simplify the technique and increase efficiency while maintaining accuracy. Proposed technique is applied to imaging red blood cells to demonstrate its effectiveness in biological applications, where precise and efficient phase imaging is needed.

2.1. Transport-of-intensity Equation

The TIE is a non-interferometric quantitative form of phase imaging derived by Teague [7]. The equation describes how phase is related to the propagation of light; in this case, how light propagates along a specific axis, here the z-axis. The general equation for the TIE is given by

Iφ=kzI,

where the right-hand side is the transverse energy and the right-hand side of Eq. (1), axial derivative can be expressed as where the right-hand side represents the transverse energy, and the axial derivative can be expressed as

Ix,y,z0Δz=Ix,y,z0+zIΔz+z2Δz22!+OΔz3,

where the expression only propagates along the z axis the light beam, represented by the expression, propagates along the z-axis. Upon approximation and neglecting the higher orders, the expression becomes discrete and can be written as

Iφk2ΔzzIx,y,zIx,y,z.

Solving the equation gives us

φ=F1k2FI1 F1 k2 FzI,

where the phase can be solved for using a Fourier-transform-based solution. Furthermore,

Ix,y,z0=Ix,y,z+Ix,y,z2.

And

Iz=kIx,y,zIx,y,z2Δz,

where I(x, y, z) and I(x, y, −z) are the focus and defocus images at a distance z from the focus z0, while I(x, y, z0) is the average. Three intensities represented in images are required for phase retrieval by the transport-of-intensity equation. Moreover, the partial derivative of the intensity ∂z I is calculated as the difference of the two recorded intensities. Two images are required for the process: One focused image and one defocused image. These images are then processed to compute the average, as described in Eq. (5). In Eq. (6), the defocused image is subtracted from the focused image, the result is divided by a constant k, and their distance. The intensities obtained from Eqs. (5) and (6) are used to retrieve the phase information in Eq. 4 [22]. The conventional approach for determining the transport of intensity requires three images: Under-focused, in-focus, and over-focused. However, single-shot methods requiring only two images is also possible, with the third intensity being the average of the two recorded images. With this, the phase can be retrieved.

2.2. Controlled Defocus for the Single-shot Method

The devised single-shot method uses a Wollaston prism to generate two beams, while introducing an optical-glass plate for controlled defocus. In this work, a 1-degree prism is selected to reduce the beam deviation when separating the beams with respect to the optical axis, as shown in Fig. 1. The WP is mounted on a rotating stage for adjustments and calibration.

Figure 1.Optical setup for ssTIE components: Incoherent light source (LED), polarizer (P), sample (S), microscope objective (MO), tube lens (TL), bandpass filter (BPF), lenses (L1, L2), Wollaston prism (WP), optical glass (OG), analyzer (A), and CCD.

For controlled defocus, an OG plate is placed in the other beam path. This is used to replace manual operation with the use of the translation stage. The OG plate has thickness of 5 mm, resulting a defocus of approximately 17 μm. By adopting a displacement of 20 μm in the translation stage, comparison with the conventional method is possible.

2.3. Experimental Setup

The ssTIE are constructed as follows: First, the LED lamp served as a light source. Next, the polarizer, an illuminated sample, and a microscope objective lens (10× magnification) are arranged in sequence. The light from the source is collimated to a tube lens (f = 200 mm). A green band-pass filter of 525-nm wavelength is arranged next to this tube lens.

Following the imaging plane, there is a 4f system comprised of two lenses, each with a focal length of 150 mm. The WP is positioned at the focus of L1 to produce the desired separation between the output beams. After the 4f system, the OG plate (5-mm thickness) and analyzer is arranged in sequence. Finally, the images are recorded using a CCD camera (8051C-USB).

Using the WP without the polarizer and analyzer, we gained two images with different background intensities as an initial results (Fig. 2). When the intensities are adjusted upon rotation, there is one specific position where the two beams have the same intensities. In contrast, regardless of the position, introducing the polarizer and analyzer provides simultaneous adjustments. If uneven beam is produced in the target position, it can be compensated by the polarizer. Also, when the final image is slightly displaced from its original position, the OG plate resulting in refraction is observed. This can be compensated by rotating the WP for image repositioning, followed by the polarizer for intensity compensation, as shown in Fig. 2(d). This technique can be applied directly to TIE phase reconstruction. The single-shot imaging technique utilizing an optical glass plate allows us to manipulate the beam and create a controlled defocus in the captured image.

Figure 2.Intensities observed based on the polarization (a)–(e), with their respective line profiles plotted against pixel distance (f)–(j).

The positions of the polarizer and analyzer are adjusted to investigate how variations in their relative angles influence the transmission of light. In one configuration the polarizer is at 10° and the analyzer at 120°, resulting in an angular difference of 110°, shown in Fig. 2(a). It is observed that the transmitted light shows the same intensity when the angular difference is 50°, shown in Fig. 2(c). The intensities are matched at the same magnitude when the optical plate is introduced, and this occurs when the difference is 66°, as shown in Fig. 2(e).

In imaging system, a 8051 camera (Thorlabs, NJ, USA) with a pixel size of 5.5 μm × 5.5 μm was used. Using a chrome target to measure the axial resolution, the known line density of the 228 lp/mm translates to 4.385 μm, giving a value of 2.3 pixels/μm. The resolution is measured to be 4.28 μm and the beam diameter is approximately 200 μm. The imaging area of the sensor covered by the beam is approximately 460 × 460 pixels.

The quantitative phase-resolution target, which has a nominal height of 50 nm and an estimated height of 56.5 nm provided by the manufacturer, is used as a standard for evaluating phase-imaging techniques (Fig. 3). The phase shift φ induced by the target is given by the equation Φ = 2πΔnt λ, where Δn is the refractive-index difference between the resolution target and the air (1.52 and 1), t is the height, and λ is the wavelength of the incident beams. This phase target enables precise comparison of the single-shot and conventional methods. According to Fig. 3(d), the average phase step of the sinusoidal grating is measured to be about 0.33 radians. Using the earlier phase equation to calculate the height gives us 62 nm.

Figure 3.Characterization using the quantitative phase resolution target. (a) Intensity pattern recorded by the camera of the single-shot system, having a pair of laterally separated images. Scale bar: 30 μm. (b) Phase reconstruction of the dashed square from (a). (c) Cropped area from (b) with a horizontal (red) and horizontal (blue) for plotting the line profile. (d) Phase cross sections along the horizontal (red) and vertical (blue) lines in (c).

We used a sample of red blood cells to test its feasibility for biomedical imaging (Fig. 4). The human blood smear (Readi-Stain® human blood film slide 313152; Carolina Biological Supply Co., NC, USA) slide is used as sample in this work. Images captured are pegged to have dimensions of 175 × 175 pixels, and the phase profiles are then computed. For this purpose, the focused and defocused images are recorded. While our current imaging system demonstrates satisfactory performance, the full width at half maximum (FWHM) computed as 4.28 μm is smaller than the pixel size (5.5 μm), indicating that the system can resolving features with a minimum size of 4.28 μm. This is a reasonable resolution for imaging the diameter of a red blood cell. Despite this, the system remains functional and provides useful imaging results. This shows that the microscope operates efficiently and effectively resolving the details such as the diameter of red blood cells. Notably, it is capable of measuring phase changes down to 0.33 radians, which corresponds to height measurement of 62 nm.

Figure 4.Phase images of red blood cells, recovered using (a) the conventional transport-of-intensity equation (TIE) method, and (b) the single-shot method using a Wollaston prism. (c) Selected areas for line profiling of (a) and (b). (d) Line profiles comparing the two techniques.

The conventional method required to capture the defocused image is set at 20 μm using the translation stage, while the single-shot method uses the two images captured on the same plane. Upon computation, our findings demonstrate that our method aligns with the conventional TIE. Initially, low spatial noise is observed in Figs. 4(a) and 4(b). This implies that the cell being observed may be positioned higher than the other cells. This also may be due to dust or dirt recorded before image processing. Also, the low spatial noise implies that higher defocus may be used to accommodate observation of the cell. Though noise still exists, our method proves to be efficient as a single-shot method.

The proposed method of phase reconstruction of red blood cells involves computing the line profile using the proposed setup. Comparing the line profile obtained from the proposed method to those from conventional techniques with the conventional method demonstrates that the proposed method achieves comparable reliability, as shown in Fig. 4(d). This approach benefits from the efficiency of single-shot techniques, which reduces the time, as it skips manual translation and the complexity involved in acquiring and analyzing phase images.

The choice of the 1-degree angle separation of the WP allows for a beam up to 200 μm in diameter. For controlled defocus, the OG plate thickness is 5 mm, which is suitable for imaging red blood cells. The defocus distance can be adjusted by changing the thickness of the OG plate [21].

Our method allows for single-shot transport-of-intensity-equation (TIE) imaging, eliminating the need for three or more image acquisitions at different axial positions. This not only reduces acquisition time because it skips translating the stage, but also minimizes potential errors related to sample movement or misalignment caused by such translation. Also, the method we propose includes a novel approach to solving intensity matching when introducing the optical-glass plate, which is necessary for phase retrieval. By rotating the polarizer and analyzer, we ensure that the captured images can achieve equal background intensities. Furthermore, the combination of these improvements makes our method applicable for imaging red blood cells. As the system has controlled defocus, the method shows potential for monitoring live cells or time-sensitive experiments, where phase imaging is essential.

In this study, we investigated the use of a Wollaston prism in a single-shot transport-of-intensity-equation (TIE) method for quantitative phase imaging. Our findings provide several important insights. First, we observed that the intensity of both beams could be simultaneously adjusted using the WP and the polarizer, and is ideal for adjustments caused by the refraction of the optical-glass plate. Second, we found that a 1-degree Wollaston prism can be used effectively for a single-shot method, offering a practical alternative to other multi-shot techniques. This method skips the need for manual translation and captures only two images, making it more efficient while maintaining performance comparable to that of existing methods. Finally, we demonstrated successful phase reconstruction in an application involving red blood cells, further validating the effectiveness and versatility of this method. To extend the research, improvements and optimization are needed. One area of focus could be the angle of separation, to determine the optimal WP angle to balance the trade-off between field of view and distortion. The WP offers practical and reliable solutions for obtaining quantitative phase information. The use of the WP makes the method easily accessible for conventional microscopes configured with a 4f system. These findings help to advance the field of quantitative phase imaging by offering a powerful tool that improves the capabilities and accuracy of TIE-based approaches.

This work was supported by the National Research Foundation of Korea (NRF) grant, funded by the Korean government (MSIT) (Grant no. RS-2023-00302281).

The authors declare that there are no conflicts of interest related to this article.

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.

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Article

Research Paper

Curr. Opt. Photon. 2024; 8(5): 502-507

Published online October 25, 2024 https://doi.org/10.3807/COPP.2024.8.5.502

Copyright © Optical Society of Korea.

Single-shot Transport-of-intensity Equation Using a Wollaston Prism for Biological Samples

Joseph Vermont Bunyi Bandoy, Cuong Manh Nguyen, An Nazmus Sakib, Suhyeon Kim, Hyuk-Sang Kwon

Department of Biomedical Science and Engineering, Gwangju Institute of Science and Technology, Gwangju 61005, Korea

Correspondence to:*hyuksang@gist.ac.kr, ORCID 0000-0002-3387-210X

Received: June 26, 2024; Revised: August 18, 2024; Accepted: August 26, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Wollaston prism (WP) has shown promise in enabling single-shot transport-of-intensity equation (ssTIE) measurements, facilitating efficient phase retrieval in microscopy. The 1-degree prism which produces the minor beam-separation angle is used to prevent distortions. An optical-glass plate is employed in the duplicated beam path to introduce defocusing. This configuration is also advantageous when aligning the beams laterally caused by the refraction of the optical-glass plate, thus allowing another method for single-shot measurements. We applied the proposed method to image the red blood cells (RBCs), demonstrating that the proposed method could be useful in various biological and medical applications.

Keywords: Microscopy, Transport of intensity equation, Wollaston prism

I. INTRODUCTION

Quantitative phase imaging (QPI) is a technique with various applications in microscopy, including label-free noninvasive capabilities in the study of cells and tissues [1]. Light’s interaction with a transparent sample causes a phase shift, and this phase shift provides essential information, as it can measure the index of refraction, morphology, thickness, and optical properties [2]. In addition, with the advancements in deep learning, QPI has been integrated with microscopy to produce images with improved resolution, and enhance the overall performance of microscopes [3].

QPI can be implemented using both interferometric and non-interferometric methods, such as digital holography (DH) [4] or transport-of-intensity equation (TIE) [5]. TIE is a non-interferometric QPI technique. In comparison, DH relies on the coherence of the light source, requires another beam path for fringe analysis, and requires phase unwrapping. This is because the phase retrieved from the hologram is bounded in the range of (−π, π), which necessitating a phase-unwrapping algorithm to obtain the true phase [6]. TIE does not require another beam which obviates the phase-unwrapping step, making reconstruction simpler.

TIE was first derived by Teague [7], where the phase retrieval was solved using the Green’s function. It was first applied in optical phase microscopy in 1998 [8]. The conventional method relies on intensity measurements at multiple distances along the propagation direction of the wave [5, 914], and recording is required for every defocus. Single-shot, on the other hand, aims to achieve phase retrieval using a single intensity image. Achieving single-shot TIE remains an area of ongoing investigation, since it eliminates the need for defocus, making it more efficient. The reconstruction of phase information from a single intensity measurement can be achieved utilizing a volume holographic microscope [15]. Another method is based on a Michelson-interferometer configuration, including two mirrors [16] or one mirror with a spatial light modulator (SLM) [17] to generate two beams for phase imaging. Using an SLM enables accurate phase retrieval by providing 2π phase modulation with linear electro-optic characteristics [18]. Furthermore, employing the Greek-ladder sieves [19] create two beams that are focused at a distance apart. In addition, the Wollaston prism (WP), a type of prism that separates orthogonal beams, can be employed in the setup. The WP is commonly rotated to adjust the intensities of the two rays. In 2021, Omar and El-Bakary [20] suggested an optical setup for the duplicated-image TIE that can accurately determine the optical anisotropy in fibers. Their method is mainly applied to determine the refractive indices of anisotropic fibers at different polarization directions, using only a single set of micrographs. However, the setup still obtains multiple images of the same object at different axial positions. To solve this, Picazo-Bueno and Micó [21] implemented a cost-effective, simple, and robust method for single-shot QPI-based TIE using an add-on optical module that can be assembled on any part of the microscope. The optical module is composed of a Stokes lens (SL), a beam-splitter cube (BSC) and an optical-glass (OG) plate, which enables intensity recording without mechanical translation. The initial results of this technique produce astigmatic images caused by the BSC, requiring the SL. Though providing good field of view, the duplicated image is flipped and has different background intensity, requiring another step for normalization.

The conventional methods for TIE typically involve capturing three images and relying on mechanical translation to achieve defocus. This approach is prone to misalignment with the sample, which can affect the accuracy of the results. On the other hand, previous single-shot methods have addressed this issue by capturing three or more images, extending the processing time and the possibility of observing image-quality variations, or increasing calibration efforts.

To address these limitations, we report a method for single-shot TIE. The system uses a Wollaston prism to produce a duplicated image, which eliminates the need for mechanical translation and only requires two images, to simplify the technique and increase efficiency while maintaining accuracy. Proposed technique is applied to imaging red blood cells to demonstrate its effectiveness in biological applications, where precise and efficient phase imaging is needed.

II. THEORY AND METHODOLOGY

2.1. Transport-of-intensity Equation

The TIE is a non-interferometric quantitative form of phase imaging derived by Teague [7]. The equation describes how phase is related to the propagation of light; in this case, how light propagates along a specific axis, here the z-axis. The general equation for the TIE is given by

Iφ=kzI,

where the right-hand side is the transverse energy and the right-hand side of Eq. (1), axial derivative can be expressed as where the right-hand side represents the transverse energy, and the axial derivative can be expressed as

Ix,y,z0Δz=Ix,y,z0+zIΔz+z2Δz22!+OΔz3,

where the expression only propagates along the z axis the light beam, represented by the expression, propagates along the z-axis. Upon approximation and neglecting the higher orders, the expression becomes discrete and can be written as

Iφk2ΔzzIx,y,zIx,y,z.

Solving the equation gives us

φ=F1k2FI1 F1 k2 FzI,

where the phase can be solved for using a Fourier-transform-based solution. Furthermore,

Ix,y,z0=Ix,y,z+Ix,y,z2.

And

Iz=kIx,y,zIx,y,z2Δz,

where I(x, y, z) and I(x, y, −z) are the focus and defocus images at a distance z from the focus z0, while I(x, y, z0) is the average. Three intensities represented in images are required for phase retrieval by the transport-of-intensity equation. Moreover, the partial derivative of the intensity ∂z I is calculated as the difference of the two recorded intensities. Two images are required for the process: One focused image and one defocused image. These images are then processed to compute the average, as described in Eq. (5). In Eq. (6), the defocused image is subtracted from the focused image, the result is divided by a constant k, and their distance. The intensities obtained from Eqs. (5) and (6) are used to retrieve the phase information in Eq. 4 [22]. The conventional approach for determining the transport of intensity requires three images: Under-focused, in-focus, and over-focused. However, single-shot methods requiring only two images is also possible, with the third intensity being the average of the two recorded images. With this, the phase can be retrieved.

2.2. Controlled Defocus for the Single-shot Method

The devised single-shot method uses a Wollaston prism to generate two beams, while introducing an optical-glass plate for controlled defocus. In this work, a 1-degree prism is selected to reduce the beam deviation when separating the beams with respect to the optical axis, as shown in Fig. 1. The WP is mounted on a rotating stage for adjustments and calibration.

Figure 1. Optical setup for ssTIE components: Incoherent light source (LED), polarizer (P), sample (S), microscope objective (MO), tube lens (TL), bandpass filter (BPF), lenses (L1, L2), Wollaston prism (WP), optical glass (OG), analyzer (A), and CCD.

For controlled defocus, an OG plate is placed in the other beam path. This is used to replace manual operation with the use of the translation stage. The OG plate has thickness of 5 mm, resulting a defocus of approximately 17 μm. By adopting a displacement of 20 μm in the translation stage, comparison with the conventional method is possible.

2.3. Experimental Setup

The ssTIE are constructed as follows: First, the LED lamp served as a light source. Next, the polarizer, an illuminated sample, and a microscope objective lens (10× magnification) are arranged in sequence. The light from the source is collimated to a tube lens (f = 200 mm). A green band-pass filter of 525-nm wavelength is arranged next to this tube lens.

Following the imaging plane, there is a 4f system comprised of two lenses, each with a focal length of 150 mm. The WP is positioned at the focus of L1 to produce the desired separation between the output beams. After the 4f system, the OG plate (5-mm thickness) and analyzer is arranged in sequence. Finally, the images are recorded using a CCD camera (8051C-USB).

III. RESULTS AND DISCUSSION

Using the WP without the polarizer and analyzer, we gained two images with different background intensities as an initial results (Fig. 2). When the intensities are adjusted upon rotation, there is one specific position where the two beams have the same intensities. In contrast, regardless of the position, introducing the polarizer and analyzer provides simultaneous adjustments. If uneven beam is produced in the target position, it can be compensated by the polarizer. Also, when the final image is slightly displaced from its original position, the OG plate resulting in refraction is observed. This can be compensated by rotating the WP for image repositioning, followed by the polarizer for intensity compensation, as shown in Fig. 2(d). This technique can be applied directly to TIE phase reconstruction. The single-shot imaging technique utilizing an optical glass plate allows us to manipulate the beam and create a controlled defocus in the captured image.

Figure 2. Intensities observed based on the polarization (a)–(e), with their respective line profiles plotted against pixel distance (f)–(j).

The positions of the polarizer and analyzer are adjusted to investigate how variations in their relative angles influence the transmission of light. In one configuration the polarizer is at 10° and the analyzer at 120°, resulting in an angular difference of 110°, shown in Fig. 2(a). It is observed that the transmitted light shows the same intensity when the angular difference is 50°, shown in Fig. 2(c). The intensities are matched at the same magnitude when the optical plate is introduced, and this occurs when the difference is 66°, as shown in Fig. 2(e).

In imaging system, a 8051 camera (Thorlabs, NJ, USA) with a pixel size of 5.5 μm × 5.5 μm was used. Using a chrome target to measure the axial resolution, the known line density of the 228 lp/mm translates to 4.385 μm, giving a value of 2.3 pixels/μm. The resolution is measured to be 4.28 μm and the beam diameter is approximately 200 μm. The imaging area of the sensor covered by the beam is approximately 460 × 460 pixels.

The quantitative phase-resolution target, which has a nominal height of 50 nm and an estimated height of 56.5 nm provided by the manufacturer, is used as a standard for evaluating phase-imaging techniques (Fig. 3). The phase shift φ induced by the target is given by the equation Φ = 2πΔnt λ, where Δn is the refractive-index difference between the resolution target and the air (1.52 and 1), t is the height, and λ is the wavelength of the incident beams. This phase target enables precise comparison of the single-shot and conventional methods. According to Fig. 3(d), the average phase step of the sinusoidal grating is measured to be about 0.33 radians. Using the earlier phase equation to calculate the height gives us 62 nm.

Figure 3. Characterization using the quantitative phase resolution target. (a) Intensity pattern recorded by the camera of the single-shot system, having a pair of laterally separated images. Scale bar: 30 μm. (b) Phase reconstruction of the dashed square from (a). (c) Cropped area from (b) with a horizontal (red) and horizontal (blue) for plotting the line profile. (d) Phase cross sections along the horizontal (red) and vertical (blue) lines in (c).

We used a sample of red blood cells to test its feasibility for biomedical imaging (Fig. 4). The human blood smear (Readi-Stain® human blood film slide 313152; Carolina Biological Supply Co., NC, USA) slide is used as sample in this work. Images captured are pegged to have dimensions of 175 × 175 pixels, and the phase profiles are then computed. For this purpose, the focused and defocused images are recorded. While our current imaging system demonstrates satisfactory performance, the full width at half maximum (FWHM) computed as 4.28 μm is smaller than the pixel size (5.5 μm), indicating that the system can resolving features with a minimum size of 4.28 μm. This is a reasonable resolution for imaging the diameter of a red blood cell. Despite this, the system remains functional and provides useful imaging results. This shows that the microscope operates efficiently and effectively resolving the details such as the diameter of red blood cells. Notably, it is capable of measuring phase changes down to 0.33 radians, which corresponds to height measurement of 62 nm.

Figure 4. Phase images of red blood cells, recovered using (a) the conventional transport-of-intensity equation (TIE) method, and (b) the single-shot method using a Wollaston prism. (c) Selected areas for line profiling of (a) and (b). (d) Line profiles comparing the two techniques.

The conventional method required to capture the defocused image is set at 20 μm using the translation stage, while the single-shot method uses the two images captured on the same plane. Upon computation, our findings demonstrate that our method aligns with the conventional TIE. Initially, low spatial noise is observed in Figs. 4(a) and 4(b). This implies that the cell being observed may be positioned higher than the other cells. This also may be due to dust or dirt recorded before image processing. Also, the low spatial noise implies that higher defocus may be used to accommodate observation of the cell. Though noise still exists, our method proves to be efficient as a single-shot method.

The proposed method of phase reconstruction of red blood cells involves computing the line profile using the proposed setup. Comparing the line profile obtained from the proposed method to those from conventional techniques with the conventional method demonstrates that the proposed method achieves comparable reliability, as shown in Fig. 4(d). This approach benefits from the efficiency of single-shot techniques, which reduces the time, as it skips manual translation and the complexity involved in acquiring and analyzing phase images.

The choice of the 1-degree angle separation of the WP allows for a beam up to 200 μm in diameter. For controlled defocus, the OG plate thickness is 5 mm, which is suitable for imaging red blood cells. The defocus distance can be adjusted by changing the thickness of the OG plate [21].

Our method allows for single-shot transport-of-intensity-equation (TIE) imaging, eliminating the need for three or more image acquisitions at different axial positions. This not only reduces acquisition time because it skips translating the stage, but also minimizes potential errors related to sample movement or misalignment caused by such translation. Also, the method we propose includes a novel approach to solving intensity matching when introducing the optical-glass plate, which is necessary for phase retrieval. By rotating the polarizer and analyzer, we ensure that the captured images can achieve equal background intensities. Furthermore, the combination of these improvements makes our method applicable for imaging red blood cells. As the system has controlled defocus, the method shows potential for monitoring live cells or time-sensitive experiments, where phase imaging is essential.

IV. CONCLUSION

In this study, we investigated the use of a Wollaston prism in a single-shot transport-of-intensity-equation (TIE) method for quantitative phase imaging. Our findings provide several important insights. First, we observed that the intensity of both beams could be simultaneously adjusted using the WP and the polarizer, and is ideal for adjustments caused by the refraction of the optical-glass plate. Second, we found that a 1-degree Wollaston prism can be used effectively for a single-shot method, offering a practical alternative to other multi-shot techniques. This method skips the need for manual translation and captures only two images, making it more efficient while maintaining performance comparable to that of existing methods. Finally, we demonstrated successful phase reconstruction in an application involving red blood cells, further validating the effectiveness and versatility of this method. To extend the research, improvements and optimization are needed. One area of focus could be the angle of separation, to determine the optimal WP angle to balance the trade-off between field of view and distortion. The WP offers practical and reliable solutions for obtaining quantitative phase information. The use of the WP makes the method easily accessible for conventional microscopes configured with a 4f system. These findings help to advance the field of quantitative phase imaging by offering a powerful tool that improves the capabilities and accuracy of TIE-based approaches.

FUNDING

This work was supported by the National Research Foundation of Korea (NRF) grant, funded by the Korean government (MSIT) (Grant no. RS-2023-00302281).

DISCLOSURES

The authors declare that there are no conflicts of interest related to this article.

DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.

Fig 1.

Figure 1.Optical setup for ssTIE components: Incoherent light source (LED), polarizer (P), sample (S), microscope objective (MO), tube lens (TL), bandpass filter (BPF), lenses (L1, L2), Wollaston prism (WP), optical glass (OG), analyzer (A), and CCD.
Current Optics and Photonics 2024; 8: 502-507https://doi.org/10.3807/COPP.2024.8.5.502

Fig 2.

Figure 2.Intensities observed based on the polarization (a)–(e), with their respective line profiles plotted against pixel distance (f)–(j).
Current Optics and Photonics 2024; 8: 502-507https://doi.org/10.3807/COPP.2024.8.5.502

Fig 3.

Figure 3.Characterization using the quantitative phase resolution target. (a) Intensity pattern recorded by the camera of the single-shot system, having a pair of laterally separated images. Scale bar: 30 μm. (b) Phase reconstruction of the dashed square from (a). (c) Cropped area from (b) with a horizontal (red) and horizontal (blue) for plotting the line profile. (d) Phase cross sections along the horizontal (red) and vertical (blue) lines in (c).
Current Optics and Photonics 2024; 8: 502-507https://doi.org/10.3807/COPP.2024.8.5.502

Fig 4.

Figure 4.Phase images of red blood cells, recovered using (a) the conventional transport-of-intensity equation (TIE) method, and (b) the single-shot method using a Wollaston prism. (c) Selected areas for line profiling of (a) and (b). (d) Line profiles comparing the two techniques.
Current Optics and Photonics 2024; 8: 502-507https://doi.org/10.3807/COPP.2024.8.5.502

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