3630
OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014
Quantitative phase imaging unit KyeoReh Lee and YongKeun Park* Department of Physics, Korea Advanced Institutes of Science and Technology, Daejeon 305-701, South Korea *Corresponding author: [email protected] Received March 3, 2014; revised May 6, 2014; accepted May 8, 2014; posted May 9, 2014 (Doc. ID 207530); published June 12, 2014 A simple and cost-effective method is presented for quantitative phase imaging. A common-path lateral phase shifting interferometer is realized through attaching a compact filter set to the output port of an existing microscope. The working principles, design criteria, and limitations are also derived and explained. In order to demonstrate the capability and applicability of the method, the optical phase images of a microsphere and individual human red blood cells are measured with high stability. © 2014 Optical Society of America OCIS codes: (090.2880) Holographic interferometry; (180.3170) Interference microscopy; (110.3175) Interferometric imaging; (170.3880) Medical and biological imaging. http://dx.doi.org/10.1364/OL.39.003630
Recent research has enabled transparent phase objects, such as biological cells and tissues to be effectively investigated using quantitative phase imaging (QPI) [1,2]. While traditional phase microscopies including phase contrast (PC) microscopy and differential interference contrast (DIC) microscopy render contrast for phase objects, QPI techniques including quantitative phase microscopy [3,4] and digital holographic microscopy [5,6] quantitatively measure the optical light field of a sample containing both amplitude and phase information. Thus, QPI techniques enable investigation of biological samples including the optical dry mass of cells [7], 3D refractive index distributions of cells [8], and light scattering spectroscopy [9]. Despite the advantages and technical advances of QPI, the translation of QPI techniques into biological and clinical laboratories is hindered due to its bulky and complicated optical setup, expensive cost for the instruments and maintenance, and incompatibility with existing microscopes. In order to address these issues, various QPI approaches have been proposed: Michelson interferometry [10], spatial light interference microscopy [11], sample field sensor [12], τ interferometry [13,14], lateral shearing interferometry [15–17], Lloyd’s mirror [18], hole array sensors [12,19], transport intensity equation (TIE) [20], and QPI algorithms from PC [21,22] or DIC images [23–28], as reviewed in Ref. [29]. Regardless of the numerous approaches, many require precise optical alignment, expensive components, and time-consuming computations that impede the expansion of QPI as a practical method. Therefore, a simple and cost-effective method for QPI is proposed in this Letter, and it is referred to as the QPI unit (QPIU). The QPIU is based on a common-path lateral phase shifting interferometer, and it can be easily added to the output port of an existing microscope as a compact filter set. The QPIU setup is presented in Fig. 1(A). The sample positioned in the microscope is illuminated with the linearly polarized light. Then, the diffracted beam of the sample is transmitted through the microscope output port, where the QPIU is attached. In order to simplify the optical setup, the QPIU is composed of three widely available optical components: a half-wave plate, a Wollaston prism (WP), and a linear polarizer. 0146-9592/14/123630-04$15.00/0
The vertically polarized beam at the output passes through the half-wave plate, resulting in the polarization being rotated by 45°. Then, the WP prism splits the beam into two beams with different propagation angles and those beams are polarized either horizontally or vertically. The linear polarizer located after the WP is aligned to have an angle of 45° with respect to both polarizations of the two separated beams. This causes the split beams to interfere with each other at an image plane where the image senor is located to record a spatially modulated interferogram. The rotation of the half-wave plate can be tuned in order to cause the light polarization to be equally weighted on the two eigen polarization states of the WP. As shown in Fig. 1(A), the polarization direction of the beam past the half-wave plate was rotated by 45°. This adjustment maximizes the contrast of the interference fringe patterns; thus, it fully utilizes the dynamic range of the image sensor. A photograph of the QPIU attached to an inverted microscopy is displayed in Fig. 1(B). In order to ensure the diffraction limited optical resolution in the QPIU and to fully utilize the pixel resolution of the image sensor, the following design criteria should be carefully considered (see Fig. 2 and its caption for the variable definitions). According to the Nyquist theorem,
Fig. 1. (A) Optical setup for the QPI unit. A unit composed of a half-wave plate (λ∕2), Wollaston prism (WP), and linear polarizer (P) is attached to the output port of an inverted microscope. The red arrows indicate the propagation directions of the beams. The white arrows indicate the polarization state of the beam. (B) Photograph of the QPIU in working conditions attached to a microscope. © 2014 Optical Society of America
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
Fig. 2. Illustration of the light fields on the image plane. The patterned objects (Σ) indicate the sample images (duplicated), the gray circles (Σc ) indicate the nonsample areas, and the black bold rectangle and smaller inner rectangles indicate the field of view and the pixel of an image sensor, respectively. fov is the field of view of the camera, CA is the final clear aperture of the optical system before the camera, θ is the separation angle of the beams split by the WP, L is the perpendicular distance between the beam separation point and the image plane, δ is the separation distance between the two duplicated images on the image plane, D is the maximum width of a sample along the direction of δ; and p is the pixel size of a camera.
an image sensor with a pixel size of p cannot access a spatial frequency higher than 1∕2p. Therefore, the carrier spatial frequency in the spatially modulated interferogram sin θ∕λ, which is determined using the separation angle between the two split beams (θ) and the wavelength of the light source (λ), should not exceed the Nyquist frequency. At the same time, the carrier spatial frequency must be sufficiently large enough not to lose the diffraction-limited resolution of a microscope. Therefore, the following condition should be satisfied: λ NA NA > sin θ >3 ; 2p M M
the tilted angle θ satisfying Eq. (1) is very small (cos θ ≥ 0.999). Separated images are identical unless samples have birefringence. For a birefringent sample, two separated images by the WP prism have orthogonal polarization states. Thus, in principle it is possible to measure the birefringence of a sample by comparing two separated polarization dependent images and appropriate illumination [32]. However, through the remainder of this Letter we do not consider the birefringence of a sample. Because tx; y spatially overlaps itself in the measured interferogram [Eq. (2)], the retrieval of the sample information tx; y is not straightforward, unless the sample region (Σ) only overlaps with the nonsample region (Σc ) in the interferogram. In order to ensure this condition, Σc must be sufficiently wide to cover the entire Σ, and δ must be larger than the maximum width of the sample (D) along the direction of separation, that is.
CA > 2D δ 2L tan θ > D
where NA and M are the numerical aperture and magnification of an objective lens, respectively. Equation (1) is a general condition for recording a spatially modulated interferogram in QPI. According to Eq. (1), the optimal θ that maximizes the capacity of objective lenses is determined to be sin−1 λ∕3p. In order to avoid the overlap between the duplicated images in the laterally shifted interferogram and to directly obtain the QPI of the sample, the design parameters in the QPIU should be chosen considering an appropriate separation distance between the two duplicated images (δ) as in Refs. [30,31]. The recorded interferogram at the image plane can be expressed as follows: 2 δ δ 2π 2π Ix; y t x − ; y ej λ sin θx t x ; y e−j λ sin θx ; 2 2 (2) where tx; y is a transparent function of the sample; x and y are the lateral coordinates, while x is set to be parallel to the direction of the separation. The elongation of images resulting from the tilted angles is ignored because
:
3
QPIU works well considering the transmission of a clear or blank area in Σc that serves as a reference beam. Thus, the proposed approach contains strong fov limitation. Because θ is defined by the WP, the distance between the beam separation point and the image plane (L) is bounded by D. Furthermore, the coherence length of the illumination (lc ) should also be considered in the QPIU. Using the WP for the separation, the optical path length difference of the two separated beams is inevitable. Accordingly, in order to construct interference patterns that cover the entire sample region, the following condition should be satisfied: lc ≥ D sin θ:
(1)
3631
(4)
In addition, in the QPIU, a distortion aberration exists in the tilted optical fields, but this can be ignored because the sample field tx; y at the image plane is significantly magnified using the microscope and the effect of the aberration on the retrieved image is negligible. That is, Eqs. (1), (3), and (4) are the design criteria for the QPIU. In order to demonstrate the capability of the QPIU, the quantitative phase images of individual microscopic objects were experimentally measured. For the illumination, a He–Ne laser (λ 633 nm; HRR050, Thorlabs Inc., USA) with a linear polarizer (LPVISE100-A, Thorlabs Inc., USA) was used. An inverted microscope (IX73, Olympus Inc., Japan) equipped with an objective lens with NA 1.4, M 100 (UPLSAPO 100XO, Olympus Inc., Japan) was used. The QPIU was constructed using a half-wave plate (WPH10M-633, Thorlabs Inc., USA), a quartz WP (68-820, Edmund Optics Inc., USA), and a linear polarizer (LPVISE100-A, Thorlabs Inc., USA). A CCD camera (Lt365R, p 4.54 μm, Lumenera Inc., USA) was used to record the interferograms. The parameters used and their dependency are summarized in Table 1. Using the QPIU, the phase images of a polystyrene microsphere with a diameter of 3 μm were imaged first. The measured quantitative phase image of the microsphere is seen clearly in Fig. 3(A). Then, individual live
OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014 Table 1. Parameters Used and Dependency
Parameters λ NA M D θ CA L δ p fov
Values
Dependency
0.633 μm 1.4 100
OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014
Quantitative phase imaging unit KyeoReh Lee and YongKeun Park* Department of Physics, Korea Advanced Institutes of Science and Technology, Daejeon 305-701, South Korea *Corresponding author: [email protected] Received March 3, 2014; revised May 6, 2014; accepted May 8, 2014; posted May 9, 2014 (Doc. ID 207530); published June 12, 2014 A simple and cost-effective method is presented for quantitative phase imaging. A common-path lateral phase shifting interferometer is realized through attaching a compact filter set to the output port of an existing microscope. The working principles, design criteria, and limitations are also derived and explained. In order to demonstrate the capability and applicability of the method, the optical phase images of a microsphere and individual human red blood cells are measured with high stability. © 2014 Optical Society of America OCIS codes: (090.2880) Holographic interferometry; (180.3170) Interference microscopy; (110.3175) Interferometric imaging; (170.3880) Medical and biological imaging. http://dx.doi.org/10.1364/OL.39.003630
Recent research has enabled transparent phase objects, such as biological cells and tissues to be effectively investigated using quantitative phase imaging (QPI) [1,2]. While traditional phase microscopies including phase contrast (PC) microscopy and differential interference contrast (DIC) microscopy render contrast for phase objects, QPI techniques including quantitative phase microscopy [3,4] and digital holographic microscopy [5,6] quantitatively measure the optical light field of a sample containing both amplitude and phase information. Thus, QPI techniques enable investigation of biological samples including the optical dry mass of cells [7], 3D refractive index distributions of cells [8], and light scattering spectroscopy [9]. Despite the advantages and technical advances of QPI, the translation of QPI techniques into biological and clinical laboratories is hindered due to its bulky and complicated optical setup, expensive cost for the instruments and maintenance, and incompatibility with existing microscopes. In order to address these issues, various QPI approaches have been proposed: Michelson interferometry [10], spatial light interference microscopy [11], sample field sensor [12], τ interferometry [13,14], lateral shearing interferometry [15–17], Lloyd’s mirror [18], hole array sensors [12,19], transport intensity equation (TIE) [20], and QPI algorithms from PC [21,22] or DIC images [23–28], as reviewed in Ref. [29]. Regardless of the numerous approaches, many require precise optical alignment, expensive components, and time-consuming computations that impede the expansion of QPI as a practical method. Therefore, a simple and cost-effective method for QPI is proposed in this Letter, and it is referred to as the QPI unit (QPIU). The QPIU is based on a common-path lateral phase shifting interferometer, and it can be easily added to the output port of an existing microscope as a compact filter set. The QPIU setup is presented in Fig. 1(A). The sample positioned in the microscope is illuminated with the linearly polarized light. Then, the diffracted beam of the sample is transmitted through the microscope output port, where the QPIU is attached. In order to simplify the optical setup, the QPIU is composed of three widely available optical components: a half-wave plate, a Wollaston prism (WP), and a linear polarizer. 0146-9592/14/123630-04$15.00/0
The vertically polarized beam at the output passes through the half-wave plate, resulting in the polarization being rotated by 45°. Then, the WP prism splits the beam into two beams with different propagation angles and those beams are polarized either horizontally or vertically. The linear polarizer located after the WP is aligned to have an angle of 45° with respect to both polarizations of the two separated beams. This causes the split beams to interfere with each other at an image plane where the image senor is located to record a spatially modulated interferogram. The rotation of the half-wave plate can be tuned in order to cause the light polarization to be equally weighted on the two eigen polarization states of the WP. As shown in Fig. 1(A), the polarization direction of the beam past the half-wave plate was rotated by 45°. This adjustment maximizes the contrast of the interference fringe patterns; thus, it fully utilizes the dynamic range of the image sensor. A photograph of the QPIU attached to an inverted microscopy is displayed in Fig. 1(B). In order to ensure the diffraction limited optical resolution in the QPIU and to fully utilize the pixel resolution of the image sensor, the following design criteria should be carefully considered (see Fig. 2 and its caption for the variable definitions). According to the Nyquist theorem,
Fig. 1. (A) Optical setup for the QPI unit. A unit composed of a half-wave plate (λ∕2), Wollaston prism (WP), and linear polarizer (P) is attached to the output port of an inverted microscope. The red arrows indicate the propagation directions of the beams. The white arrows indicate the polarization state of the beam. (B) Photograph of the QPIU in working conditions attached to a microscope. © 2014 Optical Society of America
June 15, 2014 / Vol. 39, No. 12 / OPTICS LETTERS
Fig. 2. Illustration of the light fields on the image plane. The patterned objects (Σ) indicate the sample images (duplicated), the gray circles (Σc ) indicate the nonsample areas, and the black bold rectangle and smaller inner rectangles indicate the field of view and the pixel of an image sensor, respectively. fov is the field of view of the camera, CA is the final clear aperture of the optical system before the camera, θ is the separation angle of the beams split by the WP, L is the perpendicular distance between the beam separation point and the image plane, δ is the separation distance between the two duplicated images on the image plane, D is the maximum width of a sample along the direction of δ; and p is the pixel size of a camera.
an image sensor with a pixel size of p cannot access a spatial frequency higher than 1∕2p. Therefore, the carrier spatial frequency in the spatially modulated interferogram sin θ∕λ, which is determined using the separation angle between the two split beams (θ) and the wavelength of the light source (λ), should not exceed the Nyquist frequency. At the same time, the carrier spatial frequency must be sufficiently large enough not to lose the diffraction-limited resolution of a microscope. Therefore, the following condition should be satisfied: λ NA NA > sin θ >3 ; 2p M M
the tilted angle θ satisfying Eq. (1) is very small (cos θ ≥ 0.999). Separated images are identical unless samples have birefringence. For a birefringent sample, two separated images by the WP prism have orthogonal polarization states. Thus, in principle it is possible to measure the birefringence of a sample by comparing two separated polarization dependent images and appropriate illumination [32]. However, through the remainder of this Letter we do not consider the birefringence of a sample. Because tx; y spatially overlaps itself in the measured interferogram [Eq. (2)], the retrieval of the sample information tx; y is not straightforward, unless the sample region (Σ) only overlaps with the nonsample region (Σc ) in the interferogram. In order to ensure this condition, Σc must be sufficiently wide to cover the entire Σ, and δ must be larger than the maximum width of the sample (D) along the direction of separation, that is.
CA > 2D δ 2L tan θ > D
where NA and M are the numerical aperture and magnification of an objective lens, respectively. Equation (1) is a general condition for recording a spatially modulated interferogram in QPI. According to Eq. (1), the optimal θ that maximizes the capacity of objective lenses is determined to be sin−1 λ∕3p. In order to avoid the overlap between the duplicated images in the laterally shifted interferogram and to directly obtain the QPI of the sample, the design parameters in the QPIU should be chosen considering an appropriate separation distance between the two duplicated images (δ) as in Refs. [30,31]. The recorded interferogram at the image plane can be expressed as follows: 2 δ δ 2π 2π Ix; y t x − ; y ej λ sin θx t x ; y e−j λ sin θx ; 2 2 (2) where tx; y is a transparent function of the sample; x and y are the lateral coordinates, while x is set to be parallel to the direction of the separation. The elongation of images resulting from the tilted angles is ignored because
:
3
QPIU works well considering the transmission of a clear or blank area in Σc that serves as a reference beam. Thus, the proposed approach contains strong fov limitation. Because θ is defined by the WP, the distance between the beam separation point and the image plane (L) is bounded by D. Furthermore, the coherence length of the illumination (lc ) should also be considered in the QPIU. Using the WP for the separation, the optical path length difference of the two separated beams is inevitable. Accordingly, in order to construct interference patterns that cover the entire sample region, the following condition should be satisfied: lc ≥ D sin θ:
(1)
3631
(4)
In addition, in the QPIU, a distortion aberration exists in the tilted optical fields, but this can be ignored because the sample field tx; y at the image plane is significantly magnified using the microscope and the effect of the aberration on the retrieved image is negligible. That is, Eqs. (1), (3), and (4) are the design criteria for the QPIU. In order to demonstrate the capability of the QPIU, the quantitative phase images of individual microscopic objects were experimentally measured. For the illumination, a He–Ne laser (λ 633 nm; HRR050, Thorlabs Inc., USA) with a linear polarizer (LPVISE100-A, Thorlabs Inc., USA) was used. An inverted microscope (IX73, Olympus Inc., Japan) equipped with an objective lens with NA 1.4, M 100 (UPLSAPO 100XO, Olympus Inc., Japan) was used. The QPIU was constructed using a half-wave plate (WPH10M-633, Thorlabs Inc., USA), a quartz WP (68-820, Edmund Optics Inc., USA), and a linear polarizer (LPVISE100-A, Thorlabs Inc., USA). A CCD camera (Lt365R, p 4.54 μm, Lumenera Inc., USA) was used to record the interferograms. The parameters used and their dependency are summarized in Table 1. Using the QPIU, the phase images of a polystyrene microsphere with a diameter of 3 μm were imaged first. The measured quantitative phase image of the microsphere is seen clearly in Fig. 3(A). Then, individual live
OPTICS LETTERS / Vol. 39, No. 12 / June 15, 2014 Table 1. Parameters Used and Dependency
Parameters λ NA M D θ CA L δ p fov
Values
Dependency
0.633 μm 1.4 100
