Quantitative phase imaging system with slightly-off-axis configuration and suitable for objects both larger and smaller than the size of the image sensor

Yang Yang; Zhen-Jia Cheng; Hui-Min Zhao; Qing-Yang Yue; Cheng-Shan Guo

doi:10.1364/OE.26.017199

1. Introduction

Due to the superior ability in quantitatively measuring and imaging the complex amplitude distribution (especially the phase information) of an object wavefront, digital holography (DH), as an important quantitative phase imaging technique, has drawn much attention in recent years [1–3] and have been widely applied to investigate physical, biological and chemical phenomena [4–13]. According to the relative orientation between the object and the reference beams in recording geometry, DH methods are conventionally divided into two categories: in-line and off-axis. In in-line holographic recording configurations, the intersection angle between the orientations of the object wave and the reference wave is set to be close to zero; such a configuration can let us make full use of the resolving power of the image sensor and realize image reconstruction with high spatial resolution [14–16]. The main trouble is that the reconstructed image from an in-line hologram often suffers from the noise of the autocorrelation and conjugate items, which need to be eliminated or suppressed by using phase shifting operations or other time-consuming elimination algorithms [17–27]. In off-axis DH methods, the autocorrelation and conjugate items can be well separated from the required image in spatial frequency domain and so they can be removed in image reconstruction simply by taking a digital spatial filtering to a single hologram [28], but at the cost of a high bandwidth demand on the imaging sensor. In general, to separate spatial frequencies of the object wave from those of the autocorrelation and conjugate items, it requires the bandwidth of the image sensor four times that of the object wave to be recorded.

In recent years, a novel intermediate solution between the conventional in-line and off-axis DHs, named slightly-off-axis digital holography (SODH), was proposed and developed to optimize the tradeoff between the acquisition rate and the detector bandwidth for quantitative phase microscopy [29–39]. The SODH methods do not require full separation of the spatial-frequency of the object wave from those of the autocorrelation item; it only needs to separate the spatial frequency of the object wave from those of its conjugate item, that is, it is enough to set the spatial frequency of the reference beam equal to or slightly larger than the maximum spatial frequency of the recorded object wave. Thus, the lateral resolution of the reconstructed image based on SODH methods can be largely improved. At the same time, SODH is also superior to in-line DH for observing dynamic processes, owing to its simpler phase retrieving processes, such as the retrieving methods based on Hilbert transform [29], nonlinear filtering [30] and phase derivative [35].

Although most of phase retrieving algorithms designed for SODHs are equally suitable for conventional off-axis holograms, in order to get the best resolution of the reconstructed image, the carrier frequency of the reference beam should be finely calibrated as close as possible to the maximum spatial frequency of the recorded object wave in design of the recording setups. However, most of the setups designed for SODHs are directly based on the existed off-axis holographic recording setups, in which the maximum spatial frequency of the recorded object wave and the carrier frequency of the reference wave are difficult to quantitatively determine and control and so the setups have some troubles in satisfying the optimal frequency condition for the SODHs.

In this paper, we design a system that can meet better the frequency condition for quantitative phase imaging based on SODH algorithms. In this system, the object is illuminated by a convergent spherical beam and a specially designed aperture filter is placed on the spatial frequency plane of the object wave; at the same time, a point source emitting from the edge of the aperture is taken as the reference beam, so that the optimal frequency condition for SODHs can be always guaranteed for both large and small objects as well as different magnification (or the field of view) configurations adjusted by changing the distance from the object to the aperture filter. And because the reference point source used in the system is built by one fiber branch of a 1 × 2 single-mode optical fiber splitter, the proposed system provides a low-cost way to convert a regular microscope into a slightly-off-axis holographic one. Section 2 presents the schematic layout of the system as well as a detailed description of the algorithm for extracting the complex amplitude image of the specimen from the recorded holograms. Section 3 describes and discusses the experimental demonstration of the proposed system and gives some examples of the system in imaging biological samples. Section 4 concludes the paper.

2. Principle of the system

Figure 1 shows one principle thematic of our proposed system. The system is mainly composed of a fiber-coupled laser, a 1 × 2 single-mode optical fiber splitter (OFS), a common focusing lens (FL), a specially designed aperture filter (AF) and an image sensor (IS). The beam coming from the fiber-coupled laser is split into two branches by the 1 × 2 OFS, one is firstly transformed into a convergent spherical beam by the FL and then illuminated on the tested object (O) to form the object wave, the other is directly sent by a single-mode fiber to the edge of the aperture on the AF as the reference point source for slightly-off-axis holographic recording. The AF is inserted between the specimen and the recording plane and located just at the focal plane of the convergent illumination beam. A small pinhole is set at the edge of the filtering aperture on the AF and the exit end of the reference fiber branch is fixed in this pinhole to generate the point reference beam directly emitted from this small pinhole, as illustrated in Fig. 1. Thus a slightly-off-axis hologram can be recorded at the recording plane by the IS. Because no any objective lens is inserted between the tested object and the IS, the system belongs to a lensless phase imaging system, suitable for objects both smaller and larger than the size of the effective recording region of the IS.

Fig. 1 Schematic diagram of the proposed SODH system suitable for objects both larger and smaller than the size of the image sensors.

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What follows is the principle analysis of the proposed system. Suppose λ is the wavelength of the input laser, z₁ and z₂ are, respectively, the distance from the object plane to the AF plane and the distance from the AF plane to the recording plane, and $t (x_{1}, y_{1})$ is the complex transmittance of the tested object. Under the coordinate geometry of the system as shown in Fig. 2, the reference wave directly illuminated on the recording plane and the object wave diffracted from the object plane, through the AF, to the recording plane can be respectively written as [40]:

R (x_{3}, y_{3}) = A_{r} \exp {\frac{i π}{λ z_{2}} [{(x_{3} - x_{r})}^{2} + y_{3}^{2}]},

and

\begin{matrix} O (x_{3}, y_{3}) = \frac{A_{i} \exp [i k (z_{1} + z_{2})]}{z_{1} z_{2} λ^{2}} \exp [\frac{i π}{λ z_{2}} (x_{3}^{2} + y_{3}^{2})] \iint \tilde{t} (\frac{x_{2}}{λ z_{1}}, \frac{y_{2}}{λ z_{1}}) c y l (\frac{x_{2}}{R}, \frac{y_{2}}{R}), \\ \times \exp [\frac{i π}{λ} (\frac{1}{z_{1}} + \frac{1}{z_{2}}) (x_{2}^{2} + y_{2}^{2})] \exp [\frac{- i 2 π}{λ z_{2}} (x_{2} x_{3} + y_{2} y_{3})] d x_{2} d y_{2} \end{matrix}

where

\tilde{t} (ξ, η)

is the Fourier transform (or spatial spectrum) of

t (x_{1}, y_{1})

, and

c y l (x_{2} / R, y_{2} / R)

is the transmittance function of the aperture filter (with the aperture radius of R) on AF plane; A_r and A_i are two constants dependent respectively on the input laser source and the OFS; x_r is the distance of the reference point apart from the aperture center (here this point is supposed to be located in x axis for simplicity) and it is only slightly larger than the radius R. By using variable substitution of

ξ = x_{2} / (λ z_{2})

and

η = y_{2} / (λ z_{2})

, Eq. (2) can be further expressed into

\begin{array}{l} O (x_{3}, y_{3}) = M C \exp [\frac{i π}{λ z_{2}} (x_{3}^{2} + y_{3}^{2})] \iint \tilde{t} (M ξ, M η) c y l (\frac{λ z_{2} ξ}{R}, \frac{λ z_{2} η}{R}), \\ \times \exp [i π λ M (z_{1} + z_{2}) (ξ^{2} + η^{2})] \exp [- i 2 π (x_{3} ξ + y_{3} η)] d ξ d η \\ = M C \exp [\frac{i π}{λ z_{2}} (x_{3}^{2} + y_{3}^{2})] F {\tilde{t} (M ξ, M η) c y l (\frac{λ z_{2} ξ}{R}, \frac{λ z_{2} η}{R}) \exp [i π λ M (z_{1} + z_{2}) (ξ^{2} + η^{2})]} \end{array}

in which,

F {}

denotes the Fourier transform (FT) operator, C is a complex constant, and

M = \frac{z_{2}}{z_{1}} .

is a magnification factor dependent on the distance parameters of z₁ and z₂.

Fig. 2 Coordinate geometry of our SODH system.

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Then the hologram; that is, the intensity distribution formed by coherent superimposition of the object wave and the reference wave can be recorded by the image sensor, which can be simply expressed as:

\begin{array}{l} I_{H} (x_{3}, y_{3}) = {| O (x_{3}, y_{3}) + R (x_{3}, y_{3}) |}^{2}, \\ = {| O (x_{3}, y_{3}) |}^{2} + {| R (x_{3}, y_{3}) |}^{2} + O (x_{3}, y_{3}) R^{*} (x_{3}, y_{3}) + O^{*} (x_{3}, y_{3}) R (x_{3}, y_{3}) \end{array}

where the asterisk ‘*’ denotes the complex conjugate operator.

In order to check if the recorded hologram of Eq. (5) satisfies the optimal frequency condition for reconstruction based on SODH algorithms, we can transform Eq. (5) into its spatial frequency (SF) domain by a 2D FT. After substituting Eqs. (1) and (3) into Eq. (5) and making some simplifications, the spatial spectra of the third term (corresponding to the object wave to be retrieved) and the fourth term (corresponding to the conjugate) in Eq. (5) can be respectively expressed as:

{\tilde{U}}_{3} (ξ, η) = C^{'} \tilde{t} (M ξ - \frac{M x_{r}}{λ z_{2}}, M η) c y l (\frac{λ z_{2} ξ}{R} - \frac{x_{r}}{R}, \frac{λ z_{2} η}{R}) \exp [i π λ M (z_{1} + z_{2}) [{(ξ - \frac{x_{r}}{λ z_{2}})}^{2} + η^{2}]],

and

\begin{array}{l} {\tilde{U}}_{4} (ξ, η) = \\ C^{'} {\tilde{t}}^{*} (M ξ + \frac{M x_{r}}{λ z_{2}}, M η) c y l^{*} (\frac{λ z_{2} ξ}{R} + \frac{x_{r}}{R}, \frac{λ z_{2} η}{R}) \exp [- i π λ M (z_{1} + z_{2}) [{(ξ + \frac{x_{r}}{λ z_{2}})}^{2} + η^{2}]] . \end{array}

It can be seen from Eqs. (6) and (7) that, because the reference point parameter x_r is slightly larger than the filter aperture radius R, the spatial spectrum of the object wave is well separated from that of its conjugate item, as intuitively illustrated in Fig. 3, in which ${\tilde{U}}_{1}$ and ${\tilde{U}}_{2}$ respectively indicate the spatial spectra of the first and second terms in Eq. (5). We know that, for SODHs, the spatial spectra of the object wave and its conjugate term should not overlap, but overlap of them with two autocorrelation terms is allowed [29]. So the holograms recorded based on the proposed setup shown in Fig. 1 will always keep satisfying the optimal frequency condition of SODHs whether the object is far away from or very close to the AF, and so the recorded object wave can be retrieved using the algorithms suitable for SODHs [29–36].

Fig. 3 Sketch map of the spatial spectrum of the hologram given by Eq. (5).

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From the above description we can also see that, the magnification of the image reconstructed by the hologram expressed in Eq. (5) is determined by the ratio of the distances z₁ and z₂. When $z_{1} < z_{2}$ , the magnification will be larger than 1, which is suitable for imaging the object smaller than the valid sensor size. If the tested object is one larger than the effective sensor size, we can move the object apart from the AF to get the distance z₁ larger than z₂. In this situation, the object wave can be recorded by the image sensor and an image with the magnification smaller than 1 can be reconstructed. At the same time, using the proposed system, the reconstructed image can be obtained simply by taking two Fourier transform operations.

3. Experiments and discussions

To illustrate the feasibility of our proposed system, an experimental setup was firstly, constructed according to the system schematic shown in Fig. 1. Figure 4 shows a photo of the constructed experimental setup. In our experiments, a fiber-coupled laser diode, with the central wavelength of 650 nm, is adopted as the laser source. The hologram is recorded by a CMOS image sensor with pixel size of 2.2 × 2.2 um and effective pixel number of 1940 × 1940. The other parameters of the experimental setup are taken as follows: the splitting ratio of the 1 × 2 single-mode OSF is about 50:50; the focal length of the focusing lens is 180 mm; The diameter of the filter aperture on the AF is about 5.0 mm, and the distance between the edge of the aperture and the reference point is smaller than 1.0 mm.

Fig. 4 Photo of the experimental setup.

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Figure 5 gives an example of the experiments when the object is selected as a transmittance USAF resolution target with group number of 2 to 7. Figure 5(a) is one of the recorded holograms as the object is placed before the AF with the distance of z₁ = 65 mm. The distance between the AF and the recording plane of the CMOS is set to about z₂ = 84 mm. Figure 5(b) shows the spatial spectrum of the hologram obtained via a 2D fast FT operation, in which the dashed circles indicates the part of the spatial spectrum required to be extracted for reconstruction, also corresponding to the contour edge of the filter aperture. For eliminating the noise induced by the auto-correlation terms, we can simply subtract the intensity of the tested object wave from the hologram before operating the spatial filtering, as shown in Fig. 5(c). Figures 6(a) and 6(b) show, respectively, the amplitude and phase distributions of the reconstructed image from the recorded hologram based on the SODH algorithm described in the previous section. Through comparing the line width of the reconstructed image and the original line width of the USAF target we estimated that the image magnification is about 1.3, which agrees with the theoretical prediction according to Eq. (4). Obviously, the effective recording size (about 4mm × 4mm) of the CMOS is too small to record the whole tested USAF target in this configuration. For full imaging of the target, a longer distance z₁ should be chosen in the recording process.

Fig. 5 (a) Example of the holograms recorded in the experiments; (b) spatial spectrum of the hologram obtained via a 2D fast FT; (c) spatial spectrum after subtracting the intensity of the object wave from the hologram.

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Fig. 6 Amplitude and phase distributions of reconstructed images for a USAF target with different recording parameters: (a) and (b) correspond to z₁ = 65 mm and z₂ = 84 mm, and (c) and (d) to z₁ = 210 mm and z₂ = 84 mm. The red scale bar marked in (a) and (c) are 0.56 mm and 2.0 mm, respectively.

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Figures 6(c) and 6(d) show the amplitude and phase distributions of an image reconstructed from a hologram which was recorded when the target was shifted away from the AF to the distance of z₁ = 210 mm. In this situation, the system magnification is reduced to about 0.4 and so an object with the size of about 10mm × 10mm can be recorded by the image sensor with the valid size of about 4mm × 4mm.

Figure 7 further gives some experimental results to demonstrate the feasibility of our system in recording and imaging the complex amplitude of biological samples. Figures 7(a) and 7(b) present an example of the experimental results corresponding to a wing of a small fly as the tested object. Because the object is smaller than the size of the image sensor, the distance parameters in this experiment were taken as z₁ = 65 mm and z₂ = 80 mm, and so the imaging magnification of the system is about M = 1.2. While Figs. 7(c) and 7(d) show the amplitude and phase distributions of the reconstructed image when the object is replaced by a wing of a dragonfly and the distance parameter z₁ is increased accordingly to about 400 mm and so the system magnification is reduced to about 0.2.

Fig. 7 Amplitude and phase images for biological samples with different size: (a) and (b) are the results for a small part of a fly wing, while (c) and (d) are the images for a large wing of a dragonfly. The scale bars marked in (a) and (c) are 0.6 mm and 3.7 mm, respectively.

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The above setup is only an example of our proposed system without using any objective lens between the object and the image sensor. Although such a configuration provides a convenience for recording and imaging samples that are much larger than the image sensor in size, the resolution of the system is limited by the larger interval existed between the object and the image sensor because of the insertion of the aperture and the reference source. As a diffractive imaging system, the resolution limit δ of the setup used in the above experiments can be approximately estimated by $δ = λ (z_{1} + z_{2}) / L_{s}$ (L_s indicates the size of the image sensor) [41]. Here the influence of the aperture on the imaging resolution is ignored because the size of the aperture is larger than the effective size of the image sensor in the experiments. For example, according to the experimental parameters, the estimated resolution limit of the reconstructed image shown in Figs. 6(a) and 6(b) is about 22.7 um, corresponding to an ability to resolve the lines of the fourth element of group 6 of the USAF test target, which agrees with the experimental results. To gain higher spatial resolution for imaging microbiological specimens, we can combine the system with a conventional microscope. In fact, because of the fiber-based design, the proposed system also provides a low-cost way to convert a regular microscope into a slightly-off-axis holographic one for microbiological specimens. Figure 8 gives a schematic for converting a regular microscope into a slightly-off-axis holographic one based on the principle of the proposed method. In this conversion, the light source of the conventional microscope can be simply replaced by one fiber source split from a fiber-coupled laser with a 1 × 2 OFS, and then the exit end of another fiber branch split from the same OFS can be fixed at the edge of the exit pupil of the objective lens (OL) of the microscope, without changing the optical path and adding other optical elements. In this configuration, if the illumination beam is set to be focused at the exit pupil with the reference point and the image of the object through the OL is just located at the sensor plane, Eq. (7) will be still applicable in form to this situation after taking z₁ = -z₂; but the magnification factor M will be determined by the OL parameters, instead of Eq. (4). So, high lateral resolution of the image can be achieved by use of such a pre-magnification digital holographic system [42]. For demonstrating the feasibility and convenience of this conversion, we further put a typical microscope objective (40X, 0.65) between the object and the AF in our experimental setup shown in Fig. 4 to convert the setup into a holographic microscope and used it to realize quantitative phase imaging of microbiological specimens. Figures 9(a) and 9(b) present two examples of the experimental results for quantitative phase imaging of mouse monocyte cells RAW264.7 and ascaris eggs without being stained using such a SODH microscope, it can be seen that this simply converted SODH microscope can realize fine phase contrast imaging with high spatial resolution and so it provides us a tool for analysis of the fine structure of the specimens. For example, from the retrieved phase information of the ascaris egg as shown in Fig. 9(b), we can estimate the thickness of the transparent egg cell. And the thin protein membrane outside the egg shell can be also recognizable from its phase image.

Fig. 8 Schematic for converting a regular microscope into a slightly-off-axis holographic one based on the principle of the proposed method.

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Fig. 9 Phase imaging of microbiological specimens. (a) and (b) are the reconstructed phase images of mouse monocytes cells and ascaris eggs, respectively. Scale bar: 10 um.

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4. Conclusion

In summary, we have demonstrated the feasibility of our proposed phase imaging system with slightly-off-axis configuration based on a 1 × 2 single-mode optical fiber splitter. The magnification M of the recording setup can be conveniently adjusted by changing the distance parameters z₁ and z₂ as illustrated in Fig. 2. When z₁< z₂, the magnification will be larger than one, which possesses high spatial resolution but small field of view, only suitable for recording object smaller than the size of the image sensor. If z₁> z₂, the system can achieve an ability for holographically recording and imaging an object that’s much larger than the image sensor. At the same time, a 1 × 2 single-mode optical fiber splitter is used for generating the reference and the illumination beams. Benefited from such fiber-based slightly-off-axis design, the proposed system also provides a simple and low-cost way to convert a regular microscope into a slightly-off-axis holographic one for phase imaging of microbiological specimens with high spatial resolution. We think that this design method of the system as well as the corresponding digital imaging algorithm may provide a potential approach to develop a portable phase contrast imaging system suitable for both small and large objects as well as with wide range of magnifications.

Funding

National Natural Science Foundation of China (No. 91750105).

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Quantitative phase imaging system with slightly-off-axis configuration and suitable for objects both larger and smaller than the size of the image sensor

Abstract

1. Introduction

2. Principle of the system

3. Experiments and discussions

4. Conclusion

Funding

References and links

Cited By

Figures (9)

Equations (7)

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