Edge contrast enhancement of Fresnel incoherent correlation holography (FINCH) microscopy by spatial light modulator aided spiral phase modulation

Tianxu Xu; Jiuru He; Hong Ren; Zhongchao Zhao; Guoqing Ma; Qiaoxia Gong; Shuangning Yang; Lin Dong; Fengying Ma

doi:10.1364/OE.25.029207

1. Introduction

Digital holographic microscopy (DHM) is a promising tool for obtaining a three-dimensional image of both amplitude and phase objects [1–4]. It has vast application in label-free biological cell detection, material surface measurement, dynamic 3D shape measurement and micro-optical elements detection [5–8]. In traditional digital holography, laser light is required to implement the hologram recording. Laser speckle noise will seriously interfere with the experimental results, this, however, is inevitable in coherent imaging, and there is still no good way to solve it by now. In recent years, new types of digital holography under incoherent illumination have appeared [9–11]. Among them, FINCH records holograms from incoherent light with a SLM and a charge-coupled device (CCD) [12,13]. The FINCH system is an essentially in-line incoherent interferometer. The spherical wave emitted from each point of a 3D project is split by the SLM into two beams that interfere with each other. The interfere pattern is the point source hologram, and the entire interference pattern recorded by CCD is created from incoherent summation of the point source holograms from each point source in an incoherently emitting object. The FINCH technique can be readily adapted to any standard optical imaging technique with relatively little alteration, such as telescope and microscopy [14–17].

Edge extraction is the main means of image processing and pattern recognition [18–20]. Spiral phase filter (SPF), as a new diffractive optical element, has capabilities to generate optical vortex beams with helical wavefronts [21–24], which can be used for achieving image edge enhancement and pattern recognition [25–27]. During last two decades, SPF has been widely used in many practical applications, such as spiral phase contrast microscopy [28–30], astronomical observation [31,32], and quantitative imaging [33–35], etc. In subsequent research, the SPF was introduced to in-line digital holography based on the optical vortex phase-shifting method [36]. Recently, Petr Bouchal and Zdeněk Bouchal [37] proposed a spiral contrast imaging method from standard FINCH by optical spiral recording, in which the SLM is multiplexed by a SPF and a spherical wave. The edge enhancement result is demonstrated on amplitude objects using an LED light source. This shows that the imaging characteristics of spiral FINCH system is different from those of traditional FINCH. But in [37], the imaging characteristics of the spiral FINCH was qualitatively described by analogy with those of traditional FINCH, and had not been analyzed theoretically. The precise mathematical description of the spiral PSF and the imaging effect of the system on the phase objects were also not given.

In this paper, a precise mathematical model of a spiral FINCH has been established based on wave optics, in which the SLM is space-division multiplexed by a helical lens and a conventional lens. The helical lens is a combination of a SPF and a thin lens. The point source hologram, PSF and the reconstruction distance of the system are obtained, which are different from those of traditional FINCH. The experimental point source hologram and PSF are consistent with those of computer simulation. When the system was used for imaging biological cells the effects of edge contrast enhancement and field of view (FOV) enlargement are obtained.

2. System analysis

Figure 1 shows schematic of the incoherent digital holographic microscopy system. A white-light source illuminates a 3D object, and the reflected light from the object is collected by an objective, and then propagated through a lens L and a SLM. The interference occurs on the plane of the CCD.

Fig. 1 Schematic of the incoherent digital holographic microscopy system.

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In order to evaluate the imaging quality of this system, we need to get the system’s PSF, which describes the intensity distribution in digital image of the system’s response to a point source. The SLM is space-division multiplexed by a helical lens and a conventional lens, whose reflection function can be described as the form

R (x, y) = B \exp [- \frac{i π}{λ f_{d 1}} (x^{2} + y^{2}) + i α_{j}] + B^{'} {(x^{2} + y^{2})}^{\frac{m}{2}} \exp [- \frac{i π}{λ f_{d 2}} (x^{2} + y^{2}) + i m ϕ] .

where λ is the working wavelength. α_j (j = 1, 2, 3) are constant phase shifts, which can be used to eliminate the zero-order and twin image, B and B^’ are constants, φ is the azimuthal angle on the SLM plane. The first item of Eq. (1) represents a conventional lens with focal length f_d₁, and the second item represents a helical lens which is a combination of a SPF with topological charge m, which denotes the winding number of the spiral, and a thin lens with focal length f_d₂. When the parameter m is zero, the system works in the dual lens FINCH mode, we call it the S-Pattern. For other values of m, we call it the V-Pattern.

The system consists of two parts, a microscope objective (MO) and a SLM-based interferometer. A point source located at S (0, 0,-z_o) would be imaged by the objective lens to the point P (0, 0,-z_s), a distance z_s from a spherical positive lens, with f₀ focal length, induces a diverging spherical wave on the lens plane with the form of

A {(x^{2} + y^{2} + z_{s}^{2})}^{- \frac{1}{2}} \exp [i k {(x^{2} + y^{2} + z_{s}^{2})}^{\frac{1}{2}}] .

Here A is a constant, and k = 2π/λ. In the paraxial approximation, and omitting the constant coefficients, Eq. (2) can be written as

\exp [\frac{- (x^{2} + y^{2})}{σ_{1}^{2}}] \exp [\frac{i π}{λ z_{s}} (x^{2} + y^{2})] .

Here σ₁² = 2z_s². Passing through the lens, and propagating additional distance of d till the SLM plane, the complex amplitude becomes

\exp [\frac{- (x^{2} + y^{2})}{σ_{2}^{2}}] \exp [\frac{i π}{λ f} (x^{2} + y^{2})] .

Where

σ_{2}^{2} = σ_{1}^{2} [d^{2} {(\frac{1}{f_{e}} + \frac{1}{d})}^{2} + \frac{4 d^{2}}{k^{2} σ_{1}^{4}}]

,

\frac{1}{f_{e}} = \frac{1}{z_{s}} - \frac{1}{f_{0}}

,

\frac{1}{f} = \frac{1}{d} - \frac{σ_{1}^{2}}{σ_{2}^{2}} (\frac{1}{f_{e}} + \frac{1}{d})

.

Right after the SLM, with the reflection function given in Eq. (1), the complex amplitude is

B \exp [\frac{- (x^{2} + y^{2})}{σ_{2}^{2}}] \exp [\frac{i π (x^{2} + y^{2})}{λ f_{1}} + i α_{j}] + B^{'} {(x^{2} + y^{2})}^{\frac{m}{2}} \exp [\frac{i π (x^{2} + y^{2})}{λ f_{2}} + i m ϕ] .

Where

\frac{1}{f_{1, 2}} = \frac{1}{f} - \frac{1}{f_{d 1, 2}}

.

Finally, in the CCD plane at a distance z_h from the SLM, the complex amplitude is

\begin{array}{r} 2 π B^{'} {| β |}^{- (m + 1)} [^{- \frac{k i}{z_{h}} (^{x^{2} + y^{2}) \frac{1}{2}}] m} \exp [\frac{- (x^{2} + y^{2})}{σ_{4}^{2}}] \exp [\frac{i π (x^{2} + y^{2})}{λ f_{2 h}} + i m θ + i (m + 1) δ] \\ + B \exp [\frac{- (x^{2} + y^{2})}{σ_{3}^{2}}] \exp [\frac{i π (x^{2} + y^{2})}{λ f_{1 h}} + i α_{j}] . \end{array}

Here

σ_{3}^{2} = σ_{2}^{2} [z_{h}^{2} {(\frac{1}{f_{1}} + \frac{1}{z_{h}})}^{2} + \frac{4 z_{h}^{2}}{k^{2} σ_{2}^{4}}]

,

σ_{4}^{2} = σ_{2}^{2} [z_{h}^{2} {(\frac{1}{f_{2}} + \frac{1}{z_{h}})}^{2} + \frac{4 z_{h}^{2}}{k^{2} σ_{2}^{4}}]

,

\frac{1}{f_{1 h}} = \frac{1}{z_{h}} - \frac{σ_{2}^{2}}{σ_{3}^{2}} (\frac{1}{f_{1}} + \frac{1}{z_{h}})

,

\frac{1}{f_{2 h}} = \frac{1}{z_{h}} - \frac{σ_{2}^{2}}{σ_{4}^{2}} (\frac{1}{f_{2}} + \frac{1}{z_{h}})

,

β = \frac{1}{σ_{2}^{2}} - \frac{i π}{λ} (\frac{1}{f_{2}} + \frac{1}{z_{h}})

, δ is a constant, whose value is the argument of β, and θ is the azimuthal angle on the CCD plane. The intensity of the recorded hologram is

I_{p} (x, y, m) = C_{1} (x, y, m) + C_{2} (x, y, m) \exp [\frac{i π}{λ z_{r}} (x^{2} + y^{2}) + i α_{j} - i (m + 1) δ - i m θ] + c . c . .

Where C_{1, 2} (x, y, m) is a complex function, c.c. is the complex conjugate of the second term on the right, and z_r is given by

\frac{1}{z_{r}} = \frac{1}{f_{1 h}} - \frac{1}{f_{2 h}} .

Considering that the system is shift invariant, the recorded hologram for any source point located at any point (x_s, y_s,-z_s) can be expressed as

\begin{array}{r} I_{p} (x, y, m) = D_{1} (x, y, m) + D_{2} (x, y, m) \times \exp {\frac{i π}{λ z_{r}} [(^{x - M_{T} x_{s}) 2} \\ + (^{y - M_{T} y_{s}) 2}] + i α_{j} - i (m + 1) δ - i m θ} + c . c . . \end{array}

Where D₁ = (x, y, m) is a complex function, M_T is the lateral magnification.

M_{T} = \frac{z_{h} f_{e}}{z_{s} (f_{e} + d)} .

From Eq. (9) we can see that the point source hologram has a helical wave-front structure with azimuthal angle θ and topological charge m. There is a phase singularity in the beam center, where the phase is uncertain and the light intensity remains zero because of destructive interference of the helical wave fronts. Consequently the reconstructed result shows a hollow point image.

Using a common computation routine of phase stepping, For each loading pattern, three different holograms with a different phase angle of 0°, 120° and 240° are captured and superimposed in the computer, such that the result doesn’t contain the zero-order and twin image.

\begin{array}{r} I_{F} (x, y) = I_{1} (x, y) [\exp (\pm i α_{3}) - \exp (\pm i α_{2})] \\ + I_{2} (x, y) [\exp (\pm i α_{1}) - \exp (\pm i α_{3})] \\ + I_{3} (x, y) [\exp (\pm i α_{2}) - \exp (\pm i α_{1})] . \end{array}

It follows that the three holograms are superposed to create a complex-valued hologram I_F, which contains only the useful component [12]. The PSF of the spiral FINCH I_psf can be reconstructed from I_F by calculating the Fresnel propagation formula as

I_{p s f} (x, y, m) = I_{F} (x, y, m) * \exp [\frac{i π}{λ z_{r}} (x^{2} + y^{2})] .

For a general 3D object g (x, y, z), the reconstructed image is an integral of the entire PSF given by Eq. (12), over all object intensity g (x, y, z), as follows

S (x, y, z, m) = g (x, y, z) * I_{p s f} (x, y, m) .

3. Experimental results

Schematic of the microscopic spiral imaging system based on FINCH is shown in Fig. 2. The phase masks of S and V (with topological charge m = 1) pattern are shown in the inset of Fig. 2. To get a quasi-monochromatic spatially incoherent illumination, a band pass filter with a peak wavelength of 633 nm and a bandwidth (FWHM, full width at half maximum) of 20 nm is positioned after the incoherent light source. The tested object is a USAF1951 resolution target, imaged by a 20 × , 0.4NA objective with a working distance of 5.9mm. The SLM is phase only, Holoeye Pluto, 1920 × 1080 pixels, 8μm pixel pitch. The distance between the collimation lens L (with a focal length of f₀ = 250mm) and the SLM is d = 140mm. The direction of the polarizer is adjusted to the best position since the SLM is polarized-dependent. Both S pattern and V pattern have the same focal length of f_d₁ = 375mm and f_d₂ = 385mm. For the purpose of getting a relatively perfect overlapping of interfering beams, the SLM-CCD (Hamamatsu Digital Camera C8484-05, pixel size σ_c = 6.45μm, 1344 × 1024 pixels array) distance z_h is 485mm. The superimposed complex valued holograms are digitally reconstructed in the computer. A small circular aperture with diameter of 20μm illuminated by xenon lamp (CEL-TCX250, 250W) is recorded by the system shown in Fig. 2 in V pattern. The simulated point source holograms with phase shift of 0°, 120° and 240° are shown in Figs. 3(a)-3(c), and the corresponding recorded holograms are shown in Figs. 3(e)-3(g). Considering the response sensitivity of the CCD, the simulated and recorded holograms are consistent. Figures. 3(d) and 3(h) show, respectively, the point spread functions from simulated and recorded data. Both results are consistent with each other very well, which verifies the correctness and effectiveness of the spiral imaging model in this paper.

Fig. 2 Experimental setup. L-Lens, P-Polarizer, BF-Bandpass filter.

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Fig. 3 Point source holograms and point spread functions of the small circular aperture: (a)-(c) simulated holograms from Eq. (9) with phase shift of 0°, 120° and 240°, (e)-(g) recorded holograms with phase shift of 0°, 120° and 240°, (d) and (h) point spread functions from simulated and recorded data, respectively.

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In order to verify the performance of the system on the edge contrast enhancement of the objects, two groups of experiments are carried out on the setup illustrated in Fig. 2. The experimental results of resolution target are shown in Fig. 4. Figures. 4(a) and 4(c) are the reconstructed images of S and V pattern, respectively. Comparing with the reconstructed image of S pattern, the image of V pattern shows an obvious edge contrast enhancement. The reason is that incoherent object can be regarded as a collection of many point sources, each point source is modulated by spiral FINCH system and reconstructed into a hollow point image, and thus the edge information of incoherent object is extracted due to the superposition of all hollow point images. Figures. 4(b) and 4(d) show the part magnified image of the yellow box area in Figs. 4(a) and 4(c), which demonstrate that the resolution of system could be 512 lp/mm. The edge contrast of the two pictures can also be seen in Fig. 4(e), in which the normalized intensity profiles of the identical area from Figs. 4(a) and 4(c) are depicted by the red and blue line. Then the contrast experiments are conducted on the stained and unstained biological cells. Figures. 5(a) and 5(b) show reconstructed images of stained onion cells in S and V pattern, respectively. To verify the imaging ability of the system to phase objects, another experiment using fresh human blood lymphocytes as imaging object is carried out in the system. The corresponding results of S and V pattern are shown in Figs. 5(c) and 5(d). As can be seen from the experimental results in Fig. 5, in addition to the obvious edge contrast enhancement effect, the more interesting phenomenon is that the V mode can achieve a larger FOV at the same resolution compared to the S mode, which is important for a microscopic imaging system.

Fig. 4 Experimental results of S and V pattern: (a) and (c) reconstructed images of S and V pattern; (b) and (d) part magnified images of the yellow box area in (a) and (c); (e)normalized intensity Profiles of the identical area from (a) and (c) depicted by the red and blue line.

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Fig. 5 Experimental results: (a) and (c) stained cell’s reconstructed images of S and V pattern; (c) and (d) unstained lymphocytes’ reconstructed images of S and V pattern.

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4. Conclusions

This paper has presented a detailed theoretical analysis of the imaging properties of the spiral FINCH system, which is composed of a SLM and a CCD camera. It should be noted here that the SLM isn’t indispensable, as a beam splitter and phase shift element, other cheap devices can be used instead. A precise mathematical model of the spiral imaging system has been established, in which the SLM is multiplexed by a helical lens and a conventional lens. The point source hologram, PSF and the reconstruction distance of the system are obtained, and the experimental point source hologram and PSF are compared with computer simulation and found to be in good agreement. When the system is used for imaging biological cells, the edge contrast enhancement effect and larger FOV are obtained without loss of resolution. In contrast to previous investigations of holographic microscopy, in addition to achieving edge contrast enhancement, the spiral phase modulated FINCH microscope system can reach balance between high resolution and large FOV. This effect can be used to record a series of pure phase samples such as label free living cell or intracellular tissue, which are commonly at most faintly visible in bright field mode.

Funding

National Science Foundation of China (NSFC) (61505178, 61307019, and 11504333); Natural Science Foundation of Henan Province of China (18A140032, 15A140038, and 16A140035).

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Edge contrast enhancement of Fresnel incoherent correlation holography (FINCH) microscopy by spatial light modulator aided spiral phase modulation

Abstract

1. Introduction

2. System analysis

3. Experimental results

4. Conclusions

Funding

References and links

Cited By

Figures (5)

Equations (13)

Optics Express