Coherent noise reduction in digital holographic microscopy by averaging multiple holograms recorded with a multimode laser

Feng Pan; Lizhi Yang; Wen Xiao

doi:10.1364/OE.25.021815

1. Introduction

Digital holographic microscopy (DHM) [1] is a well-known technique that can optically record and numerically reconstruct 3D information of an object. In the reflection mode, the recovered phase provides surface topography information and permits direct measurement of the object shape and the associated parameters. In the transmission mode, the recovered phase of the optical path length describes the phase function of the object. Therefore, DHM has led to the development of a huge set of applications in many fields, including quantitative microscopy in biology [2], nondestructive measurement in the industry [3], and etc. However, due to the coherent nature of the light source, the digital holograms are corrupted by a mixture of coherent noises when a laser beam illuminates an object with a rough surface or inhomogeneous interior. These noises include the parasitic interference fringes originating from the multiple reflections from different interfaces in the optical path and the unwanted scattering on the micro structures with sizes larger than the wavelength [4]. As such, the image quality and the measurement accuracy are seriously degraded in the holographic reconstructions. In order to reduce coherent noises, a number of techniques have been proposed and they can be classified into four groups: low coherence methods [5–12], multiple holograms methods [13–24], image processing methods [25–34] and hybrid methods [35].

In the first group, some approaches are based on temporally low coherent light emitted from light-emitting diodes [5, 6] or super luminescent diode [7], spatially low coherent light by use of rotating diffusers [8–10], and a random laser illumination [11]. By combining the Gabor’s holography and Zernike’s phase contrast imaging, a speckle-free phase microscopy is presented by using Halogen Lamp illumination with extremely short coherence length [12]. These light sources can guarantee that no multi-reflected-transmitted lights will interfere, and thus, remove deleterious parasitic interference fringes in the recorded holograms. Furthermore, the relationship between the degrees of spatiotemporal coherence of an illumination and the imaging quality of interferometric microscopy has been quantitatively investigated by experiments [13]. In the second group of multiple holograms approach, the reduction of coherent noise is achieved by averaging a number of reconstructed images from a set of holograms, whose noise patterns are decorrelated with each other. The decorrelation of noise can be achieved by using different wavelengths [14], using different polarization states [15, 16], varying the positions of camera [17], shifting the object slightly [18], changing illumination angle [19, 20], and applying a random phase generator, such as an optical diffuser [21, 22], a spatial light modulator [23]. As a related approach, the synthetic aperture imaging technique was also used to reduce coherent noises in DHM [24, 25]. For this strategy, the denoising performance depends on the noise decorrelation extent between the different holograms. The methods based on imaging processing were proposed with the use of digital filters [26], such as the median filter [27], the Wiener filter [28], the discrete Fourier filter [29], the wavelet filtering [30], the non-local means [31], and a multilevel two-dimensional empirical decomposition mode [32]. As an alternative approach, the coherent noise can also be suppressed by averaging a certain number of images, which are reconstructed from a single hologram with resampling masks [33, 34] or numerical jittering [35]. In these way, the coherent noise reduction can be performed at the cost of resolution and contrast of the reconstructed image. More recently, a novel approach is proposed by combining the Multi-Look Digital Hologram (MLDH) techniques and Three-Dimension Block Matching (BM3D) filtering to reduce noise in DH [36]. The method is made of two steps. Firstly the noise is reduced to the theoretical bound by averaging multiple reconstructed images with uncorrelated speckle noises. And then, based on the averaging result, a quasi noise-free reconstruction is obtained by use of the block grouping and collaborative filtering. However, the method’s capability was only proved in holographic intensity imaging, and the results are not provided for holographic phase imaging.

In this paper, we present a new approach to reduce the coherent noise by combining the low coherence recordings and multiple holograms averaging. First, by employing the periodicity of temporal coherence of multimode semiconductor laser, we record a series of holograms by changing the optical path length of the reference beam. Due to the use of low coherent light, the parasitic interference fringes caused by multiple reflections or transmissions are avoided in holograms recording. Second, we numerically reconstruct a set of intensity and phase images with uncorrelated coherent noise patterns. Third, we average the reconstructed images to suppress the coherent noise caused by the optical scattering. By performing these three steps, we can significantly improve the image quality and the measurement accuracy. The experimental results demonstrate the effectiveness of the proposed method for the reconstructed intensity and phase images.

2. Principle and method

We consider the process of digital holography recording: a reference beam R interferences with an object beam O, both beams being emitted from the same light source. The interference pattern takes the expression as

I_{H} = {| R |}^{2} + {| O |}^{2} + R^{*} O + R O^{*} .

where the superscript * denotes the complex conjugation of the wave amplitude.

Supposed that the light source is a multimode semiconductor laser and the oscillation of N longitudinal modes, the complex optical field can be written as

E = \sum_{n = - (N - 1) / 2}^{n = (N - 1) / 2} A_{n} \exp {i [(k + n Δ k) L + φ_{n}]} .

where A_n and φ_n are the optical power and phase of the nth lasing mode respectively, k = 2π/λ, λ is the center mode wavelength of laser, Δk = 2πΔλ/λ², Δλ is the wavelength interval between two adjacent modes, L is the optical transmission path.

Thus, the R and O beams can be written as, respectively

\begin{array}{l} R = R_{0} \sum_{n = - (N - 1) / 2}^{n = (N - 1) / 2} A_{n} \exp {i [(k + n Δ k) L_{R} + φ_{n}]} . \\ O = O_{0} \sum_{n = - (N - 1) / 2}^{n = (N - 1) / 2} A_{n} \exp {i [(k + n Δ k) L_{O} + φ_{n}]} . \end{array}

where R₀ is the complex amplitude of R beam, which usually is a plane or spherical wave, O₀ is the complex amplitude of O beam, which represents the surface topography for reflection imaging and the optical path difference for transmission imaging, L_R and L_O are the optical paths of R and O arms, respectively.

So, the interference term of R*O in Eq. (1), which contains the information of object, can be described as,

\begin{matrix} R^{*} O = R_{0}^{*} O_{0} \sum_{n = - (N - 1) / 2}^{n = (N - 1) / 2} \sum_{m = - (N - 1) / 2}^{m = (N - 1) / 2} A_{n} A_{m} \exp [i k (L_{O} - L_{R})] \\ \times \exp [i Δ k (m L_{O} - n L_{R})] \cdot \exp [i (φ_{m} - φ_{n})] . \end{matrix}

Because the interference of different modes does not mutually coherent, the phase different of φ_m-φ_n (n≠m) is likely to vary randomly in time. So, the interference of different modes does not contribute to the interference term, and it follows that

\begin{matrix} R^{*} O = R_{0}^{*} O_{0} \exp [i k (L_{O} - L_{R})] \sum_{n = - (N - 1) / 2}^{n = (N - 1) / 2} \sum_{m = - (N - 1) / 2}^{m = (N - 1) / 2} δ_{n, m} A_{n} A_{m} \\ \times \exp [i Δ k (n L_{O} - m L_{R})] \cdot \exp [i (φ_{m} - φ_{n})] . \end{matrix}

where δ_n,m is Kronecker delta function and described as,

δ_{n, m} = {\begin{cases} 1 (n = m) \\ 0 (n \neq m) \end{cases}

According to Eq. (5), the inference of different modes does not pay role to the holographic recording, and then each mode contributes for the recording of its respective hologram. The resulting interference pattern will be a superposition of all holograms, and the interference term can be written as

R^{*} O = R_{0}^{*} O_{0} \exp [i k (L_{O} - L_{R})] \sum_{n = - (N - 1) / 2}^{n = (N - 1) / 2} A_{n}^{2} \exp [i Δ k n (L_{O} - L_{R})] .

In order to simplify the following analysis, the value of A_n for each mode is considered equal to 1. Now, Eq. (7) can be written as

R^{*} O = R_{0}^{*} O_{0} \exp [i k (L_{O} - L_{R})] {\frac{\sin [N Δ k (L_{O} - L_{R}) / 2]}{\sin [Δ k (L_{O} - L_{R}) / 2]}} .

where the sin denotes the sine function, the term in the {·} corresponds to the visibility of interference fringes.

Based on Eq. (8), the graph in Fig. 1(b) shows the behavior of fringe visibility as a function of (L_O-L_R) with Δk = 1.046rad/mm and N = 12, which are close to the experimental condition. It can be seen that the fringe visibility repeats periodically with a spacing of 6.75 mm, which is called the interference period (IP). In order to verify this effect, a series of holograms are recorded by changing the value of L_R and keeping the value of L_O constant in the experiment setup, which is shown in the Fig. 2. Figure 1(a) represents a hologram with high fringe visibility, and Figs. 1(c), 1(d) and 1(e) show the cutouts of same region (indicated with a white rectangle) in these holograms, which are recorded with different (L_O-L_R). As can be seen, when the changing of (L_O-L_R) is non-integer multiple of the IP, the fringe visibility is low, as shown in the Figs. 1(d) and 1(e). In this case, the object information cannot be recorded in the hologram. On the other hand, when the value of (L_O-L_R) are an integer multiple of the IP, the fringe visibility are maximum, as shown in the Fig. 1(c). At these positions, the hologram has a high signal to noise ratio. Meanwhile, due to the changing of (L_O-L_R), the coherent noise in the holograms will randomly fluctuate. Thus, by digital reconstruction, a series of intensity and phase images are obtained with uncorrelated coherent noise patterns from these holograms. Moreover, thanks to the shortly coherent light, it is possible to guarantee no parasitic interference fringes caused by multiple reflections in holograms. And then, the averaging process can effectively suppress the artifact speckle pattern caused by the optical scattering on object surface or interior. Consequently, the imaging quality and the measurement accuracy can be significantly improved.

Fig. 1 The fringe visibility versus the optical path different between the R and O beams. (a): the digital hologram, (b): the calculation results, (c) (d) and (e): the experimental results, the cutouts (indicated with a white rectangle) of the holograms recorded with different (L_O-L_R).

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Fig. 2 Schematic of the experimental system. Laser, a multimode semiconductor laser; λ/2, half-wave plate; PBS, polarizing beam splitter; BE, beam expander with spatial filter; M, mirror; L, lens; O, object wave; R, reference wave; MO, microscope objective; BS, beam splitter; CMOS, camera; OPR, optical path retarder to adjust the optical path length of the reference beam.

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3. Experiments

The experimental setup is a reflective off-axis holographic interferometer with a sample pre-magnifying configuration, as depicted in Fig. 2. The source is a multimode semiconductor laser with the oscillation of 12 longitudinal modes (Laser, Oxxius LBX, with 638nm wavelength and 50mW power). The laser beam is divided into an object beam (O) and a reference beam (R) by a polarized beam splitter (PBS). A half-wave plate (λ/2) in front of the PBS is used to adjust the intensity ratio between the O and R beams. Behind the PBS, another half-wave plate (λ/2) is introduced to obtain the parallel direction of polarization in two beams. Both the beams are collimated as plane waves by the beam expanders (BE1 and BE2, composed of spatial filters and collimating lenses). In the object arm, a microscope objective (MO, Olympus UplanFLN, the magnification of 20 × , the numerical aperture of 0.4) is used to enhance the lateral resolution, which is diffraction limited and depends on the numerical aperture of the MO. The object beam is focused near the back focal plan of the MO by the condenser lens (L2) to deliver a collimated illumination wave. Then, the optical wave reflected by the object is collected by the MO and a magnified image of the object is produced behind the camera with the distance of 8 cm. In the reference arm, the optical path length of the R beam is adjusted by moving an optical path retarder (OPR, composed by the mirror of M1 and M2) to match the optical path length of the O beam. In order to improve the sampling capacity of the camera, the matching lens (L1) is optionally used to create a spherical wave with a curvature very similar to the curvature created by the MO. Finally, by calibrating the position of the OPR, the interference between the object wave emerging from the image and the spherical reference beam is accomplished with a maximal fringe contrast, and recorded by the camera (CMOS, Lumenera Lu125M, the resolution of 1024 × 1024, the pixel pitch of 6.7μm). As mentioned above, due to the periodicity of laser temporal coherence, a sequence of holograms can be recorded by continuously adjusting the OPR and keeping the position of the object with the IP.

The goal of the first experiment is to investigate the fluctuation of coherent noise with changing the value of L_R-L_O. A 1951 USAF resolution target is used as the observed object, and a sequence of holograms is recorded by keeping the L_O and changing the L_R by an integer multiple of IP. The reconstructed results are shown in Fig. 3. Figures 3(a) and 3(b) represent the amplitude images with a reference L_R and a shifted L_R, where the distance between the L_R and L_O is equal to one IP. Figures 3(c) and 3(d) represent the phase images with the reference L_R and the shifted L_R with one IP. In these figures, the enlarged cutouts of a uniform region indicated by a white rectangle show the noise patterns with more detail. Obviously, the coherent noise distribution has randomly fluctuated with the changing of L_R in both amplitude and phase images. Furthermore, in order to qualitatively discuss the statistical relation of coherent noise, the normalized cross correlations (γ) between two noise patterns is calculated by

γ = \frac{\sum_{n, m} [ϕ_{0} (n, m) - {\bar{ϕ}}_{0}] [ϕ_{L} (n, m) - {\bar{ϕ}}_{L}]}{\sqrt{\sum_{n, m} {[ϕ_{0} (n, m) - {\bar{ϕ}}_{0}]}^{2} \sum_{n, m} {[ϕ_{L} (n, m) - {\bar{ϕ}}_{L}]}^{2}}} .

where ϕ₀ and ϕ_L denotes either the amplitude or phase distribution in the reconstructed image with the reference L_R⁰ and the shifted L_R^S,

{\bar{ϕ}}_{0}

and

{\bar{ϕ}}_{L}

denote the mean values, n and m denote the pixel number of the row and column in region of interest. Based on the uniform region as mentioned above, the graph in Fig. 3(e) displays the cross correlation against the value of L_R-L_O, where the changing of L_R-L_O is expressed as the multiple of the IP. It can be seen that γ of different positions are less than 0.2. As predicted, the results show a very efficient noise decorrelation between the reconstructed images with different optical path difference between O and R beams.

Fig. 3 The experimental results of the noise decorrelation by changing the L_R and keeping L_O. (a) and (b): the amplitude images with a reference L_R and a shifted L_R, (c) and (d): the phase images with a reference L_R and a shifted L_R, (e): the normalized cross correlations (γ) of the noise patterns over the L_R-L_O, where the horizontal axis is expressed as the multiple of the interference period.

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In order to verify the proposed method, the different numbers of reconstructed images are used in the averaging process. Figures 4(a) and 4(b) represents the amplitude images from a single hologram and after 15 averaging processes, respectively. Figures 4(c) and 4(d) represent the corresponding phase images before and after the averaging process. The reduction of coherent noise is clearly visible both in the amplitude and phase images. For quantitatively comparing the effect of coherent noise reduction, the spatial standard deviation (σ) of the noise pattern can be evaluated by

σ = \sqrt{\sum_{n, m} {[ϕ (n, m) - \bar{ϕ}]}^{2} / (n \times m - 1)} .

where ϕ denotes the value distribution of the average image,

\bar{ϕ}

denotes the mean values, n and m represent the pixel number of row and column of calculation region, which is indicated with a white rectangle in Fig. 3. Figure 4(e) shows the value of σ as a function of the number of averaged images. As expected, the σ of the noise pattern in the amplitude and phase images rapidly decrease with the number of averaged images. For intensity imaging, the amplitude noise is reduced from 3.3 × 10⁻³ to 0.6 × 10⁻³ and the image quality is improved significantly. For phase measurement, the phase noise is suppressed from 11° to 2°, which corresponds to an equivalent height from 16 nm to 3 nm. Figures 5(a) and 5(b) represent the 3D surface profile of the resolution targets by using a single hologram and 15 averaging processes, respectively. Figure 5(c) shows the profiles along the straight line indicated by a white line in Fig. 5(a). These results demonstrate that the proposed method can significantly reduce the coherent noise both in amplitude and phase image.

Fig. 4 (a) and (b): the amplitude images from a single hologram and after 15 averaging process, (c) and (d): the phase images from a single hologram and after 15 averaging process, (e): the standard deviation in the amplitude and phase images over the number of reconstructed images for averaging process.

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Fig. 5 (a): 3D phase image from a single hologram, (b): phase image after 15 averaging processes, (c) phase distributions along a line before and after the averaging process.

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As an example of application, a microlens (the diameter of 90 μm and the height of 21 μm) is measured by using the proposed approach. Figures 6(a) and 6(b) present the amplitude images reconstructed from a single hologram and after averaging 15 holograms together, respectively. The amplitude noise can be seen in Fig. 6(a) and is clearly reduced in Fig. 6(b). Figures 6(c) and 6(d) show the corresponding wrapped phase images. In comparison, the phase noise in Fig. 6(c) appear significantly reduced by the averaging process. To demonstrate topography measurement, the central region of the reconstructed phase image, which is indicated by a rectangular region in Fig. 6(c), are presented in Figs. 7(a) and 7(b) with 3D perspective. And the corresponding unwrapped phase images are obtained by using a least-squares algorithm [37] and are shown in Figs. 7(c) and 7(d). Furthermore, the Figs. 7(e) and 7(f) present the compensated phase images obtained by application of a digital phase mask, which is defined by fitting the unwrapped phase data with the Zernike parameters [38]. It can be noticed that some noise points that appear in Fig. 7(e) are suppressed in Fig. 7(f). Figure 7(g) presents the phase profiles along a straight line indicated by a white line in Fig. (e) before and after the averaging process. The reduction of phase noise can clearly be observed from these phase pictures, and the accuracy of shape measurement can be significantly improved.

Fig. 6 The measurement results of a microlens. (a) and (b): the amplitude image from a single hologram and after 15 averaging process, (c) and (d): the phase image before and after the averaging process.

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Fig. 7 (a) and (b): 3D wrapped phase images before and after averaging process, (c) and (d): unwrapped phase images corresponding (a) and (b), (e) and (f): compensated phase images corresponding (c) and (d), (g): phase profiles along a straight line indicated by a white line in (e).

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4. Conclusion

This paper presents a new approach to reduce coherent noise in DHM by averaging multiple holograms recorded with a multimode laser. First, based on the periodicity of temporal coherence of multimode semiconductor laser, a series of holograms are generated and recorded by changing the optical path difference between reference and object beams. In this step, due to the shortly coherent light, the parasitic interference fringes caused by the multiple reflections or transmissions are prevented from corrupting holograms. Meanwhile, a set of object reconstructed image is obtained, which occupied different noise patterns. Afterward, the average processing can effectively suppress the speckle pattern, which comes from the optical scattering on object surface or interior. Finally, the image quality and the measurement accuracy are improved significantly. Experimental results show that the proposed method is superior to reduce the coherent noise in intensity image and suppress the phase noise in phase image. As we all know, if a set of totally uncorrelated noise patterns are added for averaging process, the standard deviation of noise pattern falls in proportion to $1 / \sqrt{N}$ , where N is the number of holograms used to averaging process [39]. According to the experimental results of the proposed method, the values of standard deviation in amplitude and phase images decrease considerably for the first few holograms but at a lesser extent for higher number of holograms. It is easy to see that the noise reduction trend is bounded by the ideal $1 / \sqrt{N}$ , as well as other multiple holograms approaches. The limit of denoising performance can be interpreted in terms of the partial correlation between the noise patterns. Although the multiple holograms methods are different from each other in noise decorrelation extents, the value of cross correlations is typically in the range of 01 to 0.2 [15–18]. Compared with other multiple hologram approaches, the proposed method can effectively suppress the parasitic fringes and speckle pattern in both intensity and phase images for DHM. Meanwhile the acquisition of multiple holograms is merely dependent on the linear motion of an optical path retarder, and the extra image registration is not required before averaging process. Although the coherent noise reduction is achieved at cost of time for capturing multiple holograms, the result is fully dependent on optical process without any loss of resolution of imaging. The proposed method is effective and feasible for inspecting the shape and surface quality of the microlens with high precision.

Funding

National Major Scientific Instruments Development Project of China (No. 2013YQ030595).

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Coherent noise reduction in digital holographic microscopy by averaging multiple holograms recorded with a multimode laser

Abstract

1. Introduction

2. Principle and method

3. Experiments

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (10)

Optics Express