## Abstract

We propose multiwavelength in-line digital holography with wavelength-multiplexed phase-shifted holograms and arbitrary symmetric phase shifts. We use phase-shifting interferometry selectively extracting wavelength information to reconstruct multiwavelength object waves separately from wavelength-multiplexed monochromatic images. The proposed technique obtains systems of equations for real and imaginary parts of multiwavelength object waves from the holograms by introducing arbitrary symmetric phase shifts. Then, the technique derives each complex amplitude distribution of each object wave selectively and analytically by solving the two systems of equations. We formulate the algorithm in the case of an arbitrary number of wavelengths and confirm its validity numerically and experimentally in the cases where the number of wavelengths is two and three.

© 2017 Optical Society of America

## 1. Introduction

Holography [1] is a technique used to record and reconstruct a wavefront diffracted from an object. Both amplitude and phase information of an object wave are recorded by utilizing interference of light, and a three-dimensional (3D) image is reconstructed by the obtained amplitude and phase distributions. The recorded information is called a “hologram”. Digital holography [2,3] is a technique used to record a digital hologram that contains an object wave, and to reconstruct both the 3D and quantitative phase images of an object using a computer. This technique has potential application to the fields of microscopy [4], particles and flow measurements [5], quantitative phase imaging [6], multimodal imaging [7], and encryption [8].

Phase-shifting digital holography [9,10] is a representative technique to capture an object wave with a full space-bandwidth product of an image sensor [11]. This digital holography has performed 3D shape measurement with multiwavelength phase unwrapping [12] and holographic 3D color imaging [13,14]. In phase-shifting color/multiwavelength digital holography, there are two types of representative implementations: time division [12] and space-division multiplexing [13,14] of multiple wavelengths. In the former, wavelength information is recorded sequentially by changing the wavelengths of light for measurement. Therefore mechanical shutters to select the measurement wavelength or operations for turning the lasers on and off are required. Furthermore, at least three phase-shifted holograms are required at each wavelength. As a result, this implementation is time consuming. In the latter, simultaneous recording of three wavelengths is possible by using a color image sensor with a Bayer color-filter array, and three phase-shifted holograms are sufficient. However, the crosstalk between object waves at multiple wavelengths occurs if the wavelength selectivity of the array is low [15]. Furthermore, recordable spatial bandwidth is restricted by the array. Space-division multiplexing decreases the field of view (FOV) and resolution, and the FOV becomes a quarter in square compared to single-wavelength phase-shifting technique. There is a tradeoff between the number of recordable wavelengths and spatial information. Therefore, it is difficult to conduct multispectral 3D imaging with a wide FOV.

Since 2013, we have presented phase-shifting digital holography selectively extracting wavelength information using wavelength-multiplexed phase-shifted holograms to obtain multiwavelength information without a color-filter array [16–19]. In this technique, multiple wavelengths are superimposed on each other in the space and spatial frequency domains and separated in the polar coordinate plane by using phase shifts, which is called phase-division multiplexing (PDM) of wavelengths. With the technique, a full space-bandwidth product can be available at each wavelength and there is no need for changing the wavelengths of the light sources. Where the number of wavelengths is *N*, 2*N* + 1 wavelength-multiplexed holograms are sufficient for 3D multiwavelength imaging, while 3*N* holograms are necessary in the time-division technique. Furthermore, we have proposed a technique that requires only 2*N* wavelength-multiplexed holograms by making the best use of 2π ambiguity of the phase [19]. Recordable wavelength bandwidth is determined by the spectral sensitivity of a monochromatic image sensor and is wider than an implementation with a color image sensor. However, up to now, phase shifts with integral multiples of 2π were needed to separately extract object waves with multiple wavelengths, and experimental demonstrations for only two wavelengths were reported in the digital holography. It is important to present a concrete image-reconstruction algorithm applicable to multicolor and multispectral 3D measurements.

In this paper, we propose multiwavelength digital holography with wavelength-multiplexed holograms and arbitrary symmetric phase shifts. The proposed technique is based on PDM of wavelengths, and arbitrary phase shifts can be adopted. We present the image reconstruction algorithm applicable when the measurement wavelengths are more than two. The proposed algorithm makes the best use of wavelength separation in the polar coordinate plane. By the technique introducing arbitrary symmetric phase shifts, multiple object waves are analytically derived from the systems of equations for real and imaginary parts of the waves. Formulation in the case of an arbitrary number of wavelengths helps to make possible multispectral holographic imaging without approximation. Our motivation of the proposal is to achieve multispectral holographic imaging analytically with small number of holograms, small phase shifts which are given by a piezo-driven mirror, no approximation, and full space-bandwidth product of a monochromatic image sensor at each wavelength, which are impossible for the conventional PDM techniques. We demonstrate two- and three-wavelength digital holographic imaging ability of the proposed technique.

## 2. Phase-shifting interferometry selectively extracting wavelength information

The concept of the proposed digital holography is based on phase-shifting interferometry selectively extracting wavelength information [16–19]. Figure 1 illustrates the schematic of the basic concept. Multiple wavelength information is superimposed in the space and spatial frequency domains, and wavelength-multiplexed phase-shifted holograms are continuously recorded by introducing different phase shifts per wavelength as shown in Fig. 1(a). Wavelength information is separated in the polar coordinate plane as illustrated in Fig. 1(b) and this separation is derived by introducing different phase shifts for respective wavelengths. Object waves with multiple wavelengths are obtained separately by phase-shifting interferometry selectively extracting wavelength information and a multiwavelength 3D image is reconstructed with diffraction integrals. It is noted that, where the number of wavelengths is *N*, 2*N* + 1 holograms are required for the initially presented PDM [16–18].

However, up to now, utilization of 2π ambiguity was needed to separately extract object waves with multiple wavelengths. As a result, large phase shifts were needed and therefore multicolor 3D imaging was not demonstrated experimentally because it was difficult to prepare a phase shifter that satisfies the need. In the case of recording holograms with three wavelengths of633, 532, and 473 nm by the conventional phase-division technique, a piezo actuator must move a mirror at 336.756 μm distance in total with one-nanometer positioning accuracy. It is difficult to prepare such a mirror with a piezo actuator to satisfy the required specification due to the effect of hysteresis, 10^{5} required dynamic range (from 10^{−4} to 10^{−9} meter) with 1 nm-order resolution, and high cost of the piezo-driven mirror. Moreover, there is a poor applicability to incoherent and low-coherence digital holography conventionally because the required moving distance becomes longer than the coherence length. Although the total movement distance is decreased by choosing the wavelengths carefully, in the case of three wavelengths of 640, 532, and 488 nm, whole movement distance is 85.120 μm with one-nanometer accuracy. Furthermore, regular phase shifts were not possible by the previously reported algorithms.

## 3. Multiwavelength digital holography with wavelength-multiplexed holograms and arbitrary symmetric phase shifts

Figure 2 illustrates an optical implementation of the proposed multiwavelength digital holography in the case where the number of wavelengths is *N*. A phase modulator such as a mirror with a piezo actuator, wave plates, or a spatial light modulator sequentially generates different phase shifts per wavelength, and a monochromatic image sensor records 2*N* + 1 wavelength-multiplexed phase-shifted holograms *I*(*x*,*y*:*α*_{11},*α*_{21}, ..., *α _{N}*

_{1}), ...,

*I*(

*x*,

*y*:

*α*

_{1 2}

_{N}_{+1},

*α*

_{2 2}

_{N}_{+1}, ...,

*α*

_{N}_{2}

_{N}_{+1}), where

*α*

_{1},

*α*

_{2}, ..., and

*α*are phase shifts at the wavelengths of

_{N}*λ*

_{1},

*λ*

_{2}, ..., and

*λ*. Where

_{N}*I*(

_{0th}*x*,

*y*) is the summation of 0th-order diffraction waves,

*A*(

_{o}*x*,

*y*) and

*A*(

_{r}*x*,

*y*) are amplitude distributions of object and reference waves,

*ϕ*(

_{o}*x*,

*y*) is a phase distribution of an object wave, then a recorded wavelength-multiplexed hologram is expressed as follows:

By using a matrix equation, 2*N* + 1 holograms *I*(*x*,*y*:*α*_{11},*α*_{21}, ..., *α _{N}*

_{1}), ...,

*I*(

*x*,

*y*:

*α*

_{1 2}

_{N}_{+1},

*α*

_{2 2}

_{N}_{+1}, ...,

*α*

_{N}_{2}

_{N}_{+1}) are expressed as

*I*(

*x*,

*y*:0, 0, ..., 0),

*I*(

*x*,

*y*:

*α*

_{11},

*α*

_{21}, ...,

*α*

_{N}_{1}),

*I*(

*x*,

*y*:-

*α*

_{11},-

*α*

_{21}, ..., -

*α*

_{N}_{1}), ...,

*I*(

*x*,

*y*:

*α*

_{1}

*,*

_{N}*α*

_{2}

*, ...,*

_{N}*α*), and

_{NN}*I*(

*x*,

*y*:-

*α*

_{1}

*,-*

_{N}*α*

_{2}

*, ..., -*

_{N}*α*), object waves are selectively extracted with simple mathematical expressions. Equation (2) is rewritten as

_{NN}*I*(

_{0th}*x*,

*y*) from the holograms with the equations below,

*N*object waves are clearly separated. This is derived by symmetric phase shifts and arbitrary shifts are applicable. When

*A*(

_{r}*x*,

*y*) is assumed as constant and phase shifts are known,

*N*variables are contained in Eqs. (4) and (5), respectively.

*N*systems of equations are obtained for real part

*Re*

_{1}(

*x*,

*y*),

*Re*

_{2}(

*x*,

*y*), ..., and

*Re*(

_{N}*x*,

*y*), and imaginary one

*Im*

_{1}(

*x*,

*y*),

*Im*

_{2}(

*x*,

*y*), ..., and

*Im*(

_{N}*x*,

*y*), respectively, and therefore

*N*complex amplitude distributions of object waves

*U*

_{1}(

*x*,

*y*) =

*A*

_{o}_{1}(

*x*,

*y*)cos

*ϕ*

_{o}_{1}(

*x*,

*y*) +

*jA*

_{o}_{1}(

*x*,

*y*)sin

*ϕ*

_{o}_{1}(

*x*,

*y*), ..., and

*U*(

_{N}*x*,

*y*) are selectively, analytically, and rigorously derived, where

*j*= (−1)

^{1/2}. LU decomposition or other mathematical techniques can solve 2

*N*systems of equations analytically with no approximations. Thus, by the proposal of PDM technique with regular, small, and arbitrary symmetric phase shifts, it is easy to conduct multispectral digital holographic imaging using wavelength-multiplexed phase-shifted holograms. As examples, we describe applications to two- and three-wavelength digital holography.

#### 3-1. Two-wavelength digital holography

Five wavelength-multiplexed phase-shifted holograms *I*_{1}(*x*,*y*), *I*_{2}(*x*,*y*), *I*_{3}(*x*,*y*), *I*_{4}(*x*,*y*), and *I*_{5}(*x*,*y*) are recorded with phase shifts (0,0), (*α*_{11},*α*_{21}), (-*α*_{11},-*α*_{21}), (*α*_{12},*α*_{22}), and (-*α*_{12},-*α*_{22}) at the wavelengths of (*λ*_{1},*λ*_{2}). From Eqs. (3)-(5) we obtain the formulas below.

*U*

_{1}(

*x*,

*y*) and

*U*

_{2}(

*x*,

*y*) are selectively extracted with the expressions:

#### 3-2. Three-wavelength digital holography

From seven wavelength-multiplexed phase-shifted holograms *I*_{1}(*x*,*y*), *I*_{2}(*x*,*y*), *I*_{3}(*x*,*y*), *I*_{4}(*x*,*y*), *I*_{5}(*x*,*y*), *I*_{6}(*x*,*y*), and *I*_{7}(*x*,*y*) and Eqs. (3)-(5), we obtain object waves at the wavelengths of *λ*_{1}, *λ*_{2}, and *λ*_{3}.

*N*object waves are derived from 2

*N*+ 1 wavelength-multiplexed phase-shifted holograms by using Eqs. (3)-(5) and solving systems of equations. In comparison with time-division technique, we can obtain a color 3D image more quickly with the PDM technique. The acceleration of the measurement speed is due to the following reasons. Phase-shifting color digital holography with time-division technique requires 3

*N*holograms. Furthermore,

*N*mechanical shutters to select a wavelength from

*N*wavelengths should be set between lasers and beam combiners. A computer to synchronize the shutters and exposure of an image sensor is required to obtain holograms without crosstalk between wavelengths. In contrast, the proposed technique can obtain multiwavelength object waves separately with 2

*N*+ 1 holograms. Therefore, measurement speed is 3

*N*/(2

*N*+ 1) times improved by the proposed technique in principle. The proposed technique can obtain four-wavelength object waves during obtaining three wavelengths by the time-division technique theoretically. As increasing the number of wavelengths recorded, the degree of acceleration approaches to 1.5. Moreover, no mechanical shutter is needed to separate wavelength information and measurement speed is not affected by the synchronization and shutters. Therefore, the proposed technique is nearly 1.5 times faster than the time-division technique theoretically, and can further accelerate the measurement speed practically.

## 4. Numerical simulations

We conducted numerical simulations to verify the effectiveness of the proposed technique in the case of *N* = 3. We assumed an optical system containing a mirror with a piezo actuator used as a phase modulator. Figure 3 shows the amplitude and phase distributions and color-synthesized image of the object wave. *λ*_{1} = 633 nm, *λ*_{2} = 532 nm, and *λ*_{3} = 473 nm were assumed as the red-, green-, and blue-wavelengths of the light sources. Respective color components of a standard image “peppers” were used as amplitude images at the wavelengths, respectively. In these simulations, distance between the object and image sensor was assumed as 250 mm, pixel pitch was 2.2 μm, number of pixels was 512 × 512, and dynamic range of the image sensor was 12 bits. We set the condition that piezo-driven mirror moved at regular distances sequentially and obtained wavelength-multiplexed phase-shifted holograms sequentially. We obtained wavelength-multiplexed holograms with various optical-path shifts of the piezo-driven mirror numerically and reconstructed the object images with the proposed technique. Then we calculated root-mean-square errors (RMSEs) and cross-correlation coefficients (CCs) of the reconstructed amplitude images to evaluate the image quality quantitatively.

Figure 4 shows the numerical results. Numerical results clarified that moving a distance of 50 nm of the mirror with a piezo actuator per phase shift was sufficient for faithful holographic multiwavelength 3D imaging. From the color-synthesized image seen in Fig. 4(d), multicolor 3D imaging ability was verified numerically. However, random noise can be seen slightly on the phase image in Fig. 4(f) because minor noise was not removed due to the small phase shifts. Such noise was successfully removed by introducing sufficient optical path shifts to separate wavelengths, as shown in Figs. 4(h)-4(j). Figure 5 indicates the graphs of the calculated RMSEs and CCs. These graphs showed its color 3D imaging ability quantitatively with less than a 100 nm shift of the mirror with a piezo actuator for each phase shift. Numerical results indicate that whole movement distance to achieve three-wavelength digital holographic imaging with high accuracy is decreased to 300 nm when recording three wavelengths of 633, 532, and 473 nm. The whole distance is further shortened to 240 nm although a little noise appears in Fig. 5. From the results, it is concluded that the whole distance of the mirror can be decreased to 1/1123 by the proposed technique in comparison to the conventional PDM. Furthermore, the minimum regular phase shifts can be estimated from the numerical results. Figures 5(a) and 5(b) indicate small phase shifts can be applied because phase shifts are calculated from the wavelengths used and moving distance of the mirror with a piezo actuator. The minimum shifts to achieve high-quality three-wavelength digital holographic imaging were estimated as 56.9, 67.7, and 76.1 degrees at the wavelengths of 633 nm, 532 nm, and 473 nm, respectively, in the case where Fresnel domain was assumed and dynamic range of a monochromatic image sensor was 12 bits. This estimation is derived from the numerical results with 50 nm regular phase shifts, as shown in Figs. 4(a)-4(g), 5(a), and 5(b). As a result, the minimum difference between the phase shifts at the wavelengths of 633 nm and 532 nm can be estimated as 10.8 degrees and that between the shifts at 532 nm and 473 nm can be as 8.44 ones. Thus, the minimum difference of phase shifts is estimated as 8.44 degrees in the case described above, which is less than π/20. The proposed PDM technique requires the wavelength separations in the polar coordinate plane and therefore the phase shifts should be different. However, the phase-shift difference of π/20 enabled selective extractions of multiple object waves. Additionally, the numerical simulations clarify that the proposed technique has the potential of the applicability to incoherent and low-coherence color digital holography because a total moving distance required for the proposed technique can be less than the coherence length of incoherent and low-coherence light. Therefore, the problems described for the previous publications are solved by the proposal. It is worthy of note that RMSEs increases and CCs decreases when the optical-path shifts of the reference arm are close to integral multiples of the halves of the respective wavelengths, which correspond to regular π or 2π phase shifts. When adopting regular phase shifts as performed in the numerical simulations, regular π phase shifts at a wavelength prevent to extract the imaginary part of the object wave at the corresponding wavelength, in the same manner of single-wavelength phase-shifting digital holography. Equation (5) indicates that sin*α* at the corresponding wavelength becomes zero when *α* is an integral multiple of π. Therefore, there is no way to retrieve imaginary part of the object wave at the wavelength and the conjugate image superimposes on the object image. Moreover, when *α* is an integral multiple of 2π, the hologram intensity at the wavelength is not changed and the object wave at the corresponding wavelength is removed by the subtraction procedures based on Eqs. (4) and (5). As a result, object information cannot be obtained. Therefore, π or 2π phase shifts should be avoided for their respective wavelengths and integral multiple of π is not suitable for regular phase shifts in the proposed technique. Thus, the validity of the proposed technique was numerically confirmed, and the guideline for phase shifting was indicated.

## 5. Experiments

We have experimented to demonstrate and validate the proposed technique. We constructed a reflection-type two-wavelength digital holography system and a transmission-type three-wavelength digital holographic microscopy system to set the cases of *N* = 2 and 3. In these experiments, we adopted a mirror with a piezo actuator as a phase modulator.

#### 5-1. Two-wavelength digital holography

Five wavelength-multiplexed phase-shifted holograms were recorded sequentially by changing the optical path of the reference arm with an in-line configuration. The oscillation wavelengths of the lasers used as light sources *λ*_{1} and *λ*_{2} were 671 nm (MSL-FN-671-100-1, CNI) and 532 nm (GLK-32300, LASOS), respectively. Coherence lengths of the lasers are longer than 10 meters. A monochromatic CMOS image sensor was used to record the grayscale holograms. The sensor has 12 bits, 2592 × 1944 pixels, and a pixel pitch of 2.2 μm. The mirror with a piezo actuator moved 70 nm for each phase shift sequentially. A miniature model of baby birds with green-colored grass was set as a color 3D object. The front side of the object was set at the distance of 115 mm from the image sensor plane. For comparison, object images were also reconstructed from a wavelength-multiplexed hologram obtained with the same in-line configuration to highlight the ability of the proposal for the wavelength separation.

Figure 6 shows the experimental results. In the same manner presented by experts in color digital holography with coherent light [20–23], we showed a photograph of the object taken with white light, monochromatic images at respective wavelengths, and its color-synthesized images in Fig. 6. As seen in Figs. 6(c) and 6(d), not only the 0th-order diffraction wave and the conjugate image, but also image components given by the crosstalk between the intensity distributions at *λ*_{1} and *λ*_{2} were reconstructed from a wavelength-multiplexed hologram. In contrast, object waves at the respective wavelengths were clearly reconstructed by the proposed technique as shown in Figs. 6(e) and 6(f). Moreover, depth information was obtained and focused images of the objects placed at different depths were successfully reconstructed as seen in Figs. 6(e)-6(h). As a result, color images at arbitrary depths were obtained, as shown in Figs. 6(i) and 6(j). Figure 6 visualizes obviously the color difference between the miniature models of the baby birds and grass, which is seen under the birds and has different color from the birds. Green-color grass was reconstructed as green color by the proposed technique with two wavelengths. Thus, the proposed technique has performed two-color 3D imaging of scattering objects. Thus, the effectiveness and color 3D imaging ability were experimentally demonstrated. Furthermore, these results clarified that the proposed technique was applicable to a scattering colored 3D object with rough surface.

#### 5-2. Three-wavelength digital holographic microscopy

We constructed a three-wavelength in-line digital holographic microscope based on the proposed technique. Figure 7 illustrates a schematic of the system. A Telecentric magnification system was designed and its magnification and theoretical resolution at the wavelength of 488 nm were 25 and 774 nm, which were calculated from the ratio of focal lengths of the lenses in the telecentric system and the numerical aperture of the microscope objective *N.A*. = 0.8. CW lasers with the oscillation wavelengths *λ*_{1} = 640 nm (RLK-4075, LASOS), *λ*_{2} = 532 nm (GLK-32300, LASOS), and *λ*_{3} = 488 nm (Sapphire 488SF-100-HS, Coherent) were used as light sources. A monochromatic CMOS image sensor used was the same as the previously described experiment. A preparation of stained mouse kidney cells was set as a specimen. A piezo-driven mirror was moved a distance of 183 nm for each phase shift. We recorded one hologram per 1.5 seconds. Seven wavelength-multiplexed phase-shifted holograms with three wavelengths were obtained sequentially. Photographs of the specimen at respective wavelengths were taken with the lasers and CMOS image sensor sequentially to investigate the quality of the images reconstructed by the proposed technique. For comparison, we reconstructed three-wavelength images using a wavelength-multiplexed hologram obtained with the same in-line configuration and using calculation of diffraction integral alone. Figure 8 shows the experimental results. In the same manner of refs [20–23], we showed monochromatic images at the respective wavelengths and the color-synthesized images of the specimen illuminated by laser beams. As seen in Figs. 8(a)-8(l), wavelength information was successfully separated and similar images at the wavelengths and its color-synthesized one of the specimen were obtained by the proposed technique, while wavelength dependency of transparency cannot be visualized from a wavelength-multiplexed in-line hologram. Furthermore, 3D imaging of stained cell nuclei with a 1 μm-order radius were achieved as shown in Figs. 8(m) and 8(n). If the object images at the respective wavelengths were successfully retrieved from wavelength-multiplexed phase-shifted holograms, intensity difference between the photographs and reconstructed intensity images at these wavelengths on the image sensor plane approaches to zero. As described in ref [24], it is important to conduct quantitative evaluations of experimental results. Therefore, we calculated root-mean-square errors (RMSEs) to evaluate the reconstructed images quantitatively. Intensity images seen in Figs. 8(a)-8(c) were used for the evaluation as the original images in this experiment. Table 1 showed the results and errors were notably decreased by using the proposed technique in comparison to the method using a wavelength-multiplexed in-line hologram and diffraction-integral calculation alone. The results indicated the ability of the proposed technique for wavelength separation and removals of unwanted image components. Thus, the effectiveness of the proposed microscopy was experimentally demonstrated.

## 6. Conclusion

We have proposed multiwavelength in-line phase-shifting digital holography that uses wavelength-multiplexed phase-shifted holograms and arbitrary symmetric phase shifts, based on phase-shifting interferometry selectively extracting wavelength information. Its effectiveness was numerically confirmed and experimentally demonstrated. Numerical results clarified that it is possible to achieve noiseless holographic multiwavelength 3D imaging with less than 100 nm movement of the mirror per phase shift while it has been difficult with such small movements by the previously reported PDM techniques [25]. Experimental results clarified its ability of color 3D imaging of scattering objects with rough surfaces and its applicability to multicolor 3D microscopy. In comparison to the time-division technique, the proposed technique is valid for implementing a simple setup and expediting the measurement because the number of images recorded decreases.

The next step of the proposed PDM technique is the extension to multispectral holographic 3D image sensing, simultaneous imaging of color and 3D shape with multiwavelength phase unwrapping, dispersion imaging of a 3D specimen by using phase images at multiple wavelengths, multicolor incoherent digital holographic imaging, and multidimensional holographic imaging. This technique has prospective applications to multispectral microscopy to observe 3D specimens with a wide field of view, color 3D image sensing, multidimensional holographic image sensors, and other multiwavelength 3D imaging applications.

## Funding

PRESTO, Japan Science and Technology Agency (JST) (JPMJPR16P8), The Nakajima Foundation (2017-2018), Konica Minolta Science and Technology Foundation (2013-2014), Research Foundation for Opt-Science and Technology (2013-2015), the Japan Society for the Promotion of Science (JSPS) (Grant-in-Aid for Young Scientists (B) 15K17474), and MEXT-Supported Program for the Strategic Research Foundation at Private Universities (2013-2018).

## Acknowledgments

We appreciate Ms. Kris Cutsail for checking the grammar of this article. We thank Dr. Sumio Nakahara and Dr. Shigeyoshi Hisada to give us optical elements and anti-vibration tables.

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