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Letter

Vol. 40, No. 12 / June 15 2015 / Optics Letters

Dual-wavelength phase-shifting digital holography selectively extracting wavelength information from wavelength-multiplexed holograms TATSUKI TAHARA,1,* RYOTA MORI,1 SHUHEI KIKUNAGA,1 YASUHIKO ARAI,1

AND

YASUHIRO TAKAKI2

1

Faculty of Engineering Science, Kansai University, 3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo 184-8588, Japan *Corresponding author: tahara@kansai‑u.ac.jp

2

Received 23 February 2015; revised 13 May 2015; accepted 16 May 2015; posted 18 May 2015 (Doc. ID 235146); published 10 June 2015

Dual-wavelength phase-shifting digital holography that selectively extracts wavelength information from five wavelength-multiplexed holograms is presented. Specific phase shifts for respective wavelengths are introduced to remove the crosstalk components and extract only the object wave at the desired wavelength from the holograms. Object waves in multiple wavelengths are selectively extracted by utilizing 2π ambiguity and the subtraction procedures based on phase-shifting interferometry. Numerical results show the validity of the proposed technique. The proposed technique is also experimentally demonstrated. © 2015 Optical Society of America OCIS codes: (090.1995) Digital holography; (090.0090) Holography; (090.1705) Color holography. http://dx.doi.org/10.1364/OL.40.002810

Holography [1,2] is a technique for recording a wavefront of an object wave and reconstructing a three-dimensional (3D) image of the object. The most remarkable feature of holography is that 3D motion-picture recording can be achieved for any ultrafast physical phenomenon [3]. Digital holography [4] is a technique for recording a complex amplitude distribution using an image sensor and reconstructing an object image using a computer. This technique has been researched for not only the observation of ultrafast phenomenon, but also quantitative phase imaging [5,6], microscopy [7,8], multimodal imaging [9,10], particle and flow measurements [11], and analysis of physical phenomenon [12]. In digital holography, phaseshifting interferometry is frequently employed to obtain a wide spatial bandwidth available for recording an object wave without the superimposition of the unwanted images, the zeroth-order diffraction wave, and conjugate image [13]. Phase-shifting digital holography has been applied to multiple wavelengths 3D image sensing, called color/multiwavelength phase-shifting digital holography [14,15]. In multiwavelength phase-shifting digital holography, multiple wavelength information is usually recorded with a color image sensor that has a 0146-9592/15/122810-04$15/0$15.00 © 2015 Optical Society of America

Bayer color filter array [14,15] or time-division technique [16]. However, in the former, the wavelength selectivity of a Bayer color filter array is low, and the crosstalk between object waves with different wavelengths occurs inherently. The brightness of the reconstructed image decreases because of the absorption of the filter array. Furthermore, the recordable spatial bandwidth is decreased by space-division multiplexing of multiple wavelengths. In the latter, although multiple wavelength information is separately recorded, a large number of holograms are required, and the multiple light sources must be turned on/off sequentially. Therefore, a longer time is required to take measurements. In this Letter, dual-wavelength digital holography for selectively extracting wavelength information is proposed as a first step for realizing a novel type of multiwavelength imaging technique. The key point is to introduce specific phase shifts against respective wavelengths to remove the crosstalk components between multiple wavelengths, and extracting only the object wave at the desired wavelength from the wavelength-multiplexed holograms. Object waves at multiple wavelengths are selectively extracted by the combination of phase-shifting interferometry and subtraction procedures. Numerical simulations and experimental results show its validity. Figure 1 illustrates the principle. The optical setup is based on phase-shifting digital holography. Multiple object and reference waves with multiple wavelengths illuminate a monochromatic image sensor simultaneously. The sensor records wavelength-multiplexed holograms I x; y∶α1 ; α2  by changing the phases of the reference waves, where α1 and α2 are the specific phase shifts at the wavelengths λ1 and λ2 , respectively. When α2 is an integral multiple of 2π, only the object wave at λ1 can be extracted from wavelength-multiplexed phaseshifted holograms by signal processing. As a result, a multiwavelength 3D image is reconstructed from monochromatic images. Note that a dual-color 3D image can be reconstructed from five holograms by the proposed technique, while six images are required for time-division technique. This is because in the case where the number of wavelengths is L, 2L  1 variables are contained in a wavelength-multiplexed hologram: the number L of object waves, L of conjugate images, and the sum

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recorded monochromatic image is the sum of I λ1 x; y∶α1  and I λ2 x; y∶α2 . When the complex amplitude distributions of object waves with different wavelengths are U λ1 x; y and U λ2 x; y, 0thx; y is the 0th-diffraction wave, Arx; y is the amplitude distribution of the reference wave, j is imaginary unit, * means complex conjugate, and M and N are integers, I x; y∶α1 ; α2  can be rewritten as follows: I x; y∶α1 ; α2   0thλ1 x; y  Arλ1 x; yfU λ1 x; y exp−jα1   U λ1 x; y expjα1 g  0thλ2 x; y  Arλ2 x; yfU λ2 x; y exp−jα2   U λ2 x; y expjα2 g: Fig. 1. Principle of the proposed dual-wavelength phase-shifting digital holography.

(2)

U λ1 x; y and U λ2 x; y are derived from five holograms as U λ1 x; y  2I x; y∶0; 0 − fI x; y∶α1 ; 2πN   Ix; y∶ − α1 ; −2πN g∕f4Arλ1 x; y1 − cos α1 g

of the 0th-order diffraction waves. Therefore, five holograms are required for solving the system of equations when L  2. Figure 2 illustrates optical implementations of the proposed digital holography. Multiple lasers irradiate beams with multiple wavelengths simultaneously. A device for shifting the phase of light, such as a piezo-driven mirror, a spatial light modulator, or wave plates, is placed in the path of the reference arm. A monochromatic image sensor sequentially records the required wavelength-multiplexed phase-shifted holograms. It is noted that the spectroscopic sensitivity of the image sensor should not be so different for each wavelength to achieve highquality multiple-wavelength imaging. The image-reconstruction algorithm is based on the theory of a previously reported conference paper [17]. A wavelengthmultiplexed phase-shifted hologram Ix; y∶α1 ; α2  is expressed as follows: Ix; y∶α1 ; α2   I λ1 x; y∶α1   I λ2 x; y∶α2 ;

(1)

 jfI x; y∶ − α1 ; −2πN  − I x; y∶α1 ; 2πN g∕4Arλ1 x; y sin α1 ;

(3)

U λ2 x; y  2I x; y∶0; 0 − fI x; y∶2πM ; α2   Ix; y∶ − 2πM ; −α2 g∕f4Arλ2 x; y1 − cos α2 g  jfI x; y∶ − 2πM ; −α2  − I x; y∶2πM ; α2 g∕4Arλ2 x; y sin α2 :

(4)

As shown in Eq. (2), intensity distribution of a hologram at a wavelength is not changed when the phase shift at the wavelength is an integral multiple of 2π, while intensity at the other wavelength is changed. If the system shown in Fig. 2(a) is used to implement the proposed technique, by moving the mirror in the reference arm with a piezo actuator at a distance Z in the depth direction, the phase shifts are α1  4πZ ∕λ1 ;

(5)

where I λ1 x; y∶α1  and I λ2 x; y∶α2  are holograms at the wavelengths of λ1 and λ2 , respectively. Equation (1) means that a

α2  4πZ ∕λ2 :

(6)

Fig. 2. Optical implementations of the proposed digital holography. Setups using (a) a piezo-driven mirror and (b) a spatial light modulator.

However, when Z is equal to M λ1 ∕2, α1 is 2πM , and α2 is 2πM λ1 ∕λ2 . As a result, the intensity I λ1 x; y∶α1  is not changed and I λ2 x; y∶α2  is, unless M λ1 ∕λ2 is an integer. α1 and α2 must not be set as π. As shown in Eqs. (2)–(4), subtraction between holograms, which is based on phase-shifting interferometry, is calculated, and the unwanted wavelength component I λ1 x; y∶α1  or I λ2 x; y∶α2  is removed. Equations (2)–(4) also indicate that dual-wavelength information is extracted selectively from five holograms. In this way, multiwavelength information can be separately extracted from 2L  1 holograms. From the extracted complex amplitude distributions on the image sensor plane, a multiwavelength 3D image is reconstructed by the calculations of diffraction integrals. Numerical simulations were conducted to verify its effectiveness. Figure 3 presents the amplitude and phase distributions of the object wave in each wavelength and one of the holograms obtained by the technique. Scattering object waves were assumed. Maximum amplitude value was 255. 640 and 532 nm were assumed as the wavelengths of the light sources. In these simulations, the distance between the object and image

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Fig. 3. Object wave for numerical simulations. Amplitude images at (a) λ  640 nm and (b) at 532 nm, respectively, (c) phase image, and (d) one of wavelength-multiplexed phase-shifted hologram obtained by the proposed technique.

sensor was assumed as 200 mm, pixel pitch as 5 μm, and number of pixels of the image sensor as 2048 × 2048. Phase shifts are calculated from Eqs. (5) and (6) under the condition of M  N  1. In the cases where the dynamic range of the image sensor was from 8 to 24 bits, the object images were reconstructed to investigate the relationship between the number of bits and image quality. Root-mean-square errors (RMSEs) of the reconstructed images were calculated in each bit resolution to evaluate their quality quantitatively. Figure 4 presents the reconstructed images at 8- and 12-bit resolutions, as an examples of the results. Faithful images were reconstructed in each resolution, and crosstalk between wavelengths was not seen. Thus, the validity of the proposed technique was numerically confirmed. Figure 5 shows the RMSEs against the bit resolution of the image sensor. The error is remarkably decreased when the bit resolution is higher than 10. These results mean that a commercially available image sensor with 8-bit resolution is applicable to the proposed technique, and 12 bits are enough to achieve high-quality imaging. We investigated the ability of the proposed technique for removing the crosstalk component I λ1 x; y∶α1  or I λ2 x; y∶α2  experimentally. The optical system shown in Fig. 6(a) was constructed and an experiment was conducted to demonstrate the proposed technique. Five wavelength-multiplexed phaseshifted holograms were recorded sequentially by using a mirror with a piezo actuator. The wavelengths of the lasers used as light sources λ1 and λ2 were 473 and 640 nm, respectively. Coherence lengths of the lasers are longer than 10 meters. A monochromatic CMOS image sensor was used to record the holograms. The sensor has 12 bits, 2592 × 1944 pixels, and a pixel pitch of 2.2 μm. From Eqs. (5) and (6), the mirror with a piezo actuator is moved at a distance Z  0 nm, 236.5 nm, and 320 nm sequentially, and phase shifts α1 ; α2  in λ1 ; λ2  are (0,0), 2πλ2 ∕λ1 ; 0, −2πλ2 ∕λ1 ; 0, 0; 2πλ1 ∕λ2 , and 0; −2πλ1 ∕λ2 . Two transparency sheets shown in Fig. 6(b) were set as a color 3D object. The logo of

Fig. 4. Numerical results. Amplitude images at λ  (a) 640 nm and (b) 532 nm reconstructed from 8-bit holograms. Amplitude images at (c) 640 nm and (d) 532 nm reconstructed from 12-bit holograms.

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Fig. 5. Root-mean-square errors in (a) amplitude and (b) phase distributions.

the International Year of Light and characters “2015” were drawn on the sheets, and blue- and red-color films were attached to the logo and characters, respectively. A red “2015” sheet and a blue logo one were set at the distances of 250 and 320 mm from the image sensor plane, respectively. For comparison, object images were also reconstructed from a wavelength-multiplexed hologram. Figure 7 shows the experimental results. Figures 7(a) and 7(b) are the photographs of the 3D object illuminated by the laser beam containing the wavelengths of 473 and 640 nm, respectively. Blue and red color films attached on the sheets absorb red and blue light, respectively. As shown in Figs. 7(c) and 7(d), not only the 0th-order diffraction wave and the conjugate image but also image components given by the crosstalk between I λ1 x; y∶α1  and I λ2 x; y∶α2  were seen from a wavelength-multiplexed hologram. In contrast, Figs. 7(e) and 7(f) show that the crosstalk components were completely removed in each wavelength by the proposed technique. As a result, a clear color 3D image was obtained, as shown in Figs. 7(g) and 7(h). Thus, both the abilities for removing the crosstalk component and imaging clear color 3D information were experimentally demonstrated. Note that the visibility of interference fringes should not be changed between recorded holograms to avoid the crosstalk. This is because the system of equations cannot be solved if

Fig. 6. Experimental setup. (a) Schematic of the optical implementation and (b) photograph of the color 3D object illuminated by both white light and laser beam containing λ1 and λ2 .

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piezo-driven mirror or a spatial light modulator should be sufficiently shorter than the coherence length of the lasers. When applying to three-wavelength digital holography, a wide range of movement with high-resolution is required for a piezo actuator to conduct phase shifts accurately. For example, up to 156 μm of optical path shift is needed for selectively extracting object waves at the wavelengths of 473, 532, and 640 nm. We have proposed dual-wavelength phase-shifting digital holography for selectively extracting multiple wavelength information by the subtraction based on phase-shifting interferometry, and numerically and experimentally investigated dual-wavelength 3D imaging ability. Numerical results clarified that noiseless dual-wavelength 3D imaging can be achieved with a 12-bit image sensor. Experimental results show the crosstalk problem can be completely solved by the signal processing of the proposed technique. In comparison to conventional time-division system [16], the proposed technique is valid for implementing a simple setup and for speeding up the measurement because the number of images recorded is decreased. If applied to parallel phase-shifting, it can be achieved to implement a single reference arm [17] and conduct highspeed recording with no crosstalk between object waves due to no color-filter array. This technique has prospective applications to multiple wavelengths 3D microscopy with a wide field of view, color 3D sensing, and other imaging applications. Japan Society for the Promotion of Science (JSPS) (15K17474); Konica Minolta Science and Technology Foundation; Ministry of Education, Culture, Sports, Science, and Technology (MEXT); Research Foundation for Opt-Science and Technology; Research Foundation of Tokyo Institute of Technology. The authors thank Kris Cutsail for checking the English grammar of this Letter. REFERENCES

Fig. 7. Object for the experiment and experimental results. Photographs of the object are illuminated by the laser beams at the wavelengths of (a) 473 nm and (b) 640 nm, respectively. (c) Image reconstructed from a wavelength-multiplexed hologram when reconstruction distance and wavelength are 320 mm and 473 nm, respectively. The areas surrounded by yellow, blue, and red rectangles are those in which the 0th-order wave, the conjugate image, and the unwanted image generated by the crosstalk, respectively. (d) Image reconstructed from a hologram when reconstruction distance and wavelength are 250 mm and 640 nm, respectively. Focused images are obtained by the proposed technique at (e) 473 nm and (f) 640 nm, respectively. Color-synthesized images focused on the distances (g) 250 mm and (h) 320 mm, respectively.

the visibilities of holograms are different from each other due to the change in the numerical model of each hologram. Therefore, the optical path shifts that are generated by a

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Dual-wavelength phase-shifting digital holography selectively extracting wavelength information from wavelength-multiplexed holograms.

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