1. INTRODUCTION

Holography [1,2] is a technique recording a complex amplitude distribution of an object wave and reconstructing a three-dimensional (3D) image of the object. Quantitative phase information can be recorded by utilizing interference of light and a reference wave is introduced to generate an interference fringe image called a “hologram.” Digital holography [3–5] sets an image sensor as a recording device and uses a computer or spatial light modulator to reconstruct the 3D object image. An image sensor captures a hologram digitally and a high-speed camera can record a motion-picture image of digital holograms without a lens. Therefore, high-speed 3D and quantitative phase motion-picture imaging can be achieved with a lensless camera by using the technique. This technique has been researched for tomographic imaging [6], microscopy [7], cell identification utilizing phase information [8], single-pixel complex amplitude imaging [9], color 3D imaging with a lensless monochromatic image sensor [10], multimodal imaging [11], analysis of physical phenomenon [12], particle measurement in 3D space [13], 3D imaging with a binary image sensor [14], and encryption of multiple 3D objects [15].

In holography, including digital holography, multiple object waves at multiple wavelengths and polarization directions can be simultaneously recorded on a 2D plane by introducing different spatial carrier frequencies per object wave [16]. Multiple spatial carrier frequencies are generated by using multiple reference arms [17], a common-path interferometer [18], or a single reference arm [19]. The technique using a reference arm is effective especially when requiring reflection-type, single-shot, multiwavelength digital holography with a compact optical system. This is because it is easily implemented with little modification of the system in both transmission- and reflection-type digital holography to increase the number of wavelengths used for recording a wavelength-multiplexed hologram. However, in this technique, most of the spatial frequency bandwidth of an image sensor is discarded to avoid the crosstalk between the object waves with different wavelengths due to a small incident angle of a reference arm. Although the space-bandwidth extension with a large angle and signal theory in multiwavelength digital holography was proposed [20], the angle was up to 15 degrees when using an image sensor whose pixel pitch is 2.2 μm. As a result, the utilization of the recordable space bandwidth was insufficient. Therefore, the numerical aperture of the system became low and fine structures of the object were not resolved clearly. However, attempting to increase the angle using an image sensor with a large pixel size is challenging, and Leclercq and Picart have tried to increase the angle [21]. A beam combiner in front of an image sensor will be removed if the angle approaches 40 degrees as an implementation of Leith–Upatnieks “analog” off-axis holography. In addition, the successful angle increase gives the improvement of spatial resolution in single-shot multiple wavelength digital holography with a single reference arm because the recordable space-bandwidth product improves without the crosstalk between object waves with multiple wavelengths.

In this paper, we propose single-shot multiwavelength digital holography using an extremely large incident angle to make the best use of the recordable space bandwidth of a monochromatic image sensor. Thanks to an extremely large angle of a reference wave against an object wave, no beam splitter in front of an image sensor is required for combining these waves, and a compact optical setup can be constructed. From a single recorded hologram, both the 3D space and multiple wavelength information of an object are reconstructed with a wide space bandwidth. We show the single-shot digital recording of multiple object waves at multiple wavelengths at an angle of 40 degrees using a single reference beam to verify its effectiveness experimentally.

2. PRINCIPLE

The proposed technique is based on spatial frequency-division multiplexing of multiple wavelengths [16–20], and is composed of utilizations of intentional aliasing, the periodicity of a digital signal [22,23], and a single reference arm to record multiple wavelengths with a compact setup. In the recording process, a monochromatic image sensor records a wavelength-multiplexed hologram using an off-axis configuration with a reference arm. In the reconstruction process, each object wave at each wavelength is selectively extracted in the spatial frequency domain by the Fourier fringe analysis [24]. After the procedures of digital holography, complex amplitude distributions at multiple wavelengths with wide spatial bandwidths are reconstructed from a single monochromatic image. Although the angle increase with undersampling has already been proposed by several research groups [21–23], this time we utilized undersampling to record a hologram using a reference beam with an even larger incident angle of more than 40 degrees. As a result, no beam combiner is required in front of the image sensor and a compact optical system such as that of the Leith–Upatnieks “analog” off-axis holography can be constructed.

Figure 1 illustrates an optical implementation in reflection digital holography. By using a single reference arm, a compact reflection-type multiwavelength digital holography system can be constructed. In addition, a more compact setup can be designed by removing a beam combiner. A large 2D spatial carrier is introduced by tilting a mirror in the path of the reference arm at a large angle. The spatial frequency of interference fringes ($\nu$) is determined by

 

Fig. 1. Optical system in reflection digital holography. Object and reference waves with multiple wavelengths illuminate a monochromatic image sensor without a beam combiner.

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From Eq. (2), the increase of the angle can effectively separate the object waves in the spatial frequency domain. When using a single reference arm, the spatial frequency difference is small and, therefore, a large angle is the key to extending the distance between object-wave spectra in the spatial frequency domain. Recordable spatial frequency bandwidth at each wavelength is extended by widening the distance, and much spatial information, such as field of view and resolution, can be recorded by improving the bandwidths, as described in Ref. [20]. After that, one should adjust the positions of the object-wave spectra according to the spatial frequency bandwidth of an image sensor to avoid the unwanted image components 0th- and minus 1st-order diffraction images.

Figure 2 illustrates the spatial frequency spectra of the recorded holograms with and without aliasing. A large angle causes aliasing although wavelength information is well separated by increasing $\theta$, as described in Figs. 2(a) and 2(b). When the sampling theorem is not satisfied, fringes cannot be recorded correctly. In digital recording, however, where $d$ is the pixel pitch of an image sensor, the spatial frequency is modulated to less than $\pm 1/(2d)$ by the periodicity of a digital signal [22,23] and an even higher spatial frequency is generated. After recording a hologram and a 2D Fourier transform (FT), replicas of the object waves appear in the spatial frequency domain due to the periodicity of digital signals. As a result, object waves are recorded even in cases where the sampling theorem is not satisfied, as shown in Fig. 2(b), and crosstalk can be avoided by widening the distance between object-wave spectra with different wavelengths. In comparison with the previously reported method [20], a much higher spatial carrier frequency is introduced and then wavelength information is better and better separated, as shown in Figs. 2(c)–2(e). The distance of object-wave spectra is extended as described in Eq. (2), and the radii of these spectra can be enlarged by adequate arrangements of the spectra under the consideration of the recordable spatial frequency bandwidth of an image sensor, as described in Fig. 2(f). The enlargements of the spectra improve the field of view and resolution and, therefore, a large angle has the ability to improve the recordable spatial bandwidth, field of view, and resolution. A recordable space-bandwidth product is improved at each wavelength and then the products are increased up to the same level as angular multiplexing [25,26], even when the wavelength difference is only 100 nm as indicated in Fig. 2(f). Furthermore, no beam combiner is required as described in a recording setup of analog holography. Note that attenuations of the amplitude distributions of object waves occur when a high spatial carrier is introduced [21,22]. An image sensor with a small fill factor should be chosen to suppress attenuations.

 

Fig. 2. Wavelength-multiplexed hologram and its spatial frequency distribution. Spatial frequency-division multiplexing of multiple wavelengths with a reference arm both (a) without aliasing and (b) with aliasing; (c)–(e) mean that the wavelength separation is well conducted as the angle $\theta$ increases; and (f) shows the recordable spatial bandwidth extension from (e).

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3. EXPERIMENTS

The effectiveness of the proposed technique was experimentally investigated. Figure 3 is a photograph of the constructed optical system based on Fig. 1, where the system does not require a beam combiner in front of the image sensor. Two CW lasers were used as light sources and the oscillation wavelengths of the lasers were ${\lambda }_1=640\mathrm{nm}$ and ${\lambda }_2=532\mathrm{nm}$. We used a monochromatic CMOS image sensor whose $d$, the number of bits, and the number of pixels were 2.2 μm, 12 bits, and $2592(\mathrm{H})\times 1944(\mathrm{V})$ to record a hologram. In general, a CMOS image sensor has a small fill factor and attenuation of interference fringes is suppressed. A Japanese coin with a diameter of 23.5 mm was set as a color object. We set the distance between the object and image sensor (z) as 460 mm. The mirror seen in Fig. 3(b) was moved along the depth direction to change the angle ${\theta }_x$, while ${\theta }_y$ was fixed. Figure 4 shows the spatial frequency images of the recorded two-wavelength-multiplexed holograms taken by the CMOS image sensor. Figure 4 indicates that object waves at two wavelengths were well separated in the spatial frequency domain as the spatial carrier frequency increased by enlarging ${\theta }_x$. Additionally, object waves were recorded even when angles ${\theta }_x$ and ${\theta }_y$ were 38.2 and 4.73 degrees, respectively. Although the spectra of $\pm 1$st-order diffraction waves were attenuated and those of the 0th-order waves were relatively amplified, these waves were successfully recorded. Thus, it was clarified that object waves at two wavelengths were successfully captured with an angle of nearly 40 degrees. Next, we set the distance $z$ as 320 mm to increase the numerical aperture of the system. As an object is put closer to an image sensor, it is shown that the resolution is improved but required spatial bandwidth is increased. Therefore, adequate separation of object waves is required and it is expected that the resolution of the reconstructed image is improved by the proposal. In this experiment, the period of interference fringes becomes 855 nm when the angle between the optical axes of object and reference waves ${\theta }_x$ and ${\theta }_y$ was 38.2 and 4.73 degrees, respectively. This period of fringes was derived from ${\lambda }_2/\sin \theta \text{and}\theta =38.5\mathrm{degrees}$ where $\theta ={({\theta }_x^2+{\theta }_y^2)}^{1/2}$. Furthermore, a wave that was diffracted from the edge of the object generated the finest fringe patterns in the experiment. The angles between the wave from the edge and reference beam ${\theta }_x^{\prime }$ and ${\theta }_y^{\prime }$ were 40.3 and 4.99 degrees, respectively; therefore, ${\theta }^{\prime }=40.6\mathrm{degrees}$. These angles were calculated from $z$ and the radius of the object. The generated finest fringes had a period of 818 nm. We investigated that a 100 nm-order interference fringe could be recorded by an image sensor with $2.2\times {10}^3\mathrm{nm}$ pixel pitch and a color 3D image was reconstructed from the recorded 2D image. Figure 5 shows the experimental results. For comparison, a color object image with $z=620\mathrm{mm}$ and 14.9 degrees [23] was also shown in Fig. 5(f). These results mean that recordable spatial frequency-bandwidth enhancement effects the image-quality improvement due to the increase of the numerical aperture of the system even when recording a fine fringe pattern. This is because the speckle size is decreased by enlarging the numerical aperture. Thus, the effectiveness of the proposed technique was experimentally demonstrated.

 

Fig. 3. Photographs of the constructed optical setup. (a) Whole system and (b) magnification of the area inside the rectangle shown in (a).

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Fig. 4. Spatial frequency distributions of recorded wavelength-multiplexed holograms with the angles of (a) 11.1 degrees, (b) 19.1 degrees, (c) 22.8 degrees, and (d) 38.2 degrees along to $x$-axis direction.

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Fig. 5. Experimental results. Photographs of the object illuminated by lasers with the wavelengths of (a) 640 nm and (b) 532 nm. (c) Spatial frequency distribution of a recorded hologram and reconstructed images at (a) 640 nm and (b) 532 nm. (f) A color-synthesized image and magnified images in the rectangle areas in the case of $z=620\mathrm{mm}$. (g) The color image synthesized from (d) and (e) and magnified images.

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4. CONCLUSION

We have proposed a single-shot multiwavelength digital holographic technique with a wide field of view, an angle of more than 40 degrees, and no beam combiner to generate interference light. Recording of fine interference fringes whose period was 818 nm and further improvement of the recordable space-bandwidth product at each wavelength were experimentally demonstrated and no half-mirror was needed for combining object and reference waves according to the proposal. This technique has prospective applications to multiwavelength microscopy for 3D moving picture recording of dynamically moving specimens with a wide field of view, highly accurate 3D shape measurement using multiwavelength phase unwrapping, quantitative phase imaging, high-speed color 3D camera for imaging of objects moving at high speeds, and other multidimensional imaging applications.