We propose multiwavelength-multiplexed phase-shifting incoherent color digital holography. In this technique, a monochrome image sensor records wavelength-multiplexed, phase-shifted, and incoherent holograms, and a phase-shifting interferometry technique selectively extracts object waves at multiple wavelengths from the several recorded holograms. Spatially incoherent light that contains multiple wavelengths illuminates objects, and multiwavelength-incoherent object waves are simultaneously obtained without using any wavelength filters. Its effectiveness is experimentally demonstrated for transparent and reflective objects.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Three-dimensional (3D) image sensing has provided useful information in many fields, such as microscopy, surface shape measurement, and surveying 3D space. 3D imaging enables the observations of physical phenomena and biological specimens without blurring, and thus the demand for 3D image-sensing techniques to obtain a 3D image with a large depth of field has increased. Wavefront sensing is a method of recording 3D space information with a 2D recording material, and holography is an actively researched wavefront-sensing technique [1,2]. The phase distribution of a light wave contains the information of a 3D location of an object, and holography acquires its complex amplitude distribution on a recording medium by utilizing the interference of light. In holography, holograms can be recorded digitally by using a photodetector or an image sensor, and a 3D image of an object can be reconstructed quantitatively by digital signal processing with a computer; this technique is called digital holography (DH) [3–7]. To date, microscopy and quantitative phase imaging [8–11], 3D displacement measurement [12,13], 3D phase tomographic imaging [14–16], simultaneous 3D imaging of multiple objects [17,18], and 3D imaging with a single pixel sensor [8,19,20] have been presented as possible applications of DH. DH frequently adopts laser light sources to obtain an interference fringe image.
Incoherent DH is a method of recording a hologram that is generated from a spatially incoherent light source [21–28]. Fluorescence light [25,26], thermal radiation , light illuminated by a light-emitting diode (LED) , and even daylight  can be recorded using incoherent DH, and thus their applications in fluorescence 3D microscopy [25,26], 3D imaging with white light , and recording of a speckleless hologram  have been realized. Recording of spatially incoherent light at multiple wavelengths enables holographic multicolor/multispectral 3D image sensing [23,29–39] and the technique is called incoherent color DH. Incoherent color DH generally adopts the temporal division technique to record wavelength information of incoherent light [30,32,33,37]. Wavelength information is separated in this technique by inserting wavelength filters in the optical path or by changing the wavelength of illumination light. As a result, the technique is time-consuming, and light intensity becomes low because of the wavelength filters. As another way, frequency-division multiplexing is adopted to record object waves at multiple wavelengths [29,31,35,39]. This multiplexing technique adopts the principle of Fourier spectrometry to obtain wavelength information. A series of incoherent holograms with a carrier wave are recorded, and a piezo-driven mirror or micrometers are used to generate carriers to object waves at multiple wavelengths and to separate these object waves in the Fourier domain. This technique has the ability to obtain wavelength information of temporally incoherent light with high wavelength resolution and is suitable for hyperspectral holographic imaging. However, in this technique, recording of a large number of incoherent holograms was required and more than 500 holograms were recorded even when obtaining three-wavelength incoherent object waves . As a novel way to record wavelength information by using a monochrome image sensor without spatial and temporal divisions, multidimensional imaging with point-spread-function (PSF) engineering was presented [34,36,38]. Particularly in the scheme, coded aperture correlation holography (COACH)  performed four-dimensional (4D) imaging that means the visualization of 3D spatial information and multiple wavelengths from wavelength-multiplexed images, by utilizing the differences of PSFs between the wavelengths . Wavelength separation is conducted from the difference of PSFs with a Fresnel lens, using correlation procedures with libraries of PSF holograms at multiple wavelengths. COACH requires the library of PSFs in advance and stability of the PSF of the system but depth resolution of the system is much improved in comparison to Fresnel incoherent correlation holography (FINCH), by utilizing PSF engineering [34,38,40].
Wavelength-selective phase-shifting interferometry (WS-PSI) [41–46] was proposed as another multiplexing technique to obtain multiwavelength object waves by using a phase shifter and a monochrome image sensor. WS-PSI adds wavelength-dependent phase shifts to wavelength-multiplexed holograms and extracts multiwavelength object waves from the recorded holograms based on phase-shifting interferometry. Experimental demonstrations of WS-PSI were initially performed with a piezo actuator as a wavelength-dependent phase shifter, using four  and five  two-wavelength-multiplexed holograms with specific phase shifts. In Ref.  we described that the least required number of wavelength-multiplexed holograms was the double of the number of central wavelengths measured with specific phase shifts. Reference  also indicated optical implementations with not only a piezo actuator but also wave plates. After that, an algorithm for WS-PSI with wavelength-dependent arbitrary phase shifts was presented with lasers  and was performed with a temporally low-coherence light source . The number of wavelength-multiplexed phase-shifted holograms required to obtain object waves selectively is 2N + 1 in WS-PSI with the algorithm of Ref. , where N is the number of central wavelengths measured. Then, image quality of WS-PSI for dynamic range of the holograms and relationship between the wavelength resolution and the dynamic range were investigated . After that, a liquid-crystal-on-silicon spatial light modulator (LCoS-SLM) was adopted as an electrically-driven phase modulator  and neither mechanically moving components nor shutters to select a wavelength of illumination light was required. Therefore, until now three-wavelength spatially coherent object waves were retrieved from wavelength-multiplexed phase-shifted seven holograms in WS-PSI [7,43,46]. The number of recorded holograms was much less than that in frequency-division multiplexing when sensing object waves at several wavelengths. Furthermore, there is no requirement for mechanically moving parts and therefore a construction of an optical system can be simplified by WS-PSI. WS-PSI can also accelerate multiwavelength holographic sensing in comparison to temporal division technique because 4N holograms were required frequently for N-wavelength DH with temporal division technique [44,46]. PSF engineering  and WS-PSI are the techniques to conduct 4D imaging from the recording of several wavelength-multiplexed images. When applying a PSF engineering technique, PSFs in 3D position at multiple wavelengths should be known by obtaining libraries of PSF holograms at multiple wavelengths in advance. Therefore, the following are expected: the number of recordings of spatially incoherent multiwavelength holograms is reduced in comparison to the temporal division and temporal division-multiplexing techniques and three-wavelength 3D imaging is conducted from the recording of seven wavelength-multiplexed images without libraries of PSF holograms. However, until now WS-PSI was neither applied to record a spatially incoherent hologram nor implemented by a single-path interferometer. Recording of spatially incoherent multiwavelength holograms is important to obtain a speckle-less holographic image and a single-path interferometer is required to record these holograms successfully.
In this article, we propose a single-path, spatially incoherent, wavelength-multiplexed, and color digital holographic sensing technique to obtain a multicolor 3D image with a small number of spatially incoherent holograms and without wavelength filters, wavelength changes of illumination light, nor libraries of PSF holograms. We adopt WS-PSI to selectively extract incoherent object waves at multiple wavelengths from wavelength-multiplexed holograms. By adopting WS-PSI to incoherent color DH, we can obtain wavelength-multiplexed, phase-shifted, and incoherent seven holograms that are sufficient to reconstruct a three-wavelength 3D image of incoherent light with a full space-bandwidth product of a monochrome image sensor. Experimental results show its effectiveness.
2. Multiwavelength-multiplexed phase-shifting incoherent color digital holography
Figure 1 shows the schematic of the proposed incoherent color DH. We call the proposed DH as multiwavelength-multiplexed phase-shifting incoherent color digital holography (MP-ICDH). Figure 1(a) shows the basic configuration of MP-ICDH. MP-ICDH can adopt a FINCH [22,38] system, and thus a single-path phase-shifting interferometer can be utilized. Wavelength-multiplexed spatially incoherent light that is irradiated from narrow-band light sources such as LEDs illuminates objects, and the light diffracted from the objects passes through a single-path interferometer. As shown in Fig. 1(b), the proposed configuration makes use of polarization to construct a single-path self-reference interferometer. In the path of the interferometer, a polarizer is set to generate linearly polarized object waves at multiple wavelengths, and a spatial light modulator (SLM) is set to display a Fresnel lens and phase shifts on the SLM plane. In other words, the SLM works as both a diffractive lens and a phase shifter. Horizontally polarized waves at multiple wavelengths are modulated by a polarization-sensitive diffractive lens. Two wavelength-multiplexed waves that differ both curvature radii of wavefronts and polarization directions are generated from a wavelength-multiplexed object wave by using the SLM. In the case where a series of phase shifts at different wavelengths are different from each other, WS-PSI [43,44] can be adopted to separate wavelength information from wavelength-multiplexed images. A polarization-sensitive electrically driven phase modulator such as a liquid-crystal-on-silicon SLM (LCoS-SLM) and a liquid crystal spatial light phase modulator can be utilized to generate phase shifts with a high wavelength dependence . Some of the available LCoS-SLMs have both a high wavelength dependence and a wide phase-modulation range , and thus a Fresnel lens function and wavelength-dependent phase shifts are simultaneously generated. Thus, the phase-modulation axis of the polarization-sensitive SLM is different from the transmission axis of the polarizer, and thus two multiwavelength object waves whose curvatures are different are generated after passing the SLM. The other polarizer aligns the polarization direction of the two waves and the two waves can interfere with each other. Finally, a monochrome image sensor records a self-interference, wavelength-multiplexed, and spatially incoherent hologram. Several wavelength-multiplexed incoherent holograms are sequentially recorded by shifting the phase of one of the two waves, using the SLM.
Figure 2 shows the optical implementations of the previously constructed DH system based on WS-PSI and the MP-ICDH system, which is illustrated to indicate the differences between the previous and proposed systems. The main differences are the spatial and temporal coherencies of light sources, the types of the interferometers, and phase-mask patterns displayed on the SLM. In the previous system, lasers with high spatial and temporal coherency are used as multiwavelength light sources. A two-arm configuration such as Michelson and Mach-Zehnder interferometers is adopted and a plane wave is introduced as a reference wave at each wavelength. On the other hand, in the proposed system multicolor LEDs can be used as light sources and a single-path interferometer such as a polarization-based FINCH system is adopted to implement a wavelength-filter-less spatially incoherent color DH system. Fresnel lens patterns with wavelength-dependent phase shifts are introduced. Spatially incoherent light sources are important to obtain a speckle-less holographic image. A single-path interferometer has a high stability against external noise such as vibration as demonstrated in Ref.  and is important to generate holograms with spatially incoherent light sources due to the extremely short optical-path-length differences between the two waves at each wavelength.
In the reconstruction procedure, one can use an algorithm of DH with WS-PSI as described in Refs. [43,44] for spatially incoherent narrow-band light generated from multiple LEDs. As theoretically described and experimentally demonstrated by FINCH [22–24], phase-shifting interferometry works for spatially incoherent phase-shifted holograms. Where I(x, y: α1j, …, αij, …, αNj) is one of the spatially-incoherent narrow-band, wavelength-multiplexed, and phase-shifted holograms, αij is a j-th phase shift at the central wavelength of λi, I0th(x, y) is the summation of the zeroth-order diffraction waves, and Ci, Aoi(x, y), and ϕoi(x, y) are a coefficient that depends on the spectral bandwidth [48,49] and distributions of amplitude and phase in an object wave at λi, respectively, the WS-PSI algorithm used to extract multiwavelength object waves selectively  is expressed by
We conducted experiments to investigate the validity of MP-ICDH. We constructed imaging-lens-less, multiwavelength-multiplexed, single-path, phase-shifting, and incoherent color digital holography systems for transparent and reflective objects with N = 2 and 3.
3.1 MP-ICDH for transparent color objects
We constructed the MP-ICDH system shown in Fig. 3 and set an LCoS-SLM (X10468-01, fabricated by Hamamatsu Photonics K.K.) to generate wavelength-dependent phase shifts and obtain a wide phase-modulation range. The phase-modulation range of this model of the SLM was twice extended and then both a diffractive lens and phase shifts were simultaneously generated. The central wavelengths of the LEDs used as light sources were λ1 = 625 and λ2 = 530 nm, and FWHMs of these LEDs were 18 and 33 nm. The LCoS-SLM set phase shifts at the wavelengths of (λ1, λ2) as (−97π/123, −100π/99), (−39π/123, −40π/99), (0, 0), (39π/123, 40π/99), and (97π/123, 100π/99). A monochrome sCMOS image sensor with 6.5 µm pixel size and 12 bits recorded wavelength-multiplexed in-line phase-shifted five incoherent holograms with 2048 × 2048 pixels. In front of the image sensor, a lens was set to collect wavelength-multiplexed light. Color objects were prepared by using transparent films and a color inkjet printer and were set in the optical path of the single-path interferometer. A green ‘T’, a red ‘H’, and a black background were drawn on the films using a color inkjet printer. Each character was 7 pt. Rectangular transparent areas were set to these objects as shown in Fig. 3. Two colored objects were 36 mm apart from each other in the depth direction. Two LEDs illuminated an object simultaneously and a monochrome image sensor recorded spatially incoherent and discrete narrow-band, wavelength-multiplexed, and phase-shifted holograms. Phase shifts at multiple wavelengths were generated sequentially by an LCoS-SLM and five grayscale holograms were obtained. Two object waves at two center wavelengths on the image sensor plane were retrieved from the recorded five holograms by applying an algorithm of WS-PSI. This time the image-reconstruction algorithm [43,44] that was described in the previous section was applied. Two-wavelength focused images of objects were reconstructed by calculating diffraction integrals. For comparison, object images were also reconstructed with the same interferometer but the two LEDs were turned on/off to record four phase-shifted incoherent holograms for each color sequentially. Object images were also obtained from a single-wavelength-multiplexed incoherent hologram to determine the wavelength selectivity of MP-ICDH. The depth resolution of the constructed MP-ICDH system is based on a FINCH system as described in Ref. . Figure 4 shows the experimental results. The reconstructed images obtained by the time-division technique in Figs. 4(a) and 4(b) did not contain any crosstalk noise between wavelengths and thus were regarded as standard images. Figure 4(c) shows a wavelength-multiplexed incoherent hologram that is recorded by a monochrome image sensor. As seen in Figs. 4(d) and 4(e), plane wavelength-dependent phase shifts were introduced to a static Fresnel phase lens pattern. In Figs. 4(f) and 4(g), not only the sum of the 0th-order diffraction waves and the conjugate images, but also image components given by the crosstalk between the intensity distributions at the two wavelengths were reconstructed from a two-wavelength-multiplexed hologram. As a result, the wavelength information of objects could not be reconstructed correctly. In contrast, color object images were clearly and successfully reconstructed by MP-ICDH, as shown in Figs. 4(h) and 4(i). We show the reconstructed images at respective wavelength components on the different depths in Figs. 4(j)–4(q). Undesired wavelength components were successfully removed by WS-PSI. Thus, it has been clarified that the proposed system can perform imaging-lens-less color 3D imaging without crosstalk between object waves at multiple wavelengths. Its image quality is close to sequential two-wavelength phase-shifting incoherent digital holography with the time-division technique.
3.2 MP-ICDH for a reflective color 3D object
We constructed an MP-ICDH system for a reflective object as shown in Fig. 5. The two LEDs, the monochrome image sensor, and the LCoS-SLM used were the same as those in the previous section. A blue LED was added to record blue incoherent holograms for the reconstruction of a multicolor 3D object image. Its central wavelength and FWHM were λ3 = 455 nm and 18 nm, respectively. The LCoS-SLM sequentially provided regular phase shifts of 40π/123, 40π/99, and 40π/78 at the wavelengths of λ1, λ2, and λ3, respectively. We set a resistor tilted to the depth direction as a reflective color 3D object. The size of the resistor was 6 mm × 3 mm. Three LEDs illuminated a reflective object simultaneously and a monochrome image sensor recorded spatially incoherent and discrete narrow-band, wavelength-multiplexed, and phase-shifted holograms seven times. Phase shifts at multiple wavelengths were generated sequentially by an LCoS-SLM and seven grayscale holograms were obtained. Three object waves at three center wavelengths on the image sensor plane were retrieved from the recorded seven holograms by applying the algorithm of WS-PSI that was described in the section 2. After that, diffraction integrals were calculated to the respective object waves. The maximum phase shift at the central wavelength of 455 nm was more than 6π due to the generations of both a Fresnel phase lens and phase shifts and thus the use of a spatial light phase modulator with a wide phase-modulation range was essential. For comparison, object images were also reconstructed with the same interferometer but the three LEDs were turned on/off to record four phase-shifted incoherent holograms for each color sequentially. Twelve holograms were recorded using phase-shifting incoherent color digital holography with time-division technique. Object images were also obtained from a single-wavelength-multiplexed incoherent hologram to show the wavelength selectivity of MP-ICDH. Figure 6 shows the experimental results. The reconstructed images obtained by the time-division technique in Figs. 6(a)–6(f) did not contain any crosstalk noise between wavelengths and thus were regarded as standard images. As seen in Figs. 6(g)–6(l), not only the sum of the zeroth-order diffraction waves and the conjugate images, but also image components given by the crosstalk between the intensity distributions at the three wavelengths were reconstructed from a three-wavelength-multiplexed hologram. As a result, wavelength information could not be reconstructed correctly. In contrast, Figs. 6(m)–6(r) that were reconstructed by MP-ICDH resembled Figs. 6(a)–6(f). Multicolor object images on the arbitrary depths were reconstructed by MP-ICDH, as shown in Figs. 6(s) and 6(t). Magnified images shown in Figs. 6(u)–6(x) indicates blur and refocused. Thus, it has been clarified that the proposed system can perform imaging-lens-less multicolor 3D imaging without crosstalk between object waves at multiple wavelengths. Its image quality is close to sequential three-wavelength four-step phase-shifting incoherent digital holography with time-division technique.
We have proposed MP-ICDH that is a single-path, self-interference, spatially incoherent, wavelength-multiplexed, and color digital holography technique to obtain a multicolor 3D image with a small number of spatially incoherent holograms and without wavelength filters nor wavelength changes of illumination light. WS-PSI enables the selective extraction of incoherent object waves from 2N + 1 wavelength-multiplexed holograms. In comparison with multicolor incoherent digital holography with Fourier spectroscopy and temporal division technique, the number of recordings of spatially incoherent color holograms is reduced by using WS-PSI. There is no requirement for mechanically moving parts and a single-path interferometer is successfully constructed employing an electrically driven phase shifter with a wide phase-modulation range. Wavelength filters are not needed to obtain an incoherent color holographic image with RGB LEDs and it is expected that light-use efficiency is improved. Experimental results show its effectiveness. Depth resolution of the constructed systems is based on FINCH . When improving the depth resolution, it is effective to combine WS-PSI and an incoherent light sensing technique with PSF engineering such as COACH. In contrast, the MP-ICDH does not require the information of libraries of PSFs at multiple wavelengths in advance. The low noise in reconstruction is conducted by applying a FINCH system due to no zero-point problem that is seen in deconvolution imaging techniques. One of the next steps is the application to hyperspectral holographic image sensing of temporally incoherent light with WS-PSI and the collaborative utilization of compressive sensing and an LCoS-SLM is considered as a solution to reduce the number of recordings drastically . The proposed single-path incoherent color digital holographic sensing system has the ability to open the applicability of multiwavelength DH to multicolor 3D sensing of stroboscopic illumination with LEDs, fluorescent lamp, and self-luminous light such as fluorescence light and chemically activated light. MP-ICDH has prospective applications in speckle-less multicolor imagers to observe 3D objects, multispectral 3D image sensing of self-luminous light, wavelength-filter-less and lens-less 3D microscopy, and other multiwavelength 3D imaging techniques.
Precursory Research for Embryonic Science and Technology (JPMJPR16P8); Japan Society for the Promotion of Science (18H01456, 19H01097).
The authors declare no conflicts of interest.
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