The polarization of light contains valuable information on the measured object. By measuring the state of polarization, stresses induced in plastic pieces can be visualized, and the orientations of polymers such as a liquid crystal, fibers in a biological specimen, and the birefringence of a sample can be measured without labeling [1–3]. The thickness of a film is measured, the distribution of a forest is identified, and the inside of river is observed clearly, exploiting the reflectance difference between horizontal and vertical polarizations [4,5]. Molecular structures are estimated by detecting the polarization of fluorescence in single-molecule fluorescence imaging [6]. The combination of a polarization measurement technique and a three-dimensional (3D) image sensing technique is required to be able to measure the polarization information on objects located at different depths simultaneously.

Incoherent digital holography (IDH) is a 3D image-sensing technique for recording an interference fringe image of an object illuminated by spatially incoherent light—such an image is called an incoherent hologram—and reconstructing the 3D image of the object from the recorded incoherent hologram [7–10]. IDH has the following attractive features: lensless 3D imaging is achieved, a deep depth range is measured, fully passive holographic 3D imaging is achieved with sunlight, and the a priori acquisition of a point spread function is not required for 3D imaging using a Gabor-zone plate pattern. In IDH, a polarimetric single-path self-interference interferometer is frequently adopted. Such an interferometer requires two linear polarizers. One of the polarizers is rotated for the imaging of linear polarization by IDH. However, the light-use efficiency of polarization-imaging IDH is decreased to less than one-quarter by using two polarizers [11]. This problem is not limited to polarization-sensitive IDH because most IDH systems are equipped with two polarizers.

I propose a polarization-filterless, polarization-sensitive, polarization-multiplexed, and phase-shifting IDH technique termed P⁴IDH. Polarization information is obtained without a polarizer in this passive 3D imaging technique, and its light-use efficiency is greatly improved in comparison with conventional IDH systems. P⁴IDH can be constructed with only two polarization-sensitive phase-only spatial light modulators (SLMs), an image sensor, and a computer. In P⁴IDH, object waves for the vertical and horizontal polarization directions are simultaneously recorded as polarization-multiplexed phase-shifted incoherent holograms. A phase-shifting interferometry technique is developed to extract the object waves selectively from the recorded polarization-multiplexed incoherent holograms. I show its effectiveness through experiments.

Figure 1 shows a schematic of P⁴IDH. In principle, the optical setup of the proposed IDH consists of an image sensor and two polarization-sensitive SLMs such as transmission liquid-crystal SLMs and liquid-crystal-on-silicon SLMs (LCoS-SLMs). A spatially incoherent light wave diffracted from an object propagates to the SLMs. The relevant SLMs modulate vertical and horizontal polarization components of the light wave. Each SLM includes a dual-focus lens with focal lengths of f ₁ and f ₂ and generates two object waves with different wavefront curvature radii. Four object waves are generated in total. Object waves with the same polarization directions form an incoherent hologram on the image sensor plane. Therefore, two self-interference incoherent holograms whose polarization directions are orthogonal, h_{p 1}(x,y;θ ₁) and h_{p 2}(x,y;θ ₂), are multiplexed on the sensor plane, where θ ₁ and θ ₂ are phase shifts for the respective polarization directions. The image sensor records polarization-multiplexed phase-shifted self-interference incoherent holograms H(x,y;θ ₁,θ ₂) by introducing θ ₁ and θ ₂ using the SLMs. It is worth noting that different phase shifts are introduced for different polarization directions in hologram recording to obtain object waves for orthogonal polarization directions through signal processing. Here, I set the two sets of object-wave information in a recorded polarization-multiplexed hologram as U_{p 1}(x,y) and U_{p 2}(x,y). Each set of object-wave information U_pn(x,y) (n = 1, 2) is generated as the incoherent sum of sub-holograms I_pn(x,y;r_o,θ_n) of multiple object points r_o= (x_o,y_o,z_o). Here, H(x,y) is expressed as

 

Fig. 1. Schematic of P⁴IDH. (a) Optical implementation and (b) image-reconstruction procedure.

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The optical system shown in Fig. 1 has merit to improve light-use efficiency. Here, where the linear polarizer’s transmittance along the transmission axis is α (≤1) and the transmittance/reflectivity of a transmissive/reflective polarization-sensitive phase-only SLM is ε (≤1), the light-use efficiency of IDH with a polarimetric single-path self-interference interferometer for polarization imaging [11] is α ² ε /4. The light-use efficiency of the proposed IDH system is ε ². The light-use efficiency is 4ε /α ² times improved. The improvement in the light-use efficiency is also expected in general phase-shifting IDH. By introducing the same phase shifts for the vertical and horizontal polarization directions, θ ₁ = θ ₂, general phase-shifting interferometry can be applied. In this case, in comparison to IDH with a polarimetric single-path self-interference interferometer [7–10], the proposed IDH system obtains 4ε /α ²-times-improved light-use efficiency using the same multi-step phase-shifting interferometry technique.

I conducted experiments to show the validity of P⁴IDH. Figure 2 shows a schematic and photograph of the constructed optical system of P⁴IDH. A red-light-emitting diode (LED), which was mounted in a four-wavelength LED head (Thorlabs, LED4D201), was used as a random-polarization light source. Two LCoS-SLMs (Hamamatsu K.K., X13138-01) with the modulation axis in the horizontal direction were set as polarization-sensitive SLMs. These SLMs displayed phase-shifted dual-focus Fresnel phase-lens patterns whose focal lengths were 10,000 mm and infinity. The two lenses were randomly distributed pixel by pixel on each SLM. The LED light with random polarization illuminated the object, a USAF1951 test target attached with linear polarization films, through a bandpass filter whose central wavelength and bandwidth were 625 nm and 2 nm, respectively. The object wave illuminated the first SLM and two light waves were generated from the horizontal polarization direction of the object wave. Then the light reflected from the first SLM passed through the 4f optical system and the light wave on the first SLM plane was imaged on the second SLM plane. The half-wave plate placed in the pass of the 4f optical system converted the vertical and horizontal polarization directions of the object wave. Therefore, the second SLM generated two light waves from the vertical polarization direction of the object wave. Four light waves illuminated the scientific complementary metal-oxide semiconductor (sCMOS) image sensor (Andor Technology, Zyla 4.2 plus). The image sensor recorded incoherent holograms of the vertical and horizontal polarization directions simultaneously. Seven polarization-multiplexed phase-shifted incoherent holograms were obtained in total using the two SLMs. The phase shifts in the horizontal and vertical polarizations of the object wave (θ ₁,θ ₂) were (3π/2,0), (π,0), (π/2,0), (0,0), (0,π/2), (0,π), and (0,3π/2). I initially examined the filterless polarization-imaging ability of P⁴IDH. Figure 3 shows experimental results. Figure 3(a) indicates that polarization information generally cannot be seen without a polarizer, although the object has polarimetric information due to the attachment of polarization films, as described in Fig. 3(b). With a linear polarizer, polarization information was visualized as shown in Figs. 3(c)–3(e). On the other hand, the polarization information was reconstructed without any polarizers by using P⁴IDH, as shown in Figs. 3(f)–3(h). Thus, the filterless polarization-imaging ability was experimentally confirmed. Next, I conducted an experiment to perform the simultaneous imaging of 3D and polarization information on reflective 3D objects by constructing the optical setup shown in Fig. 4(a). Two objects, an origami fan and an origami crane, were set at different depths, and a polarization film was placed in front of the origami fan, as shown in Fig. 4(b). The depth difference was 140 mm. The transmission axis of the film was the vertical direction. In this experiment, I set high-resolution LCoS-SLMs [13] to display the lens patterns (the lenses had focal lengths of 850 mm and infinity). Seven holograms were obtained with blue LEDs (Thorlabs, LED4D201) whose nominal wavelength and full width at half maximum were 455 nm and 18 nm, respectively. Figures 4(c)–4(k) and Visualization 1 show the experimental results. The results indicate that both 3D information and polarization information on the reflective 3D objects were successfully reconstructed. Thus, simultaneous imaging of 3D and polarization information was experimentally demonstrated.

 

Fig. 2. Constructed optical system of P⁴DHM. (a) Schematic and (b) photograph.

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Fig. 3. Experimental results of filterless polarization imaging. (a) Photograph of the object. (b) Polarization distribution of the object. Blue, red, and yellow areas mean horizontal, vertical, and random polarizations, respectively. Images obtained with a polarizer with the transmission axis in the (c) horizontal and (d) vertical directions. (e) Image obtained from the subtraction of (c) from (d). Reconstructed images of the (f) horizontal and (g) vertical polarization components. (h) Image obtained from the subtraction of (f) from (g).

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Fig. 4. Experimental results of simultaneous 3D and filterless polarization imaging. (a) Constructed optical system for reflective objects. (b) Photograph of the 3D objects and the polarization film. (c) One of the incoherent holograms. (d) Horizontal and (d) vertical polarization components of the reconstructed image of the origami fan. (f) Horizontal and (g) vertical polarization components of the reconstructed image of the origami crane. (h) Summation of (d) and (e). (i) Summation of (f) and (g). (j) Image obtained from the subtraction of (d) from (e). (k) Image obtained from the subtraction of (f) from (g). Blue and red colors in (j) and (k) mean + and −.

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It is expected that the light-use efficiency is improved in general IDH because no polarization filter is used. The proposed IDH system can also be exploited for general phase-shifting IDH. I conducted an additional experiment to observe whether or not the brightness and/or image quality of the recorded phase-shifted incoherent holograms is improved. Four-step phase-shifting interferometry was adopted and the same phase shifts for the vertical and horizontal polarization directions were set (θ ₁ = θ ₂) in the recording of four phase-shifted holograms. For comparison, four phase-shifted incoherent holograms were also obtained by Fresnel incoherent correlation holography (FINCH) with the spatial multiplexing technique [12,14,15]. This is because FINCH with spatial multiplexing uses only one polarizer and the resultant images are relatively bright compared with those of other IDH systems. The light-use efficiency in Refs. [12,14,15] becomes αε/2. The illumination-light intensity was gradually decreased by adjusting the exposure time and output of the LED light, and the change in the image quality was observed. Figure 5 shows the experimental results. Object images were successfully reconstructed by both IDH methods when holograms were recorded with sufficient light intensity, as shown in Figs. 5(a) and 5(d). However, the quality of the images reconstructed by the conventional IDH system degraded as the light intensity became limited. In contrast, the quality of the images obtained by P⁴IDH remained almost unchanged. This was because brighter holograms were recorded and the image quality was comparatively unaffected by random noise. Thus, the light-use efficiency was better, and clear images were reconstructed with high image quality using P⁴IDH under the condition of limited light intensity.

 

Fig. 5. Experimental results of four-step phase-shifting IDH with the optical setup in Fig. 2. (a)–(c) Images reconstructed by FINCH with spatial multiplexing. (d)–(f) Images reconstructed by the P⁴IDH system. Exposure time per recording of the phase-shifted hologram and the output of the LED were (a), (d) 920 µs and 3%; (b), (e) 96 µs and 1%; and (c), (f) 48 µs and 1%. The numerical focusing distance was 5 mm.

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I have proposed a phase-shifting IDH technique in which simultaneous imaging of 3D and polarization information is achieved without using any polarization filters and mechanically moving components. Bright holograms are obtained because no polarizers are used, and the image quality is better in comparison with that of images obtained by conventional IDH when the light intensity is limited. One bit of an image sensor is used for the multiplexed recording when polarization imaging with the proposed IDH, while the number of recordings is reduced. An additional optical component is needed for full-Stokes imaging. P⁴IDH enables us to detect both 3D and polarization information simultaneously without using polarization filters and will be useful for advanced analyses of specimens in microscopy, machine vision, camera, and healthcare in our daily lives.