Filter-free lens-free polarimetric incoherent digital holography

I propose an incoherent digital holography (IDH) technique in which four-dimensional (4D, three-dimensional (3D) coordinates and polarization) information is simultaneously obtained using neither polarization filters nor lenses. A filter-free lens-free self-interference incoherent interferometer for 4D imaging is designed and developed. Four-dimensional (4D) information is multiplexed in recorded phase-shifted incoherent holograms and extracted by polarization-selective phase-shifting interferometry. The validity of the proposed holography for multiplexed 4D imaging is experimentally demonstrated by the constructed filter-free lens-free self-interference IDH system and a randomly polarized light-emitting diode.


Introduction
Light contains multiple physical quantities.Physical quantities such as the amplitude, phase, wavelength, and polarization of light provide multidimensional information such as the transmittance and/or reflectance, three-dimensional (3D) coordinates, refractive index, and spectral and polarimetric properties of an object, phenomenon, or scene.The polarization of light contains valuable information on the measured object.By measuring the state of the polarization, stresses induced in plastic pieces are visualized and the orientations of polymers such as a liquid crystal, the muscle fibers in a biological specimen, and the birefringence of a sample are measured without labeling [1][2][3][4].The thickness of a film is measured, the distribution of a forest is identified, and the underwater view of a river is observed clearly, exploiting the reflectance difference between the horizontal and vertical polarizations [5,6].Molecule structures are estimated by detecting the polarization Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
of fluorescence in single-molecule fluorescence imaging [7].The combination of a polarization measurement technique and a 3D image sensing technique is required for simultaneously measuring the polarization information of objects located at different depths.
Digital holography (DH) [8,9] is a 3D image sensing technique for digitally recording an interference fringe image of an object-such an image is called a digital hologram-and reconstructing the 3D image of the object from the recorded hologram.Polarization imaging with DH has been actively researched, many optical systems have been developed [10][11][12][13][14][15], and its application to detect microplastics has been proposed [16][17][18][19].DH usually requires a laser light source to generate interference fringes.However, holography can be implemented with spatially incoherent light [20][21][22][23], and spatially incoherent holography is combined with DH, termed incoherent DH (IDH) [24][25][26][27][28]. IDH enables DH using dailyuse light and 3D imaging with incoherent light at the frame rate of an image sensor has been achieved [29][30][31].Holographic fluorescence microscopy [32][33][34], natural light holography [35], and portable hologram recorders on a moving body [36,37] have been developed.Research on polarimetric 3D imaging has also been conducted with IDH, but polarization filters were generally required [38].As a result, light-use efficiency is limited and a mechanically moving component is required to record polarization information.In another way, polarization-filterless polarimetric IDH was proposed.In the polarimetric IDH technique, polarization information is simultaneously recorded by exploiting the holographic multiplexing and is selectively reconstructed by applying the developed polarization-selective phase-shifting interferometry (PSI) [39,40], which exploits a holographic multiplexing scheme [41][42][43].However, many lenses were required in the constructed optical setups and the size of the setup was enlarged.
In this article, I propose IDH requiring neither polarization filters nor lenses for 3D and 4D (3D + polarization) imaging.A filter-free lens-free self-interference incoherent interferometer for 4D imaging is designed and developed.In the developed interferometer, spatial light modulators (SLMs), two polarization beam splitters, and polarization-selective PSI [39,40] that is based on a holographic multiplexing scheme [41][42][43] are exploited for filter-free lens-free 3D and 4D imaging.An optical setup is constructed and experimental demonstrations for 3D and 4D imaging are conducted.The validity of the proposed IDH is experimentally demonstrated for the first time.

Filter-free lens-free polarimetric incoherent digital holography (FLP-IDH)
Figure 1 illustrates a schematic of the proposed FLP-IDH.In the filter-free lens-free self-interference incoherent interferometer, horizontally and vertically polarized object waves are separated from an incoherent object wave by a polarization beam splitter.A polarization beam splitter is sometimes used as a polarization filter, but I use two polarization beam splitters as the polarization separator and combiner.Neither horizontally nor vertically polarized object waves are discarded by the polarization beam splitters in FLP-IDH and thus the splitters do not work as filters in FLP-IDH.The respective polarized object waves are modulated by SLMs, and two object waves with different wavefront curvature radii are generated for horizontal and vertical polarization directions.When using liquidcrystal-on-silicon SLMs (LCoS-SLMs), phase-shifted phase masks adopting spatial multiplexing [44] are introduced to respective SLMs to generate two waves and phase shifts.Each SLM works as both a two-wave generator and a phase shifter.Note that not only polarization-sensitive SLMs, such as LCoS-SLMs, but also nonpolarimetric phase-only and amplitudetype SLMs including digital micromirror devices can be adopted in FLP-IDH because polarized object waves are extracted by one polarization beam splitter.The other polarization beam splitter combines the polarized four object waves, which simultaneously illuminate the image sensor.The object waves whose polarization directions are aligned interfere with each other, and two self-interference incoherent holograms whose polarization directions are orthogonal, h p1 (x,y;θ 1 ) and h p2 (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 five polarization-multiplexed phase-shifted incoherent holograms H(x,y;θ 1 ,θ 2 ) by changing the phases θ 1 and θ 2 .Different phase shifts for h p1 (x,y;θ 1 ) and h p2 (x,y;θ 2 ) are introduced in the recording procedure to obtain object-wave information with orthogonal polarization directions separately in the image-reconstruction procedure.As an example, the phase shifts (θ 1 ,θ 2 ) = (0,0), (−π/2,0), (π/2,0), (0,−π/2), and (0,π/2) are set.Here, I set the two sets of object-wave information in a recorded polarizationmultiplexed hologram as U p1 (x,y) and U p2 (x,y).Each set of object-wave information U pn (x,y) (n = 1,2) is generated as the incoherent sum of subholograms 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 where h 0thn (x,y) and I 0thn (x,y) are the 0th-order diffraction waves of h pn (x,y;θ 1 ) and I pn (x,y;r o ,θ n ), respectively; i is the imaginary unit; C.C. n and c.c. n are the complex conjugates of U pn (x,y) and the second term of equation ( 3), respectively; C(r o ), C ′ (r o ), C ′′ (r o ), and C ′′′ (r o ) are coefficients; z 1 is the depth difference between an object point and the SLMs; , where λ is the wavelength of light; ; * indicates a convolution; z 2 is the depth difference between the SLMs and the image sensor; is the magnification of the IDH system; and Object-wave information is extracted from polarizationmultiplexed phase-shifted incoherent digital holograms by polarization-selective PSI [39,40] using θ 1 and θ 2 , which is the extension of a holographic multiplexing scheme to obtain 3D and wavelength information simultaneously [41][42][43].Polarization-selective PSI extracts U p1 (x,y) and U p2 (x,y) from the recorded five incoherent holograms by changing either θ 1 or θ 2 in each phase shift [40] as follows: ( In comparison to sequential recording of three phaseshifted holograms along the orthogonal polarization directions, the number of exposures is reduced from six to five, and neither linear polarizers nor mechanical shutters are required to obtain polarization information by applying polarizationselective PSI.As a result, speeding up the measurement, improvement of the light-use efficiency, and removal of mechanically moving components to record polarization information are achieved.Diffraction integrals are calculated for U p1 (x,y) and U p2 (x,y), and object waves focused on arbitrary depth planes are reconstructed.The focused images and polarization information of the objects placed on different depth planes are obtained.Therefore, filter-free lens-free holographic 4D imaging based on IDH is achieved.
Figure 2 illustrates the optical implementation adopted for the experimental demonstrations, which exploits LCoS-SLMs.An LCoS-SLM usually has a rectangle shape and modulates the phase distribution of horizontally polarized light.As shown in figure 2(a), a half-wave plate is set in front of one of the LCoS-SLMs to equalize the aperture size on the phasemodulation plane.Considering the incident angle dependence of the LCoS-SLMs and the light-use efficiency of the IDH system, triangular optical paths are adopted.Upon the construction of this implementation, each incident angle to each LCoS-SLM is set to be about 10 • to suppress both the decrease of the phase-modulation amount and the increase of unmodulated light of the LCoS-SLMs.Figures 2(b)-(d) show the phaseshifted phase masks containing the phase distributions of two spherical waves with focal lengths f 1 and f 2 , which is generated by using space-division multiplexing.Each phase mask generates two object waves with different wavefront curvature radii.The phase-shifted phase masks shown in figures 2(b)-(d) are sequentially displayed on respective SLMs.Different phase shifts for orthogonal polarization directions are introduced using respective SLMs. Figure 2(e) shows that a phase mask contains the phase distributions of a spherical wave with f 1 = 850 mm and a plane wave (f 2 = ∞) pixel by pixel.The numbers of pixels for respective phase distributions can be adjusted, and I usually set the ratio of the numbers of pixels to be 1:1.The intensity ratio is not adjusted rigorously this time, but can be changed flexibly.In figures 2(b)-(d), the phase of the spherical wave with f 1 = 850 mm is shifted.Two incoherent holograms h p1 (x,y;θ 1 ) and h p2 (x,y;θ 2 ) are spatially multiplexed on the image sensor plane and each optical axis is aligned by an optical adjustment.

Experiments
I conducted experiments to demonstrate FLP-IDH by constructing the optical setup illustrated in figure 2. Initially, I conducted an experiment to investigate the filter-free lens-free 3D imaging ability of FLP-IDH without blurring due to misalignment.This is because the horizontally and vertically polarized components of the object wave are numerically reconstructed at different 3D positions if an alignment error occurs between triangular optical paths.In this experiment, I set a nonpolarized 3D object and recorded three holograms using the IDH system shown in figure 2 to apply traditional three-step PSI.A blue light-emitting diode (LED), which was mounted in a four-wavelength LED head (Thorlabs, LED4D201) and had the nominal wavelength and full width at half-maximum of the wavelength bandwidth of 445 nm and 18 nm, respectively, was used as a spatially incoherent random-polarization light source.Two (Sony Semiconductor Solutions Corporation [45],) with the modulation axis in the horizontal direction were set as SLMs.The pixel pitch and the number of pixels of the LCoS-SLMs were 4.25 µm and 1920 × 1080, respectively.These SLMs displayed the phase-shifted phase distributions of two spherical waves whose focal lengths were 850 mm and infinity, which are shown in figures 2(b)-(d).The two spherical-wave components were randomly distributed pixel by pixel on each SLM.The distances from a polarization beam splitter to an LCoS-SLM and from the SLM to the other polarization beam splitter were 220 mm and 270 mm, respectively.A scientific complementary metal-oxide semiconductor image sensor (Andor Technology, NEO 5.5) was used.The pixel pitch and the number of pixels of the image sensor were 6.5 µm and 2560 × 2048, respectively.The stripe pattern of group 0, line 4 in a negative USAF1951 test target and an aperture (SigmaKoki, IH-36) of 2 mm diameter were set with an 82 mm depth difference, and these objects were regarded as a 3D object as shown in figure 3(a).LED light with random polarization illuminated the object, and three phase-shifted incoherent holograms with the phase shifts (−π/2, −π/2), (0,0), and (π/2,π/2) were recorded using the  LCoS-SLMs.The exposure time per recording of an incoherent hologram was 4 ms.Figures 3(b)-(f) show experimental results.Figures 3(b)-(d) show totally blurred incoherent holograms that were recorded.After recording three phase-shifted holograms and applying the image reconstruction procedure, the images numerically focused on the respective object planes were reconstructed as shown in figures 3(e) and (f). Figure 3 indicates that the focused images of the stripe pattern and the edge of the aperture were successfully reconstructed despite h p1 (x,y;θ 1 ) and h p2 (x,y;θ 2 ) being multiplexed, and the object waves for the horizontal and vertical polarization directions were simultaneously reconstructed.It was confirmed from the results that filter-free lens-free 3D imaging was successfully performed.
I conducted the experimental demonstration of FLP-IDH for filter-free lens-free 4D imaging with IDH.I set objects shown in figure 4 for the experimental demonstration.An aperture of 5 mm diameter and horizontally and vertically polarized films with markers were set at different depths, and these objects were regarded as a polarimetric 3D object.The depth difference between adjacent objects was 20 mm as shown in figure 4(a).Figure 4(b) indicates a usual camera captures only an intensity image and cannot identify the states of the polarizations of the polarimetric objects.Five polarization-multiplexed phase-shifted incoherent holograms with the phase shifts (π,0), (π/2,0), (0,0), (0,π/2), and (0,π) were recorded using the LCoS-SLMs.The exposure time per recording of a polarization-multiplexed incoherent hologram was 5 ms.The in-plane field of view (FOV) of the constructed FLP-IDH system was 8.5 × 8.5 mm 2 .The other experimental conditions were the same as those in the previous experiment.Figure 5 shows the experimental an aperture and (g), (j), (m), (p) horizontally and (h), (k), (n), (q) vertically polarized films were set.Two lines and one line were drawn in black ink on respective films as markers to show numerical refocusing.Blue and red colors of (o)-(q) mean horizontal and vertical polarizations are stronger than vertical and horizontal polarizations.
results.Figures 5(a)-(e) indicate that a totally blurred and polarization-multiplexed incoherent hologram were recorded.After recording five holograms and applying the image reconstruction procedure, the images numerically focused on the respective object planes in the horizontal and vertical polarization directions were reconstructed as shown in figures 5(f)-(q).Experimental results indicate that the polarized films whose transmission axes were orthogonal with each other were identified and their numerical refocusing was successfully achieved.This is because horizontal and vertical polarizations were identified by polarization-selective PSI and 3D information was retrieved by diffraction integrals.Figure 5 shows that the 3D and polarimetric information of the 3D object was successfully reconstructed by FLP-IDH.Numerical propagation distances for the aperture and horizontally and vertically polarized films were 460, 500, and 550 mm.The depth differences in numerical propagations between neighboring objects were 40 mm and 50 mm, while the physical depth difference between neighboring objects was 20 mm.This is because there is a nonlinear relationship between the physical depth and numerical propagation distance [30,46].Figure 5 also indicates that the focused images of the lines drawn on the polarization films as markers and the edge of the aperture were successfully obtained by numerical refocusing.Thus, the filter-free lens-free incoherent 4D imaging ability of FLP-IDH was experimentally demonstrated.

Discussions and conclusion
I discuss the ways (1) to increase the FOV for obtaining 4D information of a variety of 3D objects, (2) to obtain quantitative phase information of an object, and (3) to retrieve the polarization information of various polarimetric objects.A wide FOV is required to measure a variety of 3D objects.The FOV depends on the sensor size and the magnification M. To decrease M, z 2 should be decreased.In the experiments z 2 was more than 500 mm and it was difficult to enlarge the FOV.Therefore, to make the optical setup compact is important for improving the FOV and obtaining 4D information of a variety of 3D objects.Second, the constructed FLP-IDH system is based on self-interference holography, and it is difficult to obtain the quantitative phase information of an object.It is valid for quantitative phase imaging to adopt self-reference holography [47][48][49][50][51][52][53][54][55][56][57][58].Quantitative phase imaging of an object can be achieved by merging the current experimental design and self-reference holography.Finally, it is important to image various states of polarizations.On the present version 3D information of horizontal and vertical polarizations is visualized.It is difficult to distinguish 45 • and 135 • polarizations in the present configuration with a randomly polarized illumination.However, additional polarization information can be obtained after a variable birefringent phase modulator is inserted between the object and a polarization beam splitter.As an example, the S2 component of the Stokes parameter of an object is obtained, after both setting the variable birefringent phase modulator whose fast axis is inclined 22.5 • from the vertical axis of the polarization beam splitter and introducing π phase shift to the birefringent phase modulator, Circular polarization information (S3 component of the Stokes parameter) of an object is obtained after both setting the variable birefringent phase modulator whose fast axis is inclined 45 • from the vertical axis of the polarization beam splitter and introducing π/2 phase shift to the modulator.Detailed analysis of polarization information is important and will be conducted without any polarization filters.
I have proposed totally FLP-IDH for incoherent 4D imaging.A filter-free lens-free self-interference incoherent interferometer for 4D imaging is designed and developed.Incoherent 4D imaging with holographic multiplexing was experimentally demonstrated by constructing an optical setup.As Choi et al proposed and demonstrated motion-picture IDH with sequential PSI and two electrically controllable birefringence-mode liquid crystal cells [59], real-time IDH can be conducted by PSI using high-speed phase shifter(s).FLP-IDH 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, cameras, and healthcare in our daily lives.

Figure 2 .
Figure 2. Optical implementation of FLP-IDH, which was constructed for the experimental demonstrations.(a) Schematic of the optical setup.Phase-shifted phase masks with phase shifts of (b) −π/2, (c) 0, and (d) π/2.(e) Magnified image of an area of (d).

Figure 4 .
Figure 4. Objects for experimental demonstration of filter-free lens-free 4D imaging.(a) Top view and (b) front view.Multiple objects placed on different depths and having different polarimetric properties are regarded as a polarimetric 3D object.