## Abstract

A simple Fresnel-type self-interference incoherent digital holographic recording system is proposed. The main part of the system consists of the two linear polarizers and geometric phase lens. The geometric phase lens is employed as a polarization selective common-path interferometer. One of the polarizers is rotated by the motor and serves as a phase-shifter with the geometric phase lens, to eliminate the bias and twin image noise. A topological phase is obtained by the relative angle between the polarizer and geometric phase lens. Since this phase shifting method does not depend on the change of the optical path length, the phase shifting performance is almost constant in the broad spectral range. Using the proposed achromatic phase shifting method, a simultaneous three-color phase shifting digital hologram recording under the incoherent light source is demonstrated.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The holography is considered to be the ultimate three-dimensional (3D) recording technique, which can acquire a phase and amplitude information of the incoming wave field [1–3]. The various applications of the holography have been limited so far, since a coherent light source is required to generate an interference pattern of the incoming wavefront information. Meanwhile, the holographic recording systems under the various types of incoherent light sources are reported by employing the self-referencing technique [4–7]. Such systems are widely classified as a self-interference incoherent digital holography (SIDH). Two key components of SIDH are the wavefront modulator and phase shifter. First, the wavefront modulator divides the incoming spherical wave into two and modulates the two wavefronts differently. Please note that the two-wave interference in the SIDH does not occur between the wave that carries the object information (object wave) and the reference wave without the modulation, but between the differently modulated waves which carry the object information at the same time. There is a strong mutual coherence between wavefronts starting from the same object point at about the same time. Therefore the interference using the incoherent light source is available. Second, the phase shifting method is introduced to eliminate the bias and twin image noises which are superposed on the complex hologram information due to the nature of the interference [8]. Moving optical components in sub-wavelength units, or adjusting the optical path using retardation is a widely used phase shifting technique. The Piezo-actuator or the liquid crystal (LC) plates are the most common devices to shift the phase. The phase-only spatial light modulator (SLM) is utilized as a common-path interferometer as well as the phase shifter in the holographic system, which is often called as FINCH, that is Fresnel incoherent correlation holography [4, 9]. In this system, the quadratic phase lens pattern is displayed on the SLM. The incoming spherical wavefront, where its polarization is linear and parallel to the active axis of the SLM, is modulated by the lens on the SLM. The orthogonal component of the wave, on the other hand, is not affected by the SLM. The phase shifting is digitally achieved without any further devices, by presenting the quadratic phase profiles multiplied with the stepwise constant phase on the SLM. The optical pathway is folded once in this system because the phase-only SLM is a reflective type. On the other hand, the straight-line FINCH system is also reported by utilizing the transmissive LC GRIN lens and the variable LC waveplate [10]. Michelson interferometer consists with a beam splitter, flat and concave mirrors are used to modulate the incoming wavefront [6,11, 12]. For phase shifting, the flat mirror is driven by the Piezo-actuator configuring one of the two arms inside the interferometer. Similarly, the triangular or Mach-Zehnder interferometer is used for self-interference [5]. The SIDH systems using conventional interferometer structures only require a low-cost optics, unlike FINCH. The full-color natural light holographic camera is demonstrated with the encouraging experimental results [7].

In our previous study, the geometric phase (GP) lens is employed in the SIDH system to replace the wavefront modulators mentioned above [13]. The GP lens utilized in this study is a LC based flat lens with a thin form factor by using Pancharatnam-phase effect [14–16]. The GP lens simultaneously modulates the incoming wavefront by its positive and negative focal lengths [17–21]. The wavefront division is selective by the input polarization states. For the right-handed circularly polarized (RHCP) wave the lens serves as a convex lens, as shown in Fig. 1(a), and a concave lens for the left-handed circularly polarized (LHCP) wave (Fig. 1(b)). If the incoming wavefront is linearly polarized or unpolarized, then its power is divided almost in half and half, and both positive and negative images are formed simultaneously (Fig. 1(c)). Several methods are reported to fabricate the GP lens with high-efficiency [17, 18, 21]. The GP lens employed in our previous and present study is the off-the-shelf model from ImagineOptix [21]. The GP lens is fabricated by utilizing the analog holographic recording apparatus based on the Mach-Zehnder interferometric structure. In this fabrication system, the sample object is replaced to the lens which is used as a template. And the recording substrate is located as an analog holographic film. The recording substrate consists a substrate glass, a linear photoalignment polymer, and a polymerizable LC. The direction of LC molecules is aligned along the defined grooves formed by the linear photoalignment polymer. In the case of GP lens, the director orientations of LC molecules correspond to the quadratic phase map defined by the template lens.

The phase-only SLM is utilized to shift the phase in the previous GP lens based SIDH system. The images with the constant gray level corresponding to the 0°, 120°, 240° of phase angles are generated and displayed on the SLM. Even though the SLM is used as an early demonstration of our system, the high-end SLM is not necessary since the displayed image does not require a pixel by pixel operation.

In this paper, we present the GP lens based SIDH system, where the phase shifting is performed by the combination of the rotating linear polarizer films, GP lens, and the fixed polarizer. This optical design is simple and cost-effective, and also provides the achromatic property over the three color channels of the image sensor. The system has an entirely cylindrical symmetric single pathway along the optical axis without folding. The holograms of three color channels are simultaneously obtained with four exposures while rotating of the polarizer in sequence. Any prior knowledge or massive iterative processing is not required to initiate the system. The obtained complex-valued Fresnel type hologram is readily usable to the holographic display employing the phase-only SLM. Since both the wavefront modulation and phase shifting are performed purely by the geometric phase modulation, we would like to call this system as the geometric phase self-interference incoherent digital holography, or shortly in “GP-SIDH.” In the next section, the GP-SIDH system is briefly discussed and the phase shifting module is analyzed. In Section 3, the experimental apparatus is introduced. The achromatic performance and the target recording results are presented. The discussions are followed in the last section.

## 2. Proposed geometric phase shifter

#### 2.1. Background

The dynamic phase is usually observed by modulating the optical path length in sub-wavelength scale, whereas the geometric phase is observed without the optical path length variation. The polarization variation along the closed contour on the Poincaré sphere surface is the most representative example of the geometric phase, which is widely called as the Pancharatnam-berry phase [14,15,22]. In this case, the amount of geometric phase gain is the half of the solid angle of the cyclic path on the Poincaré sphere. Since the geometric phase is an additional phase purely obtained by the topological modulation on the Poincaré space, it is wavelength independent as long as the spectral performance of optical component is nearly constant.

The geometric phase shifters are usually operated with the combination of waveplates and the rotation angle among them. Let *P* is the linear polarizer, *Q* is the quarter-waveplate, and *H* is the half-waveplate. And any rotating part is denoted as *Q*′, for instance of rotating quarter wave plate. The optical circuit of *P* – *Q* – *H*′ – *Q* – *P* and *P* – *HQ* – (*QH*)′ – (*QH*)′ – *QH* – *P* are introduced [23], especially the latter one reveals the achromatic property. An achromatic phase shifter by rotating the linear polarizer is reported [24]. An optical circuit with the *P* –*Q* –*P*′ structure is one of the basic geometric phase shifting configuration. But to add the achromatic shifting feature on the phase shifting circuit, the *P* – *H* – *Q* – *P*′ or the *P* –*H* – *H* – *Q* – *P*′ structures are proposed and analyzed. The geometric phase shifting is introduced in the digital holographic recording. The quasi-achromatic phase shifter is implemented to achieve the full-color digital holography [25]. In this setup, the *P* – *Q* – *H*′ – *Q* – *P* structure is introduced at the input of the holographic interferometer. The systems aforementioned have the disadvantage of using multiple waveplates for an achromatic phase shift and requiring an accurate rotation angle between them.

The digital holographic systems implementing the geometric phase shifters, without mentioning the achromatic characteristics, are also reported [26, 27]. By using the *Q* – *Q′* and flat mirror circuit on the reference arm of the Michelson interferometer, which is equivalent to *Q* – *Q′* – *Q′* – *Q* circuit, the geometric phase shifting digital holographic system is reported [26]. The coherent light source is used in this system, but this system is expected to apply to the Michelson interferometer based SIDH system as well, such as the system of Kim in [7]. The SIDH system using geometric phase shifting method is also reported based on the FINCH configuration [27]. The optical circuit, in this case, is equivalent to the *P* – *Q* – *P′* structure. This study achieves the single-shot holography employing the parallel phase shifting method, where the *P′* is rotated not in temporal region but spatially divided by employing the micro-polarizer array [28].

#### 2.2. GP-SIDH

The GP-SIDH system setup employing the proposed phase shifting method is as shown in Fig. 2. Unlike the previous system setup which uses the reflective type SLM [13], all of the components in the present setup are aligned along the straight line without the folding of optical pathway or tilting. The object is regarded as a group of incoherent point light sources either by scattering or self-illuminating. A general lens collects the incoming spherical wavefront ensembles. A rotating linear polarizer and the GP lens are located in order, to serve as the geometric phase shifter. Please note that, even if the arrangement order of the GP lens and rotating polarizer is reversed, the phase shift due to the rotation of the polarizer is still observed. However, when the polarizer is behind the GP lens, a small amount of lateral displacement of interference fringe pattern due to the rotation of polarizer occurs [16]. After the GP lens, the beam is simultaneously divided into both converging and diverging waves. The resultant wavefront curvatures for both waves are large. Thus, the interference pattern after the GP lens is hard to observe. A relay lens is therefore employed to enhance the fringe pattern visibility by flattening the wavefronts. Finally, the fixed linear polarizer is located, then the image sensor follows. The two-wave interference is recorded by the image sensor, each of which has a different wavefront curvature.

The geometric phase gain according to the rotation angle Ω of the linear polarizer is analyzed using the Jones matrix calculation. To simplify the model, consider the linearly polarized wave after the first polarizer. Then the polarization states after the GP lens are the *RHCP* and *LHCP* states with equal amplitude, each of which is exp (*i*Φ)[1, *i*]* ^{T}* and exp (−

*i*Φ)[1, −

*i*]

*, respectively. Here, ±Φ is the phase modulation results due to the GP lens, which is the quadratic phase profile of positive and negative lens, and*

^{T}*T*is the transpose of the matrix. These beams propagate to the linear polarizer with arbitrary angle Ω, which is represented as [cos

^{2}Ω, cos Ω sin Ω; cos Ω sin Ω, sin

^{2}Ω] by Jones matrix representation. Then the horizontal and vertical components of the field are represented as Eq. (1).

Therefore, as the linear polarizer is rotated with angle Ω, the resultant intensity is expressed with the geometric phase gain with the amount of the twice of the rotation angle, Ω. In four-step phase shifting method, the parameter Ω is set to 0°, 45°, 90°, and 135° consecutively, which corresponds to phase shifting of intensity images in 90° step. The final hologram *U _{H}* without bias and twin image noise is obtained as Eq. (3), where

*I*is the consecutive intensity images each rotation of polarizer in Ω = 45°.

_{k}The same result is described by the topological analysis on the surface of the Poincaré sphere. The incoming wave is linearly polarized by the polarizer with an angle *δ*. The polarized beam with arbitrary rotated angle *δ* is expressed on somewhere on the equator of the Poincaré sphere, as labeled as point *A*_{1} on Fig. 3. Then this wave is divided into orthogonal circularly polarized waves while passing through the GP lens. Theses simultaneous topological conversions are marked as point *R* and *L* on both poles of the sphere. Please note that these divided twins have a mutual coherence even though they are originated from the incoherent light source. Then after the last fixed polarizer, two beams in the states of *R* and *L* are converted in the linearly polarized state again, which is labeled as point *B* on the equator of the sphere. The solid angle of the closed path *A*_{1}*RLB* is 4*δ*, and therefore the amount of geometric phase is 2*δ*. When the polarizer is rotated with an angle of Ω, where it is labeled as *A*_{2} in Fig. 3, the geometric phase gain is 2(*δ* + Ω). The phase difference 2Ω between the path *A*_{1}*RLB* and *A*_{2}*RLB* is utilized in the geometric phase shifter.

## 3. Experiments

#### 3.1. Experimental setup

Figure 4 shows the photograph of the proposed system. The focal length of 100 mm convex lens is used as an objective lens. Before the GP lens, the linear polarizer film is located. This polarizer is attached and rotated by the micro servo-motor (SG-90) which is controlled by the Arduino Uno board. The GP lens is placed from the 75 mm behind the objective lens. The GP lens in this system is the off-the-shelf model as described in the first section. The focal length of the GP lens is 100 mm at 525 nm wavelength. Both diverging and converging beams are collected and relayed by the following Nikon 50 mm f/1.4 manual focus lens. The lens is located about 55 mm after the GP lens with reversed configuration. The thickness of the lens is about 41 mm. This lens is employed to make the interference fringe radius as large as possible on the image sensor plane, but also to maintain the optical path length as short as possible in current condition. The fixed linear polarizer and image sensor are located in order. The distance between the end of the relay lens and the image sensor is about 113 mm. Therefore, the total optical pathway length required for this holographic setup is about 270 mm.

The color USB3.0 camera is used in the system (GS3-U3-60QS6C-C, FLIR). The pixel number is 2736 in horizontal and 2192 in a vertical direction. The pixel pitch is 4.54 *µm*. The 14-bit pixel depth mode is employed for hologram recording. According to the data sheet from the manufacturer, the central wavelengths of each red, green, and blue channel are estimated to 610, 535, 460 nm, respectively. Intensity information of three-color channels are recorded simultaneously per single exposure, and four images are captured in succession as the polarizer rotates.

#### 3.2. Demonstration of achromatic property

First, the achromatic performance of the proposed phase shifting configuration is demonstrated. The demonstration system is similar to the configured GP-SIDH as shown in Fig. 4, but the image sensor is placed where the GP lens image locates. Because the image of the GP lens provides high contrast Fresnel ring patterns as shown in Fig. 5(a), which is useful to observe a phase shifting effect clearly. The system is illuminated by the white light emitting diode (MNWHL4, Thorlabs). The bandpass filters are employed to limit the spectral range for the precise intensity measurement as possible. The central wavelengths of the filters are 425, 550, 650 nm, respectively, and the spectral bandwidth is equally 25 nm. In this experiment, the monochromatic recording mode of the image sensor is utilized, unlike the full-color recording in Section 3.3 and 3.4. The averaged intensity values of the central four pixels which are in the central part of the zero-order Fresnel zone pattern area are recorded in sequence while rotating the linear polarizer with the 1° step. The motor used herein to rotate the polarizer is SG-90 micro-servo motor, driven by the transmitted pulse width from the Arduino Uno board. The calibration process is operated in advance to evaluate the performance of the motor. The bandpass filter with 550 nm central wavelength is utilized in the calibration process. The initial input pulse width to the intensity variation data is as shown in Fig. 5(c), which resembles a sinusoidal curve, but reveals a thresholding level until the actuation and a slight horizontal distortion. One cycle of the sinusoidal curve is cropped, then the input signal levels are manually adjusted to follow the ideal curve. After the calibration, the intensity measurements for three different bandpass filters are operated in turn. The results are shown in Fig. 5(d), where the sinusoidal curves with the 0 to 2*π* range are observed in every input wavelength.

As a control, the phase shifting measurement using the phase-only SLM (Pluto, Holoeye) is performed, employing the system of the previous study [13]. The SLM is initially calibrated to modulate 0 to 2*π* phase angles of incoming 633 nm wavelength of light. The gray input levels from 0 to 255 are displayed on the SLM in turn, instead of rotating the polarizer. As shown in Fig. 5(d), phase modulation performance of the phase-only SLM is not stable over the various wavelength, which implies that the device setting is always required as the input wavelength is changed. Therefore, to acquire a three-color in-line hologram by using the color image sensor, 3 × *k* exposures are required where *k* is the phase shifting steps, in the system using such wavelength dependent phase shifting devices. Whereas the holographic systems with the achromatic geometric phase shifter, including the proposed method, can simultaneously obtain the three-colors by the color image sensor without the modification of phase shifting device settings for different colors.

#### 3.3. Three-color superposed reconstruction

The three color images should be superposed together with the proper reconstruction distances to reconstruct the full-color image. In the proposed system, the wavelength dependency of the GP lens is particularly problematic. Therefore, the reconstruction distances of each color channels are calculated differently from the object distance and then superposed into a single-color image. Otherwise, in the case of superposing each color images of the same reconstruction distance, the correct full-color image is not obtained as shown in Fig. 7(a). The focal length of the GP lens is dependent on the incoming wavelength, *λ _{in}*, with Eq. (4), where

*λ*is the wavelength that used in the fabrication of GP lens, and

_{ref}*f*is the corresponding focal length [29].

_{ref}A numerical reconstruction distance is derived from the *f _{gp}* which is calculated as Eq. (4). The parameters are labeled as in the Fig. 6. First, the imaging distance after the objective lens and GP lens is calculated as Eq. (5), where ${d}_{gp}^{+}$ is the imaging distance due to the positive focal length of the GP lens, and

*d*${d}_{gp}^{-}$ is vice versa.

Then, the distance after the secondary principal plane of the relay lens is derived as Eq. (6).

Finally, the numerical reconstruction distance *z _{rec}* is calculated as Eq. (7), where $\pm \mathrm{\Delta}{d}_{rl}={d}_{rl}^{\pm}(\lambda )-{d}_{rl}^{\mp}(\lambda )$.

The four-step phase shifting method is utilized in this recording process. The required phase shifting angles are 0°, 90°, 180°, and 270°. Therefore, the servo motor is calibrated to operate successive 45° stepwise rotation. The four captured images of each color channels are extracted, then processed as Eq. (3) to obtain a single complex hologram, *U _{H}*

_{(λ)}. The reconstructed wave field

*U*on (

_{R}*x, y*) domain is retrieved by Fresnel transformation under the wavelength

*λ*, by convolving the complex hologram

*U*on (

_{H}*ξ, η*) domain with the quadratic phase parameterized by the reconstruction distance

*z*(

_{rec}*λ*), which is Eq. (8).

The each reconstructed monochrome images are superposed into a single full-color image. In this system, the resolution of 2192 by 2192 pixels of 16-bit three-color channel complex hologram data is processed to yield a three-color intensity image, which consumes about 6 seconds in MATLAB with single core calculation (3.1 GHz Intel Core i5, 8 Gb memory).

The recording and digital reconstruction of the letter ’E’ are performed to demonstrate the significant differences between the general reconstruction and the object distance based estimation, which is shown in Figs. 7(a) and 7(b), respectively. The white LED light is illuminated backward of the negative target, where the diffuser sheet is attached behind. Due to the wavelength dependency of the focal length of GP lens, the reconstruction distances of each color are different (Fig. 7(c)), which is calculated using the Eq. (7), where the input parameters are as described in the experimental setup. By estimating the reconstruction distances of each color from the object distance and the wavelength dependent GP focal length, the best of focus of the retrieved hologram images are almost clearly superposed (Fig. 7(b)). The minute discrepancy due to the wavelength dependency of the sampling width is recovered by following the zero padding method [30,31].

#### 3.4. Target recording

Various three-dimensional information is recorded and reconstructed, like the system shown in Fig. 8. First, the recording of two transmissive targets is demonstrated in Fig. 9, each of which is illuminated by the white LED (MNWHL4, Thorlabs). Two sample targets are located on each side of the beam combiner, to mimic the volumetric two layer target. The negative National Bureau of Standards (NBS) 1963 chart with 4.5 cycle/mm region and the United States Air Force (USAF) 1952 resolution chart with group 1 region are employed as the targets. The diffuser sheet is attached to each target to express the self-luminous appearance. The distance between two targets is about 500 mm. The NBS 1963 chart is located at the focal plane of the objective lens, and the other one is located further.

Second, the three-dimensional reflective object is recorded. The reconstruction results are presented in Figs. 10(d)–10(f). The objects are located about 500 mm from the system input lens to record the object with a wider field of view. To illuminate a wider area, two halogen lamps (DC-950, Dolan-jenner) are employed additionally. To present a clear image output, 20 complex holograms are continuously recorded and averaged. Therefore 80 exposures are carried out. The total time consumed is about 30 seconds. The single exposure time is 0.25 second. A tapered cosine window (also known as Tukey window) is applied as an apodization filter to the complex hologram, to reduce the edge ringing effect as shown in Figs. 10(b) and 10(c) [32]. The filter equation is as described in Eq. (9), for one-dimensional case.

In Eq. (9), the parameter *x* is a sampling point, and *r* is the ratio of the tapered length compared to the total window length, which is set to 0.1 in the resultant image (Fig. 10(b)). The image is then cropped to 1900 pixels on both sides.

## 4. Discussions

The features of geometric phase shifting method herein are advantageous compared to any other techniques which mostly depending on the optical path length modulation. Firstly, because the optical path length is not modulated for phase shifting, the phase shifting performance is almost the same over the broad range of wavelength spectrum. This achromatic property brings a significant advantage to in-line digital holographic recording system, especially to achieve a full-color recording. Because the calibration process of the optical path length modulation range or the retardation range for phase shifting under the different wavelength is no longer required. Secondly, the only required optomechanical device for phase shifting is the rotating linear polarizer attached to the simple stepping motor. This combination is simple to implement and cheap, compared to widely used phase shifting devices such as Piezo-actuator, or SLM. A simple LC cell is a good candidate to replace the rotating polarizer to avoid any moving parts in the system. Lastly, owing to the all components required in this method are a transmissive-types, the holographic recording system is devised in in-line cylindrical symmetric structure, without any beam splitters or mirrors. Therefore, the system is simply aligned and has a higher light efficiency.

The further developments are required to optimize the proposed GP-SIDH system. The full-color reproduction quality from the captured hologram should be enhanced for the natural color holography. Currently, the images on Figs. 10(d)–10(f) are manually processed by the image software to provide a similar color appearance with the original objects. However, the color appearance without any post-processing is irregular and unnatural over the axial reconstruction distances, as shown in Visualization 2. The optimization of the numerical aperture of the system is required to build an almost diffraction-limited system, which is not sufficient for the current system due to the relay lens. Also, the study about the reducing the optical path length of the system should be carried out. The system without the relay lens is considered to dramatically decrease the required path length, which provides a simpler structure as well as the precision in system modeling.

## 5. Conclusion

The GP-SIDH system that is the incoherent holographic system purely operated by the geometric phase modulation is proposed. The GP lens is employed as a common path interferometer. The rotating polarizer is located in front of the GP lens to shift the phase by collaborating with the GP lens. Since the configured phase shifting system does not depend on the change of optical path length, the phase shifting performance is stable over the visible spectral range. The red, green, and blue channel holograms are simultaneously obtained by the color image sensor with the four step phase shifting method. The GP-SIDH system proposed herein is highly cost-effective and straightforward to build. The system operates without any prior knowledge. The essential components are light weighted and have a flat structure. Therefore, with further optimization and customization, the realization of the portable size holographic system is expected.

## Funding

National Research Foundation of Korea (501100003725) 2018R1A2B6005260

## Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (2018R1A2B6005260)

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