Phase shifting digital holography (PSDH) has proved to be advantageous compared to conventional digital holography (DH) in many ways. PSDH permits the usage of an on-axis configuration for recording but at the same time gets rid of the overlapping dc and conjugate images, hence utilizing the full CCD bandwidth [1]. It also enables the possibility of surface profile measurements for three-dimensional (3D) objects [2]. The key point here is the phase shifting, which is achieved through an optical path length (OPL) change in one or both the arms of the system, most of them inherited from interferometry. There are many PSDH techniques reported so far, where the phase shift is realized through piezo-electric transducer (PZT), tilting glass plate, acousto-optical modulator, moving grating, liquid crystal spatial light modulator (LCOS-SLM) and rotating waveplates or polarizers. The simplest method is by fixing a PZT to the reference mirror that produces OPL change by path length change itself [1,3,4]. Here, the speed and accuracy of the PZT limits the measurement efficiency. OPL changes can also be achieved by refractive index changes using the ordinary and extraordinary refractive indices in wave plates, which is called polarization phase shifting. Here the beams in both the arms are orthogonally polarized and pass through wave plates and polarizers. A rotation of one element by $\theta$ deg will introduce a corresponding phase shift between the sample and reference arm signals along a particular direction [5]. Rotating a polarizer is very simple compared to moving a PZT, but both suffer from the presence of moving parts which hinders the accuracy of measurements. To avoid any moving parts, the phase shifting can be done by using LCOS-SLM [6,7] or a digital micromirror device [8]. But still all the above mentioned methods make phase shifts in a time sequential manner, which makes them unsuitable for measurements on a dynamic sample. With the recent availability of polarization mask arrays, the phase shifts can be done in parallel by spatial multiplexing (with some sacrifice in spatial resolution) [9]. The polarization phase shifting technique (which is wavelength dependent) has also been improved to accommodate multiple wavelengths in DH experiments using achromatic phase shifters [10,11]. All the above mentioned techniques have a common feature in that the phase shift is achieved through OPL changes, which is generally referred to as dynamic phase ${\varphi }_D$ [12]. In this Letter, we report a phase shifting technique in digital holography that does not depend on dynamic phase but purely on geometric phase.

Berry in 1984 [13] showed that the wavefunction of a quantum system can acquire an additional phase factor known as geometric phase when the system is taken around a circuit in parameter space. Later, the geometric phase was measured by Chyba et al. using a Michelson interferometer [14]. Hariharan et al. then showed that the geometric phase ${\varphi }_G$ shows up in addition to the dynamic phase, which defines the total phase as ${\varphi }_{\text{total}}={\varphi }_D+{\varphi }_G$ [12]. Using white light, he also proved that change in geometric phase is independent of wavelength and is not the same for dynamic phase. To the best of our knowledge, a PSDH system that works on geometric phase only has not been reported. In this Letter for the first time we use the concept of pure geometric phase in Michelson interferometer geometry to realize PSDH experiments. Here we report two experiments to verify the imaging and surface shape measurement capabilities of geometric phase shifting digital holography (GPSDH). The following three major advantages can be expected from a GPSDH: (1) It is not necessary to strictly maintain orthogonal polarization states in both the arms, which gives the freedom to use samples with small amounts of polarizing property. The other PSDH methods add up phase with polarization change, whereas the proposed method only results in fringe contrast change. (2) The phase shift is independent of OPL and wavelength, which ensures more control and stability for experiments with broadband light, which offers only a small coherence length to work with. (3) The system has fewer elements, which makes it simple and cost efficient compared to other PSDH techniques, especially when using the super achromatic wave plates. The other advantages of geometric phase holography are reported by Escuti et al. [15]. We first explain the working principle of the method and then experimentally verify its potentials and applicability as a PSDH technique.

Figure 1 shows the typical experimental geometry in which two quarter wave plates (${\mathrm{QWP}}_1,{\mathrm{QWP}}_2$) are kept in one arm of the Michelson interferometer, similar to the one used by Chyba et al. [14]. In the other arm we keep the object. The light from a He–Ne laser is spatially filtered, collimated using a lens (${\mathrm{L}}_1$), and then passed through a polarizer (${\mathrm{P}}_1$), where it becomes linearly polarized. This linearly polarized beam is divided equally into both the arms by a nonpolarizing cube beam splitter. The first QWP (${\mathrm{QWP}}_1$) is fixed in the reference arm so that it makes an angle of 45° with the polarizer (${\mathrm{P}}_1$), and the second QWP (${\mathrm{QWP}}_2$) is placed behind ${\mathrm{QWP}}_1$, which is free to rotate.

 

Fig. 1. Schematic of the optical system used for PSDH measurements.

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If $E_x,E_y$ represents the $x$ and $y$ components of electric field vectors which are mutually orthogonal and are perpendicular to the direction of propagation of the reference beam, then the intensity can be written as Eq. (1) [16]:

 

Fig. 2. Closed path ABCDA represents the polarization states of the beam traveling back and forth on the reference arm, on the Poincaré sphere.

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Let the fully polarized reference beam leaving the beam splitter be represented by point $A$ in the Poincaré sphere. After passing through ${\mathrm{QWP}}_1$, the light becomes right circularly polarized, which is represented by point $B$ in Fig. 2. The beam then passes through ${\mathrm{QWP}}_2$, which is free to rotate through any angle $\psi$. Now, the beam becomes linearly polarized and reaches the point $C$, where the position of $C$ on the equator depends on the rotation angle $\psi$. After reflection from the mirror, the ray retraces its path through quarter wave plates and reaches point $A$, forming a closed loop, as shown in Fig. 2. The solid angle subtended by the closed loop on the Poincaré sphere is given by $({\varphi }_2-{\varphi }_1)(\cos {\theta }_1-\cos {\theta }_2)$.

The rotation of ${\mathrm{QWP}}_2$ by an angle of $\psi$ corresponds to the geodesic $AC$ in the Poincaré sphere given by $\mathrm{\Delta }\varphi ={\varphi }_2-{\varphi }_1$, where $\mathrm{\Delta }\varphi =\psi$. This results in a solid angle equal to $4\mathrm{\Delta }\varphi$ on the Poincaré sphere. This means the beam suffers a geometric phase shift equal to $2\psi$ when ${\mathrm{QWP}}_2$ is rotated through an angle of $\psi$ [14]. This phase ($2\psi$) is the geometric phase ${\varphi }_G$, which does not depend on the optical path length or wavelength, unlike the dynamic phase ${\varphi }_D$ [12]. Thus the phase of the reference beam suffers a pure geometric phase shift equal to $2\psi$ with respect to the object beam.

The whole process can be explained using a Jones matrix, assuming unit amplitude for the electric field and applying a rotation matrix, as shown in Eq. (3):

If $E_o{e}^{-i{\delta }_o}$ and $E_r{e}^{-i{\delta }_r}$ represent the complex wave front from the object and reference arms, respectively, then the hologram pattern formed at the detector plane can be found by Eq. (4):

The capabilities of the proposed method as a PSDH system were analyzed using two experiments, (1) to reconstruct irradiance of 3D objects without the presence of dc and conjugate terms and (2) to measure the surface profile of a known sample and quantify errors. The optical system used for the experiment is shown in Fig. 1. A He–Ne laser emitting at 632.5 nm with 8 mW is used as the light source. The detector was a Motion Pro Y4 series camera, with 13.7 μm pixel pitch, configured to acquire $768\times 768$ sized frames each at an exposure time of 33 ms. The reference mirror (M-1) is flat, and the reference beam is collimated as shown in Fig. 1. The remaining components present in the optical setup (Fig. 1) are standard products available in the market.

The first experiment is to test the imaging capabilities through reconstruction of 3D object irradiance. Two objects, “dice-1” and “dice-2” [Fig. 3(a)], were chosen and placed at a distance 81 and 91 cm, respectively, from the detector. The distance is measured with respect to the face of the dice facing the camera. The face of dice-1 inscribed with the number 1 faces the camera, and dice-2 had its face inscribed with the number 5 facing the camera. Each cubic die had a side length of 1.5 cm and is made of metal. The dice reflect less light compared to the reference mirror, and hence neutral density filters were added in the reference arm to balance intensity and increase fringe contrast. Four holograms were recorded one after the other by rotating the QWP-2 in steps of $\pi /4$. The complex amplitude of the wave front at the detector plane ($E_d$) was calculated from the four resulting holograms ($I_1,I_2,I_3$ and $I_4$) using the following relation:

 

Fig. 3. (a) Photograph of the two dice used as objects, (b) reconstruction using the intensity of one hologram, (c) reconstruction of object-1 at $z=81\mathrm{cm}$ using complex amplitude, (c) reconstruction of object-2 at $z=91\mathrm{cm}$ using complex amplitude.

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The Fresnel diffraction formula was used to propagate $E_d$ back up to distances 81 and 91 cm that correspond to the original positions of dice-1 and dice-2, respectively. Intensity was calculated from the backpropagated complex amplitude, and the results are shown in Figs. 3(c) and 3(d), respectively. In Fig. 3(c) the face of dice-1 with number 1 is in focus whereas in Fig. 3(d) the face with number 5 is in focus, which corresponds to dice-2. The Fresnel backprogagation algorithm results in a change in lateral magnification with propagation distance, and hence scales in Figs. 3(c) and 3(d) are different. Moreover, the width of both dice put together constitutes 3 cm, while the width of the complementary metal oxide semiconductor (CMOS) sensor is only 1 cm. The demagnifying property of the Fresnel algorithm is the one that made the reconstructions successful in spite of the size differences. This is also the reason behind choosing the Fresnel diffraction formula for the reconstructions in this experiment. The right side edge of dice-2 is not visible from the reconstruction shown in Fig. 3(d). This is due to the fact that a small portion of the inner edge of dice-2 was hidden from the illumination by the overlapping dice-1 (due to smaller beam size). Figure 3(b) shows the reconstruction of dice-2 again, but by using only the intensity ($I_o$) for backpropagation. It is clearly seen that the dc component is very strong and masks the necessary object information during reconstruction. These experiments demonstrate the capability of the proposed method to be used for 3D holographic imaging applications.

The second experiment was conducted to test the accuracy of the proposed method in measuring 3D surface profiles. The test sample chosen for this purpose was a concave mirror with known focal length of 10 cm that corresponds to a radius of curvature of 20 cm. The reflectivity of the concave mirror is much higher than that of the metal dice and hence the neutral density filters were removed and the exposure time was reduced to 300 μs for this experiment. The size of each frame was set to $512\times 512$ pixels, from the center of CMOS since the fringe density was too high toward the periphery, causing aliasing and making it meaningless to acquire those peripherals pixels. The concave mirror (object) was positioned 40 cm away from the detector, and four phase shifted holograms were acquired, from which the complex amplitude was obtained as explained earlier. The angular spectrum algorithm (which provides unit magnification throughout the propagation distance) was chosen for backpropagation calculations in this experiment to avoid the unwanted effects of magnification caused by the Fresnel algorithm. After backpropagation, the complex amplitude at the object surface was obtained, from which the phase profile is reconstructed. The phase profile corresponds to the optical path length difference between the object beam and reference beam (knowing the wavelength). The flat phase profile of the collimated reference beam and the uniform refractive index throughout the propagation distance makes the optical path length difference directly correspond to the surface profile of the concave mirror. The surface profile thus reconstructed is shown in Fig. 4(b). The ideal surface shape of this mirror was generated from the radius of curvature value (to be used for comparison) and is shown in Fig. 4(a). Figure 4(c) is a plot of a single line of data passing horizontally through the center of the surface profiles shown in Figs. 4(a) and 4(b). The blue curve represents the ideal values and red the measured values. Closely observing the plot reveals that both the surfaces match very closely except that the measured values are off centered by a small tilt toward the left hand side. This was due to a small tilt inherent in the positioning of the concave mirror during measurement. Figure 4(d) shows the absolute error on comparing the ideal and measured surface profiles from Figs. 4(a) and 4(b). The maximum absolute error was found to be 0.27 μm and root mean square error was calculated to be 85.6 nm from Fig. 4(d).

 

Fig. 4. (a) Simulated ideal surface profile of concave mirror of focal length ($f=10\mathrm{cm}$), (b) surface profile of mirror reconstructed from measured complex amplitude values, (c) surface profile generated from a horizontal line passing through the center of mirror surface (blue—ideal curve; red—measured curve), (d) absolute error between ideal (a) and measured (b) surfaces.

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In conclusion, we report a new method called GPSDH that uses only pure geometric phase in a Michelson geometry. The working principle was explained through mathematical derivations, and the potential for digital holographic imaging and measurements were demonstrated through experimental results. This the first report of such a method in PSDH whose main advantages stems from its ability (1) to realize error-free PSDH imaging of samples that change the polarization of the light by a small amount and (2) to offer more stability and control on experiments that demand strict path length matching, due to the use of broadband light. Some examples to mention will be (1) imaging of a sample with polarizing property, (2) white light PSDH, (3) near-field imaging with high-numerical aperture lens, and so on. It is also worth noting that geometric phase shifts can produce frequency shifts and hence can be used for Doppler phase shifting DH experiments. Hence we believe the method reported in this Letter will open up new possibilities and serve as a good choice for many digital holographic experimental studies.