We report a new optical vortex phase-shifting method for digital holography, in which an optical vortex mode is taken as the reference beam for holographic recording, and the required phase shifts are directly generated by rotating the vortex mode. In digital reconstruction, the complex amplitude of the object wave can be retrieved by use of the conventional phase shifting algorithm on condition that the digital illumination beam is replaced by an vortex beam with the same topological charge as the reference used. Both the theoretical analysis and experimental results demonstrate the feasibility of this approach.
©2004 Optical Society of America
In digital holography, holograms are captured by a CCD camera and reconstructed by a computer through diffraction integral [1, 2]. Compared with conventional holography; it saves the trouble of photographic processing and shortens the exposure time. Some other salient features include the freely focusing to yield the image at arbitrary position and the quantitative retrieving for the phase distribution of the object wave [3–6]. Lower resolution of CCD array than photographic materials, however, limits the use of an off-axis recording geometry for separating the reconstructed image from the conjugate replica and the zero diffraction order. Phase-shifting digital holography that employs the in-line setup together with phase shifts of the reference wave overcomes this issue by directly retrieving the complex amplitude in the CCD plane [6–11].
Realizing exact phase shifts is of great importance for phase-shifting digital holography [12,13]. Some phase-shifting techniques suitable for different applications have been proposed. For example, the phase shifts can be produced by moving a mirror, tilting a glass plate, moving a grating, rotating a wave plate, or using computer generated holograms [14–16].
In this letter we propose a new avenue for phase-shifting digital holography based on optical vortices. We use an optical vortex (OV) beam as the reference beam for holographic recording. This special helical wavefront can be generated by a spiral phase plate (SPP)  or a computer-generated hologram (CGH) [18, 19]. At the same time, the phase shifts required in phase-shifting algorithm can be directly produced by turning the SPP to different angles around the beam axis. Despite substituting an OV reference beam for conventional plane or spherical wave in recording a digital hologram, no special difficulties are brought for the digital reconstruction, because the OV beam is easily produced and adjusted in computer. By contrast, this substitution may be especially useful in interferometric null testing of aspheres [20, 21] and in research on dynamics of optical vortices in linear or nonlinear materials. In the following description, this method will be called OVPS (optical vortex phase-shifting) method for short.
The basic setup for the OVPS method is shown in Fig. 1. An expanded laser beam is divided by a splitter into two beams. One beam illuminates the test object to form the object beam Uo. Another beam is incident vertically on the SPP with a phase-only transmittance of exp(ilϕ), where ϕ is the azimuth angle and l is an integer known as the topological charge. The OV beam UR emerged from the SPP interferes with the object beam U0 in the CCD plane to form a hologram, which is recorded by the digital CCD camera. In order to use the phase-shifting algorithm for retrieving the complex amplitude of the object wave from the recorded holograms, some global phase shifts of the reference wave have to be introduced stepwise. This can be achieved by use of the flexibility of the SPP in the phase shift, that is, the global phase of the reference beam can be changed stepwise by rotating the SPP around its axis. On rotation of the SPP through an angle θ, the complex amplitude of the OV beam in the CCD plane can be written as
where A(r) is the amplitude, which is only dependent on the radial coordinate. We can see that the rotation of an angle θ corresponds to a definite global phase shift of Δφ = lθ. Obviously, an optimal choice of the topological charge of the SPP should be l = 1 for our purpose; in this situation the global phase shift is exactly equal to the turned angle of the SPP.
Figures 2(a), (b), (c) and (d) show the phase distributions of an OV beam with l=1, corresponding to the turned angle θ is equal to 0, π/2, π and 3π/2, respectively. Figure 2(e), (f), (g) and (h) give the interferograms formed by the above four OV beams with an oblique coherent plane wave, respectively. It can be seen from the interference fringes that the turned OV beams really have the same wave fronts except for a global phase shift in order. These phase shifts satisfy the requirements for four-frame phase-shifting digital holography . We can derive the complex amplitude distribution Uh(x,y) of the object wave on the CCD plane from four phase-shifted holograms In(x,y, θn) as
The reconstruction of the object at a distance zo is performed generally by the Fresnel transformation of the derived complex amplitude
We carried out experiments to demonstrate the method described above. The experimental setup is shown in Fig. 1. It is essentially a Mach-Zehnder interferometer. A 30-mW He-Ne laser at a wavelength of 632.8 nm was used as the light source. In principal, we could use a spiral phase plate (SPP) or a computer-generated hologram (CGH) to generate the required OV. Of course a carefully designed SPP could be a good choice in this purpose. In our experiments, however, the SPP shown in Fig.1 was replaced by a CGH of an optical vortex with l = 1 just because of the limitation of our experimental condition. This CGH was formed by interference of the required OV with an oblique plane wave in computer and was dynamically displayed onto a programmable spatial light modulator (SLM) with 1024×768 pixels (each pixel has a 18 μm × 18 μm size). In general, several reconstructed wavefronts (as different diffraction orders) emerge from the CGH. The first diffraction order of particular interest is selected by a spatial filter and is taken as the reference beam UR. Quantitative rotations of the reconstructed optical vortex can be implemented by displaying the CGHs with differently angled helical wavefront onto the SLM in a proper sequence. The interferograms formed by the vortex reference and the object waves were recorded by a CCD camera.
To quantitatively check the phase-shifting values, we first measured the global phase shifts versus the turned angle of the optical vortex by use of the Fourier method [22, 23]. Figure 3 shows our experimental result, in which the solid line is the theoretical curve and the circles are the measured values. The consistency of our experiment with the theory proves the validity of this phase-shifting method.
Next we carried out a phase-shifting digital holographic experiment. A transparency of two Chinese characters is used as the object. We adopt the four-step phase-shifting algorithm described in Eqs. (2) and (3), and the required phase shifts are generated by four CGHs with different angled optical vortices as shown in Fig. 2(a), (b), (c) and (d), respectively. After a set of holograms at four phase-shifting steps has been recorded on the CCD, the complex amplitude of the image can be calculated in terms of Eq. (2). Figure 4(a) shows the photo of the object (a Chinese word meaning “kindheartedness”). Figure 4(b) is one of four digital holograms recorded with the CCD camera. Figure 4(c) gives the image directly reconstructed by the hologram shown in Fig. 4(b). It is obvious that the reconstructed object wave is seriously contaminated by the conjugate image and the zero diffraction order. To eliminate this kind of contamination, we derive the complex amplitude of the object wave by Eq. (2). Figure 4(d) shows the final image reconstructed by the complex amplitude derived from our four digital holograms, in which the influences of the conjugate and zero diffraction parts are removed successively. This result implies that the precision of the phase shifts with the OVPS method is good enough for phase-shifting digital holography.
In digital retrieval of the complex amplitude of the object wave, we have to create an optical vortex as the illumination beam in computer. Theoretically, only when the illuminating vortex is the same as the reference vortex used in recording process, the complex amplitude of the object wave could be exactly retrieved, that is to say, the illumination vortex and the reference vortex have both the same topological charge and the same central point in the holographic plane. Keeping the same topological charge is easy. Locating the central point of the illumination vortex, however, brings a small error in general, which depends on the precision of determining the central point of the reference vortex in the CCD plane. In practical experiments, we can locate the central point of the reference vortex with the precision of two pixel spaces by use of the interference fringes between the reference vortex and a plane wave. Figures 5(a), (b), (c) and (d) give four reconstructed images of the holograms discussed in Fig. 4 when the location error is set on purpose to 8 pixels, 12 pixels, 16 pixels and 32 pixels, respectively. It can be seen that the reconstructed images keep a good quality if only the location error for a digital hologram with 1024×1024 pixels is less than 12 pixels, which implies that a location error of two pixels in practical experiments is allowed for good reconstruction in our OVPS method.
In conclusion, we presented and demonstrated the idea of optical vortex phase-shifting holography, named OVPS method, which allows us to conveniently realize arbitrary global phase shift by rotating an optical vortex around its axes. In addition, optical vortices are intriguing special optical structures with helical wave front and well-defined orbital angular momentum (OAM), which have been widely studied in different research fields [24–26]. Some of their fascinating properties have been found and many possible applications were demonstrated, which involve trapping of atoms or mesoscopic particles, driving microoptomechanical pumps, processing quantum information, generating optical vortex solitons, and so on. We believe that it is also a meaningful experiment to take an optical vortex as the reference beam in digital holographic recording. This OVPS method could offer a new approach for applications of the phase-shifting digital holography in measurement of the OAM and in research on the dynamics of the optical vortices in linear or nonlinear materials.
This work is supported in part by the NSFC under Grant Nos. 10474043, 90101030 and 10325417 (“Excellent Youth Foundation”) as well as the Research Foundation for the Doctoral Program of Higher Education under Grant No. 20030284012. Corresponding author H. T. Wang’s email address is email@example.com.
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