## Abstract

A new image reconstruction scheme for coherence holography using a modified Sagnac-type radial shearing interferometer with geometric phase shift is proposed, and the first experimental demonstration of generic Leith-type coherence holography, which reconstructs off-axis 3-D objects with depth information, is presented. The reconstructed image, represented by a coherence function, can be visualized with a controllable magnification, which opens up a new possibility for a coherence imaging microscope

©2009 Optical Society of America

## 1. Introduction

Coherence holography is an unconventional holographic technique which has the unique characteristic that a recorded object is reconstructed by the 3-D distribution of a spatial coherence function [1], rather than by the field itself as in conventional holography. Just as a computer-generated hologram can create an arbitrary 3-D optical field, the technique of coherence holography can synthesize a desired 3-D spatial coherence function [2]. Making use of this unique property, coherence holography has been applied for dispersion-free spatial coherence tomography and profilometry [3], and for the generation of coherence vortices [4]. However, all the experimental demonstrations reported in our previous papers [1-4] have been restricted to Gabor-type coherence holography for an on-axis object. A typical Gabor-type coherence hologram, realized with an incoherently illuminated Fresnel zone plate displayed on a spatial light modulator (SLM) [3], has similarity to the technique of a spatial frequency comb [5], and has been proved to be suitable for axial depth sensing based on longitudinal spatial coherence control with a variable zone plate which provides a function of coherence focusing [3,5]. Though Gabor-type coherence holography has an advantage that it can be implemented with a relatively low-resolution SLM, it inherits the same drawbacks associated with a classical Gabor hologram in conventional holography. The purpose of this paper is to propose and demonstrate, for the first time, a new technique for Leith-type coherence holography using a modified Sagnac radial shearing interferometer for the reconstruction of an off-axis coherence hologram. The proposed technique allows reconstruction of the 3-D object image with higher fidelity and stability, and better control of the magnification than in our previous experiment based on on-axis Gabor-type coherence holography which used a Michelson interferometer.

## 2. Principle

For convenience of explanation, we briefly review and summarize the essence of coherence holography; the details are found in Ref. [1]. The principle of coherence holography is based on the formal analogy between the diffraction integral and the van Cittert-Zernike theorem [6, 7]. The main difference is that the intensity transmittance * I_{S}*(

**r**) rather than the amplitude transmittance of the hologram is here made proportional to the recorded interference fringe intensity. The recording process of a coherence hologram is identical to that of a conventional hologram, in which a coherently illuminated object at point

_{S}**r**is recorded with a reference beam from a point source at

_{P}**r**, as shown in blue in Fig. 1.

_{R}However, the reconstruction process is completely different. Instead of illuminating the hologram with coherent light, we illuminate the hologram with spatially incoherent quasi-monochromatic light so that the hologram represents the irradiance distribution of a spatially incoherent extended source, as shown in red in Fig.1. In this case, the relation between the intensity transmittance of the hologram and the mutual intensity between the fields at an observation point Q and the reference source point R, which is defined by ** J**(

**r**,

_{Q}**r**)=〈

_{R}*E*(

**r**)

_{Q}*E**(

**r**)〉, is described by the van Cittert-Zernike theorem [6, 7]. By virtue of the formal analogy between the van Cittert-Zernike theorem and the diffraction formula, the mutual intensity

_{R}**(**

*J***r**,

_{Q}**r**) has the same distribution as the optical field which would be reconstructed from a conventional hologram illuminated by a coherent beam converging towards the point

_{R}**r**.

_{R}In principle, the mutual intensity can be detected by means of a Young’s interferometer as shown schematically in Fig.1, but the point probing by sequential scanning is impractical. To simultaneously reconstruct the whole 2-D image in coherence holography, we need a suitable interferometer that gives us a 2-D correlation map with correlation lengths covering the full 2-D image field. In the case of reconstruction of 3-D objects, an interferometer capable of introducing a controllable path difference between the interfering beams is used to scan the depth position. When the hologram is illuminated by a spatially incoherent light source and the light from the hologram is directed into a properly designed radial shearing interferometer which introduces an in-plane radial shear *δ*
**r**=(**r _{Q}**-

**r**) proportional to the position vector

_{R}**r**such that

*δ*

_{r}=

**r**/

*m*, the reconstructed image represented by a coherence function can be visualized as an interference fringe contrast. It should be noted that, by changing the shearing scale parameter

*m*, we can control the magnification of the reconstructed image. This gives the possibility for an unconventional coherence imaging microscope, which is endowed with an entirely new function of variable coherence zooming enabled by the controllable shearing scale parameter

*m*. In our present experiment we propose the use of a modified Sagnac radial shearing interferometer with a telescopic lens system for this purpose.

## 3. Experiments

We first demonstrate the reconstruction of off-axis 2D objects with a Sagnac radial shearing interferometer [8, 9]. Later, for 3D object reconstruction, we modify the interferometer so that we can introduce a controllable longitudinal path difference between the interfering beams. The experimental set-up shown in Fig.2 for 2D coherence holography is functionally divided into two parts. The first part is a computer-generated coherence hologram implemented by a spatial light modulator assembly to modulate the laser light intensity, followed by a rotating ground glass to destroy spatial coherence. The second part is a radial shearing interferometer to visualize a coherence image reconstructed from the coherence hologram. Linearly polarized light from a He-Ne laser passes through a half wave plate (HWP1), which rotates the orientation of polarization to control the intensity of the beam illuminating a spatial light modulator (SLM) through reflection from a polarized beam splitter (PBS1). A 5x microscope objective lens O and a lens L1 of focal length 200mm together serve as a beam expanding collimator. A half wave plate (HWP2) rotates the polarization of the collimated laser beam reflected from PBS1 to obtain maximum modulation efficiency at the SLM. The reflection type LCOS-SLM (HoloEye Model LC-R1080) placed at the focal plane of Lens L1 rotates the polarization of the incident collimated laser light according to the gray level of a computer-generated hologram displayed on the SLM.

The coherence hologram used in the experiment is a computer generated hologram of an off-axis binary object. For 2D coherence holography, we used Greek letter Ψ created on a 40×40 pixel area as the binary object. For 3D coherence holography, we used letters U, E and C with sizes about 25×25 pixels each placed at different depth locations. A conceptual diagram of the generation of the hologram is shown in Fig. 3.

For the 2D case, the optical field on the hologram *G*(*x*̂,*y*̂)=|*G*(*x*̂,*y*̂)exp[*i*Φ(*x*̂, *y*̂)] is given by the Fourier transform of the letter Ψ such that *G*(*x*̂, *y*̂)=*G*Ψ(*x*̂, *y*̂). For the 3D case, the field on the hologram is given by the superposition of the Fourier transforms of the letters U, E, and C, such that *G*(*x*̂,*y*̂)=*G _{U}*(

*x*̂,

*y*̂)exp[-

*i*Φ

_{Δz}]+

*G*(

_{E}*x*̂,

*y*̂)+

*G*(

_{C}*x*̂,

*y*̂)exp[

*i*Φ

_{Δz}], where Φ

_{Δz}is the quadratic Fresnel phase corresponding to the axial focal shift by Δ

*z*. These optical fields are calculated from the given object letters, and used to generate the coherence hologram

*H*(

*x*̑,

*y*̑). In synthesizing the coherence hologram, we removed from the interference fringe intensity the term

*G*|(

*x*̂,

*y*̂)|

^{2}which becomes the source of unwanted autocorrelation image. Instead, we enhanced the dc term

*C*to make the intensity transmittance nonnegative. We also took the square root of the modulus of the Fourier transform since the reconstruction process involves correlation of optical fields.

Figure 4 shows the coherence holograms used in the experiment.

In the experimental set up shown in Fig. 2, the polarizing beam splitter 1 (PBS1) functions as an analyzer for the light reflected from the SLM, transforming the localized polarization rotation introduced by the SLM into an intensity modulation. The effect of the SLM-induced discrete pixel structure in the modulated beam is eliminated by spatial filtering the higher order diffractions with a small circular aperture S placed in the rear focal plane of L1, and subsequently the hologram is imaged back onto a rotating ground glass by a relay lens L2 with a focal length 200mm. Thus, the hologram imaged onto the rotating ground glass acts in effect as a spatially incoherent but temporally coherent light source, which has an intensity distribution proportional to the grey level of the hologram. However, the actual ground glass does not behave as a perfect scatterer, and the light scattered by the rotating ground glass retains the directional characteristic of the incoming beam. A field lens L3 placed immediately behind the rotating ground glass directs the main lobe of the scattered light towards the optical axis of the subsequent optical setup thereby enhancing the quality of the image at the interferometer output [10].

The field distribution of the incoherently illuminated hologram is Fourier transformed by lens L4 with a focal length 250mm and introduced into the interferometer through a half wave plate 3 (HWP3). A polarizing beam splitter (PBS2) splits the incoming beam into two counter propagating beams. The telescopic system with magnification α=1.1, formed by lenses L5 (focal length 220mm) and L6 (focal length 200mm), gives a radial shear between the counter propagating beams as they travel through interferometer before they are brought back together and imaged by CCD. At any location **r** on the image plane, the interference is due to the superposition of fields at the locations **r**
*α* and **r**/*α* of the original beam [9]. Thus, at point **r** on the image plane, we have a cross-correlation of the fields between two points separated by *δ*
**r** proportional to **r** scaled by the factor (*α*-1/*α*). The resulting interference gives a 2-D correlation map that reconstructs the image as a coherence function represented by the fringe contrast. The magnification of the reconstructed image can be suitably chosen with the proper choice of the amount of shear. In our present setup with *α*=1.1, the magnification for reconstruction becomes 5.23 so that the reconstructed image size fits the field aperture of the CCD camera. The half-wave plate 3 (HWP3) is aligned such that the radially sheared interfering beams at the output of the interferometer have equal amplitudes. To detect the reconstructed coherence image represented by the fringe visibility, we need to introduce a phase shift into the common-path Sagnac interferometer. Because of its common path nature, the Sagnac interferometer is insensitive to the conventional PZT-based mechanical mirror movement. We therefore introduced the required phase shift by means of a geometric phase shifter. Inside the interferometer we made use of two quarter wave plates (QWP1 and QWP2) to turn the linearly polarized light into a circularly polarized light and back to linearly polarized light of orthogonal polarization with respect to the initial state of polarization. By rotating a half wave plate 4 (HWP4), placed between QWP1 and QWP2, we introduced a geometric phase into the counter propagating beams [11], which is used to find the fringe visibility. With an analyzer A with its axis kept at 45° to the orientation of the polarization of the two beams, interference between the two beams was achieved.

For 3D coherence holography, we modified the Sagnac radial shearing interferometer as shown in Fig. 5. The part of the experiment leading up to the rotating ground glass remains the same as in 2D coherence holography experiment.

The field distribution of the incoherently illuminated hologram is Fourier transformed by lens L4 with a focal length 500mm and introduced into the interferometer. The polarizing beam splitter 2 (PBS2) splits the incoming beam into two counter-propagating beams with orthogonal polarizations. The beams reach the non-polarization beam splitter BS at two orthogonal facets, and both get split into two and travel towards the mirrors M1 and M2. The analyzers P1 and P2 - placed with their optic axis orthogonal to each other - act like a selector switch that allows one of the beams to reach M1 and the other beam to reach M2. Mirror M2 is kept fixed, whereas M1 can be moved axially with a micrometer screw. This facilitated scanning the depth by introducing an additional path delay in one of the beams. The telescopic system with magnification *α*=1.1, formed by lenses L5 (focal length 220mm) and L6 (focal length 200mm), gives a radial shear as the light travels through the interferometer before they are brought back together and relayed by lenses L7 (focal length 500mm) and L8 (focal length 250mm) and then imaged by the CCD. PBS2 allows only those two beams that completed a round trip and got radially sheared to pass out through the interferometer exit. All the other beams generated due to unwanted reflections at BS are blocked at PBS2. At the exit of the interferometer, we made use of a quarter wave plate (QWP1), which transformed the linearly polarized beams with orthogonal polarizations into clockwise and counter-clockwise rotating circularly polarized beams. By rotating the half wave plate (HWP4), we introduced a geometric phase shift, which is used to find the fringe visibility. The other quarter plate QWP2 turned the circularly polarized beams back into linearly polarized beams with orthogonal polarizations. With an analyzer A, with its axis kept at 45° to the orientation of the polarization of the two beams, interference between the two beams was achieved.

## 4. Results

In 2D coherence holography experiment, interference images are captured with a CCD for every 5 degree rotation of the half wave plate. Fig. 6(a) shows the interference image captured by a 14-Bit cooled CCD camera (BITRAN BU-42L-14). Typically the exposure time of the CCD camera is set to about 1 second so as to destroy spatial coherence by averaging out a large number of superimposed fields created by all the possible random states of the rotating ground glass. Figs. 6(b) and 6(c) show, respectively, the phase and the contrast of the interference fringes, which were calculated using the model-based nonlinear least squares fitting method [12] which solves the over determined equations generated from the image data for the parameters of the sinusoidal fringe model. This model-based sine-curve fitting gives us a good estimate of the fringe visibility and the phase map of the interference image. The effect of noise was removed as fitting is done for the frequency determined by the phase added during each step of rotation of HWP4.

In 3D coherence holography experiment, the modified Sagnac interferometer is no longer common path and sensitive to surrounding vibrations and air turbulences. In this case, the interference images were captured by a CCD for a 22.5 degree rotation of HWP3 (phase shift of π/2). Figs. 7(a)-7(c) show the images captured when one of the beams is path-delayed by Δz=-8mm, Δz=0 and Δz=+8mm, respectively. Figs. 7(d)-7(f) show the corresponding fringe phase, and Figs. 7(g)-7(i) show the corresponding fringe visibility. As can be seen from Figs. 7(d)-7(f), the fringe phase is almost uniform at all the three locations with the only difference being that its value (initial phase) was not the same when the data were being collected for the model-based fitting. The objects U, E and C were reconstructed as fringe visibilities at the locations Δz=-8mm, Δz=0 and Δz=+8mm, respectively, as shown in the upper half of Fig. 7. Here, their conjugate images were reconstructed at the conjugate locations Δz=+8mm, Δz=0 and Δz=-8mm, respectively, as shown in the lower half.

## 4. Conclusions

In summary, we proposed and demonstrated a new scheme for reconstruction of an off-axis object by Leith-type coherence holography for the first time. The coherence hologram used in the experiment was computer-generated from a Fourier transform hologram of off-axis objects. During reconstruction, the scanning of the longitudinal (depth) direction by introducing a path delay between counter propagating beams inside the interferometer did not at all affect the radial shearing in the transverse plane. This allowed reconstructing the off-axis objects placed at different depths with the same magnification and quality, as shown in Fig. 7, which truly makes this proposed technique the most generic form of coherence holography. The choice of magnification of the reconstructed image paves the way for using it as a coherence imaging microscope.

## Acknowledgments

Part of this work was supported by Grant-in-Aid of JSPS B (2) No. 21360028.

## References and links

**1. **M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence Holography,” Opt. Express **23**, 9629–9635 (2005).
[CrossRef]

**2. **W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence function by spatial modulation of an exteded light source: a new interpretation and experimental verifications,” Appl. Opt. **41**, 1962–1971 (2002).
[CrossRef] [PubMed]

**3. **J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. **39**, 4107–4111 (2000).
[CrossRef]

**4. **W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: Local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. **96**, 073902 (2006).
[CrossRef] [PubMed]

**5. **Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express **14**, 12109–12121 (2006).
[CrossRef] [PubMed]

**6. **M. Born and E. Wolf, *Principles of Optics*, 4th ed. (Pergamon, London, 1970), *Chap. 10.*

**7. **J. W. Goodman, *Statistical Optics*, 1st ed. (Wiley, New York, 1985), *Chap. 5.*

**8. **G. Cochran, “New method of making Fresnel transforms,” J. Opt. Soc. Am. **56**, 1513–1517 (1966).
[CrossRef]

**9. **M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. **3**, 853–857 (1964).
[CrossRef]

**10. **Z. Liu, T. Gemma, J. Rosen, and M. Takeda, “An improved illumination system for spatial coherence control,” Proc. SPIE **5531**, 220–227 (2004).
[CrossRef]

**11. **P. Hariharan and M. Roy, “A geometric-phase interferometer,” J. Mod. Opt. **39**, 1811–1815 (1992).
[CrossRef]

**12. **P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas. **49**, 1189–1193(2000).
[CrossRef]