## Abstract

We propose a new technique for achromatic 3-D field correlation that makes use of the characteristics of both axial and lateral magnifications of imaging through a common-path Sagnac shearing interferometer. With this technique, we experimentally demonstrate, for the first time to our knowledge, 3-D image reconstruction of coherence holography with generic thermal light. By virtue of the achromatic axial shearing implemented by the difference in axial magnifications in imaging, the technique enables coherence holography to reconstruct a 3-D object with an axial depth beyond the short coherence length of the thermal light.

©2012 Optical Society of America

## 1. Introduction

Coherence holography is a novel unconventional holographic technique that reconstructs an object as the 3-D distribution of a spatial coherence function, rather than the optical field itself [1–4]. Because of its unique capability of controlling and synthesizing spatial coherence of quasi-monochromatic optical fields in 3-D space [5], coherence holography has been applied for dispersion-free spatial coherence tomography [6] and profilometry [7, 8], and for the generation of coherence vortices [9]. The main task in the reconstruction of coherence holography is the detection of a spatial coherence function, which requires a suitable interferometer and a fringe analysis technique to quantify the contrast and the phase of the interference fringes generated by the interferometer. In our previous paper [2], a modified Sagnac radial shearing interferometer with a geometric phase shifter was employed together with phase-shift fringe analysis for 3-D object reconstruction. There were three notable drawbacks for this scheme, viz. (1) the use of geometric phase shifter, (2) the use of rotating ground glass for incoherent illumination, and (3) the modification of Sagnac interferometer to introduce a known amount of path delay between the interfering beams that destroyed the stable common path geometry.

The unique feature of the geometric phase shifter is that it can introduce phase shifts into the Sagnac common path interferometer, which is not possible by conventional mirror shift with a piezo-electric transducer. However, the geometric phase shifter has a disadvantage that it introduces the phase shifts by mechanically rotating a wave plate, which makes the system complex and limits the speed of coherence measurement. The other possible solution is to introduce a tilt between the wave fronts of the interfering beams to generate a carrier fringe in the interferogram and use the Fourier transform method of fringe analysis to find the fringe contrast and phase [4, 10]. This allows the reconstruction of the object as a complex coherence function using only a single interferogram. The only drawback of this technique is that the resolution of the reconstructed image is limited by the finite frequency pass-band of the filtered carrier fringe pattern. The robustness of the Sagnac common path interferometer to environmental noises such as vibrations and air turbulences allowed the use of multiple interferograms. To overcome the limitations associated with the geometric phase shifter and the Fourier transform method of fringe analysis, we proposed an alternative technique called phase-shift coherence holography [3], in which a set of phase-shifted computer-generated Fourier transform holograms displayed on a spatial light modulator (SLM) can achieve the phase shift without mechanical moving components. This made the system simple and enabled faster image reconstruction while maintaining the advantage of the Sagnac common path interferometer.

In all our previous coherence holography experiments, a coherent light source like laser with a state of art optical device such as a liquid crystal SLM was used to display the hologram. The coherently illuminated hologram was then imaged onto a rotating ground glass to destroy spatial coherence and turn the hologram to a spatially incoherent source. Therefore, although the phase shifting portion of the system was made simple and free from mechanically moving components, the presence of a rotating ground glass meant that the whole system was not free from mechanically moving components. Moreover, to make a perfect incoherent illumination, a large number of independent random states have to be generated. The number of independent random states that can be generated by a rotating ground glass is limited by the finite sizes of the ground glass and the random micro structure engraved therein, and the same random pattern repeats after each rotation. Therefore, we need a huge rotating ground glass that is cumbersome and difficult to handle. The present method, we propose the use of a commercial projector for display and incoherent illumination of a coherence hologram, by which we exclude mechanically moving optical components from the system. This implementation truly meets the requirement for generic coherence holography that only an incoherent thermal light source is used to reconstruct the hologram.

The key to our new implementation is to note the fact that an axial shear associated with a radial shear in a common-path imaging Sagnac radial shearing interferometer can virtually introduce an optical path delay needed for 3-D correlation of optical field [11]. This means that the actual path delay introduced by physically stretching the arm of the interferometer in our previous implementation [2] is unnecessary. Due to the common path geometry, the reconstruction of the 3-D object is achieved even with light having low temporal coherence. Virtually there is no influence of finite temporal coherence of the light in the measured spatial coherence function because we introduce *path difference without time delay*, as will be explained in the following. With the inherent stability of a common path interferometer and controllable magnification introduced by variable shear, it functions as a robust device for correlating optical fields to detect the 3-D coherence function that represents the object recorded in the coherence hologram.

## 2. Principles

The principle of coherence holography has been described in detail in [1]. The recording process of a coherence hologram is identical to that of a conventional hologram except that the intensity transmittance${I}_{S}\left({r}_{S}\right)$, rather than the amplitude transmittance, of the hologram is 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 an object at point ${r}_{P}$ is recorded with a reference beam from a point source at${r}_{R}$.

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 Fig. 1 . In this case, the relation between the intensity transmittance of the hologram and the mutual intensity $\Gamma \left({r}_{P},{r}_{R}\right)$given by$\u3008u\left({r}_{P}\right){u}^{*}\left({r}_{R}\right)\u3009$is described by the van Cittert-Zernike theorem [12, 13]. By virtue of the formal analogy between the van Cittert-Zernike theorem and the diffraction formula, the mutual intensity$\Gamma \left({r}_{P},{r}_{R}\right)$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${r}_{R}$. In principle, the mutual intensity can be detected by means of a Young’s interferometer as shown in Fig. 1, but the point probing by sequential scanning is impractical. We assume that the Fourier transform of the incoherently illuminated hologram is stationary in space, and reduce $\Gamma \left({r}_{P},{r}_{R}\right)$to$\Gamma \left({r}_{P}-{r}_{R}\right)$. This allows simultaneous measurement of spatial coherence function as a function of${r}_{P}-{r}_{R}$with the help of an appropriate interferometer.

#### 2.1 Generation of phase-shifted coherence holograms

Let us consider a Fourier transform hologram for a 3-D object $g\left(x,y,z\right)=\left|g\left(x,y,z\right)\right|\mathrm{exp}\left[i\varphi \left(x,y,z\right)\right]$placed in the front focal region of the Fourier transform lens with focal length *f.* A conceptual diagram of the generation of the hologram is shown in Fig. 2
.

The complex amplitude at the hologram plane is given by

*f*is the distance between z = 0 and the Fourier transform lens for recording, which is made equal to the focal length of the Fourier transform lens L used in the reconstruction process. The innermost integral inside the curly brace represents the angular spectra of the object field distribution across the plane $z=z$ with their spatial frequencies represented by the coordinates $\widehat{x}$and$\widehat{y}$. The term$\mathrm{exp}\left[-i{k}_{z}\left(\widehat{x},\widehat{y}\right)z\right]$accounts for the defocus, and propagates the angular spectra of the field by distance

*z*with ${k}_{z}\left(\widehat{x},\widehat{y}\right)=\frac{2\pi}{\lambda}\sqrt{1-{\left(\frac{\widehat{x}}{f}\right)}^{2}-{\left(\frac{\widehat{y}}{f}\right)}^{2}}.$

These optical fields are calculated for the given object letters U, E and C, and used to generate the coherence holograms. A set of phase-shift holograms are generated numerically by giving known phase shifts to the object spectrum.

*m*= 0,1,2,...,N. In synthesizing the computer-generated coherence hologram, we removed from the interference fringe intensity the term ${\left|G(\widehat{x},\widehat{y})\right|}^{2}$that becomes the source of unwanted autocorrelation image. Unlike conventional holography, the intensity (rather than amplitude) transmittance of the hologram was made proportional to the interference fringe pattern such that

#### 2.2 Reconstruction using phase-shift coherence holography

The hologram is illuminated with spatially incoherent light which is modeled as an optical field with unit amplitude and instantaneous random phase ${\Phi}_{R}\left(\widehat{x},\widehat{y}\right)$in the hologram plane. The instantaneous field created on the rear focal plane of the Fourier transform lens used for reconstruction is given by

*m*, which allows one to determine the amplitude and the phase of the object $g\left(x,y,z\right)$by sinusoidal fitting algorithm.

Unlike conventional holography where a constant bias is added to the real part of the phase shifted object spectrum to make the hologram nonnegative, $\left|G\left(\widehat{x},\widehat{y}\right)\right|$is added to make $H\left(\widehat{x},\widehat{y};m\right)$positive as shown in Eq. (3). Therefore this spatially varying bias term gives rise to a distribution $2\tilde{g}\left(\Delta x,\Delta y,\Delta z\right)$around the 0th order central spot. But as mentioned earlier, this term will not vary with phase shifts and hence can be eliminated by the phase shifting technique. The advantage of adding $\left|G\left(\widehat{x},\widehat{y}\right)\right|$over adding a constant value is that the unwanted DC term $2\tilde{g}\left(0,0,0\right)$that is present everywhere in the interferogram will have a minimum value. Experimentally, this will help to improve the fringe visibility and hence enhance the quality of the reconstructed object as a coherence function.

#### 2.3 Radial and axial shearing for 3-D field correlation

To simultaneously reconstruct the whole image by coherence holography, we need a suitable interferometer that gives a 3-D correlation map with correlation lengths covering the full 3-D image field. The light from the incoherently illuminated hologram is directed into a properly designed radial shearing interferometer that can introduce a combination of radial shear and axial shear $\Delta r={r}_{2}-{r}_{1}$ proportional to the radial and axial components of the position vector $r$. Thereby the reconstructed image, which is represented by the spatial distribution of the complex coherence function, can be visualized as an interference fringe contrast and fringe phase.

As shown in Fig. 3 , telescopic lens system comprising of lenses L1 and L2 with focal lengths ${f}_{1}$ and ${f}_{2}$respectively is introduced inside a Sagnac common path interferometer. Figure 4 is drawn to analyze the radial and axial shearing properties of the common path interferometer.

Beam1 and beam 2 are the two counter propagating beams inside the interferometer shown in Fig. 3. By the telescopic imaging system the optical fields from the input locations ${r}_{1}=\left({x}_{1},{y}_{1},{z}_{1}\right)=\left(-{\alpha}^{-1}\tilde{x},-{\alpha}^{-1}\tilde{x},{\alpha}^{-2}\tilde{z}\right)$ and ${r}_{2}=\left({x}_{2},{y}_{2},{z}_{2}\right)=\left(-\alpha \tilde{x},-\alpha \tilde{x},{\alpha}^{2}\tilde{z}\right)$ are superposed at $\tilde{r}=\left(\tilde{x},\tilde{y},\tilde{z}\right)$ on the output plane of the interferometer, and detected by an image sensor; $\alpha ={f}_{2}/{f}_{1}$ is the lateral magnification of the telescopic imaging and the axial magnification is given by${\alpha}^{2}$. Because the displacement vector$\Delta r$that represents a relative 3-D shear between two optical fields is linearly related to the position vector $\tilde{r}$ as $\Delta x=\left({x}_{2}-{x}_{1}\right)=-\left(\alpha -{\alpha}^{-1}\right)\tilde{x}$, $\Delta y=\left({y}_{2}-{y}_{1}\right)=-\left(\alpha -{\alpha}^{-1}\right)\tilde{y}$ and $\Delta z=\left({z}_{2}-{z}_{1}\right)=\left({\alpha}^{2}-{\alpha}^{-2}\right)\tilde{z}$, the 3-D spatial correlation of the optical fields at the output of the interferometer will reconstruct the object as the 3-D distribution of the complex spatial coherence function. Note that the position of the output plane $\tilde{z}$ can be shifted either by varying the focus position of the camera lens that forms the conjugate image of the output plane on the image sensor or by translating the image sensor along the z axis, the latter being the case in our experiment. This in turn means that we do not need to have extra arms that introduce actual physical delay $\Delta z$ in the interferometer as in our previous implementation [2].

#### 2.4 Measurement of purely spatial coherence of optical field

Separability of spatial and temporal coherence of optical field has been a subject of critical issue. The measurement of one without being influenced by the other is indeed a difficult task because the longitudinal spatial coherence and the temporal coherence of the optical field are often tied together though the propagation of light. Figure 5 depicts the propagation of light with low temporal coherence through a Sagnac radial shearing interferometer.

To explain how the influence of temporal coherence can be excluded by this system, let us first focus our attention to the relation between the longitudinal shear $\Delta z={z}_{2}-{z}_{1}$and time delay$\Delta t$, leaving out lateral parameters. Generally, for light with finite temporal coherence, the interference between the optical fields $u\left({z}_{1},t\right)$and$u\left({z}_{2},t\right)$, which is described by the coherence function$\Gamma \left({z}_{1},{z}_{2},\Delta t=0\right)=\u3008{u}^{*}\left({z}_{1},t\right)u\left({z}_{2},t\right)\u3009$, would render interference fringes with low contrast if the time delay $\Delta z/c$introduced by path difference is greater than temporal coherence length of the source, *c* being the velocity of light. In other words, the lack of temporal coherence manifests itself as the degradation of spatial coherence through the propagation time delay introduced by the longitudinal shear. However, this influence of the short temporal coherence of the light source can be automatically corrected in our imaging Sagnac shearing interferometer. Figure 5 shows two counter propagating beams in the Sagnac interferometer with their directions of propagation made equal and with their common output plane located at$\tilde{z}=0$. By the telescopic imaging system in the Sagnac interferometer, the optical fields $u\left({z}_{1},t\right)$ and $u\left({z}_{2},t-\Delta t\right)$ are brought together to interfere on the output plane at$\tilde{z}$, conjugate to the image sensor plane. We have$\Gamma \left({z}_{1},{z}_{2},\Delta t\right)=\u3008{u}^{*}\left({z}_{1},t\right)u\left({z}_{2},t-\Delta t\right)\u3009$, with$\Delta t=\Delta z/c$. This time delay accounts for the difference in optical path lengths of the input rays to reach ${z}_{1}$ and ${z}_{2}$ from the input plane, as can be seen from the axial principal rays of the telescope. This time delay introduced in the input side perfectly compensates the time delay introduced in the output side where optical fields $u\left({z}_{1},t\right)$and $u\left({z}_{2},t\right)$are superposed. This ensures that the temporal coherence has the maximum value for any amount of correlation length$\Delta r$. This is a situation where there is path delay but no time delay between the interfering wave packets, which can be understood intuitively from the fact that the total path lengths are the same for the two counter propagating beams in the Sagnac interferometer. Because of this achromaticity of the Sagnac interferometer, the distribution of the coherence function that reconstructs the object is purely spatial. This allows reconstruction even with a broad band source like white light as the short temporal coherence of light does not influence the 3-D coherence function that reconstructs the image. In practice, however, it is recommendable to use light with narrow spectral band width to get around the chromatic aberration of the lenses inside the Sagnac interferometer.

The radial shear $\Delta x$and$\Delta y$is same for every axial shear $\Delta z$which means that reconstructed image has a lateral magnification which is independent of its axial location. Note that, since $\Delta r=\left(\Delta x,\Delta y,\Delta z\right)=\left({m}_{x}^{-1}\tilde{x},{m}_{y}^{-1}\tilde{y},{m}_{z}^{-1}\tilde{z}\right)$ with ${m}_{x}={m}_{y}=-{\left(\alpha -{\alpha}^{-1}\right)}^{-1}$ and${m}_{z}={\left({\alpha}^{2}-{\alpha}^{-2}\right)}^{-1}$, the magnification of the reconstructed image can be controlled by changing the shearing scale parameters $({m}_{x},{m}_{y},{m}_{z})$ through$\alpha $. 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 parameters.

## 3. Experiment

Coherence holograms used in the experiment is a set of numerically generated phase shifted Fourier transform holograms. The alphabets U, E and C with sizes about 80x80 pixels each kept at different depth locations were used as objects. The Fourier transform of these off-axis binary objects are used to generate the coherence hologram$H\left(\widehat{x},\widehat{y};m\right)$ synthetically as described in Eq. (3). Figure 6 shows one of the phase shifted coherence holograms used in the experiment.

The first part of the experimental set-up shown upper right in Fig. 7 is for the display of the incoherently illuminated coherence hologram, which is implemented by a commercial projector CASIO XJ-S41 DLP having a resolution of 1280x1024 pixels. The magnification of the projected hologram is adequately controlled by proper choice of lens L1. The hologram is imaged onto the front focal plane of lens L2. An interference filter T with a bandwidth of $\Delta \lambda =3nm$at $\lambda =632.8nm$is used so that the chromatic aberrations of the optical elements do not influence the reconstructed image where as the polarizer P is used to make the incoherent light linearly polarized. This gives the temporal coherence length of the light$\approx 133\mu m$, which is still significantly shorter than the longitudinal shear introduced by imaging, and serves for the validation of our system that is free from the influence of short temporal coherence.

The field distribution of the incoherently illuminated hologram is Fourier transformed by lens L2 with focal length 250mm and introduced into the interferometer through a half wave plate (HWP). A polarizing beam splitter (PBS) splits the incoming beam into two counter propagating beams that are power balanced by rotating HWP. The telescopic system with magnification $\alpha =1.1,$ formed by lenses L3 (focal length 220mm) and L4 (focal length 200mm), gives a radial and axial shear between the counter propagating beams as they travel through interferometer before they are brought back together and imaged by CCD. The resulting interference gives a 3-D field correlation distribution that reconstructs the image as a coherence function represented by fringe contrast. In our present set up with$\alpha =1.1$, the lateral and axial magnification for the reconstructed image becomes ${m}_{x}={m}_{y}=-5.23$ and ${m}_{z}=2.6$respectively. The lateral magnification is chosen in such a way that the reconstructed image size fits the field aperture of the CCD camera. With an analyzer A with its axis kept at ${45}^{0}$to the orientation of the polarization of the two beams, interference between the two orthogonally polarized beams was achieved.

## 4. Results

The coherence function was detected by applying the phase-shift technique to the Sagnac radial shearing interferometer, and the object was reconstructed as the 3-D correlation map of the fields diffracted from the hologram. The axial shear was varied at the output of the interferometer by translating the CCD camera along z direction so that the output plane that is conjugate to the CCD image sensor plane is scanned in the z direction. At each location of the CCD camera, the set of phase-shifted Fourier transform holograms was displayed sequentially with the projector and the corresponding interferograms were captured by a 14-bit cooled CCD camera (BITRAN BU-42-14).

Figures 8(a)
, 8(b) and 8(c) show one of the phase-shifted interferograms recorded at$\tilde{z}=+20mm$,$\tilde{z}=0$and $\tilde{z}=-20mm$, respectively. Figures 8(d), 8(e) and 8(f) show the corresponding fringe contrasts, and Figs. 8(g), 8(h) and 8(i) show the corresponding fringe phases calculated by the model-based nonlinear least square fitting method [14] that solves the over-determined equations generated from the image data for the parameters of the sinusoidal fringe model. The objects U, E and C were reconstructed as fringe visibilities at the locations$\tilde{z}=+20mm$,$\tilde{z}=0$and $\tilde{z}=-20mm$, respectively, as shown in the upper half of each figure in Fig. 8. In the lower half of each figure in symmetry to the center, conjugate images were reconstructed with their planes of focus appearing in the reverse order such that $\tilde{z}=-20mm$, $\tilde{z}=0$and then $\tilde{z}=+20mm$. For$\tilde{z}=+20mm$, the longitudinal correlation length is given by$\Delta z=\left({\alpha}^{2}-{\alpha}^{-2}\right)\tilde{z}=7.69mm$. This is significantly larger than the temporal coherence length of the source, which is estimated to be$\approx 133\mu m$from the spectral width of the interference filter. The fact that the images were reconstructed with sufficiently high value of spatial coherence even for the large axial depths demonstrates that the *path difference without time delay* has been achieved by virtue of the achromatic common-path imaging Sagnac interferometer.

## 5. Conclusion

In summary, we proposed and demonstrated generic coherence holography using a commercial projector with a thermal light source for illumination of the hologram and a Sagnac radial shearing interferometer for its reconstruction. The axial shear accompanying the radial shear enabled this scheme of coherence holography to reconstruct a 3-D object with an axial depth beyond the short coherence length of the thermal light. Because the Sagnac common path interferometer is very stable to environmental noise caused by vibrations and air turbulences, the reconstruction process is highly reliable. Move over the reconstruction scheme presented in this paper is completely free from any mechanically moving components.

## Acknowledgment

Part of this work was supported by Grant-in-Aid of JSPS B (2) No. 21360028, and all the authors were with the University of Electro-Communications, Tokyo, Japan, when this work was done.

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