## Abstract

The principle of unconventional holography, called coherence holography, is proposed and experimentally demonstrated for the first time. An object recorded in a hologram is reconstructed as the three-dimensional distribution of a complex spatial coherence function, rather than as the complex amplitude distribution of the optical field itself that usually represents the reconstructed image in conventional holography. A simple optical geometry for the direct visualization of the reconstructed coherence image is proposed, along with the experimental results validating the proposed principle. Coherence holography is shown to be applicable to optical coherence tomography and profilometry.

©2005 Optical Society of America

## 1. Introduction

Conventional holography is a technique that records and reconstructs the 3-D image of an object represented by an optical field distribution itself [1,2]. We propose and demonstrate a new concept of yet another type of holography, which we call coherence holography. In this technique, the information of the 3-D image is encoded into the spatial coherence function of the reconstructed optical field. Instead of detecting the image as the optical field intensity, coherence holography reconstructs the image as the degree of spatial coherence between a pair of points, of which one serves as a reference point R and the other as a probe point P on the object to be reconstructed. The proposed principle of coherence holography will open up new possibilities in recording, synthesizing, and control of a spatial coherence function for applications such as spatial coherence tomography and profilometry [3].

## 2. Principles

The principle of coherence holography is based on the formal analogy between the diffraction integral and the formula of van Cittert-Zernike theorem [4,5]. For convenience of explanation, let us first consider the case of phase-conjugate reconstruction in conventional holography, in which the illuminating light for recording and reconstruction is highly coherent both temporally and spatially. A three-dimensional object is recorded in a hologram with a spherical reference beam *u*_{R}
(**r**
_{S}) = exp(*ik*|**r**
_{S} - **r**
_{R}|)/|**r**
_{S} - **r**
_{R}| diverging from a point source R, and then the image is reconstructed by illuminating the hologram with the phase-conjugated reference beam ${u}_{R}^{\mathit{*}}$
(**r**
_{S}) = exp(-*ik*|**r**
_{S} - **r**
_{R}|)/|**r**
_{S} - **r**
_{R}| converging into the original reference source point R, as shown in Fig. 1, where **r**
_{S} and **r**
_{R} are position vectors pointing at point S on the hologram and the reference point source R, respectively. Apart from a constant factor, the reconstructed optical field at an arbitrary observation point Q is given by

$$=\int \int {I}_{A}\left({\mathbf{r}}_{\mathbf{S}}\right)\frac{\mathrm{exp}\left[ik\left(\mid {\mathbf{r}}_{\mathbf{Q}}-{\mathbf{r}}_{\mathbf{S}}\mid -\mid {\mathbf{r}}_{\mathbf{R}}-{\mathbf{r}}_{\mathbf{S}}\mid \right)\right]}{\mid {\mathbf{r}}_{\mathbf{Q}}-{\mathbf{r}}_{\mathbf{S}}\mid \bullet \mid {\mathbf{r}}_{\mathbf{R}}-{\mathbf{r}}_{\mathbf{S}}\mid}d{\mathbf{r}}_{\mathbf{S}}\phantom{\rule{.2em}{0ex}},$$

where I_{A}(**r**
_{S}) is the amplitude transmittance of the hologram, **r**
_{Q} is the position vector of the observation point Q, and the integration is taken over the hologram.

It should be noted that Eq. (1) has exactly the same form as the formula for mutual intensity given by van Cittert-Zernike theorem [4,5]:

where *I*_{I}
(**r**
_{S}) is the intensity distribution of the spatially incoherent source. The implication of this formal analogy between the formula for mutual intensity and that for the complex optical field reconstructed from a hologram with a phase-conjugated beam is the following. Referring to Fig. 1, if a hologram whose intensity transmittance *I*_{I}
(**r**
_{S}) is proportional to the recorded intensity is illuminated with quasi-monochromatic spatially incoherent light with a temporal coherence length sufficiently larger than the longitudinal depth of the 3-D object, an optical field will be generated, for which the mutual intensity between observation point Q and reference point R is equal to the optical field that would be reconstructed if the hologram with the same amplitude transmittance *I*_{A}
(**r**
_{S}) = *I*_{I}
(**r**
_{S}) were illuminated with a phase-conjugated version of the reference beam. Unlike conventional holography, the reconstructed coherence image is not directly observable. It can be visualized only as the contrast and the phase of an interference fringe pattern by using an appropriate interferometer. For this reason, we refer this technique as coherence holography to distinguish it from conventional holography. Just as a conventional computer-generated hologram (CGH) can create a three-dimensional image of a non-existing object, a computer-generated coherence hologram (CGCH) can create an optical field with a desired three-dimensional distribution of a spatial coherence function, which will open up a new possibility in the techniques for coherence control [6,7].

When the hologram is illuminated with a partially coherent light with the complex degree of coherence *μ*(**r**
_{S},**r**
_{S′}), the mutual intensity of the reconstructed field is given by [5]

where $J\left({\mathbf{r}}_{\mathbf{S}},{\mathbf{r}}_{\mathbf{S}\prime}\right)=\sqrt{{I}_{I}\left({\mathbf{r}}_{S}\right)}\sqrt{{I}_{I}\left({\mathbf{r}}_{\mathbf{S}\prime}\right)}\mu \left({\mathbf{r}}_{\mathbf{S}},{\mathbf{r}}_{\mathbf{S}\prime}\right)$ is the mutual intensity of the field immediately behind the hologram, and *I*_{I}
(**r**
_{S}) is the intensity transmittance of the hologram. If the spatial coherence is stationary and the field correlation length is shorter than the spatial structure of the hologram, then we can write *J*(**r**
_{S},**r**
_{S′})≈/*I*_{I}
(**r**
_{S})*μ*(Δ**r**
_{S}), with Δ**r**
_{S} =**r**
_{S′} -**r**
_{S}. By changing the variable of integration, we can rewrite Eq. (3) as

Noting that *I*_{A}
(**r**
_{S})=*I*_{I}
(**r**
_{S}) and substituting Eq. (1), we have

which states that if the illumination is partially coherent, the coherence function reconstructed from the coherence hologram will be blurred, being convolved by the function *μ*(Δ**r**
_{S}) of the complex degree of coherence. When the illumination is perfectly spatially incoherent, we have *μ*(Δ**r**
_{S})∝*δ*(Δ**r**
_{S}), which gives *J*(**r**
_{Q},**r**
_{R})∝*u*(**r**
_{Q},**r**
_{R}) so that recorded 3-D object is ideally reconstructed as the distribution of the mutual intensity.

We next clarify the relation of the proposed coherence holography to existing techniques of holography. Since the early time of holography, many techniques have been proposed for incoherent holography [8–12]. The main issue of incoherent holography was how to record a hologram of an incoherently illuminated or self-luminous object. As an alternative to incoherent holography, a technique based on the generalized principle of Michelson stellar interferometer was also proposed for recording an incoherent object [13,14]. Whereas incoherent holography reconstructs the object physically by illuminating the hologram with coherent light, the generalized Michelson stellar interferometer reconstructs the object numerically from the spatial coherence function detected with an interferometer. In the proposed coherence holography, however, the hologram is recorded with coherent light and is reconstructed with spatially incoherent light. In this sense, the role of coherent and incoherent light is reversed between coherence holography and incoherent holography. Whereas the generalized stellar interferometer records a coherence function and reconstructs an intensity distribution, the proposed coherence holography records the fringe intensity and reconstructs the coherence function. Therefore, the role of the intensity and the coherence function is also reversed between coherence holography and the stellar interferometer. Most closely related to the proposed coherence holography is Gamma-holography proposed by Marathay [15]. In this technique, exclusive use is made of spatially incoherent light for both recording and reconstruction of a hologram. In this sense, Gamma-holography finds its position completely opposite to conventional holography in which coherent light is used for both recording and reconstruction of a hologram. Coherence holography is somewhat similar to Gamma-holography in that the hologram is reconstructed with spatially incoherent light to produce an image as the distribution of a spatial coherence function, but differs fundamentally in that coherent light is used for recording.

## 3. Experiments

The reconstructed image encoded in the spatial coherence of the field cannot be observed directly. One could probe the coherence of the field with a Young’s double-pinhole setup, but this is a point-probing scheme that requires one to scan one of the pinholes over the 3-D space to probe the coherence of the field. We can simultaneously visualize the full-field distribution of the spatial coherence function as the contrast and the phase distribution of interference fringes by using an appropriate interferometer such as a wave-front-folding interferometer [16,17] and a triangular interferometer with a double afocal system [10]. A Michelson interferometer was used in our actual experiment, but its function can be more conveniently explained with an equivalent Fizeau interferometer shown in Fig. 2. Because of the two plane parallel plates formed by the beam splitter and the mirror, each point source S on the incoherently illuminated hologram *I*(**r**
_{S}) produces an interference fringe pattern of a Fresnel zone plate (FZP) with its central axis normal to the beam splitter and passing through the source point S at **r**
_{S}. As the result of intensity-based super position of many FZPs weighted by the irradiance of the hologram, we observe interference fringe intensity at point P given by

$$=[\int {I}_{S}\left({\mathbf{r}}_{\mathbf{S}}\right)d{\mathbf{r}}_{\mathbf{S}}]\left\{1+\mid \mu \left(\mathbf{r},\mathrm{\Delta z}\right)\mid \mathrm{cos}\left[\alpha \left(\mathrm{\Delta z}\right)-\beta \left(\mathbf{r},\mathrm{\Delta z}\right)\right]\right\},$$

where *α*(Δ*z*) is the initial phase of the FZP fringe, *μ*(**r**,Δ*z*) is a complex degree of coherence given by

$$=\int {I}_{S}\left({\mathbf{r}}_{\mathbf{S}}\right)\mathrm{exp}\left\{ik\frac{\mathrm{\Delta z}{\mid \mathbf{r}-{\mathbf{r}}_{\mathbf{S}}\mid}^{2}}{{z}^{2}}\right\}d{\mathbf{r}}_{\mathbf{S}}\u2215\int {I}_{S}\left({\mathbf{r}}_{\mathbf{S}}\right)d{\mathbf{r}}_{\mathbf{S}}.$$

It should be noted that the complex degree of coherence is given by the Fresnel transform of the incoherently illuminated hologram. If we record a Fresnel hologram with coherent light for an object at distance *z*¯ = *z*
^{2}/2Δ*z* from the hologram, and illuminate the hologram with spatially incoherent light from behind, we will observe on the beam splitter a set of interference fringe patterns whose fringe contrast and phase represent, respectively, the field amplitude and the phase of the original object recorded with coherent light.

Let us consider a special case of interest that opens up the possibility of spatial coherence tomography and profilometry [3]. Equation (7) can be rewritten as

where **r̂**_{s} = **r**
_{s}/*r*
_{SMax} is a position vector normalized with the size of the hologram *r*
_{SMax}, and tan *θ* = *r*
_{SMax}/*z*, with θ being the half angle subtended by the hologram as shown in Fig. 2. If the hologram is moved to infinity such that *z* → ∞ while keeping the angle θ constant, then **r**/*r*
_{SMax} in Eq. (8) becomes zero. This means that the spatial coherence function becomes
independent of the lateral position **r**, and varies only in the longitudinal direction as a function of Δ*z*, which enables longitudinal coherence sectioning by use of an appropriately tailored coherence function. This special optical geometry can be realized with the help of a positive lens. As shown in Fig. 3, the coherence hologram is placed in the front focal plane of the positive lens. Seen through the lens from the side of the interferometer, the virtual image of the hologram with a finite physical size *r*
_{SMax} = *f* tan *θ* looks like an infinitely large hologram with *r*
_{Smax} = ∞ located at infinity with a half angular size *θ*. By changing the computer-generated coherence hologram, one can control the longitudinal coherence function for the application to optical tomography and profilometry [3].

Shown in Fig. 4(a) is an object of letter H formed by 4x5 pixels. For this object, we generated a Fresnel hologram, with the parameters Δ*z* = 1mm and tan *θ* = 0.15, as shown in Fig. 4(d), which is an on-axis phase-only hologram because of the limited spatial resolution and dynamic range of the liquid-crystal spatial light modulator (SLM) (CRL XGA2) used for the experiment. To see the effect of the phase-only reconstruction, we first reconstructed the hologram numerically with simulated coherent light. We found that the phase-only reconstruction causes the image to be blurred, as shown in Fig. 4(b). According to the proposed principle of coherence hologram, the brightness of this somewhat blurred image of the letter H will be transformed into the fringe visibility if the hologram is illuminated with spatially incoherent light. To prove this, we displayed the hologram on the SLM and illuminated it with an extended beam from a He-Ne laser through a rotating ground glass that destroys spatial coherence. Figure 4(e) shows a fringe pattern recorded with a Nikon Macro Lens, f=55mm, F2.8, mounted on a CCD camera Sony XC55, when Δ*z* is adjusted to the design parameter of the hologram (Δ*z* = 1mm). As predicted, the fringes exhibit a high contrast in the region that corresponds to the bright part of the numerically reconstructed image of Fig. 4(b). To see this more clearly, we analyzed the fringe pattern in Fig. 4(e) by the Fourier transform technique [18], obtained the fringe contrast, and displayed it as a brightness distribution. As seen in Fig. 4(c), the contrast of the fringes observed in the image reconstructed from the incoherently illuminated coherence hologram has good correspondence to the brightness of the image numerically reconstructed from the coherently illuminated hologram.

## 4. Conclusions

In summary, we proposed the concept and the principle of an unconventional type of holography, which we called coherence holography. The unique feature of coherence holography is that, while an object is recorded with coherent light much in the same manner as in conventional holography, the recorded object is reconstructed as the 3-D distribution of a spatial coherence function from the hologram illuminated with spatially incoherent light. The computer-generated coherence hologram will opens up a new possibility for controlling the 3D coherence characteristics of the optical field. A simple optical geometry for direct visualization of the reconstructed coherence image was proposed and validated experimentally. Though we have presented the general principle of coherence holography, we also pointed out that the proposed principle includes the geometry of the hologram at infinity (realized with the help of a lens) as a special case, which can be applied to optical tomography and profilometry [3]. To our knowledge, this is the first report on the general principle and the experimental demonstrations of coherence holography.

## Acknowledgments

Part of this work was supported by Grant-in-Aid for JSPS Fellowship 15.52421 given to Wei Wang, and also by The 21st Century Center of Excellence (COE) Program on “Innovation of Coherent Optical Science” granted to The University of Electro-Communications, from Japanese Government.

## References and links

**
1
. **
D.
Gabor
, “
A new microscopic principle
,”
Nature
**
161
**
,
777
–
778
(
1948
). [CrossRef] [PubMed]

**
2
. **
E. N.
Leith
and
J.
Upatnieks
, “
Reconstructed wavefronts and communication theory
,”
J. Opt. Soc. Am.
**
52
**
,
1123
–
1130
(
1962
). [CrossRef]

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

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

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

**
6
. **
D.
Courjon
,
J.
Bulabois
, and
W. H.
Carter
, “
Use of a holographic filter to modify the coherence of a light field
,”
J. Opt. Soc. Am.
**
71
**
,
469
–
473
(
1981
). [CrossRef]

**
7
. **
D.
Mendlovic
,
G.
Shabtay
, and
A. W.
Lohmann
, “
Synthesis of spatial coherence
,”
Opt. Lett.
**
26
**
,
361
–
363
(
1999
). [CrossRef]

**
8
. **
L.
Mertz
and
N. O.
Young
, “Fresnel transformations of images,” in
*
Proceedings of Conference on Optical Instruments and Techniques
*
,
K. J.
Habell
, ed. (
Chapman and Hall, London
1961
) p.305.

**
9
. **
A. W.
Lohmann
, “
Wavefront reconstruction for incoherent object
,”
J. Opt. Soc. Am.
**
55
**
,
1555
–
1556
(
1965
). [CrossRef]

**
10
. **
G.
Cochran
, “
New method of making Fresnel transforms
,”
J. Opt. Soc. Am.
**
56
**
,
1513
–
1517
(
1966
). [CrossRef]

**
11
. **
P. J.
Peters
, “
Incoherent holography with mercury light source
,”
Appl. Phys. Lett.
**
8
**
,
209
–
210
(
1966
). [CrossRef]

**
12
. **
H. R.
Worthington
Jr.
, “
Production of holograms with incoherent illumination
,”
J. Opt. Soc. Am.
**
56
**
,
1397
–
1398
(
1966
). [CrossRef]

**
13
. **
W. H.
Carter
and
E.
Wolf
, “
Correlation theory of wavefields generated by fluctuating three-dimensional, primary, scalar sources: I. General theory
,”
Opt. Acta
**
28
**
,
227
–
244
(
1981
). [CrossRef]

**
14
. **
J.
Rosen
and
A.
Yariv
, “
General theorem of spatial coherence: application to three-dimensional imaging
,”
J. Opt. Soc. Am. A
**
13
**
,
2091
–
2095
(
1996
). [CrossRef]

**
15
. **
A. S.
Marathay
, “
Noncoherent-object hologram: its reconstruction and optical processing
,”
J. Opt. Soc. Am. A
**
4
**
,
1861
–
1868
(
1987
). [CrossRef]

**
16
. **
M. V. R. K.
Murty
, “
Interference between wavefronts rotated or reversed with respect to each other and its relation to spatial coherence
,”
J. Opt. Soc. Am.
**
54
**
,
1187
–
1190
(
1964
). [CrossRef]

**
17
. **
K.
Itoh
and
Y.
Ohtsuka
, “
Fourier-transform spectral imaging: retrieval of source information from the three-dimensional spatial coherence
,”
J. Opt. Soc. Am. A
**
3
**
,
94
–
100
(
1986
). [CrossRef]

**
18
. **
M.
Takeda
,
H.
Ina
, and
S.
Kobayashi
, “
Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry
,”
J. Opt. Soc. Am.
**
72
**
,
156
–
160
(
1982
). [CrossRef]