Photon correlation holography

Dinesh N. Naik; Rakesh Kumar Singh; Takahiro Ezawa; Yoko Miyamoto; Mitsuo Takeda

doi:10.1364/OE.19.001408

1. Introduction

Conventional holography, such as that by Leith and Upatnieks, uses coherent illumination of the hologram for reconstruction, and the information about the recoded object is present in the diffracted field itself [1]. Recently Takeda et al. proposed and demonstrated unconventional holography called coherence holography, where the hologram is illuminated with spatially incoherent light so that the object information is present in the mutual intensity or the coherence function of the field [2]. Their reconstruction scheme is based on the detection of the second order correlation of the optical field using an appropriate interferometer that measures the coherence function [3–5]. Photon correlation holography to be proposed in this paper makes use of the fourth order correlations (or intensity-based photon correlations) of the optical field to reconstruct the object using an intensity interferometer. We propose and experimentally verify the principle of photon correlation holography, which presents a yet another unconventional holographic object reconstruction scheme harnessing the correlation of photons of stochastic light.

One can find a methodological analogy in the historical development of optical imaging schemes in astronomy. In the early astronomical imaging, the image of a star was observed exclusively by direct detection of optical field intensity with the help of telescopes. Later in the early 20th century, using a Michelson stellar interferometer, stellar size was determined by measuring the mutual intensity of the field. Though the technique was susceptible to the atmospheric turbulences, it introduced the concept of imaging through the second order correlation of the field. A major breakthrough was brought about half a century ago by Hanbury Brown and Twiss who first made use of the correlations of the intensity fluctuations (or the fourth-order field correlations) of light [6]. Thereafter the development of intensity interferometer made vast strides mainly due its immense application in astronomy and astrophysics [7]. Recently the correlation of entangled photons in quantum ghost imaging schemes has helped in-depth understanding of the quantum mechanical properties of light [8]. Using classical thermal-like incoherent light, ghost imaging experiment is being performed by using the correlation of photons of incoherent light [9,10]. Since the correlation is done electronically after detection of photons as intensity, the intensity interferometers are rather simple and free from instability in phase caused by environmental noises that could mar the performance of complex-amplitude interferometers [11,12]. Even with many advantages over the conventional complex-amplitude-based interferometers, intensity interferometers sometimes suffer from low signal-to-dc ratio in the detected intensity that significantly influence the correlation when the detector has a limited dynamic range. We propose a specially designed source distribution with the help of a tailor made hologram to enhance the signal-to-dc ratio in the detected intensity signal. To our knowledge, in the existing intensity correlation techniques, the photons are correlated exclusively through time averaging or coincidence counts as time events, which gives the basic parameter that defines the extent of entanglement or correlation depending on the nature of the photons. The uniqueness of our photon correlation holography is that we correlate spatially distributed photons by spatial averaging, rather than time averaging. The advantage is that the experiment setup is rather simple; the object can be reconstructed from a single-shot spatial distribution of instantaneous light intensity detected simultaneously with an image sensor, without recourse to sequential detection of light for time correlation or coincidence counting that requires state-of-art photo detectors and/or dedicated electronics for time integration.

2. Principles

A conceptual diagram for photon correlation holography is shown in Fig. 1 . A coherently illuminated 3-D object is recorded in a hologram much in the same way as in conventional holography. A marked difference is that the hologram is illuminated through a random phase screen or a ground glass for read out. The phase screen is either static or rotates slowly so that instantaneous intensity distribution of speckles $I (r, t)$ in the observation volume can be detected by an image sensor. The unique characteristic of photon correlation holography is that the 3-D object information, recorded in the hologram and encoded into the randomly distributed optical field, is retrieved through the bias-removed correlation of the speckle intensities $〈 Δ I (r, t) Δ I (r + Δ r, t) 〉$ . As will be seen, the correlation, mathematically represented by ensemble average $〈〉$ , can be realized either by time average ${〈〉}_{T}$ or space average ${〈〉}_{S}$ or a combination of both ${〈〉}_{T S}$ depending on the experimental conditions and ergodic properties of the stochastic process generated by the phase screen. In the subsections to follow, we consider a specific system configuration for Fourier-transform holography to give a detailed account for the principle of photon correlation holography.

Fig. 1 A conceptual diagram for photon correlation holography.

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2.1 Generation of hologram

The recording process of hologram for photon correlation holography is similar to that in a conventional holography in the sense that coherent light is used. In Fig. 1, diffracted field from a coherently illuminated 3-D object is shown to interfere with the spherical waves from the reference point source at R to generate the hologram. In our work, we adopt Fourier transform geometry to synthetically generate the hologram which is represented in Fig. 2 .

Fig. 2 Geometry for generation of a Fourier-transform hologram in photon correlation holography.

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Let us consider an off-axis 3-D object expressed by a local field distribution $g (r) = g (x, y, z)$ . From the theory of diffraction based on angular-spectrum decomposition, the optical field distribution created by this 3-D object after propagated onto z = 0 plane is given by

\begin{matrix} \tilde{g} (x, y) = \iint [\int {\iint g (\tilde{x}, \tilde{y}, z) \exp [- i \frac{2 π}{λ f} (\hat{x} \tilde{x} + \hat{y} \tilde{y})] d \tilde{x} d \tilde{y}} \exp [i k_{z} (\hat{x}, \hat{y}) z] d z] \\ \times \exp [i \frac{2 π}{λ f} (x \hat{x} + y \hat{y})] d \hat{x} d \hat{y} . \end{matrix}

Hereλ is the wavelength of light, f the focal length of the Fourier transform lens L, and the range of integrations extends to

(- \infty, \infty)

with its actual range effectively set by the object size

g (\tilde{x}, \tilde{y}, z)

for the integration over

\tilde{x}, \tilde{y} and z

. 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

\hat{x}

and

\hat{y}

. The term

\exp [i k_{z} (\hat{x}, \hat{y}) z]

accounts for defocus and propagates the angular spectra of the field by distance z with

k_{z} (\hat{x}, \hat{y}) = \frac{2 π}{λ} \sqrt{1 - {(\frac{\hat{x}}{f})}^{2} - {(\frac{\hat{y}}{f})}^{2}} .

For the outermost integral with integrations over

\hat{x}

and

\hat{y}

, the actual range is effectively set by the spread of the angular-spectrum of

g (\tilde{x}, \tilde{y}, z)

.

Finally, by virtue of the Fourier transform lens, the complex amplitude to be recorded at the hologram plane becomes the Fourier spectrum of the object field

G (\hat{r}) = G (\hat{x}, \hat{y}) = \iint \tilde{g} (x, y) \exp [- i \frac{2 π}{λ f} (x \hat{x} + y \hat{y})] d x d y

In synthesizing the hologram numerically, we removed the autocorrelation term

{| G (\hat{r}) |}^{2}

which becomes the source of an unwanted autocorrelation image. Unlike conventional holography, the intensity (rather than amplitude) transmittance of the hologram was made proportional to fringe pattern such that

\begin{matrix} H (\hat{r}) \propto | G (\hat{r}) | + \frac{1}{2} G (\hat{r}) + \frac{1}{2} G^{*} (\hat{r}) \\ = | G (\hat{r}) | {1 + \cos [Φ_{G} (\hat{r})]} \end{matrix}

where

Φ_{G} (r)

is the phase of

G (\hat{r})

. The choice of reference beam having constant phase with the same amplitude distribution as the Fourier spectrum of the object is found to give better reconstruction than a plane wave with uniform amplitude. This is because the background dc in the speckle intensity is reduced in reconstruction; in other words this improves the signal-to-dc ratio which is a key factor for intensity correlation.

2.2 Reconstruction of the hologram using fourth order correlation

Referring to Fig. 3 , $U (\hat{r}, t)$ represents the field distribution at the hologram plane represented by space and time coordinates $\hat{r}$ and t, respectively, whereas $H (\hat{r})$ represents the intensity transmittance of the hologram. The hologram is illuminated with laser light through a moving random phase screen, which physically simulates spatially incoherent pseudo thermal light with unit amplitude and instantaneous random phase $Φ_{R} (\hat{r}, t)$ in the hologram plane.

Fig. 3 Geometry for reconstruction of hologram in photon correlation holography.

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The instantaneous field created at the rear focal plane of the Fourier transform lens L is given by

u (r; t) = \int \sqrt{H (\hat{r})} \exp [i Φ_{R} (\hat{r}, t)] \exp (i \frac{2 π}{λ f} r \cdot \hat{r}) d \hat{r}

where the square root transforms the intensity transmittance of the hologram to amplitude transmittance. The vector

\hat{r}

represents the spatial frequency coordinate in the hologram plane. The resulting speckle intensity observed by detector is given by

I (r, t) = u^{*} (r, t) u (r, t) = {| u (r, t) |}^{2}

We assume that the complex optical field $u (r, t)$ can be regarded as a stationary stochastic process (SP) at least in the limited 3-D volume of our interest within which the field intensity is practically detectable by experiment. Though the detected intensity $I (r, t)$ is random in nature and gives no directly observable reconstructed image by itself, the information of the recorded object is encoded in its cross-covariance. To calculate the cross-covariance function, we define the fourth-order moment of $u (r, t)$ representing the intensity correlation as

\begin{matrix} m_{4} (r, r + Δ r, t) = 〈 I (r, t) I (r + Δ r, t) 〉 \\ = 〈 u^{*} (r, t) u (r, t) u (r + Δ r, t) u^{*} (r + Δ r, t) 〉 \end{matrix}

where

〈 ... 〉

denotes ensemble average. Though

m_{4}

is formally represented as

m_{4} (r, r + Δ r, t)

in Eq. (6) to include non-stationary processes,

m_{4}

in our case is independent of tandrdue to the assumption of stationarity of the field

u (r, t)

both in time and space. We will see that it is a function of

Δ r

only, the difference of the spatial coordinates. The cross-covariance of

I (r, t)

is defined as

\begin{matrix} C (r, r + Δ r, t) = 〈 Δ I (r, t) Δ I (r + Δ r, t) 〉 \\ = m_{4} (r, r + Δ r, t) - {\bar{I}}^{2} \end{matrix}

where

Δ I (r, t) = I (r, t) - \bar{I}

and

\bar{I} = 〈 I (r, t) 〉 = \int H (\hat{r}) d \hat{r}

. Using Eq. (4), Eq. (6) can be rewritten as

\begin{matrix} m_{4} (r, r + Δ r, t) = 〈 \int \int \int \int \sqrt{H ({\hat{r}}_{1}) H ({\hat{r}}_{2}) H ({\hat{r}}_{3}) H ({\hat{r}}_{4})} \\ \times \exp {i [Φ_{R} ({\hat{r}}_{2}, t) - Φ_{R} ({\hat{r}}_{1}, t) + Φ_{R} ({\hat{r}}_{4}, t) - Φ_{R} ({\hat{r}}_{3}, t)]} \\ \times \exp [i \frac{2 π}{λ f} ({\hat{r}}_{2} - {\hat{r}}_{1} + {\hat{r}}_{4} - {\hat{r}}_{3}) \cdot r] \exp [i \frac{2 π}{λ f} ({\hat{r}}_{4} - {\hat{r}}_{3}) \cdot Δ r] d {\hat{r}}_{1} d {\hat{r}}_{2} d {\hat{r}}_{3} d {\hat{r}}_{4} 〉 \end{matrix}

Under the assumption of Ergodicity in time and space, the ensemble average can be replaced either by time averaging based on integration over tor by space averaging based on integration overr.

2.3 Ensemble average replaced by time average

In Eq. (8) variable t appears only in the instantaneous random phase $Φ_{R} (\hat{r}, t)$ introduced by the moving random phase screen. Therefore we can write

\begin{matrix} m_{4} (r, r + Δ r) = \int \int \int \int \sqrt{H ({\hat{r}}_{1}) H ({\hat{r}}_{2}) H ({\hat{r}}_{3}) H ({\hat{r}}_{4})} \\ \times {〈 \exp {i [Φ_{R} ({\hat{r}}_{2}, t) - Φ_{R} ({\hat{r}}_{1}, t) + Φ_{R} ({\hat{r}}_{4}, t) - Φ_{R} ({\hat{r}}_{3}, t)]} 〉}_{T} \\ \times \exp [i \frac{2 π}{λ f} ({\hat{r}}_{2} - {\hat{r}}_{1} + {\hat{r}}_{4} - {\hat{r}}_{3}) \cdot r] \exp [i \frac{2 π}{λ f} ({\hat{r}}_{4} - {\hat{r}}_{3}) \cdot Δ r] d {\hat{r}}_{1} d {\hat{r}}_{2} d {\hat{r}}_{3} d {\hat{r}}_{4}, \end{matrix}

where

{〈 ... 〉}_{T}

denotes time average performed by integration over t. Assuming ideal spatially incoherent illumination with pseudo thermal light that obeys Gaussian statistics, we have

{〈 \exp [i Φ_{R} (\hat{r}, t)] 〉}_{T} = 0

, and

{〈 \exp {i [Φ_{R} ({\hat{r}}_{i}, t) - Φ_{R} ({\hat{r}}_{j}, t)]} 〉}_{T} = δ ({\hat{r}}_{i} - {\hat{r}}_{j})

. The assumption of Gaussian statistics reduces the fourth order averaging process in Eq. (9) into products of second order averages.

\begin{array}{l} {〈 \exp {i [Φ_{R} ({\hat{r}}_{2}, t) - Φ_{R} ({\hat{r}}_{1}, t) + Φ_{R} ({\hat{r}}_{4}, t) - Φ_{R} ({\hat{r}}_{3}, t)]} 〉}_{T} \\ = {〈 \exp {i [Φ_{R} ({\hat{r}}_{2}, t) - Φ_{R} ({\hat{r}}_{1}, t)]} 〉}_{T} {〈 \exp {i [Φ_{R} ({\hat{r}}_{4}, t) - Φ_{R} ({\hat{r}}_{3}, t)]} 〉}_{T} \\ + {〈 \exp {i [Φ_{R} ({\hat{r}}_{4}, t) - Φ_{R} ({\hat{r}}_{1}, t)]} 〉}_{T} {〈 \exp {i [Φ_{R} ({\hat{r}}_{2}, t) - Φ_{R} ({\hat{r}}_{3}, t)]} 〉}_{T} \\ = δ ({\hat{r}}_{2} - {\hat{r}}_{1}) δ ({\hat{r}}_{4} - {\hat{r}}_{3}) + δ ({\hat{r}}_{4} - {\hat{r}}_{1}) δ ({\hat{r}}_{2} - {\hat{r}}_{3}) \end{array}

This implies that non-zero contribution to

m_{4} (r, r + Δ r)

in Eq. (9) comes only from the coordinates in its spectral domain that satisfy either of the two conditions

\begin{matrix} {\hat{r}}_{2} - {\hat{r}}_{1} = 0 & and & {\hat{r}}_{4} - {\hat{r}}_{3} = 0 \end{matrix}

\begin{matrix} {\hat{r}}_{4} - {\hat{r}}_{1} = 0 & and & {\hat{r}}_{2} - {\hat{r}}_{3} = 0 \end{matrix}

Substituting Eq. (10) into Eq. (9), we have the fourth-order moment of

u (r, t)

as

\begin{matrix} m_{4} (Δ r) = {[\int H (\hat{r}) d \hat{r}]}^{2} + \int H ({\hat{r}}_{1}) \exp [i \frac{2 π}{λ f} {\hat{r}}_{1} \cdot Δ r] d {\hat{r}}_{1} \times \int H ({\hat{r}}_{2}) \exp [- i \frac{2 π}{λ f} {\hat{r}}_{2} \cdot Δ r] d {\hat{r}}_{2} \\ = {\bar{I}}^{2} + {| \int H (\hat{r}) \exp [i \frac{2 π}{λ f} \hat{r} \cdot Δ r] d \hat{r} |}^{2} \end{matrix}

Note that the fourth-order moment does not depend on r, which indicates stationarity of the statistical process. Hence from Eq. (7), we have for the cross-covariance

C (Δ r) = {| \int H (\hat{r}) \exp [i \frac{2 π}{λ f} \hat{r} \cdot Δ r] d \hat{r} |}^{2}

2.4 Ensemble average replaced by space average

If the optical field is Ergodic in space, ensemble average can be replaced by space average. We can write Eq. (8) as

\begin{matrix} m_{4} (r, r + Δ r) = \int \int \int \int \sqrt{H ({\hat{r}}_{1}) H ({\hat{r}}_{2}) H ({\hat{r}}_{3}) H ({\hat{r}}_{4})} \\ \times \exp {i [Φ_{R} ({\hat{r}}_{2}, t) - Φ_{R} ({\hat{r}}_{1}, t) + Φ_{R} ({\hat{r}}_{4}, t) - Φ_{R} ({\hat{r}}_{3}, t)]} \\ \times {〈 \exp [i \frac{2 π}{λ f} ({\hat{r}}_{2} - {\hat{r}}_{1} + {\hat{r}}_{4} - {\hat{r}}_{3}) \cdot r] 〉}_{S} \exp [i \frac{2 π}{λ f} ({\hat{r}}_{4} - {\hat{r}}_{3}) \cdot Δ r] d {\hat{r}}_{1} d {\hat{r}}_{2} d {\hat{r}}_{3} d {\hat{r}}_{4}, \end{matrix}

where

{〈 ... 〉}_{S}

denotes space average performed by integration overrin observation space. In this case, the space average results in a delta function

δ ({\hat{r}}_{2} - {\hat{r}}_{1} + {\hat{r}}_{4} - {\hat{r}}_{3})

. Therefore

\begin{matrix} m_{4} (Δ r, t) = \int \int \int \int \sqrt{H ({\hat{r}}_{1}) H ({\hat{r}}_{2}) H ({\hat{r}}_{3}) H ({\hat{r}}_{4})} \\ \times \exp {i [Φ_{R} ({\hat{r}}_{2}, t) - Φ_{R} ({\hat{r}}_{1}, t) + Φ_{R} ({\hat{r}}_{4}, t) - Φ_{R} ({\hat{r}}_{3}, t)]} \\ \times δ ({\hat{r}}_{2} - {\hat{r}}_{1} + {\hat{r}}_{4} - {\hat{r}}_{3}) \exp [i \frac{2 π}{λ f} ({\hat{r}}_{4} - {\hat{r}}_{3}) \cdot Δ r] d {\hat{r}}_{1} d {\hat{r}}_{2} d {\hat{r}}_{3} d {\hat{r}}_{4} \end{matrix}

The condition

{\hat{r}}_{2} - {\hat{r}}_{1} + {\hat{r}}_{4} - {\hat{r}}_{3} = 0

defines the stationary manifold in

\hat{r}

domain [13,14]. The integrand

m_{4} (Δ r, t)

is equal to zero outside the stationary manifold. In our model we assume the stochastic process to be Gaussian. In Ref [13,14], it is shown that, for a continuous-time real and strictly stationary stochastic process that obeys normal or Gaussian statistics, its trispectrum (the Fourier spectrum of

m_{4} (Δ r, t)

) in

\hat{r}

domain has support only on normal manifolds (NM) which are the three subsets of the stationary manifold. The condition for the normal manifold is given by

\begin{matrix} {\hat{r}}_{2} - {\hat{r}}_{1} = 0 & and & {\hat{r}}_{4} - {\hat{r}}_{3} = 0 \end{matrix}

\begin{matrix} {\hat{r}}_{1} + {\hat{r}}_{3} = 0 & and & {\hat{r}}_{2} + {\hat{r}}_{4} = 0 \end{matrix}

\begin{matrix} {\hat{r}}_{4} - {\hat{r}}_{1} = 0 & and & {\hat{r}}_{2} - {\hat{r}}_{3} = 0 \end{matrix}

The normal manifold condition given in Eq. (18) will contribute only if the spectrum has Hermitian symmetry which is the case for a real valued signal. In our case, the field

u (r, t)

is complex valued as its spectrum

U (\hat{r}, t) = \sqrt{H (\hat{r})} \exp [i Φ_{R} (\hat{r}, t)]

is no longer Hermitian due to the presence of random phase

Φ_{R} (\hat{r}, t)

. The lack of Hermitian symmetry in the spectrum reduces the non-vanishing support of normal manifold conditions to only Eq. (17) and Eq. (19) which are same as Eq. (11) and Eq. (12). Using these two conditions, Eq. (16) can be rewritten as

m_{4} (Δ r) = {\bar{I}}^{2} + {| \int H (\hat{r}) \exp [i \frac{2 π}{λ f} \hat{r} \cdot Δ r] d \hat{r} |}^{2}

Hence the cross-covariance of the intensity is given by

C (Δ r) = {| \int H (\hat{r}) \exp [i \frac{2 π}{λ f} \hat{r} \cdot Δ r] d \hat{r} |}^{2}

The implication of Eq. (14) and Eq. (21) is that, by calculating the cross-covariance of intensity,

C (Δ r)

, one can reconstruct the Fourier transform hologram as if the hologram were read out with coherent light in conventional holography. The cross-covariance can be obtained either by time average or by space average even though the two approaches are poles apart. It should be noted that

C (Δ r)

represents only the intensity of the reconstructed object and the phase information is lost, which differentiates photon correlation holography from coherence holography [2] and conventional holography [1] that reconstruct the object either by complex coherence function or complex optical field itself.

3. Simulation

As shown in Fig. 2, two letters $Γ and
O$ placed at different depth locations were used as a 3-D object. Each of the letters was created on a 10x10 pixel area and kept at off axis locations at $z = 0$ and $z = 2 mm$ planes, respectively. The pixel pitch in the object plane as well as in the hologram plane was chosen to be $7.86 μ m$ , and the wavelength of light was $λ = 632.8 nm$ .

3.1 Reconstruction using time average

In the numerical generation of a hologram and its reconstruction using time average, a Fourier transform lens L of focal length 25mm is chosen for the sake of computational efficiency. A set of independent random phase screens is introduced in the hologram plane to simulate the illumination from a pseudo thermal light source. The spatial correlation length of the random phase introduced in the hologram plane (a pixel pitch in our model) is much smaller than the finest fringes of the hologram shown in Fig. 4(a) . Under this condition, we can treat the field at the hologram plane as nearly delta correlated. For the hologram illuminated through each of the random phase screens, corresponding instantaneous intensity $I (r, t)$ is calculated by taking inverse Fourier transform of the field on the hologram plane, which numerically simulates the role played by the Fourier transform lens L. By sequentially changing the random phase screen, we generated a temporally fluctuating intensity signal at a location r in the inverse Fourier transform plane. To perform the correlation, the point at the origin ( $r = 0$ ) of the inverse Fourier transform plane was chosen as a reference point. The cross-covariance $C (Δ r)$ was calculated by correlating $Δ I (r = 0, t)$ with $Δ I (r, t)$ through time integration. Since we chose the origin $r = 0$ to be the reference point, we have $Δ r = r$ . In general, due to the stationarity of the field in space, any location r could be a candidate for the reference point.

Fig. 4 (a) Hologram generated by Eq. (3) for simulated reconstruction using time average; (b), (c) and (d) are the 3-D object images reconstructed from the time average cross-covariance of the intensity variations between the reference point at the origin (for which $r = 0$ and $z=0mm$ ) and other points on the three planes for which $z= - 2mm$ , $z=0mm$ and $z=2mm$ , respectively.

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A total of about $4 \times 10^{6}$ independent random phase screens were used to generate temporally fluctuating intensity signal $I (r, t)$ for correlation. The choice of the number $4 \times 10^{6}$ is to make it same as the number of pixels to be averaged in the spatial average to be discussed in the next subsection. Figure 4(a) shows the hologram generated for simulating reconstruction using time average. Figures 4(b)–4(d) show the simulated results for the cross-covariance function expressed by $C (Δ x, Δ y, Δ z = - 2 mm)$ , $C (Δ x, Δ y, Δ z = 0)$ and $C (Δ x, Δ y, Δ z = 2 mm)$ that reconstruct the letter O, the letter Γ and their conjugate images, respectively. Note that 3-D depth information is preserved like conventional holography.

3.2 Reconstruction using space average

In this case, the focal length of the lens L is chosen to be 200mm for computational efficiency. Therefore the hologram generated in this case is scaled accordingly. Nevertheless, since the same lens L is used in performing inverse Fourier transform for reconstruction, the size of the reconstructed image remains the same as that in the previous case. For reconstruction by spatial averaging, $I (r; t)$ is calculated for the hologram illuminated through a fixed single random phase screen. This simulates the illumination of hologram by a time-frozen pseudo thermal light source. In this case $C (Δ r)$ is calculated from the snap-shot 3-D intensity distribution of speckles by correlating spatially fluctuating $I (r; t)$ with $I (r + Δ r; t)$ distributed in a 3-D volume that is sectioned along z direction with a 2-D image area having resolution of 2048x2048 pixels. Since the two letters O and Γ chosen as object are confined to $z = 0$ and　 $z = 2$ mm planes, their reconstruction can be achieved by correlating the intensity distributions between any pair of planes separated by $Δ z = 2 mm$ within the stationary region.

Figure 5(a) shows the hologram generated for simulating reconstruction using space average. Figures 5(b)–5(d) show the simulated results for the space average cross-covariance function $C (Δ x, Δ y, Δ z = - 2 mm)$ , $C (Δ x, Δ y, Δ z = 0)$ and $C (Δ x, Δ y, Δ z = 2 mm)$ , respectively; again the letter O, the letter Γ and their conjugate images are reconstructed like conventional holography. The total of about $4 \times 10^{6}$ pixels participated in the averaging process.

Fig. 5 (a) Hologram generated by Eq. (3) for simulated reconstruction using space average; (b), (c) and (d) are the 3-D object images reconstructed from the space average cross-covariance of the intensity distributions between the two planes for which $Δ z = - 2 mm$ , $Δ z = 0 mm$ and $Δ z = 2 mm$ , respectively.

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4. Experiment

As shown mathematically in Eq. (14) and Eq. (21) and numerically confirmed by Fig. 4 and Fig. 5, the object recorded in the hologram (which is read out through a random phase screen) can be reconstructed from the cross-covariance of intensity distributions by replacing ensemble average either by time average or by space average. The fact that both cases yielded identical results justifies the assumptions of stationarity and Ergodicity made in our model. The time average approach for reconstruction is computationally cumbersome because it requires a huge number of random numbers for the realization of the time sequence of great many random phase screens. Moreover, the experimental realization of the huge number of independent random phase screens using a rotating ground glass has a clear limitation caused by periodic rotation. On the other hand, the space average approach is computationally quick and efficient because it requires only a single-shot 3-D intensity distribution. More specifically, for experimental object reconstruction using space average, only a static ground glass is needed to generate an instantaneous field of pseudo thermal light. Having clear advantages over time average process, we experimentally demonstrate the object reconstruction using space average.

Figure 6 shows the experimental set up for photon correlation holography for reconstruction using space average. A 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 the SLM through reflection from a polarized beam splitter (PBS). A 5x microscope objective lens O and a lens L1 of focal length 200mm together serve as a beam expanding collimator. Half wave plate2 (HWP2) rotates the polarization of the collimated laser beam reflected from PBS to obtain maximum intensity modulation efficiency at the SLM. For the numerical generation of hologram described by Eq. (3), two letters $Γ and
O$ , each created on a 20x20 pixel area and kept at off axis locations at $z = 0$ and $z = 5 mm$ planes respectively, were used as object. The pixel pitch in the object plane is $7.4 μ m$ (adjusted to the pixel pitch of CCD camera). The pixel pitch of the Spatial light modulator LCOS-SLM (HoloEye Model LC-R1080) used to display the hologram is $8.1 μ m$ . The hologram displayed on the SLM is imaged onto a static ground glass with magnification of 1.5. Hence for the numerical synthesis, the pixel pitch in the hologram plane becomes $12.15 μ m$ . The focal length of the lens L is 500mm which is same as L3, the one to be used for reconstruction.

Fig. 6 The experimental setup for photon correlation holography.

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The amplitude transmittance of the hologram was made $\sqrt[4]{H (\hat{r})}$ instead of theoretical $\sqrt{H (\hat{r})}$ just for the technical reason to adaptively enhance its high spatial frequency components that tend to take lower values due to non-linearity of SLM display. The enhancement of higher spatial frequency can be seen in the hologram shown in Fig. 7 . The reflection type SLM placed at the focal plane of Lens L1 rotates the polarization of the incident collimated laser light according to the gray level of the computer-generated hologram displayed on it. PBS 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 static ground glass by a relay lens L2 with a focal length 300mm. The field distribution due to scattering by the ground glass is Fourier transformed by lens L3 with a focal length 500mm and the speckle intensity is recorded on a 14-Bit cooled CCD camera (BITRAN BU-42L-14) having the image resolution of 2048x2048 pixels. With the help of a stepper motor stage, the camera can be moved along z direction to scan the speckle along z direction. Figures 8(a) –8(l) show magnified images of the 128x128 pixel area selected from the center of the full-field speckle intensity images recorded at the planes from $z = - 6 mm$ to $z = 5mm$ with steps of 1mm. Figure 8(m) shows the full-field speckle intensity recorded at $z = 0$ plane with the CCD camera having total image resolution of 2048x2048 pixels, and the yellow square represents the location of the image area shown in Figs. 8(a)–8(l). A careful look at Figs. 8(a)–8(l) reveals how the recorded speckle intensity is gradually changing as we translate the CCD camera in the longitudinal direction with the stepper motor stage.

Fig. 7 Hologram generated by Eq. (3) for experiment using space average.

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Fig. 8 (a)-(l) Magnified images of the 128x128 pixel area selected from the center of the full-field speckle intensity image having resolution of 2048x2048 pixels, recorded at the planes from $z = - 6 mm$ to $z = 5mm$ with steps of 1mm. (m) The full-field speckle intensity recorded at $z = 0$ plane with the CCD camera having total image resolution of 2048x2048 pixels. The yellow square in the center indicates the location of the magnified image area shown in (a)-(l).

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Because of the spatial stationarity, the reconstruction of the part of object at a specific z plane can be achieved by correlating the intensity at any two planes separated by $Δz=z$ in the stationary region. To show the reconstruction of the 3-D object, we correlated the intensity at the plane $z = 0$ with a set of planes ranging from $z = - 6 mm$ to $z = 5 mm$ . Figures 9(a) –9(l) show, respectively, the experimental results for the cross-covariance functions given by $C (Δ x, Δ y, Δ z)$ with $Δ z$ being varied from $Δ z = - 6 mm$ to $Δ z = 5 mm$ with steps of 1mm. Figure 9(b) represents the reconstructed image O, Fig. 9(g) represents the reconstructed image Γ with its conjugate image, whereas Fig. 9(l) represents the reconstructed conjugate image of O. Note that the function of focusing can be achieved by intensity correlation and 3-D depth information is preserved like conventional holography even though the phase information about the object is not present.

Fig. 9 (a)-(l) The 3-D object images reconstructed from the space average cross-covariance of the intensity distributions $C (Δ x, Δ y, Δ z)$ with $Δ z$ varied from $Δ z = - 6 mm$ to $Δ z = 5 mm$ with steps of 1mm. (b) represents the reconstructed image O, (g) represents the reconstructed image Γ with its conjugate image and (l) represents the reconstructed conjugate image of O. The focusing function is achieved by intensity correlation.

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The letters O, Γ and their conjugate images are reconstructed as predicted by theory and numerical simulations. About $3.9 \times 10^{6}$ pixels participated in the spatial averaging during the process of correlation. In all the figures showing the reconstructed object, the high value zero-th order central peak is chopped at a convenient level to clearly see the reconstructed objects at the off-axis locations. The result presented in this paper involving spatial average is achieved through the integrations over x and y coordinates during the process of averaging. The result could be further enhanced by performing integration along z coordinate also in the stationary region.

5. Conclusions

We proposed and experimentally demonstrated 3D object reconstruction by photon correlation holography for the first time to our knowledge. Ensemble average realized using space average made the reconstruction process quick and efficient. With the proposed holographic technique, one can encode arbitrary object information in correlations of photons of stochastic light. Synthetically generated hologram that enhances the signal to background dc ratio in the detected intensity could find application in other intensity correlation techniques.

Acknowledgement

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

References and links

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Photon correlation holography

Abstract

1. Introduction

2. Principles

2.1 Generation of hologram

2.2 Reconstruction of the hologram using fourth order correlation

2.3 Ensemble average replaced by time average

2.4 Ensemble average replaced by space average

3. Simulation

3.1 Reconstruction using time average

3.2 Reconstruction using space average

4. Experiment

5. Conclusions

Acknowledgement

References and links

Cited By

Figures (9)

Equations (21)

Optics Express