Real-time coherence holography

Dinesh N. Naik; Takahiro Ezawa; Yoko Miyamoto; Mitsuo Takeda

doi:10.1364/OE.18.013782

1. Introduction

The generation of quasi monochromatic optical fields with desired spatial coherence distribution has attracted great deal of attention in recent years [1–11]. A solution has emerged in the form of coherence holography, a rather new and unconventional holographic technique with a unique characteristic that a recorded object wave can be reconstructed with a 3-D coherence function [1]. Because of its unique capability of controlling and synthesizing spatial coherence of quasi-monochromatic optical fields in 3-D space, coherence holography has been applied for dispersion-free spatial coherence tomography and profilometry [5,7,8,12], and for the generation of coherence vortices [5]. Recently, coherence holography capable of reconstructing a 3-D object wave with a computer generated coherence hologram has been presented [2,3]. In the present work, coherence holography capable of real-time recording and reconstruction is proposed and the experimental demonstration with a generic Leith-type coherence holography is presented. The generation of coherence hologram is done optically in real-time using a Mach-Zehnder interferometer and its reconstruction is achieved using a Sagnac radial shearing interferometer [13,14]. The characteristic that the magnification of the reconstructed object wave is determined by the amount of the radial shear introduced to the interfering beams endows this system with a unique function as a coherence imaging microscope with variable magnification. The main task in the reconstruction of coherence holography is the detection of a spatial coherence function, which requires an appropriate 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 geometric phase shifter was employed for the phase-shift fringe analysis. 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 shifts 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. To make the reconstruction process simpler and faster, a spatial carrier frequency is introduced, and the Fourier transform method of fringe-pattern analysis [16] is used to quantify the coherence function from a single fringe pattern obtained with the Sagnac interferometer. Unlike the real-time holographic interferometry which compares wavefront from an object under test with a reconstructed wavefront from its hologram recorded earlier, the real-time coherence holography is aimed to encode an arbitrary optical field distribution into a corresponding distribution of coherence function and visualize it in real-time, all optically in the form of the contrast and the phase of a spatial carrier fringe pattern.

2. Principles

For convenience of explanation, we briefly review and summarize the essence of coherence holography; the details are found in Ref. [1]. Because of the formal analogy between the diffraction integral and the formula of van Cittert-Zernike theorem [17,18], when a hologram is illuminated with a spatially incoherent light, the coherence function or the mutual intensity between a pair of points separated by a length $δ r = r_{2} - r_{1}$ on the image plane is proportional to the optical field which would be observed at a point $r = δ r$ by conventional holography if the hologram were illuminated by a coherent light source. In coherence holography, the generation of hologram is identical to that of conventional hologram in which the optical field from a coherently illuminated object is superposed with a reference beam and the interference fringes are recorded. In real-time coherence holography, no use is made of devices, such as CCD and SLM, for recording and display of a hologram. Instead, a holographic fringe pattern due to the interference between object and reference beams is created directly on a rotating ground glass. The spatial coherence of the interfering field is destroyed completely by the rotating ground glass so that the hologram is represented in real time by the irradiance distribution of a spatially incoherent extended source. In effect this extended source serves as an incoherently illuminated hologram which reconstructs the recorded object wave as a spatial coherence function. In the present experiment we adopted the geometry of a lensless Fourier transform hologram as shown in Fig. 1 .

Fig. 1 Geometry for real-time coherence holography.

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In general, for an off-axis object $g_{o} (x, y, z)$ , the complex amplitude at the hologram plane is given by

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

\begin{matrix} where \tilde{g} (x, y) = \iint [\int {\iint g_{o} (\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}

represents the optical field distribution of a 3D object propagated onto z = 0 plane. Hereλ is the wavelength of light, and f is the distance between z = 0 and the hologram plane, 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

\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}} .

Upon interference with a reference spherical wave $U_{r} (\hat{x}, \hat{y}) = \exp [i \frac{π}{λ f} ({\hat{x}}^{2} + {\hat{y}}^{2})]$ originated from point R with the same sphericity as the waves from $\tilde{g} (x, y)$ , the object field given by Eq. (1) creates the intensity distribution known as a lensless Fourier transform hologram [15];

\begin{matrix} {| H (\hat{x}, \hat{y}) |}^{2} \propto 1 + {| U (\hat{x}, \hat{y}) |}^{2} - \iint \tilde{g} (x, y) \exp [- i \frac{2 π}{λ f} (x \hat{x} + y \hat{y})] d x d y \\ - \iint {\tilde{g}}^{*} (x, y) \exp [i \frac{2 π}{λ f} (x \hat{x} + y \hat{y})] d x d y, \end{matrix}

where

H (\hat{x}, \hat{y}) = U (\hat{x}, \hat{y}) + U_{r} (\hat{x}, \hat{y}) .

In conventional Fourier transform holography, we reconstruct the recorded object wave by illuminating the hologram with coherent light and taking an inverse Fourier transform optically with a Fourier transform lens of focal length f:

g' (x, y; z) = \iint {| H (\hat{x}, \hat{y}) |}^{2} \exp [i k_{z} (\hat{x}, \hat{y}) z] \exp [i \frac{2 π}{λ f} (x \hat{x} + y \hat{y})] d \hat{x} d \hat{y} .

In coherence holography, we display the interference fringes directly on a rotating ground glass, which destroys the spatial coherence by adding a random phase $Φ_{R} (\hat{x}, \hat{y})$ to the field in that plane. Temporal fluctuation of the random phase is implicit in $Φ_{R} (\hat{x}, \hat{y})$ though a time variable does not appear explicitly in its notation. The scattered optical field is given by

g (x, y; z) = \iint H (\hat{x}, \hat{y}) \exp [i Φ_{R} (\hat{x}, \hat{y})] \exp [i k_{z} (\hat{x}, \hat{y}) z] \exp [i \frac{2 π}{λ f} (x \hat{x} + y \hat{y})] d \hat{x} d \hat{y} .

This field by itself does not reconstruct the object wave because the phase has been scrambled. We correlate the field to find the coherence function $Γ (Δ x, Δ y; Δ z)$ between a pair of points:

\begin{matrix} Γ (Δ x, Δ y; Δ z) = 〈 g^{*} (x_{1}, y_{1}; z_{1}) g (x_{2}, y_{2}; z_{2}) 〉 \\ = \iint \iint H^{*} ({\hat{x}}_{1}, {\hat{y}}_{1}) H ({\hat{x}}_{2}, {\hat{y}}_{2}) 〈 \exp [- i Φ_{R} ({\hat{x}}_{1}, {\hat{y}}_{1})] \exp [i Φ_{R} ({\hat{x}}_{2}, {\hat{y}}_{2})] 〉 \\ \begin{matrix}  \end{matrix} \times \exp [- i k_{z} ({\hat{x}}_{1}, {\hat{y}}_{1}) z_{1}] \exp [i k_{z} ({\hat{x}}_{2}, {\hat{y}}_{2}) z_{2}] \\ \begin{matrix}  \end{matrix} \times \exp [- i \frac{2 π}{λ f} (x_{1} {\hat{x}}_{1} + y_{1} {\hat{y}}_{1})] \exp [i \frac{2 π}{λ f} (x_{2} {\hat{x}}_{2} + y_{2} {\hat{y}}_{2})] d {\hat{x}}_{1} d {\hat{y}}_{1} d {\hat{x}}_{2} d {\hat{y}}_{2} \\ = \iint {| H ({\hat{x}}_{1}, {\hat{y}}_{1}) |}^{2} \exp [i k_{z} ({\hat{x}}_{1}, {\hat{y}}_{1}) Δ z] \exp [i \frac{2 π}{λ f} ({\hat{x}}_{1} Δ x + {\hat{y}}_{1} Δ y)] d {\hat{x}}_{1} d {\hat{y}}_{1}, \end{matrix}

where < > denotes ensemble average,

Δ x = x_{2} - x_{1}, Δ y = y_{2} - y_{1}, Δ z = z_{2} - z_{1},

and use has been made of the assumption

〈 \exp [- i Φ_{R} ({\hat{x}}_{1}, {\hat{y}}_{1})] \exp [i Φ_{R} ({\hat{x}}_{2}, {\hat{y}}_{2})] 〉 = δ ({\hat{x}}_{1} - {\hat{x}}_{2}, {\hat{y}}_{1} - {\hat{y}}_{2})

for an ideal rotating ground glass. From the fact that Eq. (6) is identical to Eq. (4), now it can be clearly seen that the 3-D coherence function of the optical field from the incoherently illuminated hologram reconstructs the object wave in the exactly same manner as a coherently illuminated hologram reconstructs the object wave in conventional holography. In this case the resolution of the reconstructed object wave in real-time coherence holography will be same as that in conventional holography because of the exact formal analogy between the formulas. In principle the mutual intensity between any two points can be detected by means of a Young’s interferometer, but such sequential point probing is impractical. To reconstruct the image in coherence holography simultaneously we need a suitable interferometer that gives us a 2D correlation map in real time for the entire correlation lengths covering the full image field. For this purpose we use a Sagnac radial shearing interferometer, which gives the interferogram

I (x, y; z) = 〈 {| g (x_{1}, y_{1}; z_{1}) + g (x_{2}, y_{2}; z_{2}) |}^{2} 〉 = 2 Γ (0, 0, 0) + 2 Re {Γ (Δ x, Δ y; Δ z)},

where we have assumed that the reconstructed random field is statistically stationary such that

\begin{array}{l} Γ (0, 0, 0) = 〈 g^{*} (x_{1}, y_{1}; z_{1}) g (x_{1}, y_{1}; z_{1}) 〉 = 〈 g^{*} (x_{2}, y_{2}; z_{2}) g (x_{2}, y_{2}; z_{2}) 〉 \\ Γ (Δ x, Δ y, Δ z) = 〈 g^{*} (x_{1}, y_{1}; z_{1}) g (x_{2}, y_{2}; z_{2}) 〉 . \end{array}

As will be described in the next section, the radial shear is generated by a telescopic optical system with beam magnification α such that $x = α x_{1} = α^{- 1} x_{2}$ and $y = α y_{1} = α^{- 1} y_{2}$ , which gives $Δ x = (α - α^{- 1}) x$ and $Δ y = (α - α^{- 1}) y$ . The reconstructed object wave is scaled by a factor ${(α - α^{- 1})}^{- 1}$ . This indicates a possibility of a coherence-zooming microscope with variable magnification. Unlike a conventional microscope, the magnification of the coherence microscope becomes infinity when the magnification of the telescopic system becomes unity $α = 1$ . Although the raw image reconstructed as the coherence function can be observed in real time in the form of the distribution of the fringe contrast and the phase variation in the interferogram expressed by Eq. (7), the quantification of the coherence function requires fringe analysis.

3. Experiments

In our experiment, a lensless Fourier transform hologram of a transmission object is generated with the help of a Mach-Zehnder interferometer as shown in the left half of Fig. 2 . For convenience, we chose a 2-D transmissive object and placed it in the z = 0 plane so that $\tilde{g} (x, y) = g_{o} (x, y, z = 0) .$ Three letters U E and C, each being about a millimeter in size and placed at off axis positions in z = 0 plane, were used as the object. The vibration angle of linearly polarized light from He-Ne laser is adjusted to an appropriate angle by a half wave plate 1 (HWP1) so that a polarizing beam splitter 1 (PBS1) splits the incoming light into two orthogonally polarized beams with an adequate power ratio. A half wave plate 2 (HWP2) inside the Mach-Zehnder interferometer rotates the polarization to make it same as that of the other beam. One of the beams collimated by lenses L2 and L3 coherently illuminates the object, and the other beam generates a point source with the help of lens L1. Seen from the output side of the Mach-Zehnder interferometer, an off-axis object with a point source at the centre is observed, which forms the geometry for recording a lensless Fourier transform hologram. Instead of recording interference fringes of the hologram, they are directly displayed on a rotating ground glass so that the hologram is represented in real time by the irradiance distribution of a spatially incoherent extended source. The relative intensity of the superposing fields can be controlled by rotating the half wave plate 1 (HWP1) kept at the input of the Mach-Zehnder interferometer.

Fig. 2 The experimental set up for real-time coherence holography.

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The reconstruction part of the experiment is similar to one described in [2] in the case of reconstruction of 2-D objects. The field distribution of the incoherently illuminated hologram is Fourier transformed by lens L4 with a focal length 300mm and introduced into the interferometer through a half wave plate 3 (HWP3). Since the focal length of L4 is different from the distance between object plane and the ground glass plane, the reconstructed object wave is scaled accordingly. A polarizer P placed after the ground glass nulls any depolarization of the scattered light due to birefringence or multiple scattering in the ground glass material. 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. The resulting interference gives a 2-D correlation map that reconstructs the image as a coherence function represented by the fringe contrast. In our present setup with α = 1.1, the magnification for reconstruction becomes ${(α - α^{- 1})}^{- 1}$ = 5.23 so that the reconstructed image size fits the field aperture of the CCD camera. The half-wave plate 3 (HWP3) balances the amplitudes of the radially sheared interfering beams at the output of the interferometer. With an analyzer A with its axis kept at $45^{o}$ to the orientation of the polarization of the two beams, interference between the two beams was achieved.

4. Result

To observe the reconstructed raw image represented by fringe contrast in real-time and to quantify the coherence function by the Fourier transform method of fringe pattern analysis, we introduced a small tilt into one of the mirrors in the Sagnac interferometer and generated an interferogram with a spatial carrier frequency.

Shown in Fig. 3(a) is the interferogram captured by a 14-Bit cooled CCD camera (BITRAN BU-42L-14) at the interferometer output. 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 superposed fields created by all the possible random states of the rotating ground glass. A closer look at the interferogram reveals that high contrast fringes are observed at locations corresponding to the letters U, E and C, and the central area which corresponds to the zero-th order noise in conventional holography. Figure 3(b) shows the Fourier transform spectra of the interferogram. The spectrum around one of the 1st order spectral peaks on the carrier frequency location was band-pass filtered, brought to centre, and then inverse Fourier transformed to obtain the complex amplitude of the fringe pattern which gives the coherence function $Γ (Δ x, Δ y, Δ z = 0)$ . The modulus of the coherence function obtained by the Fourier transform method is shown in Fig. 3(c), which demonstrate the validity of the proposed principle. The resolution of the reconstructed object wave will be limited by the spatial carrier frequency in the interference image.

Fig. 3 Reconstructed images: (a) Raw intensity image resulted from shearing interference in real time described by Eq. (7); (b) Fourier spectrum of the interference image and (c) Contrast image representing the coherence function described by Eq. (8).

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5. Conclusions

We proposed and experimentally demonstrated object wave reconstruction using coherence holography in real-time. Because the Sagnac common path interferometer is very stable to environmental noises caused by vibrations and air turbulences, the reconstruction process is highly reliable. In our present experiment we have used a 2-D object because the lensless Fourier transform method is used for the generation of hologram. But the reconstruction method using Sagnac radial shearing interferometer can also reconstruct wavefronts from 3-D objects without any further modification. For 3-D object wave reconstruction, we only need to focus the CCD camera onto a different depth location at the output of Sagnac radial shearing interferometer. As the radial shear introduced in the transverse direction does not depend on depth, the object wave reconstructed at different depths has the same magnification.

Acknowledgement

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

References and links

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Real-time coherence holography

Abstract

1. Introduction

2. Principles

3. Experiments

4. Result

5. Conclusions

Acknowledgement

References and links

Cited By

Figures (3)

Equations (8)

Optics Express