Vectorial coherence holography

Rakesh Kumar Singh; Dinesh N. Naik; Hitoshi Itou; Yoko Miyamoto; Mitsuo Takeda

doi:10.1364/OE.19.011558

1. Introduction

Coherence Holography (CH) is a technique of unconventional holography that can control a spatial coherence function using an incoherently illuminated hologram [1,2]. An object coherently recorded in a hologram or numerically recorded in a computer generated hologram (CGH) is reconstructed in terms of the distribution of a coherence function in contrast to conventional holography where object is reconstructed in terms of optical field [1–6]. Making use of CH as a means to control 2D and 3D spatial coherence functions, various applications have been proposed and demonstrated, among which are profilometry based on longitudinal spatial coherence [3,4], dispersion-free spatial coherence tomography [5], and generation of coherence vortex [6]. A relevant technique of variable coherence tomography for estimating statistical properties of the random media has also been investigated [7]. Coherence plays a crucial role in imaging [8–10], and an illuminating system with controllable coherence has been proposed [11]. However, applications of these techniques are all restricted to scalar optical fields.

Recently importance of the joint effect of polarization and coherence has been recognized in the context of imaging [12], as well as in the cases that deal with inhomogeneously polarized fields [13–19]. Great efforts have been made to unify the theory of coherence and polarization [13]. Single-point quantities such as a polarization matrix and Stokes parameters are not suitable to characterize such fields. To extend the concept of single-point quantities to two-point quantities, a coherence-polarization matrix [17] and generalized Stokes parameters [20] have been introduced. Other efforts of generalization include introduction of the degree of coherence for electromagnetic field [19], definition of complex degree of polarization [21], and extension of van Cittert Zernike theorem to a vectorial regime [22–24]. Despite their importance, most of these efforts have been intended for analysis and characterization of statistical vector fields, rather than for synthesis and control of the statistical vector fields. With increasing interest in the synergetic effect of polarization and coherence in various applications such as microscopy, lithography and tomography, it is highly desired to have a means to control both coherence and polarization properties of the source. Significant steps in this direction have been initiated recently in vectorial regime [12,25,26]. Synthesis of electromagnetic Gaussian Schell-model source is recently proposed by making use of spatially incoherent source and generalized van-Cittert Zernike theorem [26]. The primary spatially incoherent source, characterized by position dependent polarization matrix, is realized by two suitably amplitude modulated mutually un-correlated laser sources with help of a Mach-Zehnder interferometer. To implement the idea of controlled synthesis of statistical vector fields, we propose a new technique based on CH.

The purpose of this paper is to extend CH technique to vectorial regime. This extension provides a new technique to control both coherence and polarization properties of the field. To implement the extension of coherence holography to the vectorial regime, we use a pair of coherence holograms each of which is assigned to one of the orthogonal polarization components of the vectorial field. Whereas CH synthesizes and controls a coherence function as a reconstructed image, the proposed vectorial coherence holography (VCH) synthesizes and controls elements of coherence-polarization matrix using properly designed holograms. The principle for controlling the elements of coherence-polarization matrix can be explained on the basis of generalized van Cittert Zernike theorem for electromagnetic wave [21–23]. In statistical optics, ensemble average is usually replaced by time average under the assumption of ergodicity. However, it is also known that space average can replace ensemble average for a Gaussian model under ergodicity [27,28]. Recently we have reported a technique based on space averaging as a substitute of time averaging for the reconstruction of an object using intensity correlation [28]. This property of the Gaussian model gives liberty to adopt an experimental strategy for space averaging as replacement of ensemble averaging. Exploiting this characteristic, we have adopted a space-average model suitable for experiment. Spatially stationary random Gaussian electromagnetic fields were generated and detected so that ensemble average is replaced by space average. We demonstrate by experiment that desired spatial structures recorded in VCH are reconstructed in the components of a coherence-polarization matrix for two different examples of objects.

2. Principle

The principle of CH has been explained in detail in [1,2]. However, for convenience of explanation, we briefly review the essence of the basic principle of CH. In the scalar regime, a coherently recorded hologram is read out with incoherent light, and the object is reconstructed as the 3-D distribution of a coherence function, which is detected by using an appropriate interferometer. The principle of coherence holography is based on the similarity between the diffraction formula and van Cittert Zernike theorem. However extension of CH technique to vectorial regime demands control of all the four components of the coherence polarization matrix. One straight forward extension of CH to vectorial regime is based on using two holograms, each of which is assigned to one of the orthogonal polarization components of the object field. Figure 1 shows an example of recording $E_{x}$ and $E_{y}$ field components of an object $g_{x} (r)$ and $g_{y} (r)$ independently in two Fourier transform holograms $H_{x} (\hat{r})$ and $H_{y} (\hat{r})$ . The object consists of two numerals “0” and “1” emitting, respectively, x-polarized light and y-polarized light with their Fourier spectra $G_{x} (\hat{r})$ and $G_{y} (\hat{r})$ . Each of the holograms is read out with spatially incoherent light with the same state of polarization as in the recording process. Note that illumination strategy is devised in such a way that orthogonal polarization components pick up their corresponding transmittance of the holograms independent of others. Presence of two holograms for orthogonally polarized components and their read out with spatially incoherent lights of the corresponding states of polarization distinguish this technique from scalar coherence holography. Turning now our attention to the reconstruction process of the Fourier transform coherence holograms, let us denote the complex amplitudes of the orthogonal polarization components immediately behind the hologram as

E_{i} (\hat{r}) = H_{i} (\hat{r}) \exp [i ϕ_{i} (\hat{r})] \begin{matrix} (Subscript: i = x, y) \end{matrix},

where

H_{i} (\hat{r}) = | H_{i} (\hat{r}) | \exp [i φ_{i} (\hat{r})]

with

| H_{i} (\hat{r}) |

and

φ_{i} (\hat{r})

being the amplitude transmittance of the hologram and a deterministic phase of the read out light, respectively, and

ϕ_{i} (\hat{r})

is a random phase introduced to destroy spatial coherence by making light pass through a ground glass in our experiment. The scattered field is optically Fourier transformed with a Fourier transform lens of focal length f so that the fields on the observation becomes

Fig. 1 (a) Holographic recording of E _x component of the object field: (b) Holographic recording of E _y component of the object field. Two holograms are used to reconstruct desired images in the components of a coherence-polarization matrix.

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{\tilde{E}}_{i} (\tilde{r}) = \int E_{i} (\hat{r}) \exp (- i \frac{2 π \tilde{r} \cdot \hat{r}}{λ f}) d \hat{r} = \int H_{i} (\hat{r}) \exp [i ϕ_{i} (\hat{r})] \exp (- i \frac{2 π \tilde{r} \cdot \hat{r}}{λ f}) d \hat{r}

Second-order correlation properties of the electromagnetic field can be characterized with the help of a 2x2 coherence-polarization matrix whose elements are written as

\begin{array}{l} W_{i j} ({\tilde{r}}_{1}, {\tilde{r}}_{2}) = 〈 {\tilde{E}}_{i}^{*} ({\tilde{r}}_{1}) {\tilde{E}}_{j} ({\tilde{r}}_{2}) 〉 = \iint 〈 E_{i}^{*} ({\hat{r}}_{1}) E_{j}^{} ({\hat{r}}_{2}) 〉 \exp [- i \frac{2 π}{λ f} ({\tilde{r}}_{2} \cdot {\hat{r}}_{2} - {\tilde{r}}_{1} \cdot {\hat{r}}_{1})] d {\hat{r}}_{1} d {\hat{r}}_{2} \\ \begin{matrix}  \end{matrix} = \iint H_{i}^{*} ({\hat{r}}_{1}) H_{j}^{} ({\hat{r}}_{2}) 〈 \exp [- i ϕ_{i} ({\hat{r}}_{1})] \exp [i ϕ_{j} ({\hat{r}}_{2})] 〉 \exp [- i \frac{2 π}{λ f} ({\tilde{r}}_{2} \cdot {\hat{r}}_{2} - {\tilde{r}}_{1} \cdot {\hat{r}}_{1})] d {\hat{r}}_{1} d {\hat{r}}_{2}, \end{array}

where < > stands for ensemble average. We assume that x and y components pass through the same ground glass which is free from birefringence, i.e.

ϕ_{i} (\hat{r}) = ϕ_{j} (\hat{r})

, and model the ground glass as a delta-correlated phase screen, i.e.

〈 \exp [- i ϕ_{i} ({\hat{r}}_{1})] \exp [i ϕ_{j} ({\hat{r}}_{2})] 〉 = δ ({\hat{r}}_{2} - {\hat{r}}_{1})

. Then Eq. (3) transforms into van Cittert-Zernike theorem for electromagnetic field

W_{i j} (Δ \tilde{r}) = \int W_{i j}^{H} (\hat{r}) \exp (- i \frac{2 π}{λ f} Δ \tilde{r} \cdot \hat{r}) d \hat{r} = \int H_{i}^{*} (\hat{r}) H_{j}^{} (\hat{r}) \exp (- i \frac{2 π}{λ f} Δ \tilde{r} \cdot \hat{r}) d \hat{r},

where

Δ \tilde{r} = {\tilde{r}}_{2} - {\tilde{r}}_{1}

, and

W_{i j}^{H} (\hat{r}) = H_{i}^{*} (\hat{r}) H_{j}^{} (\hat{r})

is elements of a 2x2 polarization matrix on the hologram plane, which are real for

i = j

and can be complex for

i \neq j

. Off-diagonal elements of the polarization matrix, which may be regarded as a special form of a coherence-polarization matrix with its two observation points degenerated into a single point, represent correlation between orthogonal polarization components at the hologram plane. Similarly to the case of scalar CH, van Cittert-Zernike theorem for electromagnetic field expressed by Eq. (4) serves as the basis for designing VCH for synthesizing a vectorial field with a desired coherence-polarization matrix.

As is often the case, when the field is stationary and ergodic in time, ensemble average < > is replaced by time average < >_T. We can also think of a similar characteristic of random fields in space domain. Under the assumption of spatial ergodicity, one may replace ensemble averages < > with space averages < >_R. Depending on the averaging process, different experimental schemes can be adopted for the investigation of the stochastic field. As an alternative to generating a temporally fluctuating random field by using a rotating ground glass and taking its time average, one can generate a spatially fluctuating random field by passing light through a static ground glass and take its space average. Assuming that the field is stationary and ergodic in space and noting that ${\tilde{r}}_{2} = {\tilde{r}}_{1} + Δ \tilde{r}$ , we rewrite Eq. (3) in terms of spatial average with respect to variable ${\tilde{r}}_{1}$ over the observation plane:

\begin{array}{l} W_{i j} (Δ \tilde{r}) = {〈 {\tilde{E}}_{i}^{*} ({\tilde{r}}_{1}) {\tilde{E}}_{j} ({\tilde{r}}_{1} + Δ \tilde{r}) 〉}_{R} = \int {\tilde{E}}_{i}^{*} ({\tilde{r}}_{1}) {\tilde{E}}_{j} ({\tilde{r}}_{1} + Δ \tilde{r}) d {\tilde{r}}_{1} \\ \begin{matrix} \begin{array}{l} = \iint H_{i}^{*} ({\hat{r}}_{1}) \exp [- i ϕ_{i} ({\hat{r}}_{1})] H_{j}^{} ({\hat{r}}_{2}) \exp [i ϕ_{j} ({\hat{r}}_{2})] \\ \begin{matrix}  \end{matrix} \times {\int \exp [- i \frac{2 π}{λ f} ({\hat{r}}_{2} - {\hat{r}}_{1}) \cdot {\tilde{r}}_{1}] d {\tilde{r}}_{1}} \exp (- i \frac{2 π}{λ f} Δ \tilde{r} \cdot {\hat{r}}_{2}) d {\hat{r}}_{1} d {\hat{r}}_{2}, \end{array} \end{matrix} \\ \begin{matrix} = \int H_{i}^{*} ({\hat{r}}_{1}) H_{j}^{} ({\hat{r}}_{1}) \exp (- i \frac{2 π}{λ f} Δ \tilde{r} \cdot {\hat{r}}_{1}) d {\hat{r}}_{1} \end{matrix} = \int W_{i j}^{H} ({\hat{r}}_{1}) \exp (- i \frac{2 π}{λ f} Δ \tilde{r} \cdot {\hat{r}}_{1}) d {\hat{r}}_{1} \end{array}

where use has been made of the relation

\int \exp [- i \frac{2 π}{λ f} ({\hat{r}}_{2} - {\hat{r}}_{1}) \cdot {\tilde{r}}_{1}] d {\tilde{r}}_{1} = δ ({\hat{r}}_{2} - {\hat{r}}_{1})

and the fact that x and y components pass through the same ground glass which is free from birefringence, i.e.

ϕ_{i} (\hat{r}) = ϕ_{j} (\hat{r})

. Note that the result of Eq. (5) based on spatial average is same as Eq. (4) based on ensemble average. This justifies the use of spatial average in our experiment to be described in the next section.

3. Experiments

Different experimental schemes can be devised to observe a coherence-polarization matrix depending on the implementation of ensemble average either by time averaging or space averaging. To gain advantage of a single-shot recording of a static random field, we used space averaging as a substitute for the ensemble average, and the spatial distribution of the static random vector field is detected using a specially designed polarization interferometer. An experimental scheme for generation of scattered electromagnetic field and its detection is shown in Fig. 2 . A linearly polarized beam from a He-Ne laser at wavelength 633nm is oriented at $45^{\circ}$ by half-wave plate HWP1, spatially filtered with microscope objective O₁ and pinhole S and subsequently collimated by lens L₁. The collimated beam splits into two arms by non-polarizing beam splitter BS₁. The first arm of the interferometer (indicated by a blue line) produces two orthogonally polarized and mutually tilted reference beams with the help of a triangular Sagnac polarization interferometer geometry with a telescopic system formed by lens L2 and L3. The beam linearly polarized at $45^{\circ}$ enters polarization beam splitter PBS1, and splits into two beams which counter propagate in the Sagnac interferometer and exit from polarization beam splitter PBS1 as a pair of collimated and orthogonally polarized reference beams with an appropriate amount of tilt controlled by telescopic system L2 and L3 and mirrors M3 and M4. The second arm of the interferometer (indicated by a red line) generates statistically polarized fields with a desired coherence-polarization matrix by means of VCH.

Fig. 2 Experimental set-up for generation of random electromagnetic field and its detection using interferomteric method. O1 stands for microscope objective, S spatial filter and other terms have their usual meaning.

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Desired control of coherence-polarization matrix is realized as follows. Two holograms of the orthogonally polarized objects, as described earlier in Fig. 1, are placed next to each other and imaged on the ground glass using triangular Sagnac geometry with lens L₄.

Working principle of the imaging geometry is shown in Fig. 3 . Images of the two holograms are laterally shifted on the ground glass and their separation is controlled by mirror M₆. Holograms of the orthogonal polarization components are placed in such a way that both orthogonally polarized beams carry replicas of both holograms as images on the ground glass. Tilt of mirror M₆ is adjusted in such a way that both polarization components pick up holograms of different polarization components in overlapping area. Only the light inside this overlapping area is allowed to pass through the ground glass by a small aperture, and consequently holograms of both orthogonal polarization components experience same random structure of the ground glass. Scattering of the beam through the random glass plate produces a speckle pattern with Gaussian statistics. The split ratio of the amplitudes between the orthogonal polarization components on the hologram is adjusted by rotating half wave plate HWP2.

Fig. 3 Triangular Sagnac geometry used to image orthogonal polarization components on ground glass plane.

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The coherence holograms used in the experiment are computer generated holograms (CGHs) of off-axis objects. Both orthogonally polarized objects are of size 40X40 pixels and Fourier transform holograms are created numerically as shown in Fig. 1. In our experiments, we have selected two different kinds of objects such as numerals 0 and 1, and vortices with helical phase structure and a dark core. The complex amplitudes of the vortices are expressed by $r \exp (- r^{2} / w^{2}) \exp (i m θ)$ , where θ is the azimuthal angle on the transverse plane. The parameter w represents the size of the dark core, which is selected to be 12 pixels. The helical phase structure of the vortices is represented by integer m called topological charge. The values of m are chosen to be 1 and −1 for x and y polarized objects, respectively. Two different off-axis positions are selected for the case of the numeral objects. For example the objects 0 and 1 are placed at two corners in the first and second quadrants of the rectangular window with size 162X162 pixels. Optical field of the object on the hologram plane is obtained by Fourier transform and this is expressed as:

G_{i} (\hat{x}, \hat{y}) = | G_{i} (\hat{x}, \hat{y}) | \exp [i ψ_{i} (\hat{x}, \hat{y})]

Here

i = (x, y)

stands for the x- or y-polarization component of the orthogonally polarized objects, and ψ the phase of the diffracted field on the Fourier plane. In synthesizing the computer generated holograms (CGH), we remove from the interference fringe intensity the term

{| G (\hat{x}, \hat{y}) |}^{2}

which becomes the source of an unwanted autocorrelation image [29]. In contrast to conventional holography, the intensity (rather than amplitude) transmittance of the hologram is made proportional to the interference fringe pattern, such that

H_{i} (\hat{x}, \hat{y}) \propto \sqrt{| G_{i} (\hat{x}, \hat{y}) | [1 + \cos ψ_{i} (\hat{x}, \hat{y})]}

The two CGHs for the orthogonal polarization components are printed on transparent film sheets and used in experiment. The two holograms for the orthogonal polarization components are placed next to each other and are imaged by lens L4 through the Sagnac geometry with PBS2 in such a way that the two orthogonally polarized beams form mutually shifted replicas of the holograms on the ground glass, as shown in Fig. 3. The orthogonally polarized lights, which are allowed to pass through the aperture with the information about the two holograms, are scattered by the static ground glass to produce a random electromagnetic field in the detector plane. The orthogonal polarization components of the scattered field are detected using a spatial-frequency multiplex polarization interferometer combined with the Fourier fringe analysis [30,31]. Referring to Fig. 2, the two orthogonally polarized reference beams interfere with the corresponding polarization components of the scattered field with different tilt angles to produce a spatial-frequency multiplexed fringe pattern composed of two sets of interference fringes with a different spatial carrier frequency for each one of the polarization components. Making use of Fourier fringe analysis, the spectrum for each polarization component is discriminated in the spatial frequency domain and the information about the amplitude and phase is retrieved in the signal domain. A calibration arm is used to produce one extra beam with a known standard polarization state of 45 degree linear polarization for the purpose of calibration of systematic errors in polarization measurement. Two set of spatial frequency multiplexed interference fringe patterns for orthogonal polarization components are recorded for the combinations of the sample-and-reference and standard-and-reference fields. The complex amplitudes of the orthogonally polarized components of the light from the sample were calibrated against those for the known standard to compensate the systematic errors introduced by imperfect optical elements and misalignments. This process also removes the errors in the estimation of the spatial carrier frequency associated with Fourier fringe analysis. Complex field of orthogonal polarization components of the speckle field is obtained with respect to components of

45^{\circ}

calibration beam. Lens L6 and L7 are used for imaging interference pattern onto CCD plane with 2x magnification. The detailed description on the measurement technique has been reported elsewhere [31].

4. Results

The two sets of spatial frequency multiplexed interference fringe pattern for the orthogonal polarization components corresponding to the sample-and-reference fields and the standard-and-reference fields were recorded by a 14 bit cooled CCD camera (BITRAN BU-42L-14). The complex fields of orthogonally polarized components of the scattered fields were obtained by calibrating the measured values against those of the known standard beam with a 45 degree linearly polarized light. The amplitude and phase of the elements of coherence-polarization matrix are shown in Figs. 4 and 5 , respectively, for the objects representing numerals and vortices. Physical dimension in Figs. 4 and 5 is expressed in unit of pixel number where the unit pixel size is 7.4 micron with origin located at (150,150). The elements of the coherence-polarization matrices were calculated from the correlations between the 2-D distributions of the random polarization components based on space averaging.

Fig. 4 Elements of coherence-polarization matrix: amplitude distribution of (a) $W_{x x} (Δ \tilde{r})$ (b) $W_{y y} (Δ \tilde{r})$ (c) $W_{x y} (Δ \tilde{r})$ (d) $W_{y x} (Δ \tilde{r})$ ; phase distribution of (e) $W_{x x} (Δ \tilde{r})$ (f) $W_{y y} (Δ \tilde{r})$ (g) $W_{x y} (Δ \tilde{r})$ , and (h) $W_{y x} (Δ \tilde{r})$ .

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Fig. 5 Elements of coherence-polarization matrix: amplitude distribution of (a) $W_{x x} (Δ \tilde{r})$ (b) $W_{y y} (Δ \tilde{r})$ (c) $W_{x y} (Δ \tilde{r})$ (d) $W_{y x} (Δ \tilde{r})$ ; phase distribution of (e) $W_{x x} (Δ \tilde{r})$ (f) $W_{y y} (Δ \tilde{r})$ (g) $W_{x y} (Δ \tilde{r})$ , and (h) $W_{y x} (Δ \tilde{r})$ .

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Numerals 0 and 1 were used as x and y polarized objects for orthogonally polarized CGHs and the objects were reconstructed in the form of the spatial distributions of the elements of the coherence-polarization matrix as shown in Fig. 4. Due to Eq. (5) for the reconstruction based on spatial averaging and Eq. (8) for the hologram transmittance

W_{i i}^{H} ({\hat{r}}_{1}) = H_{i} ({\hat{x}}_{1}, {\hat{y}}_{1}) H_{i}^{*} ({\hat{x}}_{1}, {\hat{y}}_{1}) = | G_{i} ({\hat{x}}_{1}, {\hat{y}}_{1}) | + (1 / 2) [G_{i} ({\hat{x}}_{1}, {\hat{y}}_{1}) + G_{i}^{*} ({\hat{x}}_{1}, {\hat{y}}_{1})],

the x and y polarized objects are reconstructed as the 2-D amplitude distributions of

W_{x x} (Δ \tilde{r})

and

W_{y y} (Δ \tilde{r})

, respectively, as shown in Figs. 4(a) and 4(b). Their phase structures are shown in Figs. 4(e) and 4(f). Figures 4 (c) and 4(g) represent, respectively, the amplitude and phase distributions of

W_{x y} (Δ \tilde{r})

, while those for

W_{y x} (Δ \tilde{r})

are shown in Figs. 4(d) and 4(h), respectively. In this case, the interpretation of the reconstructed image is not straightforward because the hologram transmittance involves a complex cross term with a phase which is determined by the difference between the readout beams for the x and y components

W_{x y}^{H} ({\hat{r}}_{1}) = H_{x} ({\hat{x}}_{1}, {\hat{y}}_{1}) H_{y}^{*} ({\hat{x}}_{1}, {\hat{y}}_{1}) = | H_{x} ({\hat{x}}_{1}, {\hat{y}}_{1}) | | H_{y} ({\hat{x}}_{1}, {\hat{y}}_{1}) | \exp {i [φ_{x} ({\hat{x}}_{1}, {\hat{y}}_{1}) - φ_{y} ({\hat{x}}_{1}, {\hat{y}}_{1})]},

Because of the multiplication of the hologram transmittance in the Fourier domain, the reconstructed images of $W_{x y} (Δ \tilde{r})$ and $W_{y x} (Δ \tilde{r})$ can be interpreted as a cross-correlation between the images that would be reconstructed from the individual holograms $H_{x} ({\hat{x}}_{1}, {\hat{y}}_{1})$ and $H_{y} ({\hat{x}}_{1}, {\hat{y}}_{1})$ . Strictly, the images for $W_{x x} (Δ \tilde{r})$ and $W_{y y} (Δ \tilde{r})$ were reconstructed from the holograms ${| H_{x} ({\hat{x}}_{1}, {\hat{y}}_{1}) |}^{2}$ and ${| H_{y} ({\hat{x}}_{1}, {\hat{y}}_{1}) |}^{2}$ , and not resulted from the holograms $H_{x} ({\hat{x}}_{1}, {\hat{y}}_{1})$ and $H_{y} ({\hat{x}}_{1}, {\hat{y}}_{1})$ . Nonetheless they allow one to estimate the images of $W_{x y} (Δ \tilde{r})$ and $W_{y x} (Δ \tilde{r})$ as being approximated by the cross-correlations between the images for $W_{x x} (Δ \tilde{r})$ and $W_{y y} (Δ \tilde{r})$ . Due to the cross-correlation with the central peaks in $W_{x x} (Δ \tilde{r})$ and $W_{y y} (Δ \tilde{r})$ , weak images of the numerals 0 and 1 are observed in Fig. 4(c) and Fig. 4(d). Unlike the case of $W_{x x}^{H} ({\hat{r}}_{1})$ and $W_{y y}^{H} ({\hat{r}}_{1})$ , the complex characteristic of the transmittance function $W_{x y}^{H} ({\hat{r}}_{1})$ plays significant role in the reconstruction of non-diagonal components of the coherence-polarization matrix. Note that Figs. 4(c) and 4(d) represent shifts of the images off from the center, which is due to the residual linear phase term $φ_{x} ({\hat{x}}_{1}, {\hat{y}}_{1}) - φ_{y} ({\hat{x}}_{1}, {\hat{y}}_{1})$ in the complex $W_{x y}^{H} ({\hat{r}}_{1})$ . This linear phase term originates from the fact that, in imaging two holograms onto the ground glass, two spherical wavefronts with a radius of curvature f are mutually translated by $2 Δ {\hat{x}}_{1}$ and $2 Δ {\hat{y}}_{1}$ as shown is Fig. 3, so that

\begin{array}{l} φ_{x} ({\hat{x}}_{1}, {\hat{y}}_{1}) - φ_{y} ({\hat{x}}_{1}, {\hat{y}}_{1}) = k {[{({\hat{x}}_{1} + Δ {\hat{x}}_{1})}^{2} + {({\hat{y}}_{1} + Δ {\hat{y}}_{1})}^{2}] - [{({\hat{x}}_{1} - Δ {\hat{x}}_{1})}^{2} + {({\hat{y}}_{1} - Δ {\hat{y}}_{1})}^{2}]} / 2 f \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \begin{matrix} \approx k ({\hat{x}}_{1} Δ {\hat{x}}_{1} + {\hat{y}}_{1} Δ {\hat{y}}_{1}) / f \end{matrix} \end{array}

The linear phase produces polarization modulation on the random glass plate and consequently introduces the positional shifts in non-diagonal terms of coherence-polarization matrix. Presence of residual linear phase is transformed into position shift in Fourier plane and this shift is equal to 0.11 mm in our results for beam with 7.5 mm. However this position shift vanishes for full overlapping of the orthogonal components on the random glass plane, because this corresponds to the case for which $Δ {\hat{x}}_{1} = 0$ and $Δ {\hat{y}}_{1} = 0$ . This means that complex characteristics of $W_{x y}^{H} ({\hat{r}}_{1})$ ceases to exist for full overlapping case. Figures 4(g) and 4(h) represent phase structure of $W_{x y} (Δ \tilde{r})$ and $W_{y x} (Δ \tilde{r})$ respectively. Reverse in positional shift in $W_{y x} (Δ \tilde{r})$ in comparison to $W_{x y} (Δ \tilde{r})$ arises due to conjugation of phase term in the transmittance function.

We use same technique to create vortices in the elements of coherence-polarization matrix. Note that, unlike the case of numerals, two vortices for orthogonal polarization components are positioned at the same location in generating CGH. However the topological charges for these two vortices are 1 and −1. Using the space averaging process as mentioned previously, objects are reconstructed in the spatial distribution of the corresponding elements of the coherence-polarization matrix as shown in Fig. 5. Figures 5(a) and 5(b) represents vortices with opposite unit topological charges in the amplitude distributions of $W_{x x} (Δ \tilde{r})$ and $W_{y y} (Δ \tilde{r})$ .

Their phase structures, as encircled by white ring, are shown in Figs. 5(e) and 5(f) respectively. Whereas Figs. 5(c) and 5(d) represent properties of the optical field immediately behind the hologram. The positional shift in the structure of non-diagonal terms of coherence matrix also observed as noticed in Fig. 4 because of residual linear phase term in $W_{x y}^{H} ({\hat{r}}_{1})$ and $W_{y x}^{H} ({\hat{r}}_{1})$ . Results of $W_{x y} (Δ \tilde{r})$ and $W_{y x} (Δ \tilde{r})$ possesses two lobes structure for x and y polarized objects with unit and opposite topological charge. These results due to the cross-correlation of the x and y polarized field, which leads to two side lobes for this particular condition. Position and number of lobes in non-diagonal terms of the polarization- coherence matrix is controlled by topological charge of the vortex encoded into holograms.

5. Conclusions

In conclusion, we have extended coherence holography technique to vectorial regime and orthogonally polarized objects are reconstructed in the corresponding elements of the coherence-polarization matrix. More flexible control in the non-diagonal components of the coherence polarization matrix is possible by exploiting the liberty in the complex characteristics of the transmittance function on source plane

Acknowledgments

Part of this work was supported by JSPS Fellowship 21・09071, and Grant-in-Aid of JSPS B (2) No. 21360028.

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Vectorial coherence holography

Abstract

1. Introduction

2. Principle

3. Experiments

4. Results

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (11)

Optics Express