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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 12 — Jun. 6, 2011
  • pp: 11558–11567
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Vectorial coherence holography

Rakesh Kumar Singh, Dinesh N. Naik, Hitoshi Itou, Yoko Miyamoto, and Mitsuo Takeda  »View Author Affiliations


Optics Express, Vol. 19, Issue 12, pp. 11558-11567 (2011)
http://dx.doi.org/10.1364/OE.19.011558


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Abstract

Extension of coherence holography to vectorial regime is investigated. A technique for controlling and synthesizing optical fields with desired elements of coherence-polarization matrix is proposed and experimentally demonstrated. The technique uses two separate coherence holograms, each of which is assigned to one of the orthogonal polarization components of the vectorial fields.

© 2011 OSA

1. Introduction

Coherence Holography (CH) is a technique of unconventional holography that can control a spatial coherence function using an incoherently illuminated hologram [1

1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629. [CrossRef] [PubMed]

,2

2. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633. [CrossRef] [PubMed]

]. 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

1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629. [CrossRef] [PubMed]

6

6. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006). [CrossRef] [PubMed]

]. 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

3. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000). [CrossRef]

,4

4. P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. 48(34), H40–H47 (2009). [CrossRef] [PubMed]

], dispersion-free spatial coherence tomography [5

5. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109. [CrossRef] [PubMed]

], and generation of coherence vortex [6

6. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006). [CrossRef] [PubMed]

]. A relevant technique of variable coherence tomography for estimating statistical properties of the random media has also been investigated [7

7. E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29(11), 1233–1235 (2004). [CrossRef] [PubMed]

]. Coherence plays a crucial role in imaging [8

8. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951). [CrossRef]

10

10. H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

], and an illuminating system with controllable coherence has been proposed [11

11. A. W. McCollough and G. M. Gallatin, “Illumination system with spatially controllable partial coherence compensation for line width variance in a photolithographic system,” US Patent 6628370 B1 (2003).

]. 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

12. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-25-20418. [CrossRef] [PubMed]

], as well as in the cases that deal with inhomogeneously polarized fields [13

13. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

19

19. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-4-1063. [CrossRef] [PubMed]

]. Great efforts have been made to unify the theory of coherence and polarization [13

13. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

]. 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

17. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

] and generalized Stokes parameters [20

20. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). [CrossRef] [PubMed]

] have been introduced. Other efforts of generalization include introduction of the degree of coherence for electromagnetic field [19

19. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-4-1063. [CrossRef] [PubMed]

], definition of complex degree of polarization [21

21. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004). [CrossRef] [PubMed]

], and extension of van Cittert Zernike theorem to a vectorial regime [22

22. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000). [CrossRef]

24

24. T. Shirai, “Some consequences of the van Cittert-Zernike theorem for partially polarized stochastic electromagnetic fields,” Opt. Lett. 34(23), 3761–3763 (2009). [CrossRef] [PubMed]

]. 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

12. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-25-20418. [CrossRef] [PubMed]

,25

25. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]

,26

26. M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009). [CrossRef]

]. Synthesis of electromagnetic Gaussian Schell-model source is recently proposed by making use of spatially incoherent source and generalized van-Cittert Zernike theorem [26

26. M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009). [CrossRef]

]. 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

21. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004). [CrossRef] [PubMed]

23

23. A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express 17(3), 1746–1752 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-3-1746. [CrossRef] [PubMed]

]. 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

27. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002). [CrossRef]

,28

28. D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011). [CrossRef] [PubMed]

]. 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

28. D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011). [CrossRef] [PubMed]

]. 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

1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629. [CrossRef] [PubMed]

,2

2. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633. [CrossRef] [PubMed]

]. 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
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.
shows an example of recording Ex and Ey field components of an objectgx(r)andgy(r)independently in two Fourier transform holograms Hx(r^)and Hy(r^). The object consists of two numerals “0” and “1” emitting, respectively, x-polarized light and y-polarized light with their Fourier spectra Gx(r^)andGy(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
Ei(r^)=Hi(r^)exp[iϕi(r^)](Subscript: i=x,y),
(1)
whereHi(r^)=|Hi(r^)|exp[iφi(r^)] with |Hi(r^)| andφi(r^) being the amplitude transmittance of the hologram and a deterministic phase of the read out light, respectively, and ϕi(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

E˜i(r˜)=Ei(r^)exp(i2πr˜r^λf)dr^=Hi(r^)exp[iϕi(r^)]exp(i2πr˜r^λf)dr^
(2)

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
Wij(r˜1,r˜2)=E˜i*(r˜1)E˜j(r˜2)=Ei*(r^1)Ej(r^2)exp[i2πλf(r˜2r^2r˜1r^1)]dr^1dr^2=Hi*(r^1)Hj(r^2)exp[iϕi(r^1)]exp[iϕj(r^2)]exp[i2πλf(r˜2r^2r˜1r^1)]dr^1dr^2,
(3)
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. ϕi(r^)=ϕj(r^), and model the ground glass as a delta-correlated phase screen, i.e. exp[iϕi(r^1)]exp[iϕj(r^2)]=δ(r^2r^1). Then Eq. (3) transforms into van Cittert-Zernike theorem for electromagnetic field
Wij(Δr˜)=WijH(r^)exp(i2πλfΔr˜r^)dr^=Hi*(r^)Hj(r^)exp(i2πλfΔr˜r^)dr^,
(4)
where Δr˜=r˜2r˜1, and WijH(r^)=Hi*(r^)Hj(r^)is elements of a 2x2 polarization matrix on the hologram plane, which are real for i=jand can be complex for ij. 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.

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
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.
. A linearly polarized beam from a He-Ne laser at wavelength 633nm is oriented at 45by 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 45enters 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.

Working principle of the imaging geometry is shown in Fig. 3
Fig. 3 Triangular Sagnac geometry used to image orthogonal polarization components on ground glass plane.
. 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.

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
Fig. 4 Elements of coherence-polarization matrix: amplitude distribution of (a) Wxx(Δr˜) (b) Wyy(Δr˜) (c) Wxy(Δr˜) (d) Wyx(Δr˜); phase distribution of (e) Wxx(Δr˜) (f) Wyy(Δr˜) (g) Wxy(Δr˜), and (h) Wyx(Δr˜).
and 5
Fig. 5 Elements of coherence-polarization matrix: amplitude distribution of (a) Wxx(Δr˜) (b) Wyy(Δr˜) (c) Wxy(Δr˜) (d) Wyx(Δr˜); phase distribution of (e) Wxx(Δr˜) (f) Wyy(Δr˜) (g) Wxy(Δr˜), and (h) Wyx(Δr˜).
, 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.

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
WiiH(r^1)=Hi(x^1,y^1)Hi*(x^1,y^1)=|Gi(x^1,y^1)|+(1/2)[Gi(x^1,y^1)+Gi*(x^1,y^1)],
(9)
the x and y polarized objects are reconstructed as the 2-D amplitude distributions ofWxx(Δr˜)and Wyy(Δr˜), 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 Wxy(Δr˜), while those for Wyx(Δr˜)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

WxyH(r^1)=Hx(x^1,y^1)Hy*(x^1,y^1)=|Hx(x^1,y^1)||Hy(x^1,y^1)|exp{i[φx(x^1,y^1)φy(x^1,y^1)]},
(10)

Because of the multiplication of the hologram transmittance in the Fourier domain, the reconstructed images of Wxy(Δr˜)and Wyx(Δr˜)can be interpreted as a cross-correlation between the images that would be reconstructed from the individual holograms Hx(x^1,y^1) and Hy(x^1,y^1). Strictly, the images for Wxx(Δr˜)and Wyy(Δr˜)were reconstructed from the holograms |Hx(x^1,y^1)|2 and|Hy(x^1,y^1)|2, and not resulted from the holograms Hx(x^1,y^1)and Hy(x^1,y^1). Nonetheless they allow one to estimate the images of Wxy(Δr˜)and Wyx(Δr˜) as being approximated by the cross-correlations between the images for Wxx(Δr˜)andWyy(Δr˜). Due to the cross-correlation with the central peaks in Wxx(Δr˜)and Wyy(Δr˜), weak images of the numerals 0 and 1 are observed in Fig. 4(c) and Fig. 4(d). Unlike the case of WxxH(r^1)and WyyH(r^1), the complex characteristic of the transmittance function WxyH(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 φx(x^1,y^1)φy(x^1,y^1)in the complexWxyH(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 2Δx^1and 2Δy^1as shown is Fig. 3, so that

φx(x^1,y^1)φy(x^1,y^1)=k{[(x^1+Δx^1)2+(y^1+Δy^1)2][(x^1Δx^1)2+(y^1Δy^1)2]}/2fk(x^1Δx^1+y^1Δy^1)/f
(11)

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 Δx^1=0andΔy^1=0. This means that complex characteristics of WxyH(r^1)ceases to exist for full overlapping case. Figures 4(g) and 4(h) represent phase structure of Wxy(Δr˜) and Wyx(Δr˜) respectively. Reverse in positional shift in Wyx(Δr˜)in comparison to Wxy(Δr˜)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 Wxx(Δr˜)and Wyy(Δr˜).

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 WxyH(r^1) andWyxH(r^1). Results of Wxy(Δr˜)and Wyx(Δr˜)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.

References and links

1.

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629. [CrossRef] [PubMed]

2.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633. [CrossRef] [PubMed]

3.

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000). [CrossRef]

4.

P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt. 48(34), H40–H47 (2009). [CrossRef] [PubMed]

5.

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109. [CrossRef] [PubMed]

6.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006). [CrossRef] [PubMed]

7.

E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29(11), 1233–1235 (2004). [CrossRef] [PubMed]

8.

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951). [CrossRef]

9.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953). [CrossRef]

10.

H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).

11.

A. W. McCollough and G. M. Gallatin, “Illumination system with spatially controllable partial coherence compensation for line width variance in a photolithographic system,” US Patent 6628370 B1 (2003).

12.

D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-25-20418. [CrossRef] [PubMed]

13.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

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J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts-Company, 2006).

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J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett. 34(16), 2429–2431 (2009). [CrossRef] [PubMed]

16.

J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-19-20105. [CrossRef] [PubMed]

17.

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]

18.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

19.

J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-4-1063. [CrossRef] [PubMed]

20.

O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). [CrossRef] [PubMed]

21.

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004). [CrossRef] [PubMed]

22.

F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000). [CrossRef]

23.

A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express 17(3), 1746–1752 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-3-1746. [CrossRef] [PubMed]

24.

T. Shirai, “Some consequences of the van Cittert-Zernike theorem for partially polarized stochastic electromagnetic fields,” Opt. Lett. 34(23), 3761–3763 (2009). [CrossRef] [PubMed]

25.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]

26.

M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009). [CrossRef]

27.

M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002). [CrossRef]

28.

D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011). [CrossRef] [PubMed]

29.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010). [CrossRef] [PubMed]

30.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

31.

D. N. Naik, R. K. Singh, H. Itou, Y. Miyamoto, and M. Takeda, “A highly stable interferometric technique for polarization measurement,” 156, Photonics (December 2010), IIT Guwahati, India.

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(090.2880) Holography : Holographic interferometry
(260.5430) Physical optics : Polarization

ToC Category:
Holography

History
Original Manuscript: March 31, 2011
Revised Manuscript: May 16, 2011
Manuscript Accepted: May 16, 2011
Published: May 31, 2011

Citation
Rakesh Kumar Singh, Dinesh N. Naik, Hitoshi Itou, Yoko Miyamoto, and Mitsuo Takeda, "Vectorial coherence holography," Opt. Express 19, 11558-11567 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11558


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References

  1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-23-9629 . [CrossRef] [PubMed]
  2. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-13-10633 . [CrossRef] [PubMed]
  3. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000). [CrossRef]
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  5. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14(25), 12109–12121 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12109 . [CrossRef] [PubMed]
  6. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett. 96(7), 073902 (2006). [CrossRef] [PubMed]
  7. E. Baleine and A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29(11), 1233–1235 (2004). [CrossRef] [PubMed]
  8. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. Lond. A Math. Phys. Sci. 208(1093), 263–277 (1951). [CrossRef]
  9. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953). [CrossRef]
  10. H. H. Hopkins, “Image formation with coherent and partially coherent light,” Photograph. Sci. Eng. 21, 114–122 (1977).
  11. A. W. McCollough and G. M. Gallatin, “Illumination system with spatially controllable partial coherence compensation for line width variance in a photolithographic system,” US Patent 6628370 B1 (2003).
  12. D. P. Brown and T. G. Brown, “Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging,” Opt. Express 16(25), 20418–20426 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-25-20418 . [CrossRef] [PubMed]
  13. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
  14. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts-Company, 2006).
  15. J. Sorrentini, M. Zerrad, and C. Amra, “Statistical signatures of random media and their correlation to polarization properties,” Opt. Lett. 34(16), 2429–2431 (2009). [CrossRef] [PubMed]
  16. J. Broky and A. Dogariu, “Complex degree of mutual polarization in randomly scattered fields,” Opt. Express 18(19), 20105–20113 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-19-20105 . [CrossRef] [PubMed]
  17. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998). [CrossRef]
  18. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]
  19. J. Tervo, T. Setala, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-24-4-1063 . [CrossRef] [PubMed]
  20. O. Korotkova and E. Wolf, “Generalized stokes parameters of random electromagnetic beams,” Opt. Lett. 30(2), 198–200 (2005). [CrossRef] [PubMed]
  21. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004). [CrossRef] [PubMed]
  22. F. Gori, M. Santarsiero, R. Borghi, and G. Piquero, “Use of the van Cittert-Zernike theorem for partially polarized sources,” Opt. Lett. 25(17), 1291–1293 (2000). [CrossRef]
  23. A. S. Ostrovsky, G. Martínez-Niconoff, P. Martínez-Vara, and M. A. Olvera-Santamaría, “The van Cittert-Zernike theorem for electromagnetic fields,” Opt. Express 17(3), 1746–1752 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-3-1746 . [CrossRef] [PubMed]
  24. T. Shirai, “Some consequences of the van Cittert-Zernike theorem for partially polarized stochastic electromagnetic fields,” Opt. Lett. 34(23), 3761–3763 (2009). [CrossRef] [PubMed]
  25. G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002). [CrossRef]
  26. M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Synthesis of electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 26(6), 1437–1443 (2009). [CrossRef]
  27. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213(4-6), 201–221 (2002). [CrossRef]
  28. D. N. Naik, R. K. Singh, T. Ezawa, Y. Miyamoto, and M. Takeda, “Photon correlation holography,” Opt. Express 19(2), 1408–1421 (2011). [CrossRef] [PubMed]
  29. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett. 35(10), 1728–1730 (2010). [CrossRef] [PubMed]
  30. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
  31. D. N. Naik, R. K. Singh, H. Itou, Y. Miyamoto, and M. Takeda, “A highly stable interferometric technique for polarization measurement,” 156, Photonics (December 2010), IIT Guwahati, India.

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