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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 18 — Aug. 27, 2012
  • pp: 19658–19669
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Coherence holography by achromatic 3-D field correlation of generic thermal light with an imaging Sagnac shearing interferometer

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


Optics Express, Vol. 20, Issue 18, pp. 19658-19669 (2012)
http://dx.doi.org/10.1364/OE.20.019658


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Abstract

We propose a new technique for achromatic 3-D field correlation that makes use of the characteristics of both axial and lateral magnifications of imaging through a common-path Sagnac shearing interferometer. With this technique, we experimentally demonstrate, for the first time to our knowledge, 3-D image reconstruction of coherence holography with generic thermal light. By virtue of the achromatic axial shearing implemented by the difference in axial magnifications in imaging, the technique enables coherence holography to reconstruct a 3-D object with an axial depth beyond the short coherence length of the thermal light.

© 2012 OSA

1. Introduction

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 shift 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. The other possible solution is to introduce a tilt between the wave fronts of the interfering beams to generate a carrier fringe in the interferogram and use the Fourier transform method of fringe analysis to find the fringe contrast and phase [4

4. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010). [CrossRef] [PubMed]

, 10

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

]. This allows the reconstruction of the object as a complex coherence function using only a single interferogram. The only drawback of this technique is that the resolution of the reconstructed image is limited by the finite frequency pass-band of the filtered carrier fringe pattern. The robustness of the Sagnac common path interferometer to environmental noises such as vibrations and air turbulences allowed the use of multiple interferograms. To overcome the limitations associated with the geometric phase shifter and the Fourier transform method of fringe analysis, we proposed an alternative technique called phase-shift coherence holography [3

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

], in which a set of phase-shifted computer-generated Fourier transform holograms displayed on a spatial light modulator (SLM) can achieve the phase shift without mechanical moving components. This made the system simple and enabled faster image reconstruction while maintaining the advantage of the Sagnac common path interferometer.

2. Principles

The principle of coherence holography has been described in detail in [1

1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005). [CrossRef] [PubMed]

]. The recording process of a coherence hologram is identical to that of a conventional hologram except that the intensity transmittanceIS(rS), rather than the amplitude transmittance, of the hologram is made proportional to the recorded interference fringe intensity. The recording process of a coherence hologram is identical to that of a conventional hologram, in which an object at point rP is recorded with a reference beam from a point source atrR.

However, the reconstruction process is completely different. Instead of illuminating the hologram with coherent light, we illuminate the hologram with spatially incoherent quasi-monochromatic light so that the hologram represents the irradiance distribution of a spatially incoherent extended source, as shown in Fig. 1
Fig. 1 Geometry of Fourier transform coherence holography using a commercial projector for display of hologram.
. In this case, the relation between the intensity transmittance of the hologram and the mutual intensity Γ(rP,rR)given byu(rP)u*(rR)is described by the van Cittert-Zernike theorem [12

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

, 13

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

]. By virtue of the formal analogy between the van Cittert-Zernike theorem and the diffraction formula, the mutual intensityΓ(rP,rR)has the same distribution as the optical field which would be reconstructed from a conventional hologram illuminated by a coherent beam converging towards the pointrR. In principle, the mutual intensity can be detected by means of a Young’s interferometer as shown in Fig. 1, but the point probing by sequential scanning is impractical. We assume that the Fourier transform of the incoherently illuminated hologram is stationary in space, and reduce Γ(rP,rR)toΓ(rPrR). This allows simultaneous measurement of spatial coherence function as a function ofrPrRwith the help of an appropriate interferometer.

2.1 Generation of phase-shifted coherence holograms

Let us consider a Fourier transform hologram for a 3-D object g(x,y,z)=|g(x,y,z)|exp[iϕ(x,y,z)]placed in the front focal region of the Fourier transform lens with focal length f. A conceptual diagram of the generation of the hologram is shown in Fig. 2
Fig. 2 Generation of a Fourier transform coherence hologram.
.

The complex amplitude at the hologram plane is given by
G(x^,y^){g(x,y,z)exp[i2πλf(xx^+yy^)]dxdy}exp[ikz(x^,y^)z]dz,
(1)
Hereλ is the wavelength of light, and f is the distance between z = 0 and the Fourier transform lens for recording, 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 x^andy^. The termexp[ikz(x^,y^)z]accounts for the defocus, and propagates the angular spectra of the field by distance z with kz(x^,y^)=2πλ1(x^f)2(y^f)2.

These optical fields are calculated for the given object letters U, E and C, and used to generate the coherence holograms. A set of phase-shift holograms are generated numerically by giving known phase shifts to the object spectrum.
G(x^,y^;m)=G(x^,y^)exp(i2πm/N),
(2)
where m = 0,1,2,...,N. In synthesizing the computer-generated coherence hologram, we removed from the interference fringe intensity the term |G(x^,y^)|2that becomes the source of unwanted autocorrelation image. Unlike conventional holography, the intensity (rather than amplitude) transmittance of the hologram was made proportional to the interference fringe pattern such that

H(x^,y^;m)|G(x^,y^)|+12G(x^,y^)exp(i2mπN)+12G*(x^,y^)exp(i2mπN)|G(x^,y^)|{1+cos[Φ(x^,y^)+2mπN]}
(3)

2.2 Reconstruction using phase-shift coherence holography

The hologram is illuminated with spatially incoherent light which is modeled as an optical field with unit amplitude and instantaneous random phase ΦR(x^,y^)in the hologram plane. The instantaneous field created on the rear focal plane of the Fourier transform lens used for reconstruction is given by
u(x,y,z;m)=H(x^,y^;m)exp[iΦR(x^,y^)]exp[ikz(x^,y^)z]exp[i2πλf(xx^+yy^)]dx^dy^,
(4)
where the square root transforms the intensity transmittance of the hologram to amplitude transmittance. Here kz(x^,y^)=2πλ1(x^f)2(y^f)2, where λ is the wavelength of light. Note that an instantaneous time parameter is implicitly included in ΦR(x^,y^) and u(x,y,z;m). Though u(x,y,z;m) is a spatially random field, and gives no directly observable image by itself, the information of the recorded object is encoded in its coherence function. To detect the coherence function, u(x,y,z;m) is directed into an interferometer. We correlate the field with a shearing interferometer and find the mutual intensity or coherence function Γ(Δx,Δy,Δz)between a pair of points. The field intensity at the output of the shearing interferometer is given by
I(Δx,Δy,Δz;m)=|u(x1,y1,z1;m)+u(x2,y2,z2;m)|2=2Γ(0,0,0;m)+2Re[Γ(Δx,Δy,Δz;m)],
(5)
where <...> denotes ensemble average, and
Γ(0,0,0;m)=u*(x1,y1,z1;m)u(x1,y1,z1;m)=u*(x2,y2,z2;m)u(x2,y2,z2;m)=H(x^,y^;m)dx^dy^
(6)
Γ(Δx,Δy,Δz;m)=u*(x1,y1,z1;m)u(x2,y2,z2;m)=H(x^,y^;m)exp[ikz(x^,y^)Δz]exp[i2πλf(x^Δx+y^Δy)]dx^dy^
(7)
In arriving at Eq. (6) and Eq. (7), we have made use of the fact that the hologram is illuminated by spatially incoherent quasi-monochromatic light with high temporal coherence, for which we haveexp[iΦR(x^,y^)]exp[iΦR(x^',y^')]=δ(x^x^',y^y^'). Note that the random field u(x,y,z;m) is assumed to be stationary both in time and space, so that the mutual intensity depends on the difference of the coordinatesΔx=x2x1,Δy=y2y1and Δz=z2z1. On substituting Eq. (3) into Eq. (7), we have
Γ(Δx,Δy,Δz;m)=g˜(Δx,Δy,Δz)+12g(Δx,Δy,Δz)exp(i2mπN)+12g*(Δx,Δy,Δz)exp(i2mπN)
(8)
where
g˜(Δx,Δy,Δz)=|G(x^,y^)|exp[ikz(x^,y^)Δz]exp[i2πλf(x^Δx+y^Δy)]dx^dy^.
(9)
Finally the field intensity at the output of the shearing interferometer is given by
I(Δx,Δy,Δz;m)=Re{2g˜(0,0,0)+2g˜(Δx,Δy,Δz)+g(0,0,0)exp(i2mπN)+g(Δx,Δy,Δz)exp(i2mπN)+g*(0,0,0)exp(i2mπN)+g*(Δx,Δy,Δz)exp(i2mπN)}
(10)
The first term generates a uniform background. The third and the fifth terms also generate uniform backgrounds which is harmful because they vary with phase shift. However, these terms can be eliminated by placing the object far off from the optical axis such thatg(0,0,0)=0. The second term generates an unwanted distribution with a peak in the center, but the influence of this term can be removed by the phase shift technique because it does not change with phase shift. The fourth and the sixth terms represent the desired image and its conjugate image, respectively. They can be retrieved individually by the phase shift technique because their spatial overlap can be avoided by the off-axis placement of the object. The field intensity I(Δx,Δy,Δz;m)varies sinusoidally with the phase shift introduced by the parameter m, which allows one to determine the amplitude and the phase of the object g(x,y,z)by sinusoidal fitting algorithm.

Unlike conventional holography where a constant bias is added to the real part of the phase shifted object spectrum to make the hologram nonnegative, |G(x^,y^)|is added to make H(x^,y^;m)positive as shown in Eq. (3). Therefore this spatially varying bias term gives rise to a distribution 2g˜(Δx,Δy,Δz)around the 0th order central spot. But as mentioned earlier, this term will not vary with phase shifts and hence can be eliminated by the phase shifting technique. The advantage of adding |G(x^,y^)|over adding a constant value is that the unwanted DC term 2g˜(0,0,0)that is present everywhere in the interferogram will have a minimum value. Experimentally, this will help to improve the fringe visibility and hence enhance the quality of the reconstructed object as a coherence function.

2.3 Radial and axial shearing for 3-D field correlation

To simultaneously reconstruct the whole image by coherence holography, we need a suitable interferometer that gives a 3-D correlation map with correlation lengths covering the full 3-D image field. The light from the incoherently illuminated hologram is directed into a properly designed radial shearing interferometer that can introduce a combination of radial shear and axial shear Δr=r2r1 proportional to the radial and axial components of the position vector r. Thereby the reconstructed image, which is represented by the spatial distribution of the complex coherence function, can be visualized as an interference fringe contrast and fringe phase.

As shown in Fig. 3
Fig. 3 Radial and axial shearing Sagnac common path interferometer.
, telescopic lens system comprising of lenses L1 and L2 with focal lengths f1 and f2respectively is introduced inside a Sagnac common path interferometer. Figure 4
Fig. 4 Telescopic system for 3-D optical field correlation.
is drawn to analyze the radial and axial shearing properties of the common path interferometer.

2.4 Measurement of purely spatial coherence of optical field

Separability of spatial and temporal coherence of optical field has been a subject of critical issue. The measurement of one without being influenced by the other is indeed a difficult task because the longitudinal spatial coherence and the temporal coherence of the optical field are often tied together though the propagation of light. Figure 5
Fig. 5 Longitudinal shear with time delay compensation.
depicts the propagation of light with low temporal coherence through a Sagnac radial shearing interferometer.

To explain how the influence of temporal coherence can be excluded by this system, let us first focus our attention to the relation between the longitudinal shear Δz=z2z1and time delayΔt, leaving out lateral parameters. Generally, for light with finite temporal coherence, the interference between the optical fields u(z1,t)andu(z2,t), which is described by the coherence functionΓ(z1,z2,Δt=0)=u*(z1,t)u(z2,t), would render interference fringes with low contrast if the time delay Δz/cintroduced by path difference is greater than temporal coherence length of the source, c being the velocity of light. In other words, the lack of temporal coherence manifests itself as the degradation of spatial coherence through the propagation time delay introduced by the longitudinal shear. However, this influence of the short temporal coherence of the light source can be automatically corrected in our imaging Sagnac shearing interferometer. Figure 5 shows two counter propagating beams in the Sagnac interferometer with their directions of propagation made equal and with their common output plane located atz˜=0. By the telescopic imaging system in the Sagnac interferometer, the optical fields u(z1,t) and u(z2,tΔt) are brought together to interfere on the output plane atz˜, conjugate to the image sensor plane. We haveΓ(z1,z2,Δt)=u*(z1,t)u(z2,tΔt), withΔt=Δz/c. This time delay accounts for the difference in optical path lengths of the input rays to reach z1 and z2 from the input plane, as can be seen from the axial principal rays of the telescope. This time delay introduced in the input side perfectly compensates the time delay introduced in the output side where optical fields u(z1,t)and u(z2,t)are superposed. This ensures that the temporal coherence has the maximum value for any amount of correlation lengthΔr. This is a situation where there is path delay but no time delay between the interfering wave packets, which can be understood intuitively from the fact that the total path lengths are the same for the two counter propagating beams in the Sagnac interferometer. Because of this achromaticity of the Sagnac interferometer, the distribution of the coherence function that reconstructs the object is purely spatial. This allows reconstruction even with a broad band source like white light as the short temporal coherence of light does not influence the 3-D coherence function that reconstructs the image. In practice, however, it is recommendable to use light with narrow spectral band width to get around the chromatic aberration of the lenses inside the Sagnac interferometer.

The radial shear ΔxandΔyis same for every axial shear Δzwhich means that reconstructed image has a lateral magnification which is independent of its axial location. Note that, since Δr=(Δx,Δy,Δz)=(mx1x˜,my1y˜,mz1z˜) with mx=my=(αα1)1 andmz=(α2α2)1, the magnification of the reconstructed image can be controlled by changing the shearing scale parameters (mx,my,mz) throughα. This gives the possibility for an unconventional coherence imaging microscope, which is endowed with an entirely new function of variable coherence zooming enabled by the controllable shearing scale parameters.

3. Experiment

Coherence holograms used in the experiment is a set of numerically generated phase shifted Fourier transform holograms. The alphabets U, E and C with sizes about 80x80 pixels each kept at different depth locations were used as objects. The Fourier transform of these off-axis binary objects are used to generate the coherence hologramH(x^,y^;m) synthetically as described in Eq. (3). Figure 6
Fig. 6 One of the phase shifted coherence holograms used in the experiment.
shows one of the phase shifted coherence holograms used in the experiment.

The first part of the experimental set-up shown upper right in Fig. 7
Fig. 7 Experimental set up for generic coherence holography.
is for the display of the incoherently illuminated coherence hologram, which is implemented by a commercial projector CASIO XJ-S41 DLP having a resolution of 1280x1024 pixels. The magnification of the projected hologram is adequately controlled by proper choice of lens L1. The hologram is imaged onto the front focal plane of lens L2. An interference filter T with a bandwidth of Δλ=3nmat λ=632.8nmis used so that the chromatic aberrations of the optical elements do not influence the reconstructed image where as the polarizer P is used to make the incoherent light linearly polarized. This gives the temporal coherence length of the light133μm, which is still significantly shorter than the longitudinal shear introduced by imaging, and serves for the validation of our system that is free from the influence of short temporal coherence.

The field distribution of the incoherently illuminated hologram is Fourier transformed by lens L2 with focal length 250mm and introduced into the interferometer through a half wave plate (HWP). A polarizing beam splitter (PBS) splits the incoming beam into two counter propagating beams that are power balanced by rotating HWP. The telescopic system with magnification α=1.1, formed by lenses L3 (focal length 220mm) and L4 (focal length 200mm), gives a radial and axial 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 3-D field correlation distribution that reconstructs the image as a coherence function represented by fringe contrast. In our present set up withα=1.1, the lateral and axial magnification for the reconstructed image becomes mx=my=5.23 and mz=2.6respectively. The lateral magnification is chosen in such a way that the reconstructed image size fits the field aperture of the CCD camera. With an analyzer A with its axis kept at 450to the orientation of the polarization of the two beams, interference between the two orthogonally polarized beams was achieved.

4. Results

The coherence function was detected by applying the phase-shift technique to the Sagnac radial shearing interferometer, and the object was reconstructed as the 3-D correlation map of the fields diffracted from the hologram. The axial shear was varied at the output of the interferometer by translating the CCD camera along z direction so that the output plane that is conjugate to the CCD image sensor plane is scanned in the z direction. At each location of the CCD camera, the set of phase-shifted Fourier transform holograms was displayed sequentially with the projector and the corresponding interferograms were captured by a 14-bit cooled CCD camera (BITRAN BU-42-14).

Figures 8(a)
Fig. 8 (a), (b) and (c) are the interference fringes at depthsz˜=20mm,z˜=0and z˜=+20mm, respectively; (d), (e) and (f) shows the corresponding fringe visibilities; (g), (h) and (i) shows the corresponding fringe phases.
, 8(b) and 8(c) show one of the phase-shifted interferograms recorded atz˜=+20mm,z˜=0and z˜=20mm, respectively. Figures 8(d), 8(e) and 8(f) show the corresponding fringe contrasts, and Figs. 8(g), 8(h) and 8(i) show the corresponding fringe phases calculated by the model-based nonlinear least square fitting method [14

14. P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas. 49(6), 1189–1193 (2000). [CrossRef]

] that solves the over-determined equations generated from the image data for the parameters of the sinusoidal fringe model. The objects U, E and C were reconstructed as fringe visibilities at the locationsz˜=+20mm,z˜=0and z˜=20mm, respectively, as shown in the upper half of each figure in Fig. 8. In the lower half of each figure in symmetry to the center, conjugate images were reconstructed with their planes of focus appearing in the reverse order such that z˜=20mm, z˜=0and then z˜=+20mm. Forz˜=+20mm, the longitudinal correlation length is given byΔz=(α2α2)z˜=7.69mm. This is significantly larger than the temporal coherence length of the source, which is estimated to be133μmfrom the spectral width of the interference filter. The fact that the images were reconstructed with sufficiently high value of spatial coherence even for the large axial depths demonstrates that the path difference without time delay has been achieved by virtue of the achromatic common-path imaging Sagnac interferometer.

5. Conclusion

In summary, we proposed and demonstrated generic coherence holography using a commercial projector with a thermal light source for illumination of the hologram and a Sagnac radial shearing interferometer for its reconstruction. The axial shear accompanying the radial shear enabled this scheme of coherence holography to reconstruct a 3-D object with an axial depth beyond the short coherence length of the thermal light. Because the Sagnac common path interferometer is very stable to environmental noise caused by vibrations and air turbulences, the reconstruction process is highly reliable. Move over the reconstruction scheme presented in this paper is completely free from any mechanically moving components.

Acknowledgment

Part of this work was supported by Grant-in-Aid of JSPS B (2) No. 21360028, and all the authors were with the University of Electro-Communications, Tokyo, Japan, when this work was done.

References and links

1.

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express 13(23), 9629–9635 (2005). [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). [CrossRef] [PubMed]

3.

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

4.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express 18(13), 13782–13787 (2010). [CrossRef] [PubMed]

5.

W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. 41(10), 1962–1971 (2002). [CrossRef] [PubMed]

6.

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). [CrossRef] [PubMed]

7.

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

8.

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

9.

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]

10.

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

11.

M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt. 3(7), 853–857 (1964). [CrossRef]

12.

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

13.

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

14.

P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas. 49(6), 1189–1193 (2000). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(090.0090) Holography : Holography
(100.3010) Image processing : Image reconstruction techniques

ToC Category:
Holography

History
Original Manuscript: June 6, 2012
Manuscript Accepted: July 24, 2012
Published: August 13, 2012

Citation
Dinesh N. Naik, Takahiro Ezawa, Rakesh Kumar Singh, Yoko Miyamoto, and Mitsuo Takeda, "Coherence holography by achromatic 3-D field correlation of generic thermal light with an imaging Sagnac shearing interferometer," Opt. Express 20, 19658-19669 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-19658


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References

  1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express13(23), 9629–9635 (2005). [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. Express17(13), 10633–10641 (2009). [CrossRef] [PubMed]
  3. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett.35(10), 1728–1730 (2010). [CrossRef] [PubMed]
  4. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express18(13), 13782–13787 (2010). [CrossRef] [PubMed]
  5. W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt.41(10), 1962–1971 (2002). [CrossRef] [PubMed]
  6. 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. Express14(25), 12109–12121 (2006). [CrossRef] [PubMed]
  7. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt.39(23), 4107–4111 (2000). [CrossRef] [PubMed]
  8. P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt.48(34), H40–H47 (2009). [CrossRef] [PubMed]
  9. 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]
  10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982). [CrossRef]
  11. M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt.3(7), 853–857 (1964). [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Chap. 10.
  13. J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5.
  14. P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas.49(6), 1189–1193 (2000). [CrossRef]

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