Abstract

We present a digital holographic microscope that permits one to image polarization state. This technique results from the coupling of digital holographic microscopy and polarization digital holography. The interference between two orthogonally polarized reference waves and the wave transmitted by a microscopic sample, magnified by a microscope objective, is recorded on a CCD camera. The off-axis geometry permits one to reconstruct separately from this single hologram two wavefronts that are used to image the object-wave Jones vector. We applied this technique to image the birefringence of a bent fiber. To evaluate the precision of the phase-difference measurement, the birefringence induced by internal stress in an optical fiber is measured and compared to the birefringence profile captured by a standard method, which had been developed to obtain high-resolution birefringence profiles of optical fibers.

© 2005 Optical Society of America

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    [CrossRef]

2004

T. Colomb, E. Cuche, F. Montfort, P. Marquet, C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[CrossRef]

2003

2002

2001

2000

1999

1995

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

1994

1982

1966

1965

Ahn, T.-J.

Alexeenko, I.

Beghuin, D.

Chu, P. L.

Colomb, T.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[CrossRef]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, C. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002).
[CrossRef] [PubMed]

Coppola, G.

Cuche, E.

Dahlgren, P.

De Nicola, S.

Delacrétaz, G.

Depeursinge, C.

Dubois, F.

El-Diasty, F.

F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibers,” J. Opt. A: Pure Appl. Opt. 1, 197–200 (1999).
[CrossRef]

Ferraro, P.

Finizio, A.

Gabor, D.

Goss, W. P.

Han, W.-T.

Indebetouw, G.

Kaneko, T.

Kim, D. Y.

Kim, Y. H.

Kischel, P.

Klysubun, P.

Legros, J.-C.

Lohmann, A. W.

Marquet, P.

Mei, G.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Minetti, C.

Monnom, O.

Montfort, F.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[CrossRef]

Ohtsuka, Y.

Oka, K.

Oldenbourg, R.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

Osten, W.

Paek, U.-C.

Park, Y.

Pedrini, G.

Petrov, V. D.

Pierattini, G.

Salathé, R. P.

Tiziani, H. J.

Whitbread, T.

Yourassowsky, C.

Appl. Opt.

P. L. Chu, T. Whitbread, “Measurement of stresses in optical fiber and preform,” Appl. Opt. 21, 4241–4245 (1982).
[CrossRef] [PubMed]

Y. Ohtsuka, K. Oka, “Contour mapping of the spatiotemporal state of polarization of light,” Appl. Opt. 33, 2633–2636 (1994).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, P. Dahlgren, C. Depeursinge, G. Delacrétaz, R. P. Salathé, “Simultaneous amplitude and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
[CrossRef]

E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

Y. Park, T.-J. Ahn, Y. H. Kim, W.-T. Han, U.-C. Paek, D. Y. Kim, “Measurement method for profiling the residual stress and the strain-optic coefficient of an optical fiber,” Appl. Opt. 41, 21–26 (1999).
[CrossRef]

T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, C. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002).
[CrossRef] [PubMed]

F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J.-C. Legros, P. Kischel, “Pattern recognition with a digital holographic microscope working in partially coherent illumination,” Appl. Opt. 41, 4108–4119 (2002).
[CrossRef] [PubMed]

G. Pedrini, I. Alexeenko, W. Osten, H. J. Tiziani, “Temporal phase unwrapping of digital hologram sequences,” Appl. Opt. 42, 5846–5854 (2003).
[CrossRef] [PubMed]

A. W. Lohmann, “Reconstruction of vectorial wavefronts,” Appl. Opt. 4, 1667–1668 (1965).
[CrossRef]

J. Microsc.

R. Oldenbourg, G. Mei, “New polarized light microscope with precision universal compensator,” J. Microsc. 180, 140–147 (1995).
[CrossRef] [PubMed]

J. Opt. A: Pure Appl. Opt.

F. El-Diasty, “Interferometric determination of induced birefringence due to bending in single-mode optical fibers,” J. Opt. A: Pure Appl. Opt. 1, 197–200 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Technol.

Opt. Commun.

T. Colomb, E. Cuche, F. Montfort, P. Marquet, C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (10)

Fig. 1
Fig. 1

(a) Experimental setup. Oin illuminating wave, O object wave; R1 and R2, polarized reference waves; Pol. α, polarizer oriented at α; λ/2, half-wave plate; M, mirror; BS, beam splitter; L, lens with focal length fL; MO, the microscope objective, and CCD, the change-coupled device. (b) Detail showing the off-axis geometry at the incidence on the CCD. The xoyo plane is parallel to the CCD camera. R1 in the yozo plane and R2 in the xozo plane are coming from different spatial directions. (c) Detail showing the ray tracing in the object arm. The collimated beam focalized with the lens L illuminates a small portion of the sample placed between the lens L and its focal plane. The MO magnifies the transmitted beam to produce a divergent beam that covers the entire chip area of the CCD.

Fig. 2
Fig. 2

Hologram of the nonstripped bent fiber. The magnifying glass permits one to visualize the two different curved fringe patterns corresponding to the interference of the object wave with the two orthogonally polarized reference waves.

Fig. 3
Fig. 3

Two-dimensional Fourier spectrum of the hologram presented in Fig. 2. ZO is the frequencies associated with the zero order of diffraction, and P indicates the contributions produced by parasitic interferences. R1*O and R1O* are the frequencies associated with the horizontal polarization component corresponding, respectively, to the virtual and real images; R2*O and R2O* are the frequencies associated with the vertical polarization component.

Fig. 4
Fig. 4

Filtered two-dimensional Fourier spectra. Selection of the virtual image (a) for the horizontal polarization component, (b) for the vertical polarization component.

Fig. 5
Fig. 5

Amplitude and phase reconstructions for an unbent fiber [(a)–(d)] and for a bent fiber [(e)–(h)]. (i) is the cladding region with focusing core in the middle, (j) is the coating region, and (k) is the refractive-index liquid region. (a), (e) |o1|; (b), (f) |o2|; (c), (g) phase |o1|; (d), (h) phase |o2|. Circles indicate positions of inhomogeneity in the azimuth due to some dust particles. The center of curvature is on the right of bent-fiber reconstructed images.

Fig. 6
Fig. 6

SOP images reconstructed from images of Fig. 5: (a), (c) the azimuth and (b), (d) the phase difference for an unbent fiber (a), (b) and a bent fiber (c), (d). Mean profiles of Fig. 7 are defined along the major axis of dashed dark rectangles.

Fig. 7
Fig. 7

Graphs of mean profiles defined by rectangle of Fig. 6: (a) the azimuth, (b) the phase difference.

Fig. 8
Fig. 8

Reconstructed phase and phase-difference images. (a) phase (o1), (b) phase (o2), (c) phase difference. The phase-difference standard deviation in the rectangle area is about 6 degrees.

Fig. 9
Fig. 9

Comparison between the phase-difference measurements performed with the reference method (solid curve) and performed with the Pol-DHM.

Fig. 10
Fig. 10

|o1| image reconstruction of standard USAF target hologram recorded with a 20× MO. The smallest elements correspond to 228 line pairs/mm.

Equations (15)

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O = ( o 1 exp ( ϕ o ) o 2 exp ( ϕ o + Δ φ o ) 0 ) exp [ i ( k o · r ) ] = ( o 1 o 2 0 ) exp [ i ( k o · r + ϕ o ) ] , R 1 = ( r 1 0 0 ) exp [ i k 1 · r ] , R 2 = ( 0 r 2 0 ) exp [ i k 2 · r ] ,
k o = 2 π λ [ 0 0 1 ] , k 1 = 2 π λ [ 0 sin ( θ 1 ) cos ( θ 1 ) ] , k 2 = 2 π λ [ - sin ( θ 2 ) 0 cos ( θ 2 ) ] ,
I H ( x , y ) = ( R 1 + R 2 + O ) · ( R 1 + R 2 + O ) * = R 1 2 + R 2 2 + O 2 + R 1 O * + R 2 O * + R 1 * O + R 2 * O .
I H ( k , l ) = k Δ x - Δ x / 2 k Δ x + Δ x / 2 l Δ y - Δ y / 2 l Δ y + Δ y / 2 I H ( x , y ) d x d y ,
I H j ( x , y ) = R j * O ,
Ψ j = R j I H .
Ψ j ( ξ , η ) = A exp [ i π λ d ( ξ 2 + η 2 ) ] × R j I H ( x , y ) exp [ i π λ d ( x 2 + y 2 ) ] exp [ i 2 π λ d ( x ξ + y η ) ] d x d y .
Ψ j ( m , n ) = A exp [ i π λ d ( m 2 Δ ξ 2 + n 2 Δ η 2 ) ] × DFT { R D j ( k , l ) I H ( k , l ) exp [ i π λ d ( k 2 Δ x 2 + l 2 Δ y 2 ) ] } m , n ,
R D j = exp [ i ( k D j x k Δ x + k D j y l Δ y ) ] .
Φ j ( m , n ) = exp [ - i π λ ( m 2 Δ ξ 2 p ξ j + n 2 Δ η 2 p η j ) ] ,
Ψ j ( m , n ) = A Φ j ( m , n ) exp [ i π λ d ( m 2 Δ ξ 2 + n 2 Δ η 2 ) ] × DFT { R D j ( k , l ) I H ( k , l ) exp [ i π λ d ( k 2 Δ x 2 + l 2 Δ y 2 ) ] } m , n .
Ψ j = R D j R j * O .
Ψ j = r j o j = r j o j ;             phase ( Ψ j ) = phase ( o j ) - phase ( r j ) + ϕ o .
tan ( ɛ ) = Ψ 2 Ψ 1 = o 2 o 1 , Δ φ o = phase ( Ψ 2 ) - phase ( Ψ 1 ) + Δ φ R ,
I ( x , z ) = I max sin 2 [ θ + Δ θ ( x , z ) ] .

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