Abstract

The generalized phase contrast (GPC) method is explored for improving the accuracy in quantitative reconstruction of two-dimensional phase distribution from images of semi-transparent objects viewed with a common-path interferometer (CPI). We propose a novel optical scheme for highly accurate determination of the object-dependent complex synthetic reference wave (SRW) in a CPI. Using a simple 4f imaging optical setup, GPC provides an analytic model of the SRW profile that is shown here to increase phase measurement accuracy over the entire output aperture. The improved accuracy due to the GPC model can exceed one order of magnitude compared to that of the conventional plane wave model of the reference beam. Furthermore, we describe a novel method for accurate derivation of the strength of the phase object’s zero spatial frequency component based on the intensity of the traditionally ignored halo region encompassing the interferogram. Combining this information with three inteferometric measurements, full-field phase images with unconstrained phase strokes are obtained accurately.

© 2008 Optical Society of America

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References

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  1. F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [Crossref]
  2. T. Noda and S. Kawata, “Separation of phase and absorption images in phase-contrast microscopy,” J. Opt. Soc. Am. A 9, 924–931 (1992).
    [Crossref]
  3. H. Kadono, M. Ogusu, and S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
    [Crossref]
  4. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 21, 2503–2505 (2004).
    [Crossref]
  5. N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. 46, 1836–1842 (2007).
    [Crossref] [PubMed]
  6. J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40, 268–282 (2001).
    [Crossref]
  7. J. Glückstad and P. C. Mogensen, “Analysis of wavefront sensing using a common path interferometer architecture,” in Proceedings 2 International workshop on adaptive optics for industry and medicine, Durham (GB), 12–16 July 1999, World Scientific, Singapore, 241–246 (1999).
  8. C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
    [Crossref]
  9. J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. 32, 3281–3283 (2007).
    [Crossref] [PubMed]
  10. C. S. Guo, X. Liu, J. L. He, and H. T. Wang, “Optimal annulus structures of optical vortices,” Opt. Express 12, 4625–4634 (2004).
    [Crossref] [PubMed]
  11. S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express 14, 3792–3805 (2006).
    [Crossref] [PubMed]
  12. S. Wolfling, E. Lanzmann, M. Israeli, N. Ben-Yosef, and Y. Arieli, “Spatial phase-shift interferometry - a wavefront analysis technique for three-dimensional topometry,” J. Opt. Soc. Am. A 22, 2498–2509 (2005).
    [Crossref]
  13. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31, 775–777 (2006).
    [Crossref] [PubMed]

2007 (3)

2006 (2)

2005 (1)

2004 (2)

C. S. Guo, X. Liu, J. L. He, and H. T. Wang, “Optimal annulus structures of optical vortices,” Opt. Express 12, 4625–4634 (2004).
[Crossref] [PubMed]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 21, 2503–2505 (2004).
[Crossref]

2001 (1)

1994 (1)

H. Kadono, M. Ogusu, and S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[Crossref]

1992 (1)

1934 (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Alonzo, C. A.

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[Crossref]

J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. 32, 3281–3283 (2007).
[Crossref] [PubMed]

Arieli, Y.

Badizadegan, K.

N. Lue, W. Choi, G. Popescu, T. Ikeda, R. R. Dasari, K. Badizadegan, and M. S. Feld, “Quantitative phase imaging of live cells using fast Fourier phase microscopy,” Appl. Opt. 46, 1836–1842 (2007).
[Crossref] [PubMed]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 21, 2503–2505 (2004).
[Crossref]

Ben-Yosef, N.

Bernet, S.

Choi, W.

Dasari, R. R.

Deflores, L. P.

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 21, 2503–2505 (2004).
[Crossref]

Feld, M. S.

Fürhapter, S.

Glückstad, J.

J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. 32, 3281–3283 (2007).
[Crossref] [PubMed]

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[Crossref]

J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40, 268–282 (2001).
[Crossref]

J. Glückstad and P. C. Mogensen, “Analysis of wavefront sensing using a common path interferometer architecture,” in Proceedings 2 International workshop on adaptive optics for industry and medicine, Durham (GB), 12–16 July 1999, World Scientific, Singapore, 241–246 (1999).

Guo, C. S.

He, J. L.

Ikeda, T.

Israeli, M.

Iwai, H.

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 21, 2503–2505 (2004).
[Crossref]

Jesacher, A.

Kadono, H.

H. Kadono, M. Ogusu, and S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[Crossref]

Kawata, S.

Lanzmann, E.

Liu, X.

Lue, N.

Maurer, C.

Mogensen, P. C.

J. Glückstad and P. C. Mogensen, “Optimal phase contrast in common-path interferometry,” Appl. Opt. 40, 268–282 (2001).
[Crossref]

J. Glückstad and P. C. Mogensen, “Analysis of wavefront sensing using a common path interferometer architecture,” in Proceedings 2 International workshop on adaptive optics for industry and medicine, Durham (GB), 12–16 July 1999, World Scientific, Singapore, 241–246 (1999).

Noda, T.

Ogusu, M.

H. Kadono, M. Ogusu, and S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[Crossref]

Palima, D.

Popescu, G.

Ritsch-Marte, M.

Rodrigo, P. J.

J. Glückstad, D. Palima, P. J. Rodrigo, and C. A. Alonzo, “Laser projection using generalized phase contrast,” Opt. Lett. 32, 3281–3283 (2007).
[Crossref] [PubMed]

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[Crossref]

Toyooka, S.

H. Kadono, M. Ogusu, and S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[Crossref]

Vaughan, J. C.

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 21, 2503–2505 (2004).
[Crossref]

Wang, H. T.

Wolfling, S.

Zernike, F.

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

New J. Phys. (1)

C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Photon-efficient grey-level image projection by the generalized phase contrast method,” New J. Phys. 9, 132 (2007).
[Crossref]

Opt. Commun. (1)

H. Kadono, M. Ogusu, and S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Physica (1)

F. Zernike, “Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Other (1)

J. Glückstad and P. C. Mogensen, “Analysis of wavefront sensing using a common path interferometer architecture,” in Proceedings 2 International workshop on adaptive optics for industry and medicine, Durham (GB), 12–16 July 1999, World Scientific, Singapore, 241–246 (1999).

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Figures (9)

Fig. 1.
Fig. 1.

Common-path interferometer – a 4f imaging system with a phase contrast filter (PCF).

Fig. 2.
Fig. 2.

Modified common-path interferometer with a dynamic PCF implemented with a spatial light modulator (SLM). Additional half mirror (HM2), pinhole (PH), lens (L2) enables the measurement of the complex synthetic reference wave by a Shark-Hartmann sensor (SH).

Fig. 3.
Fig. 3.

Amplitude profiles of the SRW for θ=π obtained by FFT-based simulation (squares) and by GPC model (red curve) for aperture sizes corresponding to (a) η=0.20, (b) η=0.41 and (c) η=0.64. Corresponding aperture-truncated input fields are also plotted (blue curve). FFT-calculated output interferograms for (d) η=0.20, (e) η=0.41 and (f) η=0.64

Fig. 4.
Fig. 4.

SRW amplitude profiles for θ=π obtained by FFT-based simulation (squares) and by GPC model (red curve) for an input π-phase disc of different fill factor and aperture size combinations (a) η=0.41, F=0.1, (b) η=0.51, F=0.1, (c) η=0.64, F=0.1, (d) η=0.41, F=0.2, (e) η=0.51, F=0.2, and (f) η=0.64, F=0.2. Corresponding aperture-truncated input fields are also plotted (blue curve).

Fig. 5.
Fig. 5.

The three interferograms obtained with PCF shifts (a) θ=θ 0=0, (b) θ 1=π/2, (c)θ 2=π, and (d) the halo intensity I 2(r′>R′). η=0.4 is used.

Fig. 6.
Fig. 6.

Surface plots showing (a) the phase reconstruction and (b) the residual error.

Fig. 7.
Fig. 7.

Interferograms for an object consisting of alternating π/2 and -π/2 phase discs obtained with PCF shifts (a) θ 0=0, (b) θ 1=π/2, (c) θ 2=π and plots comparing the residual phase error obtained when the (d) planar and (e) GPC model of SRW are assumed. η=0.4 is used.

Fig. 8.
Fig. 8.

Interferograms for an obstructed helical phase of charge =10 obtained with PCF shifts (a) θ 0=0, (b) θ 1=π/2, (c) θ 2=π and plots comparing the residual phase error obtained when the (d) planar and (e) GPC model of SRW are assumed. η=0.4 is used.

Fig. 9.
Fig. 9.

Maximum peripheral phase error as a function of the topological charge of a centrally obstructed vortex phase object.

Equations (19)

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H ( f X , f Y ) = { 1 + [ exp ( j θ ) 1 ] circ ( ρ ρ 0 ) for circular geometry 1 + [ exp ( j θ ) 1 ] rect ( f X f 0 ) rect ( f Y f 0 ) for square geometry ,
h ( x , y ) = { δ ( x , y ) + [ exp ( j θ ) 1 ] ρ 0 J 1 ( 2 π ρ 0 r ) r circular δ ( x , y ) + [ exp ( j θ ) 1 ] f 0 2 sinc ( f 0 x ) sinc ( f 0 y ) square ,
I ( x , y ) = u ( x , y ) h ( x , y ) 2
= { u ( x , y ) + [ exp ( j θ ) 1 ] { u ( x , y ) [ ρ 0 J 1 ( 2 π ρ 0 r ) r ] } 2 circular u ( x , y ) + [ exp ( j θ ) 1 ] { u ( x , y ) [ f 0 2 sinc ( f 0 x ) sinc ( f 0 y ) ] } 2 square ,
I ( x , y ) { u ( x , y ) + U ( 0 , 0 ) [ exp ( j θ ) 1 ] g C ( r ) 2 circular u ( x , y ) + U ( 0 , 0 ) [ exp ( j θ ) 1 ] g S ( x , y ) 2 square ,
U ( f X = 0 , f Y = 0 ) = U ( 0 , 0 ) exp ( j ϕ U ) = Γ u ( x , y ) exp [ j ϕ ( x , y ) ] d x d y Γ d x d y ,
g C ( r ) = 2 π R 0 ρ 0 J 1 ( 2 π ρ R ) J 0 ( 2 π ρ r ) d ρ
g S ( x , y ) = g X ( x ) g Y ( y )
= L 2 f 0 2 f 0 2 sinc ( L f X ) exp ( j 2 π f X x ) d f X f 0 2 f 0 2 sinc ( L f Y ) exp ( j 2 π f Y y ) d f Y .
I ( r > R ) U ( 0 , 0 ) 2 exp ( j θ ) 1 2 [ g C ( r > R ) ] 2 circular ,
I ( x > L , y > L ) U ( 0 , 0 ) 2 exp ( j θ ) 1 2 [ g X ( x > L ) ] 2 [ g Y ( y > L ) ] 2 square .
U ( 0 , 0 ) 2 = 1 4 sin ( θ 2 ) m , n I ( x m 2 + y n 2 > R ) m , n [ g C ( x m 2 + y n 2 ) > R ] 2 .
I ( x , y ) u ( x , y ) 2 + 4 U ( 0 , 0 ) 2 sin 2 ( θ 2 ) [ g C ( r ) ] 2
+ 4 U ( 0 , 0 ) sin ( θ 2 ) u ( x , y ) g C ( r ) cos [ ϕ ( x , y ) ϕ U ( θ + π ) 2 ] .
I 0 ( x , y ) u ( x , y ) 2 ,
I 1 ( x , y ) u ( x , y ) 2 + 2 U ( 0 , 0 ) 2 [ g C ( r ) ] 2
+ 2 U ( 0 , 0 ) u ( x , y ) g C ( r ) { sin [ ϕ ( x , y ) ] cos [ ϕ ( x , y ) ] } ,
I 2 ( x , y ) u ( x , y ) 2 + 4 U ( 0 , 0 ) 2 [ g C ( r ) ] 2 4 U ( 0 , 0 ) u ( x , y ) g C ( r ) cos [ ϕ ( x , y ) ] ,
tan [ ϕ ( x , y ) ] = 2 I 1 ( x , y ) I 2 ( x , y ) I 0 ( x , y ) I 0 ( x , y ) I 2 ( x , y ) + 4 U ( 0 , 0 ) 2 [ g C ( r ) ] 2 .

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