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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

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

2007 (3)

2006 (2)

2005 (1)

2004 (2)

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

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

2001 (1)

1994 (1)

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

1992 (1)

1934 (1)

Zernike, F.  (1934). Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode. Physica 1, 689-704.
[CrossRef]

Alonzo,

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

Alonzo, P.J.

Arieli, N.

Badizadegan, J.C.

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

Badizadegan, R.R.

Ben-Yosef, M.

Bernet,

Choi, N.

Dasari, H.

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

Dasari, T.

Deflores, G.

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

Feld, K.

Feld, R.R.

Popescu, G. , Ikeda, T. , Dasari, R.R. , and Feld, M.S.  (2006). Diffraction phase microscopy for quantifying cell structure and dynamics. Opt. Lett. 31, 775-777.
[CrossRef] [PubMed]

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

Fürhapter, A.

Glückstad,

Glückstad, P.J.

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

Guo,

He, X.

Ikeda, G.

Israeli, E.

Iwai, K.

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

Jesacher, S.

Kadono,

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

Kawata, T.

Lanzmann, S.

Liu, C.S.

Lue,

Maurer, S.

Mogensen, J.

Noda,

Ogusu, H.

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

Palima, J.

Popescu,

Popescu, G. , Ikeda, T. , Dasari, R.R. , and Feld, M.S.  (2006). Diffraction phase microscopy for quantifying cell structure and dynamics. Opt. Lett. 31, 775-777.
[CrossRef] [PubMed]

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

Popescu, W.

Ritsch-Marte, C.

Rodrigo, C.A.

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

Rodrigo, D.

Toyooka, M.

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

Vaughan, L.P.

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

Wang, J.L.

Wolfling,

Zernike,

Zernike, F.  (1934). Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode. Physica 1, 689-704.
[CrossRef]

Appl. Opt. (2)

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

N. J. Phys. (1)

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

Opt. Commun. (1)

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

Opt. Express (2)

Opt. Lett. (3)

Physica (1)

Zernike, F.  (1934). Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode. Physica 1, 689-704.
[CrossRef]

Other (1)

Glückstad, J. , and Mogensen, P.C.  "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).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

Equations on this page are rendered with MathJax. Learn more.

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 .

Metrics