Abstract

Interference microscopy using spatial Fourier filtering with a vortex phase element leads to interference fringes that are spirals rather than closed rings. Depressions and elevations in the optical thickness of the sample can be distinguished immediately by the sense of rotation of the spirals. This property allows an unambiguous reconstruction of the object's phase profile from one single interferogram. We investigate the theoretical background of “spiral interferometry” and suggest various demodulation techniques based on the processing of one single interferogram or multiple interferograms.

© 2006 Optical Society of America

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References

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  1. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
    [CrossRef]
  2. Z. Jaroszewicz and A. Kolodziejczyk, "Zone plates performing generalized Hankel transforms and their metrological applications," Opt. Commun. 102, 391-396 (1993).
    [CrossRef]
  3. G. A. Swartzlander, "Peering into darkness with a vortex spatial filter," Opt. Lett. 26, 497-499 (2001).
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral phase contrast imaging in microscopy," Opt. Express 13, 689-694 (2005).
    [CrossRef] [PubMed]
  7. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, "Shadow effects in spiral phase contrast microscopy," Phys. Rev. Lett. 94, 233902 (2005).
    [CrossRef] [PubMed]
  8. M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
    [CrossRef]
  9. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral interferometry," Opt. Lett. 30, 1953-1955 (2005).
    [CrossRef] [PubMed]
  10. D.W.Robinson and G.T.Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (IOP, 1993).
  11. S. Sundbeck, I. Gruzberg, and D. G. Grier, "Structure and scaling of helical modes of light," Opt. Lett. 30, 477-479 (2005).
    [CrossRef] [PubMed]
  12. K. G. Larkin, D. J. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1862-1870 (2001).
  13. J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: Geometric Phases in Physics," Am. J. Phys. 65, 180-185 (1997).
    [CrossRef]
  14. A. Y. M. Ng, C. W. See, and M. G. Somekh, "Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator," J. Microsc. (Oxford) 214, 334-340 (2004).
    [CrossRef]
  15. G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
    [CrossRef]
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

2005

2004

K. Crabtree, J. A. Davis, and I. Moreno, "Optical processing with vortex-producing lenses," Appl. Opt. 43, 1360-1367 (2004).
[CrossRef] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
[CrossRef]

A. Y. M. Ng, C. W. See, and M. G. Somekh, "Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator," J. Microsc. (Oxford) 214, 334-340 (2004).
[CrossRef]

2001

2000

1997

J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: Geometric Phases in Physics," Am. J. Phys. 65, 180-185 (1997).
[CrossRef]

1993

Z. Jaroszewicz and A. Kolodziejczyk, "Zone plates performing generalized Hankel transforms and their metrological applications," Opt. Commun. 102, 391-396 (1993).
[CrossRef]

1992

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Anandan, J.

J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: Geometric Phases in Physics," Am. J. Phys. 65, 180-185 (1997).
[CrossRef]

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
[CrossRef]

Bernet, S.

Bone, D. J.

Campos, J.

Christian, J.

J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: Geometric Phases in Physics," Am. J. Phys. 65, 180-185 (1997).
[CrossRef]

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
[CrossRef]

Cottrell, D. M.

Crabtree, K.

Davis, J. A.

Develis, J. B.

G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

Fürhapter, S.

Grier, D. G.

Gruzberg, I.

Jaroszewicz, Z.

Z. Jaroszewicz and A. Kolodziejczyk, "Zone plates performing generalized Hankel transforms and their metrological applications," Opt. Commun. 102, 391-396 (1993).
[CrossRef]

Jesacher, A.

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Kolodziejczyk, A.

Z. Jaroszewicz and A. Kolodziejczyk, "Zone plates performing generalized Hankel transforms and their metrological applications," Opt. Commun. 102, 391-396 (1993).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
[CrossRef]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1862-1870 (2001).

McNamara, D. E.

Moreno, I.

Ng, A. Y. M.

A. Y. M. Ng, C. W. See, and M. G. Somekh, "Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator," J. Microsc. (Oxford) 214, 334-340 (2004).
[CrossRef]

Oldfield, M. A.

Reynolds, G. O.

G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

Ritsch-Marte, M.

See, C. W.

A. Y. M. Ng, C. W. See, and M. G. Somekh, "Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator," J. Microsc. (Oxford) 214, 334-340 (2004).
[CrossRef]

Sheppard, C. J. R.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
[CrossRef]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Somekh, M. G.

A. Y. M. Ng, C. W. See, and M. G. Somekh, "Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator," J. Microsc. (Oxford) 214, 334-340 (2004).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Sundbeck, S.

Swartzlander, G. A.

Thompson, B. J.

G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Wanelik, K.

J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: Geometric Phases in Physics," Am. J. Phys. 65, 180-185 (1997).
[CrossRef]

Am. J. Phys.

J. Anandan, J. Christian, and K. Wanelik, "Resource Letter GPP-1: Geometric Phases in Physics," Am. J. Phys. 65, 180-185 (1997).
[CrossRef]

Appl. Opt.

J. Microsc. (Oxford)

M. R. Arnison, K. G. Larkin, C. J. R. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. (Oxford) 214, 7-12 (2004).
[CrossRef]

A. Y. M. Ng, C. W. See, and M. G. Somekh, "Quantitative optical microscope with enhanced resolution using a pixelated liquid crystal spatial light modulator," J. Microsc. (Oxford) 214, 334-340 (2004).
[CrossRef]

J. Mod. Opt.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, "The phase rotor filter," J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

Z. Jaroszewicz and A. Kolodziejczyk, "Zone plates performing generalized Hankel transforms and their metrological applications," Opt. Commun. 102, 391-396 (1993).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, "Shadow effects in spiral phase contrast microscopy," Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

Other

G. O. Reynolds, J. B. Develis, and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (SPIE, 1989).
[CrossRef]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

D.W.Robinson and G.T.Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (IOP, 1993).

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Figures (8)

Fig. 1
Fig. 1

Schematic vortex filter setup. The spiral filter is modified such that its central area is assigned a constant phase value. The gray tones of the filter correspond to respective phase shifts. Note that the coordinate system ( x , y ) is mirrored compared with the ( x , y ) system.

Fig. 2
Fig. 2

Comparison of filter kernels K and K V for f = 0.1 m and ρ m a x = 0.01 m . (a) and (b) show the absolute values, (c) and (d) the phase of the kernels. Note that the axis scaling for (c) and (d) has been changed with respect to (a) and (b) for better visualization; (e) and (f) show the real parts of the cross section defined by ϕ = π 2 .

Fig. 3
Fig. 3

Graphical scheme to explain the convolution process for the case of a pure amplitude object. The result at a certain location is derived by shifting the (mirrored) kernel to this point and integrating over the product of the shifted kernel with E i n ( x , y ) .

Fig. 4
Fig. 4

(a) Gaussian-shaped elevation as phase object. (b) After the filter process; a phase factor proportional to the geometrical direction of the local phase gradient has been added.

Fig. 5
Fig. 5

Demodulation using contour lines. (a) “Classical” closed-fringe interferogram of a deformation in a plastic film, (b) corresponding spiral interferogram, (c) and (d) single contour line raw and after preprocessing, respectively. The reconstructed three-dimensional shape is shown in (e) and (f).

Fig. 6
Fig. 6

Demodulation based on center lines. (a) “Skeleton” of the spiral fringe pattern, which roughly consists of connected intensity maxima, (b) and (c) reconstructed three-dimensional shapes.

Fig. 7
Fig. 7

Quantitative error analysis of the single-image-based demodulation methods. The input object (a) of the simulation consists of two Gaussian-shaped phase deformations, (b) and (c) show the results of the contour line and center line reconstruction methods, respectively, together with their difference from the original phase sample. Note that the axis scalings of the error plots are different from the scalings of the phase object graphs.

Fig. 8
Fig. 8

Principle of multi-image demodulation. The mean value of three “complexified” images is proportional to the spiral filtered object field, but without its zeroth order. After restoration of the missing field amplitude, the spiral back transformation is accomplished. Finally, the original phase distribution is restored by using a standard phase unwrapping algorithm.

Equations (25)

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E o u t ( x , y ) = ( E i n * K V ) ( x , y ) = x = y = E i n ( x , y ) K V ( x x , y y ) d x d y .
K V ( r , ϕ ) = 2 π λ f i exp ( i ϕ ) ρ = 0 ρ m a x ρ J 1 ( 2 π λ f r ρ ) d ρ .
K ( r , ϕ ) = 2 π λ f ρ = 0 ρ m a x ρ J 0 ( 2 π λ f r ρ ) d ρ = ρ m a x r J 1 ( 2 π λ f ρ m a x r ) .
K ̃ V ( r , ϕ ) = { 1 N exp ( i ϕ ) for R 1 < r < R 2 0 elsewhere ,
E o u t ( P ) = 1 N ϕ P = 0 2 π r P = R 1 R 2 exp ( i ϕ P ) E ̂ i n ( r P , ϕ P ) × r P d r P d ϕ P ,
E ̂ i n ( x P , y P ) E ̂ i n ( 0 ) exp [ i ψ ̂ i n ( 0 ) ] + exp [ i ψ ̂ i n ( 0 ) ] [ g A m ( 0 ) r P + i E ̂ i n ( 0 ) g P h ( 0 ) r P ] .
E o u t ( P ) exp [ i ψ i n ( P ) ] g A m ( P ) exp [ i δ A m ( P ) ] + i E i n ( P ) g P h ( P ) exp [ i δ P h ( P ) ] .
ψ i n = α t a n up to multiples of 2 π .
h = α t a n Δ n λ 2 π up to multiples of Δ n λ ,
I ( x , y ) = A o b j ( x , y ) exp [ i ψ o b j ( x , y ) ] + A r e f exp ( i ψ r e f ) 2 .
I ¯ c ( x , y ) = 1 n j = 1 n I c j = A r e f A o b j ( x , y ) exp [ i ψ o b j ( x , y ) ] ,
K V ( x , y ) = 1 λ f A p e r t u r e exp [ i θ ( μ , ν ) ] exp ( i 2 π λ f ( x μ + y ν ) ) d μ d ν ,
x = r cos ϕ , y = r sin ϕ , μ = ρ cos θ , ν = ρ sin θ ,
K V ( r , ϕ ) = exp ( i ϕ ) λ f ρ = 0 ρ m a x θ = 0 2 π ρ exp ( i θ ) exp ( i 2 π λ f r ρ cos θ ) d ρ d θ .
K V ( r , ϕ ) = 2 π λ f i exp ( i ϕ ) ρ = 0 ρ max ρ J 1 ( 2 π λ f r ρ ) d ρ ,
J n ( z ) = i n 2 π θ = 0 2 π exp ( i z cos θ ) exp ( i n θ ) d θ .
K V ( r , ϕ ) = i exp ( i ϕ ) π ρ m a x 2 r [ J 1 ( 2 π λ f r ρ ) H 0 ( 2 π λ f r ρ ) J 0 ( 2 π λ f r ρ ) H 1 ( 2 π λ f r ρ ) ] ,
E o u t ( P ) E i n ( P ) N ϕ P = 0 2 π r P = R 1 R 2 exp ( i ϕ P ) r P d r P d ϕ P + exp [ i ψ i n ( P ) ] g A m ( P ) N exp ( i ϕ P ) r P 2 cos [ ϕ P δ A m ( P ) ] d r P d ϕ P + i E i n ( P ) g P h ( P ) N exp ( i ϕ P ) r P 2 cos [ ϕ P δ P h ( P ) ] d r P d ϕ P .
E o u t ( P ) exp [ i ψ i n ( P ) ] g A m ( P ) 2 N { exp [ i δ A m ( P ) ] ϕ P r P exp ( i 2 ϕ P ) r P 2 d r P d ϕ P + exp [ i δ A m ( P ) ] ϕ P r P r P 2 d r P d ϕ P } + i E i n ( P ) g P h ( P ) 2 N { exp [ i δ P h ( P ) ] ϕ P r P exp ( i 2 ϕ P ) r P 2 d r P d ϕ P + exp [ i δ P h ( P ) ] ϕ P r P r P 2 d r P d ϕ P } .
E o u t ( P ) 1 3 R 2 3 R 1 3 R 2 2 R 1 2 { exp [ i ψ i n ( P ) ] g A m ( P ) exp [ i δ A m ( P ) ] + i E i n ( P ) g P h ( P ) exp [ i δ P h ( P ) ] } .
I = A o b j exp ( i ψ o b j ) + A r e f exp ( i ψ r e f ) 2 = A o b j 2 + A r e f 2 + A o b j A r e f { exp [ i ( ψ o b j ψ r e f ) ] + exp [ i ( ψ o b j ψ r e f ) ] } ,
I c = ( A o b j 2 + A r e f 2 ) exp ( i ψ r e f ) + A o b j A r e f exp ( i ψ o b j ) + A o b j A r e f exp ( i ψ o b j ) exp ( i 2 ψ r e f ) .
I ¯ c = 1 3 ( A o b j 2 + A r e f 2 ) [ exp ( i ψ 1 ) + exp ( i ψ 2 ) + exp ( i ψ 3 ) ] + A o b j A r e f exp ( i ψ o b j ) + 1 3 A o b j A r e f exp ( i ψ o b j ) [ exp ( i 2 ψ 1 ) + exp ( i 2 ψ 2 ) + exp ( i 2 ψ 3 ) ] .
I ¯ = 1 3 n I n = A o b j 2 + A r e f 2 ,
A r e f = [ I ¯ 2 ± ( I ¯ 2 ) 2 I ¯ c 2 ] 1 2 .

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