Abstract

We describe a new wavefront analysis method, in which certain wavefront manipulations are applied to a spatially defined area in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing methods, making this method a spatial phase-shift interferometry method. We demonstrate the system’s dependence on a defined spatial Airy number, which is the ratio of the characteristic dimension of the manipulated area and the Airy disk diameter of the optical system. We analytically obtain the resulting intensity data of the optical setup and develop various methods to accurately reconstruct the inspected wavefront out of the data. These reconstructions largely involve global techniques, in which the entire wavefront’s pattern affects the reconstruction of the wavefront in any given position. The method’s noise sensitivity is analyzed, and actual reconstruction results are presented.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. C. Woods, A. H. Greenaway, “Wave-front sensing by use of a Green’s function solution to the intensity transport equation,” J. Opt. Soc. Am. A 20, 508–512 (2003).
    [CrossRef]
  2. E. Lopez-Lago, R. de-la-Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A, Pure Appl. Opt. 4, 299–302 (2002).
    [CrossRef]
  3. M. A. Vorontsov, E. W. Justh, L. A. Beresnev, “Adaptive optics with advanced phase-contrast techniques. I. High-resolution wave-front sensing,” J. Opt. Soc. Am. A 18, 1289–1299 (2001).
    [CrossRef]
  4. J. H. Burning, D. R. Herriot, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  5. K. Creath, “Phase measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
  6. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36, 8098–8115 (1997).
    [CrossRef]
  7. F. Zernike, “Diffraction theory of knife-edge test and its improved form, the phase-contrast,” Mon. Not. R. Astron. Soc. 94, 371 (1934).
  8. M. Pluta, “Stray-light problem in phase contrast microscopy or toward highly sensitive phase contrast devices: a review,” Opt. Eng. (Bellingham) 32, 3199–3214 (1993).
    [CrossRef]
  9. C. S. Anderson, “Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances,” Appl. Opt. 34, 7474–7485 (1995).
    [CrossRef] [PubMed]
  10. T. Noda, S. Kawata, “Separation of phase and absorption images in phase-contrast microscopy,” J. Opt. Soc. Am. A 9, 924–931 (1992).
    [CrossRef]
  11. R. Liang, J. K. Erwin, M. Mansuripur, “Variation on Zernike’s phase-contrast microscope,” Appl. Opt. 39, 2152–2158 (2000).
    [CrossRef]
  12. H. Kadono, M. Ogusu, S. Toyooka, “Phase shifting common path interferometer using a liquid-crystal phase modulator,” Opt. Commun. 110, 391–400 (1994).
    [CrossRef]
  13. C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
    [CrossRef] [PubMed]
  14. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
    [CrossRef]
  15. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  16. S. Wolfling, N. Ben-Yosef, Y. Arieli, “A generalized method for wavefront analysis,” Opt. Lett. 29, 462–464 (2004).
    [CrossRef] [PubMed]
  17. S. Wolfling, D. Banitt, N. Ben-Yosef, Y. Arieli, “Innovative metrology method for the 3D measurement of MEMS structures,” in Proc. SPIE 5343, 255–263 (2004).
    [CrossRef]
  18. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [CrossRef] [PubMed]
  19. A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer-Verlag, 2000).
  20. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

2004 (2)

S. Wolfling, D. Banitt, N. Ben-Yosef, Y. Arieli, “Innovative metrology method for the 3D measurement of MEMS structures,” in Proc. SPIE 5343, 255–263 (2004).
[CrossRef]

S. Wolfling, N. Ben-Yosef, Y. Arieli, “A generalized method for wavefront analysis,” Opt. Lett. 29, 462–464 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

E. Lopez-Lago, R. de-la-Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A, Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

2001 (1)

2000 (1)

1997 (1)

1996 (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

1995 (1)

1994 (2)

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

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
[CrossRef] [PubMed]

1993 (1)

M. Pluta, “Stray-light problem in phase contrast microscopy or toward highly sensitive phase contrast devices: a review,” Opt. Eng. (Bellingham) 32, 3199–3214 (1993).
[CrossRef]

1992 (2)

1978 (1)

1974 (1)

1934 (1)

F. Zernike, “Diffraction theory of knife-edge test and its improved form, the phase-contrast,” Mon. Not. R. Astron. Soc. 94, 371 (1934).

Anderson, C. S.

Arieli, Y.

S. Wolfling, D. Banitt, N. Ben-Yosef, Y. Arieli, “Innovative metrology method for the 3D measurement of MEMS structures,” in Proc. SPIE 5343, 255–263 (2004).
[CrossRef]

S. Wolfling, N. Ben-Yosef, Y. Arieli, “A generalized method for wavefront analysis,” Opt. Lett. 29, 462–464 (2004).
[CrossRef] [PubMed]

Banitt, D.

S. Wolfling, D. Banitt, N. Ben-Yosef, Y. Arieli, “Innovative metrology method for the 3D measurement of MEMS structures,” in Proc. SPIE 5343, 255–263 (2004).
[CrossRef]

Ben-Yosef, N.

S. Wolfling, D. Banitt, N. Ben-Yosef, Y. Arieli, “Innovative metrology method for the 3D measurement of MEMS structures,” in Proc. SPIE 5343, 255–263 (2004).
[CrossRef]

S. Wolfling, N. Ben-Yosef, Y. Arieli, “A generalized method for wavefront analysis,” Opt. Lett. 29, 462–464 (2004).
[CrossRef] [PubMed]

Beresnev, L. A.

Brangaccio, D. J.

Burning, J. H.

Creath, K.

C. R. Mercer, K. Creath, “Liquid-crystal point-diffraction interferometer,” Opt. Lett. 19, 916–918 (1994).
[CrossRef] [PubMed]

K. Creath, “Phase measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1988), Vol. 26, pp. 349–393.
[CrossRef]

de-la-Fuente, R.

E. Lopez-Lago, R. de-la-Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A, Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

Erwin, J. K.

Fienup, J. R.

Gallagher, J. E.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

Greenaway, A. H.

Herriot, D. R.

Justh, E. W.

Kadono, H.

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

Kawata, S.

Liang, R.

Lopez-Lago, E.

E. Lopez-Lago, R. de-la-Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A, Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

Mansuripur, M.

Mercer, C. R.

Noda, T.

Ogusu, M.

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

Paxman, R. G.

Phillion, D. W.

Pluta, M.

M. Pluta, “Stray-light problem in phase contrast microscopy or toward highly sensitive phase contrast devices: a review,” Opt. Eng. (Bellingham) 32, 3199–3214 (1993).
[CrossRef]

Quarteroni, A.

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer-Verlag, 2000).

Ragazzoni, R.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

Rosenfeld, D. P.

Sacco, R.

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer-Verlag, 2000).

Saleri, F.

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer-Verlag, 2000).

Schulz, T. J.

Toyooka, S.

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

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

Vorontsov, M. A.

White, A. D.

Wolfling, S.

S. Wolfling, D. Banitt, N. Ben-Yosef, Y. Arieli, “Innovative metrology method for the 3D measurement of MEMS structures,” in Proc. SPIE 5343, 255–263 (2004).
[CrossRef]

S. Wolfling, N. Ben-Yosef, Y. Arieli, “A generalized method for wavefront analysis,” Opt. Lett. 29, 462–464 (2004).
[CrossRef] [PubMed]

Woods, S. C.

Zernike, F.

F. Zernike, “Diffraction theory of knife-edge test and its improved form, the phase-contrast,” Mon. Not. R. Astron. Soc. 94, 371 (1934).

Appl. Opt. (4)

J. Mod. Opt. (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289–293 (1996).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

E. Lopez-Lago, R. de-la-Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A, Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

J. Opt. Soc. Am. A (4)

Mon. Not. R. Astron. Soc. (1)

F. Zernike, “Diffraction theory of knife-edge test and its improved form, the phase-contrast,” Mon. Not. R. Astron. Soc. 94, 371 (1934).

Opt. Commun. (1)

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

Opt. Eng. (Bellingham) (1)

M. Pluta, “Stray-light problem in phase contrast microscopy or toward highly sensitive phase contrast devices: a review,” Opt. Eng. (Bellingham) 32, 3199–3214 (1993).
[CrossRef]

Opt. Lett. (3)

Proc. SPIE (1)

S. Wolfling, D. Banitt, N. Ben-Yosef, Y. Arieli, “Innovative metrology method for the 3D measurement of MEMS structures,” in Proc. SPIE 5343, 255–263 (2004).
[CrossRef]

Other (3)

A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (Springer-Verlag, 2000).

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, 1996).

K. Creath, “Phase measurements interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, 1988), Vol. 26, pp. 349–393.
[CrossRef]

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 (8)

Fig. 1
Fig. 1

General characteristic optical setup of the SPS interferometry technique.

Fig. 2
Fig. 2

Four-pad simulative object of different heights and intensity reflectances: ( reflectance , height ) = ( 1 , 40 nm ) , ( 0.2 , 100 nm ) , ( 0.28 , 80 nm ) , ( 0.76 , 0 nm ) . (a) Three-dimensional view of the object. (b) Reflectance map.

Fig. 3
Fig. 3

SPS simulative intensities. (a) Standard imaging ( k = 1 , θ = 0 ). (b) Phase-manipulated intensities ( k = 0.45 , θ = π ). Observe the halo around the field stop in (b).

Fig. 4
Fig. 4

Various profiles of the functions C ± , C = S , and B k = R k . (a) Reconstructed functions C (dashed curve) and C + (dotted curve) for the basic example. (b) Low-pass C (solid curve) and attenuated high-pass B k (dashed–dotted curve) for the basic example. Note that C (solid curve) can be reconstructed out of the two local functions C ± in (a) using a smoothness argument. (c) Same as (a) but CCD noise was added to the simulation. Observe that a simple smoothness argument cannot be used anymore to reconstruct C. Note also that the noise is amplified in areas where C ± are close to each other. (d) Low-pass C (solid curve) and attenuated high-pass B k (dashed–dotted curve) of a perfectly flat mirror. Note that, within the field stop, both C and B are smooth.

Fig. 5
Fig. 5

Two profiles: (a) phase of the low-pass α for the basic example [the phase is expressed in nanometers using formula (1)], (b) residual error after the global polynomial approximation algorithm is applied.

Fig. 6
Fig. 6

Average within the field stop of the contrast modulation map M ( x , y ) of Eq. (32) for the simulative target described in Fig. 2.

Fig. 7
Fig. 7

(a) Residual height map error (scale in nanometers) (b) Profile of the error map.

Fig. 8
Fig. 8

Chrome-plated standard of 88 nm step height, as measured by the SPS method. (a)–(c) Three intensities are recorded by the CCD camera, where the X and Y axes are in CCD pixels and the bar is in gray levels. (d) The resulting height map of the inspected object, as reconstructed by the system, is presented as a 3D view, where the X and Y axes are in micrometers and the bar is the height in nanometers. (e) A profile across the sample indicates its step height.

Equations (46)

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

ϕ ( x , y ) = 4 π h ( x , y ) λ mod 2 π ,
H ( x , y ) = k exp ( i θ ) G Δ ( x , y ) + [ 1 G Δ ( x , y ) ] = G Δ ( x , y ) ( k exp ( i θ ) 1 ) + 1 ,
[ k exp ( i θ ) 1 ] S ( x , y ) + f ( x , y ) .
S ( x , y ) f ( x , y ) g ( x , y )
g ( x , y ) = F 1 [ G Δ ( u λ L , v λ L ) ] ,
g ( x , y ) = 1 2 ( δ λ L ) 2 π J 1 ( π δ λ L x 2 + y 2 ) π δ λ L x 2 + y 2 ,
g ( x , y ) = ( δ λ L ) 2 sinc ( δ λ L x ) sinc ( δ λ L y ) .
I θ ( x , y ) = [ k exp ( i θ ) 1 ] S ( x , y ) + f ( x , y ) 2 .
I 0 ( x , y ) = f ( x , y ) 2 = A 2 ( x , y ) .
I θ ( x , y ) = k exp ( i θ ) 1 2 S ( x , y ) 2 .
I θ ( x , y ) = k exp ( i θ ) S ( x , y ) + R ( x , y ) 2 .
g ̃ ( x ̃ , y ̃ ) = A ( D ) g ( A ( D ) x ̃ , A ( D ) y ̃ ) = F 1 [ G Δ ( u λ L , v λ L ) ] = F 1 [ G Δ ( λ L A ( D ) u ̃ , λ L A ( D ) v ̃ ) ] .
G Δ ( λ L A ( D ) u ̃ , λ L A ( D ) v ̃ ) = G Δ ̃ ( λ L A ( D ) A ( Δ ) u ̃ , λ L A ( D ) A ( Δ ) v ̃ ) .
SAN = A ( Δ ) A ( D ) λ L .
g ̃ ( x ̃ , y ̃ ) = F 1 [ G Δ ̃ ( u ̃ SAN , v ̃ SAN ) ] .
I θ ( x , y ) = ( k exp ( i θ ) 1 ) C ( x , y ) exp [ i ψ ( x , y ) ] + A ( x , y ) 2 ,
I θ ( x , y ) = β 0 ( x , y ) + β c ( x , y ) cos θ + β s ( x , y ) sin θ ,
β 0 ( x , y ) = A 2 ( x , y ) + ( 1 + k 2 ) C 2 ( x , y ) 2 A ( x , y ) C ( x , y ) cos ψ ( x , y ) ,
β c ( x , y ) = 2 k A ( x , y ) C ( x , y ) cos ψ ( x , y ) 2 k C 2 ( x , y ) ,
β s ( x , y ) = 2 k A ( x , y ) C ( x , y ) sin ψ ( x , y ) .
I θ ( x , y ) = ( k 2 C 2 ( x , y ) + B 2 ( x , y ) ) + 2 k C ( x , y ) B ( x , y ) cos [ θ + η ( x , y ) α ( x , y ) ] .
β 0 ( x , y ) = k 2 C 2 ( x , y ) + B 2 ( x , y ) ,
β c ( x , y ) = 2 k C ( x , y ) B ( x , y ) cos [ η ( x , y ) α ( x , y ) ] ,
β s ( x , y ) = 2 k C ( x , y ) B ( x , y ) sin [ η ( x , y ) α ( x , y ) ] .
ψ ( x , y ) = ϕ ( x , y ) α ( x , y ) = arg { 1 + B ( x , y ) C ( x , y ) exp i [ η ( x , y ) α ( x , y ) ] } .
C 2 ( x , y ) = 1 1 + k 2 β 0 ( x , y ) = 1 2 k β c ( x , y ) ,
X 2 β 0 X + β c 2 + β s 2 4 = 0 .
C ± ( x , y ) = 1 k 2 { β 0 ( x , y ) ± [ β 0 2 ( x , y ) β c 2 ( x , y ) β s 2 ( x , y ) ] 1 2 } 1 2 .
A ( x , y ) = [ β 0 ( x , y ) + 1 k β c ( x , y ) + ( 1 k 2 ) C 2 ( x , y ) ] 1 2 .
A ( x , y ) = [ β 0 ( x , y ) + β c ( x , y ) ] 1 2 ,
ψ ( x , y ) = arg [ β c ( x , y ) + 2 k C 2 ( x , y ) + i β s ( x , y ) ] ,
Γ ( x , y ) A ( x , y ) exp [ i ψ ( x , y ) ] C ( x , y ) .
S = ( Γ S ) g .
s = L s ,
S = ( Γ S ) g n .
M ( x , y ) = Max θ I θ ( x , y ) Min θ I θ ( x , y ) Max θ I θ ( x , y ) + Min θ I θ ( x , y ) = 2 k C ( x , y ) B ( x , y ) k 2 C 2 ( x , y ) + B 2 ( x , y ) .
M ( x , y ) = { [ β c ( x , y ) β 0 ( x , y ) ] 2 + [ β s ( x , y ) β 0 ( x , y ) ] 2 } 1 2 .
P [ 1 cos θ 1 sin θ 1 1 cos θ N sin θ N ] .
I ( x , y ) P β ( x , y ) 2 = Min V I ( x , y ) P V 2 .
θ p = θ 0 + 2 π ( p 1 ) N , 1 p N .
1 M 2 ( x , y ) β 0 2 ( x , y ) = d ( x , y ) .
S ( x , y ) = f ( s , t ) g ( x s , y t ) d s d t .
g n ( x , y , s , t ) = i + j n g i , j ( s , t ) x i y j ,
U P T P = [ N c j s j c j c j 2 c j s j s j c j s j s j 2 ] .
Q ( N ) = ( c j ) 2 c j 2 + ( s j ) 2 s j 2 2 c j s j c j s j ,
Q ( N ) ( c j c j 2 s j s j 2 ) 2 0 .

Metrics