Abstract

Among several techniques, phase shifting interferometry can be implemented with a grating used as a beam divider to attain several interference patterns around each diffraction order. Because each pattern has to show a different phase-shift, a suitable shifting technique must be employed. Phase gratings are attractive to perform the former task due to their higher diffraction efficiencies. But as is very well known, the Fourier coefficients of only-phase gratings are integer order Bessel functions of the first kind. The values of these real-valued functions oscillate around zero, so they can adopt negative values, thereby introducing phase shifts of π at certain diffraction orders. Because this almost trivial fact seems to have been overlooked in the literature regarding its practical implications, in this communication such phase shifts are stressed in the description of interference patterns obtained with grating interferometers. These patterns are obtained by placing two windows in the object plane of a 4f system with a sinusoidal grating/grid in the Fourier plane. It is shown that the corresponding experimental observations of the fringe modulation, as well as the corresponding phase measurements, are all in agreement with the proposed description. A one-shot phase shifting interferometer is finally proposed taking into account these properties after proper incorporation of modulation of polarization.

© 2008 Optical Society of America

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References

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  1. R. M. A. Azzam, "Division-of-Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light," Opt. Acta 29, 685-689 (1982).
    [CrossRef]
  2. • V. Ronchi, "Forty Years of History of a Grating Interferometer," Appl. Opt. 3, 437-451 (1964).
    [CrossRef]
  3. A. Cornejo-Rodríguez, "Ronchi Test," in Optical Shop Testing, D. Malacara ed., (Wiley, New York, 1992)
  4. E. S. Barrekette, H. Freitag, "Diffraction by a Finite Sinusoidal Phase Grating," IBM Journal, 345-349 (1963).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics, 2nd edition, (McGraw-Hill, 1988).
  6. F. Kneubühl, "Diffraction Grating Spectroscopy," Appl. Opt. 8, 505-519 (1969).
    [CrossRef] [PubMed]
  7. V. Arrizón and D. Sánchez-De-La-Llave, "Common-Path Interferometry with One-Dimensional Periodic Filters," Opt. Lett. 29, 141-143 (2004).
    [CrossRef] [PubMed]
  8. C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizon, "Optical Tomography of Transparent Objects with Phase-Shifting Interferometry and Stepping-Wise Shifted Ronchi Ruling," J. Opt. Soc. Am. A 23, 298-305 (2006).
    [CrossRef]
  9. C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizon, "Common-Path Phase-Shifting Interferometer with Binary Grating," Opt. Commun. 264, 13-17 (2006).
    [CrossRef]
  10. G. Rodriguez-Zurita, C. Meneses-Fabian, N. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, "One-Shot Phase-Shifting Phase-Grating Interferometry with Modulation of Polarization: case of four interferograms," Opt. Express 16, 7806-7817 (2008).
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  11. D. A. Thomas and J. C. Wyant, "High Efficiency Grating Lateral Shear Interferometer," Opt. Eng. 15, 477 (1976).
  12. P. W. Ramijan, "Processing Stereo Photographs by Optical Subtraction," Ph. D. Thesis, University of Rochester (1978).
  13. J. Schwieder, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital Wave-Front Measuring Interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983).
    [CrossRef]
  14. T. Kreis, "Digital Holographic Interference-Phase Measurement Using the Fourier-Transform Method," J. Opt. Soc. Am. A 3, 847-855 (1986).
    [CrossRef]
  15. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
    [CrossRef]
  16. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, "Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer," Appl. Opt. 44, 6861-6868 (2005).
    [CrossRef] [PubMed]
  17. J. C. Wyant, "Double Frequency Grating Lateral Shear Interferometer," Appl. Opt. 12, 2057-2060 (1973).
    [CrossRef] [PubMed]

2008

2006

2005

2004

1999

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

1986

1983

1982

R. M. A. Azzam, "Division-of-Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light," Opt. Acta 29, 685-689 (1982).
[CrossRef]

1976

D. A. Thomas and J. C. Wyant, "High Efficiency Grating Lateral Shear Interferometer," Opt. Eng. 15, 477 (1976).

1973

1969

1964

Arrizon, V.

Arrizón, V.

Azzam, R. M. A.

R. M. A. Azzam, "Division-of-Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light," Opt. Acta 29, 685-689 (1982).
[CrossRef]

Barrekette, E. S.

E. S. Barrekette, H. Freitag, "Diffraction by a Finite Sinusoidal Phase Grating," IBM Journal, 345-349 (1963).
[CrossRef]

Barrientos-García, B.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Brock, N.

Burow, R.

Elssner, K.-E.

Freitag, H.

E. S. Barrekette, H. Freitag, "Diffraction by a Finite Sinusoidal Phase Grating," IBM Journal, 345-349 (1963).
[CrossRef]

Grzanna, J.

Hayes, J.

Kneubühl, F.

Kreis, T.

Meneses-Fabian, C.

Merkel, K.

Millerd, J.

Moore, A. J.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

North-Morris, M.

Novak, M.

Pérez-López, C.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Robledo-Sánchez, C.

Rodriguez-Zurita, G.

Ronchi, V.

Sánchez-De-La-Llave, D.

Schwieder, J.

Spolaczyk, R.

Thomas, D. A.

D. A. Thomas and J. C. Wyant, "High Efficiency Grating Lateral Shear Interferometer," Opt. Eng. 15, 477 (1976).

Toto-Arellano, N.

Tschudi, T

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Vázquez-Castillo, J. F.

Wang, L.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Wyant, J.

Wyant, J. C.

D. A. Thomas and J. C. Wyant, "High Efficiency Grating Lateral Shear Interferometer," Opt. Eng. 15, 477 (1976).

J. C. Wyant, "Double Frequency Grating Lateral Shear Interferometer," Appl. Opt. 12, 2057-2060 (1973).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Acta

R. M. A. Azzam, "Division-of-Amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of all Four Stokes Parameters of Light," Opt. Acta 29, 685-689 (1982).
[CrossRef]

Opt. Commun.

C. Meneses-Fabian, G. Rodriguez-Zurita, and V. Arrizon, "Common-Path Phase-Shifting Interferometer with Binary Grating," Opt. Commun. 264, 13-17 (2006).
[CrossRef]

Opt. Eng.

D. A. Thomas and J. C. Wyant, "High Efficiency Grating Lateral Shear Interferometer," Opt. Eng. 15, 477 (1976).

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T , Tschudi, "Spatial Phase-Stepped Interferometry using a Holographic Optical Element," Opt. Eng. 38, 2069-2074 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Other

A. Cornejo-Rodríguez, "Ronchi Test," in Optical Shop Testing, D. Malacara ed., (Wiley, New York, 1992)

E. S. Barrekette, H. Freitag, "Diffraction by a Finite Sinusoidal Phase Grating," IBM Journal, 345-349 (1963).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd edition, (McGraw-Hill, 1988).

P. W. Ramijan, "Processing Stereo Photographs by Optical Subtraction," Ph. D. Thesis, University of Rochester (1978).

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Figures (8)

Fig. 1.
Fig. 1.

TWPGI with phase periodic element G i of period d. i=1 one-dimensional case (grating), i=2 two-dimensional case (grid).

Fig. 2.
Fig. 2.

Amplitude signs of diffraction orders (hypothetical grating) in the image plane of a TWPGI resulting from windows displacement of ± x0/2. Upper left: windows configuration.

Fig. 3.
Fig. 3.

Bessel domain for the ninth lobule of J 0 (left). Corresponding Fourier spectrum of the phase grating (right).

Fig. 4.
Fig. 4.

(a) π-phase distribution of diffraction orders of grids. The dashed lines enclose diffraction orders of indexes 0,n or m,0. (b) TWPGI order superposition: Configuration W 1: interference pattern signs for windows displaced along the horizontal axis. Configuration W 2: interference patterns signs for displaced windows along a line at 45°. Respective displacements of diffraction patterns are remarked in dashed lines. An explanation of dot and cross patterns to be found in the text.

Fig. 5.
Fig. 5.

Experimental results with a phase-grating. Each image was subject to the same scaling process (from 0 to 255).

Fig. 6.
Fig. 6.

Experimental patters for a phase-grid (composite image, windows configuration W 2).

Fig. 7.
Fig. 7.

Polarizing filters array for 90° phase stepping.

Fig. 8.
Fig. 8.

Upper row: phase dot. Four 90° phase-shifted interferograms and unwrapped phase. Lower row: phase step. Four 90° phase-shifted interferograms and unwrapped phase.

Tables (1)

Tables Icon

Table 1. Phase shifts of the 16 patterns within the dotted square of Fig. 6, as measured by the method from Ref.14 (from the left, above).

Equations (24)

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G 1 ( μ , ς ) = e i 2 π · A g sin [ 2 π F 0 μ ] = q = J q ( 2 π A g ) · e i 2 π · q F 0 μ ,
G ˜ 1 ( x , y ) = q = J q ( 2 π A g ) δ ( x q F 0 , y )
G 2 ( μ , ς ) = e i 2 π A g sin [ 2 π · X 0 μ ] e i 2 π A g sin [ 2 π · Y 0 ς ] = q = J q ( 2 π A g ) e i 2 π · q F 0 μ r = J r ( 2 π A g ) e i 2 π · r F 0 ς ,
G ˜ 2 ( x , y ) = q = q = r = r = J q ( 2 π A g ) J r ( 2 π A g ) δ ( x q F 0 , y r F 0 ) ,
t 1 ( x , y ) = w ( x + x 0 2 , y ) + w ' ( x x 0 2 , y )
t f 1 ( x , y ) = t 1 ( x , y ) * G ˜ 1 ( x , y )
= w ( x + x 0 2 , y ) * q = J q ( 2 π A g ) δ ( x q F 0 , y )
+ [ w ( x x 0 2 , y ) e i ϕ ( x x 0 2 , y ) ] * q = J q ( 2 π A g ) δ ( x q F 0 , y ) .
t f 1 ( x , y ) = q = [ J q ( 2 π A g ) + J q 1 ( 2 π A g ) e i ϕ ( x x 0 [ q 1 2 ] , y ) ] w ( x x 0 [ q 1 2 ] , y ) .
m q = 2 J q J q 1 J q 2 + J q 1 2 .
t f 2 ( x , y ) = t 2 ( x , y ) * G ˜ 2 ( x , y )
= q = w ( x + x 0 2 , y + x 0 2 ) * J q ( 2 π A g ) J r ( 2 π A g ) δ ( x q F 0 , y r F 0 )
+ q = w ' ( x x 0 2 , y x 0 2 ) * J q ( 2 π A g ) J r ( 2 π A g ) δ ( x q F 0 , y r F 0 ) .
t f 2 ( x , y ) = q = r = J q ( 2 π A g ) J r ( 2 π A g ) w ( x + x 0 2 [ 1 2 q ] , y + x 0 2 [ 1 2 r ] ) +
q = r = J q 1 ( 2 π A g ) J r 1 ( 2 π A g ) w ' ( x x 0 2 [ 1 2 q ] , y x 0 2 [ 1 2 r ] ) ,
t f 2 ( x , y ) =
q = r = [ J q ( 2 π A g ) J r ( 2 π A g ) + J q 1 ( 2 π A g ) J r 1 ( 2 π A g ) e i ϕ { ( x x 0 [ q 1 2 ] ) , ( y x 0 [ r 1 2 ] ) } ]
× w ( x x 0 [ q 1 2 ] , y x 0 [ r 1 2 ] ) .
m qr = 2 J q J q 1 J r J r 1 ( J q J r ) 2 + ( J q 1 J r 1 ) 2 .
J T 2 = A ( ξ , α ' ) { ( J q 2 + J q 1 2 ) + 2 J q J q 1 cos [ ξ ( ψ , α ' ) ϕ ( x , y ) ] }
ξ ( ψ , α ' ) = Arc Tan [ sin [ 2 ψ ] · sin [ α ' ] + sin 2 [ ψ ] · sin [ 2 α ' ] cos 2 [ ψ ] + sin 2 [ ψ ] · cos [ 2 α ' ] + sin [ 2 ψ ] · cos [ α ' ] ] ,
A ( ψ , α ' ) = 1 + sin [ 2 ψ ] cos [ α ' ] .
J i 2 = A ( ξ , α ' ) { ( J q 2 + J q 1 2 ) + 2 J q J q 1 cos [ ξ ( ψ i , α ' ) ϕ ( x , y ) ] } ,
tan ϕ = J 1 2 · J 3 2 J 2 2 J 4 2

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