Abstract

Calibrated multichannel electronic interferometry is an electro-optic technique for performing phase shifting of transient phenomena. The design of an improved system for calibrated multichannel electronic interferometry is discussed. This includes a computational method for alignment of three phase-shifted interferograms and determination of the pixel correspondence. During calibration the phase, modulation, and bias of the optical system are determined. These data are stored electronically and used to compensate for errors associated with the path differences in the interferometer, the separation of the phase-shifted interferograms, and the measurement of the phase shift.

© 1995 Optical Society of America

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References

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  1. W. Jüptner, T. M. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 22–29 (1983).
  2. P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).
  3. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
    [CrossRef]
  4. D. W. Watt, C. M. Vest, “Digital interferometry for flow visualization,” Exp. Fluids 5, 401–406 (1987).
    [CrossRef]
  5. T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Digital phase-stepping holographic interferometry in measuring 2-D density fields,” Exp. Fluids 9, 231–235 (1990).
    [CrossRef]
  6. D. M. Shough, O. Y. Kwon, “Phase-shifting pulsed-laser interferometer,” in Phase Conjugation, Beam Combining, and Diagnostics, I. Abramowitz, R. A. Fisher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.739, 174–180 (1987).
  7. M. Kujawińska, D. W. Robinson, “Multichannel phase-stepped holographic interferometry,” Appl. Opt. 27, 312–320 (1988).
    [CrossRef]
  8. M. Kujawińska, A. Spik, D. W. Robinson, “Quantitative analysis of transient events by ESPI,” in Interferometry ’89, Z. Jaroszewicz, M. Pluta, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1121, 416–423 (1989).
  9. T. D. Upton, D. W. Watt, “Calibrated multichannel electronic interferometry for quantitative flow visualization,” Exp. Fluids 14, 271–276 (1993).
    [CrossRef]
  10. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 5, pp. 264–284.
  11. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1979), Chap. 5, pp. 109–116.
  12. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 3, pp. 24–48.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 4–25.
  14. C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  15. R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
    [CrossRef]
  16. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  17. R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).
  18. J. P. Holman, Experimental Methods for Engineers (McGraw-Hill, New York, 1989), Chap. 3, pp. 41–49.

1993 (1)

T. D. Upton, D. W. Watt, “Calibrated multichannel electronic interferometry for quantitative flow visualization,” Exp. Fluids 14, 271–276 (1993).
[CrossRef]

1990 (2)

C. P. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
[CrossRef]

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Digital phase-stepping holographic interferometry in measuring 2-D density fields,” Exp. Fluids 9, 231–235 (1990).
[CrossRef]

1988 (1)

1987 (1)

D. W. Watt, C. M. Vest, “Digital interferometry for flow visualization,” Exp. Fluids 5, 401–406 (1987).
[CrossRef]

1985 (2)

P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

1983 (1)

1982 (1)

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 3, pp. 24–48.

Brophy, C. P.

Burow, R.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

Dändliker, R.

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Elssner, K.-E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 4–25.

Grzanna, J.

Hariharan, P.

P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1979), Chap. 5, pp. 109–116.

Holman, J. P.

J. P. Holman, Experimental Methods for Engineers (McGraw-Hill, New York, 1989), Chap. 3, pp. 41–49.

Jüptner, W.

W. Jüptner, T. M. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 22–29 (1983).

Kreis, T. M.

W. Jüptner, T. M. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 22–29 (1983).

Kreitlow, H.

W. Jüptner, T. M. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 22–29 (1983).

Kujawinska, M.

M. Kujawińska, D. W. Robinson, “Multichannel phase-stepped holographic interferometry,” Appl. Opt. 27, 312–320 (1988).
[CrossRef]

M. Kujawińska, A. Spik, D. W. Robinson, “Quantitative analysis of transient events by ESPI,” in Interferometry ’89, Z. Jaroszewicz, M. Pluta, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1121, 416–423 (1989).

Kwon, O. Y.

D. M. Shough, O. Y. Kwon, “Phase-shifting pulsed-laser interferometer,” in Phase Conjugation, Beam Combining, and Diagnostics, I. Abramowitz, R. A. Fisher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.739, 174–180 (1987).

Lanen, T. A. W. M.

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Digital phase-stepping holographic interferometry in measuring 2-D density fields,” Exp. Fluids 9, 231–235 (1990).
[CrossRef]

Merkel, K.

Nebbeling, C.

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Digital phase-stepping holographic interferometry in measuring 2-D density fields,” Exp. Fluids 9, 231–235 (1990).
[CrossRef]

Robinson, D. W.

M. Kujawińska, D. W. Robinson, “Multichannel phase-stepped holographic interferometry,” Appl. Opt. 27, 312–320 (1988).
[CrossRef]

M. Kujawińska, A. Spik, D. W. Robinson, “Quantitative analysis of transient events by ESPI,” in Interferometry ’89, Z. Jaroszewicz, M. Pluta, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1121, 416–423 (1989).

Schwider, J.

Shough, D. M.

D. M. Shough, O. Y. Kwon, “Phase-shifting pulsed-laser interferometer,” in Phase Conjugation, Beam Combining, and Diagnostics, I. Abramowitz, R. A. Fisher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.739, 174–180 (1987).

Spik, A.

M. Kujawińska, A. Spik, D. W. Robinson, “Quantitative analysis of transient events by ESPI,” in Interferometry ’89, Z. Jaroszewicz, M. Pluta, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1121, 416–423 (1989).

Spolaczyk, R.

Thalmann, R.

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Upton, T. D.

T. D. Upton, D. W. Watt, “Calibrated multichannel electronic interferometry for quantitative flow visualization,” Exp. Fluids 14, 271–276 (1993).
[CrossRef]

van Ingen, J. L.

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Digital phase-stepping holographic interferometry in measuring 2-D density fields,” Exp. Fluids 9, 231–235 (1990).
[CrossRef]

Vest, C. M.

D. W. Watt, C. M. Vest, “Digital interferometry for flow visualization,” Exp. Fluids 5, 401–406 (1987).
[CrossRef]

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 5, pp. 264–284.

Watt, D. W.

T. D. Upton, D. W. Watt, “Calibrated multichannel electronic interferometry for quantitative flow visualization,” Exp. Fluids 14, 271–276 (1993).
[CrossRef]

D. W. Watt, C. M. Vest, “Digital interferometry for flow visualization,” Exp. Fluids 5, 401–406 (1987).
[CrossRef]

Willemin, J.-F.

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1979), Chap. 5, pp. 109–116.

Appl. Opt. (2)

Exp. Fluids (3)

D. W. Watt, C. M. Vest, “Digital interferometry for flow visualization,” Exp. Fluids 5, 401–406 (1987).
[CrossRef]

T. A. W. M. Lanen, C. Nebbeling, J. L. van Ingen, “Digital phase-stepping holographic interferometry in measuring 2-D density fields,” Exp. Fluids 9, 231–235 (1990).
[CrossRef]

T. D. Upton, D. W. Watt, “Calibrated multichannel electronic interferometry for quantitative flow visualization,” Exp. Fluids 14, 271–276 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

R. Dändliker, R. Thalmann, J.-F. Willemin, “Fringe interpolation by two-reference-beam holographic interferometry: reducing sensitivity to hologram misalignment,” Opt. Commun. 42, 301–306 (1982).
[CrossRef]

Opt. Eng. (2)

R. Dändliker, R. Thalmann, “Heterodyne and quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 824–831 (1985).

P. Hariharan, “Quasi-heterodyne holographic interferometry,” Opt. Eng. 24, 632–638 (1985).

Other (9)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
[CrossRef]

D. M. Shough, O. Y. Kwon, “Phase-shifting pulsed-laser interferometer,” in Phase Conjugation, Beam Combining, and Diagnostics, I. Abramowitz, R. A. Fisher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.739, 174–180 (1987).

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 5, pp. 264–284.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1979), Chap. 5, pp. 109–116.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 3, pp. 24–48.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 4–25.

J. P. Holman, Experimental Methods for Engineers (McGraw-Hill, New York, 1989), Chap. 3, pp. 41–49.

W. Jüptner, T. M. Kreis, H. Kreitlow, “Automatic evaluation of holographic interferograms by reference beam phase shifting,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 22–29 (1983).

M. Kujawińska, A. Spik, D. W. Robinson, “Quantitative analysis of transient events by ESPI,” in Interferometry ’89, Z. Jaroszewicz, M. Pluta, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1121, 416–423 (1989).

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Figures (9)

Fig. 1
Fig. 1

Schematic of a calibrated multichannel electronic interferometer.

Fig. 2
Fig. 2

Optimum distance from the gratings to the camera.

Fig. 3
Fig. 3

Displacement of the diffracted orders in a flat beam splitter.

Fig. 4
Fig. 4

Imaging with a negative meniscus lens.

Fig. 5
Fig. 5

Displacement of the diffracted orders for a misaligned beam.

Fig. 6
Fig. 6

Common types of misalignment determined by the use of spatial cross correlation with 2× magnification in the Fourier domain: (a) a one-pixel error in both the pixel correspondence and alignment of the diffraction plane, (b) a one-pixel error in the pixel correspondence, (c) a one-pixel error in alignment of the diffraction plane, (d) a well-aligned beam.

Fig. 7
Fig. 7

Calibration of a multichannel electronic interferometer: (a) phase, (b) modulation, (c) bias.

Fig. 8
Fig. 8

Sequential phase maps of a transient buoyant thermal plume.

Fig. 9
Fig. 9

Unwrapped phase of the plume shown in Fig. 8.

Equations (37)

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I ( x , y ) = a ( x , y ) { 1 + m ( x , y ) cos [ Θ ( x , y ) ] } ,
I i ( x , y ) = a ( x , y ) { 1 + m ( x , y ) cos [ Θ ( x , y ) + Ψ i ] } ,
t ( y ) = J 0 ( M / 2 ) + J 1 ( M / 2 ) exp [ i ( 2 π f g y + Ψ ) ] + J 1 ( M / 2 ) exp [ - i ( 2 π f g y + Ψ - π ) ] ,
Δ Ψ = 2 π f g Δ y ,
α k = sin - 1 ( k λ f g ) ,
z g = w c - z bs tan { sin - 1 [ n air n bs sin ( α 1 ) ] } tan ( α 1 ) + z bs ,
d k = z bs ( cos ( k ) tan { sin - 1 [ n air n bs sin ( k ) ] } - sin ( k ) ) ,
k = bs + α k
k = cos - 1 [ cos ( bs ) cos ( α k ) ] ,
OPD = n bs z bs cos { sin - 1 [ n air n bs sin ( k ) ] } - n air z bs cos ( k ) .
s i 2 = f 2 ( 1 + f 2 f 1 ) - f 2 2 f 1 2 s o 1 ,
s o 1 < f 1 ( 1 + f 1 f 2 ) .
M T = - f 2 f 1 .
M T = w c w t ,
f 2 f 1 = w c w t .
d 2 = f m ( d 1 - f 1 ) d 1 - f 1 - f m + f 2 ,
s i 2 = f 2 [ 1 + f 2 ( d 1 - f 1 - f m ) ( d 1 - f m ) f 1 f m 2 ] - f 2 2 ( d 1 - f 1 - f m ) 2 f 1 2 f m 2 s o 1 .
s o 1 < f 1 2 f m 2 + f 1 f 2 ( d 1 - f 1 - f m ) ( d 1 - f m ) f 2 ( d 1 - f 1 - f m ) 2 .
M T = f 2 ( d 1 - f 1 - f m ) f 1 f m .
α k = sin - 1 [ k λ f g + sin ( δ in ) ] - δ in ,
d k = z bs ( cos ( k ) tan { sin - 1 [ n air n bs sin ( k ) ] } - sin ( k ) ) ,
k = cos - 1 [ cos ( α k + δ in ) cos ( δ out ) ] ,
OPD = n air [ z g - z bs cos ( k ) - z g - z bs cos ( α k ) ] + n bs ( z bs cos { sin - 1 [ n air n bs sin ( k ) ] } - z bs cos { sin - 1 [ n air n bs sin ( α k ) ] } ) .
ϕ f g ( X , Y ) = - - f * ( x , y ) g ( x + X , y + Y ) d x d y ,
V Ψ = Ψ 2 π V 2 π ,
Ψ = ( N - 1 ) N π ,
I 1 ( x , y ) = a ( x , y ) { 1 + m ( x , y ) cos [ Θ ( x , y ) ] } ,
I 2 ( x , y ) = a ( x , y ) { 1 + m ( x , y ) cos [ Θ ( x , y ) + Ψ ] } ,
I 3 ( x , y ) = a ( x , y ) { 1 + m ( x , y ) cos [ Θ ( x , y ) + 2 Ψ ] } ,
I 4 ( x , y ) = a ( x , y ) { 1 + m ( x , y ) cos [ Θ ( x , y ) + 3 Ψ ] } .
Ψ = cos - 1 [ ( I 4 - I 1 ) - ( I 3 - I 2 ) 2 ( I 3 - I 2 ) ] .
Θ ( x , y ) = tan - 1 ( [ cos ( Ψ ) - 1 ] { [ 2 cos ( Ψ ) + 1 ] I 1 - [ 2 cos ( Ψ ) + 2 ] I 2 + I 3 } [ 1 - cos 2 ( Ψ ) ] 1 / 2 { [ 2 cos ( Ψ ) - 1 ] I 1 - [ 2 cos ( Ψ ) ] I 2 + I 3 } )
a ( x , y ) m ( x , y ) = I 2 - I 1 [ cos ( Ψ ) - 1 ] cos ( Θ ) - sin ( Ψ ) sin ( Θ )
a ( x , y ) m ( x , y ) = I 3 - I 2 [ cos ( Ψ ) - 1 ] [ 2 cos ( Ψ ) + 1 ] cos ( Θ ) - sin ( Ψ ) [ 2 cos ( Ψ ) + 1 ] sin ( Θ )
a ( x , y ) = I 1 - a m cos ( Θ )
Θ ( x , y ) = tan - 1 [ ( I 2 - I 3 ) cos ( Ψ 1 ) + ( I 3 - I 1 ) cos ( Ψ 2 ) + ( I 1 - I 2 ) cos ( Ψ 3 ) ( I 2 - I 3 ) sin ( Ψ 1 ) + ( I 3 - I 1 ) sin ( Ψ 2 ) + ( I 1 - I 2 ) sin ( Ψ 3 ) ] .
Θ ( x , y ) = tan - 1 { [ a 1 m 1 a 1 ( I 2 a 2 - I 3 a 3 ) ] cos ( Ψ 1 ) + [ a 2 m 2 a 2 ( I 3 a 3 - I 1 a 1 ) ] cos ( Ψ 2 ) + [ a 3 m 3 a 3 ( I 1 a 1 - I 2 a 2 ) ] cos ( Ψ 3 ) [ a 1 m 1 a 1 ( I 2 a 2 - I 3 a 3 ) ] sin ( Ψ 1 ) + [ a 2 m 2 a 2 ( I 3 a 3 - I 1 a 1 ) ] sin ( Ψ 2 ) + [ a 3 m 3 a 3 ( I 1 a 1 - I 2 a 2 ) ] sin ( Ψ 3 ) } .

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