Abstract

Common-path interferometers have been used to perform phase visualization for over 40 years. A number of techniques have been proposed, including dark central ground, phase contrast (π/2 and π), and field-absorption interferometers. The merits of the interferometers have been judged ad hoc by either tests with a small number of phase objects or by computer simulation. Three standardized criteria, which consolidate the work of others, are proposed to evaluate common-path interferometers: fringe visibility, fringe irradiance, and fringe accuracy. The interferometers can be described as one generic class of Fourier-plane filters and can be analyzed for all input conditions. Closed-form expressions are obtained for visibility and irradiance under the forced condition that little inaccuracy is tolerated. This analysis finds that the π-phase-contrast interferometer is a good choice if the optical phase disturbance is at least 2π; for smaller disturbances, the π/2 filter selected by Zernike is near optimum. It is shown mathematically that the resulting fringe visibility is highly object dependent, and good results are not ensured. By allowing the optical beam to be 50% larger than the phase object, the interferometer performs well under all conditions. With this approach and a combination π-phase/field-absorption filter, interference fringe visibility is greater than 0.8 for all phase objects.

© 1995 Optical Society of America

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References

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  1. R. Anderson, J. Milton, C. Anderson, “Common path interferometers for flow visualization: a review,” AIAA 95-0481 (American Institute of Aeronautics and Astronautics, New York, 1995).
  2. R. C. Anderson, S. Lewis, “Flow visualization by dark central ground interferometer,” Appl. Opt. 24, 3687 (1985).
    [CrossRef] [PubMed]
  3. R. C. Anderson, J. E. Milton, “A large aperture inexpensive interferometer for routine flow measurements,” in Proceedings of the International Congress on Instrumentation in Aerospace Simulation Facilities (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 394–399.
    [CrossRef]
  4. L. Carr, M. Chandrasekhara, S. Ahmed, N. Brock, “A study of dynamic stall using real time interferometry,” AIAA 91-0007 (American Institute of Aeronautics and Astronautics, New York, 1991).
  5. L. Carr, M. Chandrasekhara, N. Brock, “A quantitative study of unsteady compressible flow on an oscillating airfoil,” AIAA-91-1683 (American Institute of Aeronautics and Astronautics, New York, 1991).
  6. M. Taylor, “Phase contrast flow visualization,” M.S. thesis (College of Engineering, University of Florida, Gainesville, Fla., 1980).
  7. S. F. Erdmann, “Ein neues, sehr einfaches Interferometer zum Erhalt quantitativ auswertbarer Strömungsbilder,” Appl. Sci. Res. Sect. B 2, 1–50 (1951).
  8. R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).
  9. R. N. Smartt, “Special applications of the point-diffraction interferometer,” in Interferometry, G. W. Hopkins, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 192, 35–40 (1979).
  10. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part I,” Physica 9, 686–698 (1942; F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chaps. 5–7.

1985 (1)

1975 (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

1951 (1)

S. F. Erdmann, “Ein neues, sehr einfaches Interferometer zum Erhalt quantitativ auswertbarer Strömungsbilder,” Appl. Sci. Res. Sect. B 2, 1–50 (1951).

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part I,” Physica 9, 686–698 (1942; F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
[CrossRef]

Ahmed, S.

L. Carr, M. Chandrasekhara, S. Ahmed, N. Brock, “A study of dynamic stall using real time interferometry,” AIAA 91-0007 (American Institute of Aeronautics and Astronautics, New York, 1991).

Anderson, C.

R. Anderson, J. Milton, C. Anderson, “Common path interferometers for flow visualization: a review,” AIAA 95-0481 (American Institute of Aeronautics and Astronautics, New York, 1995).

Anderson, R.

R. Anderson, J. Milton, C. Anderson, “Common path interferometers for flow visualization: a review,” AIAA 95-0481 (American Institute of Aeronautics and Astronautics, New York, 1995).

Anderson, R. C.

R. C. Anderson, S. Lewis, “Flow visualization by dark central ground interferometer,” Appl. Opt. 24, 3687 (1985).
[CrossRef] [PubMed]

R. C. Anderson, J. E. Milton, “A large aperture inexpensive interferometer for routine flow measurements,” in Proceedings of the International Congress on Instrumentation in Aerospace Simulation Facilities (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 394–399.
[CrossRef]

Brock, N.

L. Carr, M. Chandrasekhara, N. Brock, “A quantitative study of unsteady compressible flow on an oscillating airfoil,” AIAA-91-1683 (American Institute of Aeronautics and Astronautics, New York, 1991).

L. Carr, M. Chandrasekhara, S. Ahmed, N. Brock, “A study of dynamic stall using real time interferometry,” AIAA 91-0007 (American Institute of Aeronautics and Astronautics, New York, 1991).

Carr, L.

L. Carr, M. Chandrasekhara, S. Ahmed, N. Brock, “A study of dynamic stall using real time interferometry,” AIAA 91-0007 (American Institute of Aeronautics and Astronautics, New York, 1991).

L. Carr, M. Chandrasekhara, N. Brock, “A quantitative study of unsteady compressible flow on an oscillating airfoil,” AIAA-91-1683 (American Institute of Aeronautics and Astronautics, New York, 1991).

Chandrasekhara, M.

L. Carr, M. Chandrasekhara, N. Brock, “A quantitative study of unsteady compressible flow on an oscillating airfoil,” AIAA-91-1683 (American Institute of Aeronautics and Astronautics, New York, 1991).

L. Carr, M. Chandrasekhara, S. Ahmed, N. Brock, “A study of dynamic stall using real time interferometry,” AIAA 91-0007 (American Institute of Aeronautics and Astronautics, New York, 1991).

Erdmann, S. F.

S. F. Erdmann, “Ein neues, sehr einfaches Interferometer zum Erhalt quantitativ auswertbarer Strömungsbilder,” Appl. Sci. Res. Sect. B 2, 1–50 (1951).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chaps. 5–7.

Lewis, S.

Milton, J.

R. Anderson, J. Milton, C. Anderson, “Common path interferometers for flow visualization: a review,” AIAA 95-0481 (American Institute of Aeronautics and Astronautics, New York, 1995).

Milton, J. E.

R. C. Anderson, J. E. Milton, “A large aperture inexpensive interferometer for routine flow measurements,” in Proceedings of the International Congress on Instrumentation in Aerospace Simulation Facilities (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 394–399.
[CrossRef]

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

R. N. Smartt, “Special applications of the point-diffraction interferometer,” in Interferometry, G. W. Hopkins, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 192, 35–40 (1979).

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Taylor, M.

M. Taylor, “Phase contrast flow visualization,” M.S. thesis (College of Engineering, University of Florida, Gainesville, Fla., 1980).

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part I,” Physica 9, 686–698 (1942; F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
[CrossRef]

Appl. Opt. (1)

Appl. Sci. Res. Sect. B (1)

S. F. Erdmann, “Ein neues, sehr einfaches Interferometer zum Erhalt quantitativ auswertbarer Strömungsbilder,” Appl. Sci. Res. Sect. B 2, 1–50 (1951).

Jpn. J. Appl. Phys. (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part I,” Physica 9, 686–698 (1942; F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects Part II,” Physica 9, 974–986 (1942).
[CrossRef]

Other (7)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chaps. 5–7.

R. Anderson, J. Milton, C. Anderson, “Common path interferometers for flow visualization: a review,” AIAA 95-0481 (American Institute of Aeronautics and Astronautics, New York, 1995).

R. N. Smartt, “Special applications of the point-diffraction interferometer,” in Interferometry, G. W. Hopkins, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 192, 35–40 (1979).

R. C. Anderson, J. E. Milton, “A large aperture inexpensive interferometer for routine flow measurements,” in Proceedings of the International Congress on Instrumentation in Aerospace Simulation Facilities (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 394–399.
[CrossRef]

L. Carr, M. Chandrasekhara, S. Ahmed, N. Brock, “A study of dynamic stall using real time interferometry,” AIAA 91-0007 (American Institute of Aeronautics and Astronautics, New York, 1991).

L. Carr, M. Chandrasekhara, N. Brock, “A quantitative study of unsteady compressible flow on an oscillating airfoil,” AIAA-91-1683 (American Institute of Aeronautics and Astronautics, New York, 1991).

M. Taylor, “Phase contrast flow visualization,” M.S. thesis (College of Engineering, University of Florida, Gainesville, Fla., 1980).

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Figures (15)

Fig. 1
Fig. 1

Diagram of a common-path interferometer.

Fig. 2
Fig. 2

Irradiance Patterns: (a) phase-distorted beam interfered with perfect reference beam, (b) output interferogram for incorrect filter size.

Fig. 3
Fig. 3

Filter for dark central ground decomposed into the field and the central dot. Note that the dot does not have to be located in the center of the field.

Fig. 4
Fig. 4

Plot of the amplitude and the phase of the reference beam: (a) amplitude of reference beam when θ(x, y) = 0, (b) phase of reference beam when θ(x, y) = 0, (c) amplitude of reference beam when θ(x, y) = phase object, (d) phase of reference beam when θ(x, y) = phase object.

Fig. 5
Fig. 5

Maximum visibility as a function of filter size and b value.

Fig. 6
Fig. 6

Fringe visibility as a function of b for several values of r(0, 0). This assumes a filter spot size of μ = 0.4. DCG, dark central ground; PC, phase change.

Fig. 7
Fig. 7

Visibility for phase objects that have ±θ0 phase change. Visibility is plotted as a function of the central dot phase for a phase-contrast filter.

Fig. 8
Fig. 8

Output interferograms for a sinusoidal input object with ±5° variation: (a) central dark ground filter, (b) π/2-phase-contrast filter, (c) 8π/9-phase-contrast filter, (d) π-phase-contrast filter.

Fig. 9
Fig. 9

Visibility and irradiance as a function of α: (a) maximum-visibility/intensity, (b) for quadratic-phase object shown in Fig. 2.

Fig. 10
Fig. 10

Output interferogram for quadratic-phase function with a central field absorption with α = 0, i.e., central dark ground.

Fig. 11
Fig. 11

Visibility/maximum intensity plots as a function of phase of the central spot: (a) maximum visibility/intensity, (b) for quadratic-phase object shown in Fig. 2.

Fig. 12
Fig. 12

Interferogram obtained from field-absorption method. Total light throughput is less than 1%.

Fig. 13
Fig. 13

Relationship between filter a 2 and b values for two filter types.

Fig. 14
Fig. 14

Interference of outer field interferometer with perfect reference beam.

Fig. 15
Fig. 15

Visibility for different fractional areas.

Equations (28)

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s ( x , y ) = circ [ ( x 2 + y 2 ) 1 / 2 R 0 ] g ( x , y ) exp [ j θ ( x , y ) ] ,
circ [ ( x 2 + y 2 ) 1 / 2 R 0 ] = 1 ,             ( x 2 + y 2 ) 1 / 2 R 0 = 0 ,             ( x 2 + y 2 ) 1 / 2 > R 0 .
I ( x , y ) = 1 + r 2 ( x , y ) + 2 r ( x , y ) × cos [ θ ( x , y ) - Ω ( x , y ) ] + n ( x , y ) ,
S ( ξ , γ ) = c - - s ( x , y ) exp [ j 2 π λ F ( x ξ + y γ ) ] d x d y , = J { circ [ ( x 2 + y 2 ) 1 / 2 R 0 ] } J { g ( x , y ) exp [ j θ ( x , y ) ] } ,
H ( ξ , γ ) = a ( 1 + b exp ( j ϕ ) circ { [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 W f } ) ,
O ( ξ , γ ) = a ( S ( ξ , γ ) + b exp ( j ϕ ) S ( ξ , γ ) × circ { [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 W f } ) .
o ( x , y ) = a [ s ( - x , - y ) + b exp ( j ϕ ) r ( x , y ) ] ,
r ( x , y ) = J ( S ( ξ , γ ) circ { [ ( ξ - ξ 0 ) 2 + ( γ - θ 0 ) 2 ] 1 / 2 W f } ) ,
o ( x , y ) a [ s ( - x , - y ) + b exp ( j ϕ ) r 0 ] ,
I ( x , y ) a 2 ( 1 + b 2 r 0 2 + 2 b r 0 × cos { θ ( - x , - y ) - ϕ - ang [ r ( x , y ) ] } ) ,
V = I max - I min I max + I min = 2 b r 0 1 + b 2 r 0 2 ,
I max = 2 a 2 b r 0 .
r 0 = 1 λ F A = π W r 2 S ( ξ , γ ) circ { [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 W f } d ξ d γ ,
r ( x , y ) S ( ξ 0 , γ 0 ) exp [ j 2 π [ x ξ 0 + y γ 0 ) ] λ F A = π W f 2 d ξ d γ = S ( ξ 0 , γ 0 ) π W f 2 exp [ j 2 π ( x ξ 0 + y γ 0 ) ] λ F ,
W f = μ 1.22 λ F 2 R 0 ,
S ( ξ 0 , γ 0 ) = 1 λ F A = π R 0 2 s ( x , y ) exp [ - j 2 π ( x ξ 0 + y γ 0 ) ] d x d y π R 0 2 λ F .
0 r ( 0 , 0 ) 3.5 μ 2 .
0 V 7 b μ 2 1 + 12.25 b 2 μ 4 .
I ( x , y ) a 2 { 1 + b 2 r 0 2 + 2 b r 0 × cos [ θ ( - , - y ) - ϕ - Θ ] } ,
I ( x , y ) a 2 { 1 + b 2 r 0 2 + 2 b cos [ θ ( - x , - y ) - ϕ ] } .
V = I ( + θ 0 ) - I ( - θ 0 ) I ( + θ 0 ) + I ( - θ 0 ) ,
H caf ( ξ , γ ) = 1 ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 > R 0 , = α ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 R 0 ,
H PC ( ξ , γ ) = 1 ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 > R 0 = exp ( j γ ) ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 R 0 ,
H FA ( ξ , γ ) = α ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 > R 0 = 1 ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 R 0 ,
H C ( ξ , γ ) = α ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 > R 0 = exp ( j π ) ,             [ ( ξ - ξ 0 ) 2 + ( γ - γ 0 ) 2 ] 1 / 2 R 0 ,
S A 2 ( 0 , 0 ) = 1 λ F A 2 = f A s ( x , y ) d x d y = π f R 0 2 Λ F ,
3.5 μ 2 f r ( 0 , 0 ) 3.5 μ 2 .
7 b f μ 2 1 + 12.26 ( b f ) 2 μ 4 V 7 b μ 2 12.25 b 2 μ 4 .

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