Abstract

Least-square error criteria are used to fit 1-D interference fringe pattern irradiance data to a physically meaningful function of the form I(x) = B(x) + E(x) cos[P(x)], where B(x), E(x), and P(x) are low-order polynomials. This procedure is intended to complement digital fringe recognition by providing a method for smoothing and interpolating among fringe position data when the number of fringes is small, there are more than ten irradiance measurements per fringe, and accurate phase values are needed at arbitrary locations in the field.

© 1983 Optical Society of America

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References

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  1. H. E. Cline, A. S. Holik, W. E. Lorensen, Appl. Opt. 21, 4481 (1982).
    [CrossRef] [PubMed]
  2. M. J. D. Powell, Comput. J. 7, 155 (1974).
    [CrossRef]
  3. L. H. Taylor, G. B. Brandt, Exp. Mech. 12, 543 (1972).
    [CrossRef]
  4. G. B. Brandt, L. H. Taylor, Proc. Symposium on Engineering Applications of Holography Soc. Photo-Opt. Instrum. Eng. 123 (1972).
  5. J. E. Sollid, Opt. Eng. 14, 460 (1975).
    [CrossRef]
  6. R. Berggren, Opt. Spectra 12, 22 (1970).
  7. E. R. Freniere, O. E. Toler, R. Race, Opt. Eng. 20, 253 (1981).
  8. D. H. McLain, Comput. J. 17, 318 (1974).
    [CrossRef]

1982

1981

E. R. Freniere, O. E. Toler, R. Race, Opt. Eng. 20, 253 (1981).

1975

J. E. Sollid, Opt. Eng. 14, 460 (1975).
[CrossRef]

1974

D. H. McLain, Comput. J. 17, 318 (1974).
[CrossRef]

M. J. D. Powell, Comput. J. 7, 155 (1974).
[CrossRef]

1972

L. H. Taylor, G. B. Brandt, Exp. Mech. 12, 543 (1972).
[CrossRef]

G. B. Brandt, L. H. Taylor, Proc. Symposium on Engineering Applications of Holography Soc. Photo-Opt. Instrum. Eng. 123 (1972).

1970

R. Berggren, Opt. Spectra 12, 22 (1970).

Berggren, R.

R. Berggren, Opt. Spectra 12, 22 (1970).

Brandt, G. B.

L. H. Taylor, G. B. Brandt, Exp. Mech. 12, 543 (1972).
[CrossRef]

G. B. Brandt, L. H. Taylor, Proc. Symposium on Engineering Applications of Holography Soc. Photo-Opt. Instrum. Eng. 123 (1972).

Cline, H. E.

Freniere, E. R.

E. R. Freniere, O. E. Toler, R. Race, Opt. Eng. 20, 253 (1981).

Holik, A. S.

Lorensen, W. E.

McLain, D. H.

D. H. McLain, Comput. J. 17, 318 (1974).
[CrossRef]

Powell, M. J. D.

M. J. D. Powell, Comput. J. 7, 155 (1974).
[CrossRef]

Race, R.

E. R. Freniere, O. E. Toler, R. Race, Opt. Eng. 20, 253 (1981).

Sollid, J. E.

J. E. Sollid, Opt. Eng. 14, 460 (1975).
[CrossRef]

Taylor, L. H.

L. H. Taylor, G. B. Brandt, Exp. Mech. 12, 543 (1972).
[CrossRef]

G. B. Brandt, L. H. Taylor, Proc. Symposium on Engineering Applications of Holography Soc. Photo-Opt. Instrum. Eng. 123 (1972).

Toler, O. E.

E. R. Freniere, O. E. Toler, R. Race, Opt. Eng. 20, 253 (1981).

Appl. Opt.

Comput. J.

M. J. D. Powell, Comput. J. 7, 155 (1974).
[CrossRef]

D. H. McLain, Comput. J. 17, 318 (1974).
[CrossRef]

Exp. Mech.

L. H. Taylor, G. B. Brandt, Exp. Mech. 12, 543 (1972).
[CrossRef]

Opt. Eng.

E. R. Freniere, O. E. Toler, R. Race, Opt. Eng. 20, 253 (1981).

J. E. Sollid, Opt. Eng. 14, 460 (1975).
[CrossRef]

Opt. Spectra

R. Berggren, Opt. Spectra 12, 22 (1970).

Proc. Symposium on Engineering Applications of Holography Soc. Photo-Opt. Instrum. Eng.

G. B. Brandt, L. H. Taylor, Proc. Symposium on Engineering Applications of Holography Soc. Photo-Opt. Instrum. Eng. 123 (1972).

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Figures (4)

Fig. 1
Fig. 1

Measured irradiance vs position for the fringe pattern evaluated in the example.

Fig. 2
Fig. 2

Initial guess for the fringe pattern phase distribution. A third-order polynomial was fitted to the fringe centers by linear regression analysis. The coefficients from this curve fit are used as the initial guess for the nonlinear regression analysis.

Fig. 3
Fig. 3

Irradiance distribution function that resulted from the nonlinear regression analysis superimposed on the measured irradiance data.

Fig. 4
Fig. 4

Fringe pattern phase function that resulted from the nonlinear regression analysis compared with the initial guess phase function that was based on fringe center position data.

Tables (1)

Tables Icon

Table I Irradiance in Arbitrary Units and Phase in Radians at Fringe Centers

Equations (11)

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I ( x ) = B ( x ) + E ( x ) cos [ P ( x ) ] ,
U 1 ( x ) = A 1 ( x ) exp [ i ϕ , ( x ) ] ,
U 2 ( x ) = A 2 ( x ) exp [ i ϕ 2 ( x ) ] ,
I ( x ) = | U ( x ) | 2 = | U 1 ( x ) + U 2 ( x ) | 2
Î ( x i ) = B ̂ ( x i ) + Ê ( x i ) cos [ P ̂ ( x i ) ] ,
e i = Î ( x i ) I ( x i ) ,
E = i ( e i ) 2 ,
P ̂ ( x i ) = 2.80 + 31.0 x 27.9 x 2 2.83 x 3 .
B ̂ ( x i ) = 0.414 0.0689 x .
Ê ( x i ) = 0.228 0.0753 x .
P ̂ ( x i ) = 2.46 + 36.2 x 37.0 x 2 + 1.99 x 3 .

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