Abstract

An inversion algorithm, involving a linear least-squares method, is developed to analyze the intensity data obtained by application of the traveling knife-edge method. Through use of simulated intensity data, diameters can be measured to an accuracy of 0.05%. The method also yields good results when it is applied to He–Ne laser beam data.

© 1983 Optical Society of America

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References

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  1. G. Rinker, G. Bohannon, IEEE Trans. Plasma Sci. PS-8, 55 (1980).
    [CrossRef]
  2. E. H. A. Granneman, M. J. van der Wiel, Rev. Sci. Instrum. 46, 332 (1975).
    [CrossRef]
  3. S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
    [CrossRef]
  4. Y. C. Kiang, R. W. Lang, Appl. Opt. 22, 1296 (1983).
    [CrossRef] [PubMed]
  5. J. G. Edwards, H. R. Gallantree, R. M. Quilliam, J. Phys. E 10, 699 (1977).
    [CrossRef]
  6. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, J. M. Franke, Appl. Opt. 10, 2775 (1971).
    [CrossRef] [PubMed]
  7. M. Mauck, Appl. Opt. 18, 599 (1979).
    [CrossRef] [PubMed]
  8. Y. Suzaki, A. Tachibana, Appl. Opt. 14, 2809 (1975).
    [CrossRef] [PubMed]
  9. Laser Handbook, (Metrologic Instruments, Bellmar, N.J.1979), p. 28.
  10. H. D. Brunk, An Introduction to Mathematical Statistics (Blaisdell, Waltham, Mass., 1965), p. 389.

1983 (1)

1980 (2)

G. Rinker, G. Bohannon, IEEE Trans. Plasma Sci. PS-8, 55 (1980).
[CrossRef]

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

1979 (1)

1977 (1)

J. G. Edwards, H. R. Gallantree, R. M. Quilliam, J. Phys. E 10, 699 (1977).
[CrossRef]

1975 (2)

E. H. A. Granneman, M. J. van der Wiel, Rev. Sci. Instrum. 46, 332 (1975).
[CrossRef]

Y. Suzaki, A. Tachibana, Appl. Opt. 14, 2809 (1975).
[CrossRef] [PubMed]

1971 (1)

Arnaud, J. A.

Bohannon, G.

G. Rinker, G. Bohannon, IEEE Trans. Plasma Sci. PS-8, 55 (1980).
[CrossRef]

Brunk, H. D.

H. D. Brunk, An Introduction to Mathematical Statistics (Blaisdell, Waltham, Mass., 1965), p. 389.

de la Claviere, B.

Edwards, J. G.

J. G. Edwards, H. R. Gallantree, R. M. Quilliam, J. Phys. E 10, 699 (1977).
[CrossRef]

Franke, E. A.

Franke, J. M.

Gallantree, H. R.

J. G. Edwards, H. R. Gallantree, R. M. Quilliam, J. Phys. E 10, 699 (1977).
[CrossRef]

Granneman, E. H. A.

E. H. A. Granneman, M. J. van der Wiel, Rev. Sci. Instrum. 46, 332 (1975).
[CrossRef]

Hubbard, W. M.

Kiang, Y. C.

Klein, M. P.

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

Lang, R. W.

Mandeville, G. D.

Mauck, M.

Quilliam, R. M.

J. G. Edwards, H. R. Gallantree, R. M. Quilliam, J. Phys. E 10, 699 (1977).
[CrossRef]

Rinker, G.

G. Rinker, G. Bohannon, IEEE Trans. Plasma Sci. PS-8, 55 (1980).
[CrossRef]

Sorscher, S. M.

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

Suzaki, Y.

Tachibana, A.

van der Wiel, M. J.

E. H. A. Granneman, M. J. van der Wiel, Rev. Sci. Instrum. 46, 332 (1975).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Plasma Sci. (1)

G. Rinker, G. Bohannon, IEEE Trans. Plasma Sci. PS-8, 55 (1980).
[CrossRef]

J. Phys. E (1)

J. G. Edwards, H. R. Gallantree, R. M. Quilliam, J. Phys. E 10, 699 (1977).
[CrossRef]

Rev. Sci. Instrum. (2)

E. H. A. Granneman, M. J. van der Wiel, Rev. Sci. Instrum. 46, 332 (1975).
[CrossRef]

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

Other (2)

Laser Handbook, (Metrologic Instruments, Bellmar, N.J.1979), p. 28.

H. D. Brunk, An Introduction to Mathematical Statistics (Blaisdell, Waltham, Mass., 1965), p. 389.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the experimental configuration.

Fig. 2
Fig. 2

Integrated Gaussian profile for β = ( 2 ) 1. Solid dots indicate experimental data points, and the solid line represents the best-fit profile.

Fig. 3
Fig. 3

Integrated Gaussian profile for β = 0.177. Solid dots indicate experimental data points, and solid line represents the best-fit profile.

Fig. 4
Fig. 4

Integrated Gaussian profile for the He–Ne laser beam. A single translation step corresponds to 25.4 μm. Solid dots indicate experimental data points, and the solid line represents the best-fit profile.

Fig. 5
Fig. 5

Schematic flow chart for the computer algorithm.

Equations (24)

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G ( x , y ) = I 0 exp [ β 2 r 2 ( x , y ) ] ,
R ( x b ) = x b G ( x , y ) dxdy ,
R ( x b ) = I 0 ( π / β 2 ) 1 / 2 x b exp [ β 2 ( x x 0 ) 2 ] dx .
R ( x b ) = R ( x b ) / R ( ) = β π 1 / 2 x b exp [ β 2 ( x x 0 ) 2 ] dx ,
N ( z ) = ( 2 π ) 1 / 2 z exp ( n 2 / 2 ) d n .
n 2 / 2 = β 2 ( x x 0 ) 2
n = 2 β x 2 β x 0
d n = 2 β dx .
x = n 2 β + x 0 ,
x b = z 2 β + x 0 .
x 10 = 1.28 2 β + x 0 ;
x 90 = 1.28 2 β + x 0 ,
x 0 = ( x 10 + x 90 ) / 2
β 1 = 0.552 ( x 10 x 90 ) .
f ( x ) = 1 1 + exp ( z )
P ( z ) = i = 0 m a i z i
f ( z ) = 1 1 + exp [ P ( z ) ] .
P ( z ) = log [ f 1 ( z ) 1 ]
a 0 = 6.71387 × 10 3 , a 1 = 1.55115 , a 2 = 5.13306 × 10 2 , a 3 = 5.49164 × 10 2 ,
R ( x b ) = f ( z )
f ( z ) R ( x b ) .
R ( x b ) = x b G ( x ) dx = x b G ( x ) dx = G ( x ) dx x b G ( x ) dx = 1 R ( x b ) .
f ( z ) = 1 f ( z ) .
d = 2 2 β 1 .

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