Abstract

This paper describes numerical procedures for data reduction of full spheres from interferometric data taken at various positions around the surface of the sphere. The technique allows the use of practical f/No. optics, incomplete coverage or overlap of the interferograms, and differences in optical alignment of each interferogram.

© 1987 Optical Society of America

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References

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  1. J. A. Lipa, G. J. Siddal, “High Precision Measurement of Gyro Rotor Sphericity,” Precis. Eng. 26, 123 (1980).
    [CrossRef]
  2. J. Thunen, C. Kwon, “Full Aperture Testing with Subaperture Test Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).
  3. C. Kim, “Polynomial Fit of Interferograms,” Ph.D. Dissertation, U. Arizona (1982).
  4. C. Kim, “Polynomial Fit of Interferograms,” Appl. Opt. 21, 4521 (1982).
    [CrossRef] [PubMed]
  5. W. W. Chow, G. N. Lawrence, “Method for Subaperture Testing Interferogram Reduction,” Opt. Lett. 8, 468 (1983).
    [CrossRef] [PubMed]
  6. G. N. Lawrence, W. W. Chow, “Influence of Higher Order Noise in Wavefront Reconstruction,” Proc. Soc. Photo. Opt. Instrum. Eng. 440, 30 (1983).
  7. T. Stuhlinger, “The Testing of Large Telescope Systems Using Multiple Subapertures,” Proc. Soc. Photo. Opt. Instrum. Eng. 440, 30 (1984).
  8. S. C. Jensen, W. W. Chow, G. N. Lawrence, “Subaperture Testing Approaches: a Comparison,” Appl. Opt. 23, 740 (1984).
    [CrossRef] [PubMed]
  9. J. E. Negro, “Subaperture Optical System Testing,” Appl. Opt. 23, 1921 (1984).
    [CrossRef] [PubMed]
  10. G. N. Lawrence, W. W. Chow, “Wave-Front Tomography by Zernike Polynomial Decomposition,” Opt. Lett. 9, 267 (1984).
    [CrossRef] [PubMed]
  11. WAP is a fringe reduction program developed by BSC Optics, 1824 Rita N.E., Albuquerque, NM 87106.
  12. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).
  13. R. Day, T. Beery, G. Lawrence, “Interferometric Measurements of Full Spheres,” presented at the Fourth Annual Target Fabrication Specialists Meeting28 Mar. 1985).
  14. R. Day, G. Lawrence, “Advancements in the Interferometric Measurements of Full Spheres,” presented at the Fifth Annual Target Fabrication Specialists Meeting (28 Mar. 1986).
  15. R. Day, G. Lawrence, T. Beery, “Optical Sphericity Measurements on Full Spheres,” Proc. Soc. Photo. Opt. Instrum. Eng. 661, 000 (1986).
  16. R. A. Williams, O. Y. Kwon, “Multiple Subaperture Interferometric Testing of Full Spheres,” presented at Optical Society of America Annual Meeting, 1986.
  17. O. Kwon, Lockheed Palo Alto Research Laboratory; private communication.

1986 (1)

R. Day, G. Lawrence, T. Beery, “Optical Sphericity Measurements on Full Spheres,” Proc. Soc. Photo. Opt. Instrum. Eng. 661, 000 (1986).

1984 (4)

1983 (3)

J. Thunen, C. Kwon, “Full Aperture Testing with Subaperture Test Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

W. W. Chow, G. N. Lawrence, “Method for Subaperture Testing Interferogram Reduction,” Opt. Lett. 8, 468 (1983).
[CrossRef] [PubMed]

G. N. Lawrence, W. W. Chow, “Influence of Higher Order Noise in Wavefront Reconstruction,” Proc. Soc. Photo. Opt. Instrum. Eng. 440, 30 (1983).

1982 (1)

1980 (1)

J. A. Lipa, G. J. Siddal, “High Precision Measurement of Gyro Rotor Sphericity,” Precis. Eng. 26, 123 (1980).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

Beery, T.

R. Day, G. Lawrence, T. Beery, “Optical Sphericity Measurements on Full Spheres,” Proc. Soc. Photo. Opt. Instrum. Eng. 661, 000 (1986).

R. Day, T. Beery, G. Lawrence, “Interferometric Measurements of Full Spheres,” presented at the Fourth Annual Target Fabrication Specialists Meeting28 Mar. 1985).

Chow, W. W.

Day, R.

R. Day, G. Lawrence, T. Beery, “Optical Sphericity Measurements on Full Spheres,” Proc. Soc. Photo. Opt. Instrum. Eng. 661, 000 (1986).

R. Day, G. Lawrence, “Advancements in the Interferometric Measurements of Full Spheres,” presented at the Fifth Annual Target Fabrication Specialists Meeting (28 Mar. 1986).

R. Day, T. Beery, G. Lawrence, “Interferometric Measurements of Full Spheres,” presented at the Fourth Annual Target Fabrication Specialists Meeting28 Mar. 1985).

Jensen, S. C.

Kim, C.

C. Kim, “Polynomial Fit of Interferograms,” Appl. Opt. 21, 4521 (1982).
[CrossRef] [PubMed]

C. Kim, “Polynomial Fit of Interferograms,” Ph.D. Dissertation, U. Arizona (1982).

Kwon, C.

J. Thunen, C. Kwon, “Full Aperture Testing with Subaperture Test Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

Kwon, O.

O. Kwon, Lockheed Palo Alto Research Laboratory; private communication.

Kwon, O. Y.

R. A. Williams, O. Y. Kwon, “Multiple Subaperture Interferometric Testing of Full Spheres,” presented at Optical Society of America Annual Meeting, 1986.

Lawrence, G.

R. Day, G. Lawrence, T. Beery, “Optical Sphericity Measurements on Full Spheres,” Proc. Soc. Photo. Opt. Instrum. Eng. 661, 000 (1986).

R. Day, G. Lawrence, “Advancements in the Interferometric Measurements of Full Spheres,” presented at the Fifth Annual Target Fabrication Specialists Meeting (28 Mar. 1986).

R. Day, T. Beery, G. Lawrence, “Interferometric Measurements of Full Spheres,” presented at the Fourth Annual Target Fabrication Specialists Meeting28 Mar. 1985).

Lawrence, G. N.

Lipa, J. A.

J. A. Lipa, G. J. Siddal, “High Precision Measurement of Gyro Rotor Sphericity,” Precis. Eng. 26, 123 (1980).
[CrossRef]

Negro, J. E.

Siddal, G. J.

J. A. Lipa, G. J. Siddal, “High Precision Measurement of Gyro Rotor Sphericity,” Precis. Eng. 26, 123 (1980).
[CrossRef]

Stuhlinger, T.

T. Stuhlinger, “The Testing of Large Telescope Systems Using Multiple Subapertures,” Proc. Soc. Photo. Opt. Instrum. Eng. 440, 30 (1984).

Thunen, J.

J. Thunen, C. Kwon, “Full Aperture Testing with Subaperture Test Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

Williams, R. A.

R. A. Williams, O. Y. Kwon, “Multiple Subaperture Interferometric Testing of Full Spheres,” presented at Optical Society of America Annual Meeting, 1986.

Appl. Opt. (3)

Opt. Lett. (2)

Precis. Eng. (1)

J. A. Lipa, G. J. Siddal, “High Precision Measurement of Gyro Rotor Sphericity,” Precis. Eng. 26, 123 (1980).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. Thunen, C. Kwon, “Full Aperture Testing with Subaperture Test Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

Proc. Soc. Photo. Opt. Instrum. Eng. (3)

R. Day, G. Lawrence, T. Beery, “Optical Sphericity Measurements on Full Spheres,” Proc. Soc. Photo. Opt. Instrum. Eng. 661, 000 (1986).

G. N. Lawrence, W. W. Chow, “Influence of Higher Order Noise in Wavefront Reconstruction,” Proc. Soc. Photo. Opt. Instrum. Eng. 440, 30 (1983).

T. Stuhlinger, “The Testing of Large Telescope Systems Using Multiple Subapertures,” Proc. Soc. Photo. Opt. Instrum. Eng. 440, 30 (1984).

Other (7)

WAP is a fringe reduction program developed by BSC Optics, 1824 Rita N.E., Albuquerque, NM 87106.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

R. Day, T. Beery, G. Lawrence, “Interferometric Measurements of Full Spheres,” presented at the Fourth Annual Target Fabrication Specialists Meeting28 Mar. 1985).

R. Day, G. Lawrence, “Advancements in the Interferometric Measurements of Full Spheres,” presented at the Fifth Annual Target Fabrication Specialists Meeting (28 Mar. 1986).

R. A. Williams, O. Y. Kwon, “Multiple Subaperture Interferometric Testing of Full Spheres,” presented at Optical Society of America Annual Meeting, 1986.

O. Kwon, Lockheed Palo Alto Research Laboratory; private communication.

C. Kim, “Polynomial Fit of Interferograms,” Ph.D. Dissertation, U. Arizona (1982).

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Figures (5)

Fig. 1
Fig. 1

Area of sphere illuminated by interferometric testing of an f/0.75 lens.

Fig. 2
Fig. 2

Typical pattern of subapertures on a spherical surface.

Fig. 3
Fig. 3

Coordinate systems for conversion from subaperture to global coordinates.

Fig. 4
Fig. 4

Computer-generated interferograms and the result after data reduction for case 1.

Fig. 5
Fig. 5

Computer-generated interferograms for case 5.

Tables (4)

Tables Icon

Table I First Nine Spherical Harmonic Functions with Aberration Identification

Tables Icon

Table II Lowest Four Spherical Modes and Corresponding Zernike Aberrations

Tables Icon

Table III Test Cases for Computer-Generated Interferograms

Tables Icon

Table IV Results of Computer-Generated Test Cases

Equations (40)

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W s ( ϴ , Φ ) = n = 0 m = n m = n a n m S n m ( ϴ , Φ ) ,
( 1 ) m exp ( i m Φ ) cos ( m Φ ) for m > 0 , ( 1 ) m exp ( i m Φ ) sin ( m Φ ) for m < 0 .
W s ( ϴ , Φ ) n = 0 N S m = n m = n a n m S n m ( ϴ , Φ ) ,
W a ( ϴ , Φ ) = W l ( ϴ , Φ ) + W s ( ϴ , Φ ) ,
W l ( ϴ , Φ ) k = 1 K n = 1 4 b n Z n ( ϴ , Φ ) χ k ( ϴ , Φ ) ,
W s ( ϴ , Φ ) n = 2 N S m = n m = n a n m S n m ( ϴ , Φ ) ,
J = 4 K + ( N S + 1 ) 2 4
i = 1 I Z i b i k j = 1 J S j a j ,
b i k = j = 1 J Z i S j k a j ,
Z i Z j k = δ i j ,
b = Ga ,
G = [ g 111 g 121 g 1 J 1 g I 11 g I 21 g 112 g I 12 g 11 K g I 1 K g I J K ]
b = [ b 11 b I 1 b 12 b I 2 b 1 K b I K ] , a = [ a 1 a 2 a J ] .
G 1 = ( G t G ) 1 G t ,
I * K J .
a = G 1 b .
f 1 f 2 k = 0 1 0 2 π f 1 ( r , ψ ) f 2 ( r , ψ ) r d ψ d r ,
Z i S j k = 0 1 0 2 π Z i ( r , ψ ) S j ( ϴ , Φ ) r d ψ d r .
ϴ = r ρ k , Φ = ψ + π 2 ,
X = sin ( ϴ ) cos ( Φ ) , Y = sin ( ϴ ) sin ( Φ ) , Z = cos ( ϴ ) .
[ X Y Z ] = R z ( Φ k ) R y ( ϴ k ) [ X Y Z ] ,
R y ( ϴ k ) = [ cos ( ϴ k ) 0 sin ( ϴ k ) 0 1 0 sin ( ϴ k ) 0 cos ( ϴ k ) ] , R z ( Φ k ) = [ cos ( Φ k ) sin ( Φ k ) 0 sin ( Φ k ) cos ( Φ k ) 0 0 0 1 ] .
ϴ = cos 1 ( Z ) , Φ = cos 1 [ Y sin ϴ ] , if ϴ 0 , Φ = 0 , if ϴ = 0 .
I = G 1 G ,
b = G ( N S = 3 ) a 1 , a 2 = G 1 ( N S = 2 ) b .
a 2 = G 1 ( N S = 2 ) G ( N S = 3 ) a 1 .
G ( N S = 2 ) = G 2 , G ( N S = 3 ) = [ G 2 G 3 ] ,
G 1 ( N S = 2 ) = ( G 2 t G 2 ) 1 G 2 t = G 2 1 .
a 2 = [ I 2 G 2 1 G 3 ] a 1 ,
order order 2 3 a 1 = [ 11111 00000000 ] ,
order order 2 3 a 1 = [ 00000 11111111 ] ,
Y 00 ( ϴ , Φ ) = 1 4 π
Y 11 ( ϴ , Φ ) = 3 8 π sin ϴ cos Φ
Y 10 ( ϴ , Φ ) = 3 4 π cos ϴ
Y 11 ( ϴ , Φ ) = 3 8 π sin ϴ sin Φ
Y 22 ( ϴ , Φ ) = 5 96 π 3 sin ( 2 ϴ ) cos ( 2 Φ )
Y 21 ( ϴ , Φ ) = 5 24 π 3 sin ϴ cos Φ
Y 20 ( ϴ , Φ ) = 5 4 π ( 3 2 cos ( 2 Φ ) 1 2 )
Y 21 ( ϴ , Φ ) = 5 24 π 3 sin ϴ sin Φ
Y 22 ( ϴ , Φ ) = 5 96 π 3 sin ( 2 ϴ ) sin ( 2 Φ )

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