Abstract

Several low-order Zernike modes are photographed for visualization. These polynomials are extended to include both circular and annular pupils through a Gram-Schmidt orthogonalization procedure. Contrary to the traditional understanding, the classical least-squares method of determining the Zernike coefficients from a sampled wave front with measurement noise has been found numerically stable. Furthermore, numerical analysis indicates that the so-called Gram-Schmidt method and the least-squares method give practically identical results. An alternate method using the orthogonal property of the polynomials to determine their coefficients is also discussed.

© 1980 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.
  2. S. N. Bezdidko, Sov. J. Opt. Technol. 41, 425 (1974).
  3. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [CrossRef]
  4. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  5. J. Y. Wang, J. Opt. Soc. Am. 67, 383 (1977).
    [CrossRef]
  6. L. C. Bradley, J. Herrmann, Appl. Opt. 13, 331 (1974).
    [CrossRef] [PubMed]
  7. M. P. Rimmer, C. M. King, D. G. Fox, Appl. Opt. 11, 2790 (1972).
    [CrossRef] [PubMed]
  8. J. S. Loomis, FRINGE User's Manual (Optical Sciences Center, U. Arizona, 1976).
  9. D. Malacara, Optical Shop-Testing (Wiley, New York, 1978), Appendix 2.
  10. E. R. Freniere, O. E. Toler, R. Race, Proc. Soc. Photo-Opt. Instrum. Eng. 171 (1979).
  11. R. Cubalchini, J. Opt. Soc. Am. 69, 972 (1979).
    [CrossRef]
  12. G. E. Forsythe, J. Soc. Ind. Math. 5, 74 (1957).
    [CrossRef]
  13. A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965), pp. 399–401.
  14. B. Noble, Applied Linear Algebra (Prentice-Hall, New York, 1969), Chap. 8.
  15. H. Sumita, Jpn. J. Appl. Phys. 8, 1027 (1969).
    [CrossRef]
  16. J. H. Wilkinson, Rounding Errors in Algebraic Processes (Prentice-Hall, New York, 1963).
  17. D. Dutton, A. Cornejo, M. Latta, Appl. Opt. 7, 125 (1968).
    [CrossRef] [PubMed]
  18. M. P. Rimmer, J. C. Wyant, Appl. Opt. 14, 142 (1975).
    [PubMed]

1979 (2)

E. R. Freniere, O. E. Toler, R. Race, Proc. Soc. Photo-Opt. Instrum. Eng. 171 (1979).

R. Cubalchini, J. Opt. Soc. Am. 69, 972 (1979).
[CrossRef]

1977 (1)

1976 (1)

1975 (1)

1974 (2)

S. N. Bezdidko, Sov. J. Opt. Technol. 41, 425 (1974).

L. C. Bradley, J. Herrmann, Appl. Opt. 13, 331 (1974).
[CrossRef] [PubMed]

1972 (1)

1969 (1)

H. Sumita, Jpn. J. Appl. Phys. 8, 1027 (1969).
[CrossRef]

1968 (1)

1965 (1)

1957 (1)

G. E. Forsythe, J. Soc. Ind. Math. 5, 74 (1957).
[CrossRef]

Bezdidko, S. N.

S. N. Bezdidko, Sov. J. Opt. Technol. 41, 425 (1974).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.

Bradley, L. C.

Cornejo, A.

Cubalchini, R.

Dutton, D.

Forsythe, G. E.

G. E. Forsythe, J. Soc. Ind. Math. 5, 74 (1957).
[CrossRef]

Fox, D. G.

Freniere, E. R.

E. R. Freniere, O. E. Toler, R. Race, Proc. Soc. Photo-Opt. Instrum. Eng. 171 (1979).

Fried, D. L.

Herrmann, J.

King, C. M.

Latta, M.

Loomis, J. S.

J. S. Loomis, FRINGE User's Manual (Optical Sciences Center, U. Arizona, 1976).

Malacara, D.

D. Malacara, Optical Shop-Testing (Wiley, New York, 1978), Appendix 2.

Noble, B.

B. Noble, Applied Linear Algebra (Prentice-Hall, New York, 1969), Chap. 8.

Noll, R. J.

Race, R.

E. R. Freniere, O. E. Toler, R. Race, Proc. Soc. Photo-Opt. Instrum. Eng. 171 (1979).

Ralston, A.

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965), pp. 399–401.

Rimmer, M. P.

Sumita, H.

H. Sumita, Jpn. J. Appl. Phys. 8, 1027 (1969).
[CrossRef]

Toler, O. E.

E. R. Freniere, O. E. Toler, R. Race, Proc. Soc. Photo-Opt. Instrum. Eng. 171 (1979).

Wang, J. Y.

Wilkinson, J. H.

J. H. Wilkinson, Rounding Errors in Algebraic Processes (Prentice-Hall, New York, 1963).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.

Wyant, J. C.

Appl. Opt. (4)

J. Opt. Soc. Am. (4)

J. Soc. Ind. Math. (1)

G. E. Forsythe, J. Soc. Ind. Math. 5, 74 (1957).
[CrossRef]

Jpn. J. Appl. Phys. (1)

H. Sumita, Jpn. J. Appl. Phys. 8, 1027 (1969).
[CrossRef]

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

E. R. Freniere, O. E. Toler, R. Race, Proc. Soc. Photo-Opt. Instrum. Eng. 171 (1979).

Sov. J. Opt. Technol. (1)

S. N. Bezdidko, Sov. J. Opt. Technol. 41, 425 (1974).

Other (6)

J. S. Loomis, FRINGE User's Manual (Optical Sciences Center, U. Arizona, 1976).

D. Malacara, Optical Shop-Testing (Wiley, New York, 1978), Appendix 2.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 9.2.

J. H. Wilkinson, Rounding Errors in Algebraic Processes (Prentice-Hall, New York, 1963).

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965), pp. 399–401.

B. Noble, Applied Linear Algebra (Prentice-Hall, New York, 1969), Chap. 8.

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Figures (4)

Fig. 1
Fig. 1

Expansion coefficients of R n m ( r ) in terms of Q n m ( r ): (a) diagonal terms arid (b) off-diagonal terms.

Fig. 2
Fig. 2

Comparison of mode shapes between R n m ( r ) and Q n m ( r ) for β = 0.5: (a) R n m ( r ) and (b) Q n m ( r ).

Fig. 3
Fig. 3

Distribution of eigenvalues for Zernike polynomial and monomial expansions using eighty points uniform sampling.

Fig. 4
Fig. 4

Distribution of eigenvalues for several sampling methods.

Tables (4)

Tables Icon

Table I Relation Among Zernike Polynomial Indices

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Table II Photographs of Zernike Polynomials

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Table III Condition Number for Several Sets of Samples

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Table IV Computed Zernike Coefficient a8 Using Method of Orthogonalization (True Value a8 = 0.70711)

Equations (47)

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Z even = [ 2 ( n + 1 ) ] 1 / 2 R n m ( r ) cos m θ Z odd = [ 2 ( n + 1 ) ] 1 / 2 R n m ( r ) sin m θ } m 0 Z j = [ ( n + 1 ) ] 1 / 2 R n m ( r ) m = 0 } ,
R n m ( r ) = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! r n 2 s .
R n m ( r ) = R n m ( r ) .
0 1 R n m ( r ) R n m ( r ) rdr = δ n n 2 ( n + 1 ) ,
d 2 r W ( r ) Z j ( r ) Z j ( r ) = δ j j ,
W ( r ) = { 1 / π for | r | 1 0 for | r | > 1
R n m ( r ) = γ nm m Q m m ( r ) + γ nm + 2 m Q m + 2 m ( r ) + + γ nn m Q n m ( r ) ,
γ nn m = 2 [ 2 ( n + 1 ) ] 1 / 2 × { β 1 [ R n m ( r ) j = 0 n m 2 γ nm + j m Q m + j m ( r ) ] 2 rdr } 1 / 2 ,
γ nm + j m = 2 ( m + j + 1 ) β 1 R n m ( r ) Q m + j m ( r ) rdr .
Q n m ( r ) = Q n m ( r ) ,
β 1 Q n m ( r ) Q n m ( r ) rdr = δ n n 2 ( n + 1 ) ,
d 2 r W ( r ) Z j ( r ) Z j ( r ) = δ j j ,
W ( r ) = { 1 / π β | r | 1 0 otherwise .
Q n n ( r ) = { β 1 [ R n n ( r ) ] 2 rdr } 1 / 2 R n n ( r ) .
Φ ( r ) = j = 1 a j Z j ( r ) ,
i = 1 ( 2 π λ ) 2 [ Φ 2 ¯ ( Φ ¯ ) 2 ] ,
Φ n ¯ = 1 π 0 1 0 2 π Φ n rdrd θ , n = 1 , 2 .
i = 1 ( 2 π / λ ) 2 j = 2 a j 2 .
j = 2 a j Z j ( r i ) = φ i i = 1 , 2 , , M ,
Za = Φ .
Δ = i = 1 [ a j Z j ( r i ) φ i ] 2 .
Z T Za = Z T Φ ,
a = ( Z T Z ) 1 Z T Φ .
i = 1 M g μ ( r i ) g ν ( r i ) = δ μ ν .
Δ = i = 1 M [ φ i j = 1 b j g j ( r i ) ] 2 .
b j = i = 1 M φ i g j ( r i ) .
Z j ( r ) = k = 1 j α jk g k ( r ) .
α jj 2 = i = 1 M [ Z j ( r i ) ] 2 k = 1 j 1 α jk 2 ,
α jk = i = 1 M Z j ( r i ) g k ( r i ) , j k .
j = 1 N a j Z j ( r ) = k = 1 N b k g k ( r ) .
[ α 11 α 21 α 31 . . . α N 1 0 α 22 α 32 . . . α N 2 0 0 α 33 . . . α N 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . α NN ] [ a 1 a 2 a 3 . . . . a N ] = [ b 1 b 2 b 3 . . . . b N ]
α a = b .
a N = b N / α NN ,
a j = b j k = j + 1 N α kj a k j = 1 , 2 , , N 1 .
α T α a = α T b .
α T α a = Z T ( Φ + ) .
a = U Λ 1 U T Z T ( Φ + ) .
δ j i = 1 M Z ji i / λ j j = 1 , 2 , , N .
φ ( x , y ) = i = 0 n j = 0 i B ij x i y i j .
C ( Z ) = ( λ 1 / λ N ) 1 / 2 ,
i = 1 M z j 2 ( r j θ i ) .
0 1 0 2 π Z j ( r , θ ) Z j ( r , θ ) rdrd θ = π δ j j .
| i = 1 M Z ji i | ( i = 1 M Z ji 2 ) 1 / 2 ( i = 1 M i 2 ) 1 / 2 ,
i = 1 M Z ji 2 π / Δ A M / R 2 ,
i = 1 M i 2 M e 2 .
δ i Me / λ j j = 1 , 2 , , N .
a j = 1 π 0 1 0 2 π φ ( r , θ ) Z j ( r , θ ) rdrd θ j = 1 , 2 , , N .

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