Abstract

The set of Fourier series is discussed following some discussion of Zernike polynomials. Fourier transforms of Zernike polynomials are derived that allow for relating Fourier series expansion coefficients to Zernike polynomial expansion coefficients. With iterative Fourier reconstruction, Zernike representations of wavefront aberrations can easily be obtained from wavefront derivative measurements.

© 2006 Optical Society of America

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References

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  1. R. K. Tyson, Opt. Lett. 7, 262 (1982).
    [CrossRef] [PubMed]
  2. G. Conforti, Opt. Lett. 8, 390 (1983).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Sec. 9.2.
  4. F. Roddier and C. Roddier, Appl. Opt. 30, 1325 (1991).
    [CrossRef] [PubMed]
  5. L. A. Poyneer, M. Troy, B. Macintosh, and D. T. Gavel, Opt. Lett. 28, 798 (2003).
    [CrossRef] [PubMed]
  6. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972), formulas 9.1.44, 9.1.45, and 11.4.6.
  7. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [CrossRef]
  8. Equation in Ref. has an extra multiplication factor of 1/pi.
  9. G.-m. Dai, J. Opt. Soc. Am. A 13, 1218 (1996).
    [CrossRef]
  10. V. N. Mahajan, in Proc. SPIE 5173, 1 (2003).
    [CrossRef]

2003 (2)

1996 (1)

1991 (1)

1983 (1)

G. Conforti, Opt. Lett. 8, 390 (1983).
[CrossRef]

1982 (1)

1976 (1)

1972 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972), formulas 9.1.44, 9.1.45, and 11.4.6.

1965 (1)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Sec. 9.2.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972), formulas 9.1.44, 9.1.45, and 11.4.6.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Sec. 9.2.

Conforti, G.

G. Conforti, Opt. Lett. 8, 390 (1983).
[CrossRef]

Dai, G.-m.

Gavel, D. T.

Macintosh, B.

Mahajan, V. N.

V. N. Mahajan, in Proc. SPIE 5173, 1 (2003).
[CrossRef]

Noll, R. J.

Poyneer, L. A.

Roddier, C.

Roddier, F.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972), formulas 9.1.44, 9.1.45, and 11.4.6.

Troy, M.

Tyson, R. K.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Sec. 9.2.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (3)

Proc. SPIE (1)

V. N. Mahajan, in Proc. SPIE 5173, 1 (2003).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1965), Sec. 9.2.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972), formulas 9.1.44, 9.1.45, and 11.4.6.

Equation in Ref. has an extra multiplication factor of 1/pi.

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Figures (2)

Fig. 1
Fig. 1

Reconstruction error as a fraction of input RMS as a function of the number of discrete points. Zernike orders of 6, 8, and 10 were used.

Fig. 2
Fig. 2

Example of wavefront contour maps (left panel) before and (right panel) after reconstruction. The wavefront reconstruction error is 3%.

Tables (1)

Tables Icon

Table 1 Comparison of Calculated and Input Wavefronts for the First Six Orders of Zernike Coefficients ( N = 2000 )

Equations (26)

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W ( R r , θ ) = i = 0 c i Z i ( r , θ ) ,
c i = 1 π P ( r ) W ( R r , θ ) Z i ( r , θ ) d r ,
Z i ( r , θ ) = R n m ( r ) Θ m ( θ ) ,
R n m ( r ) = s = 0 ( n m ) 2 ( 1 ) s n + 1 ( n s ) ! r n 2 s s ! [ ( n + m ) 2 s ] ! [ ( n m ) 2 s ] ! ,
Θ m ( θ ) = { 2 cos m θ ( m > 0 ) 1 ( m = 0 ) 2 sin m θ ( m < 0 ) } .
U i ( k , ϕ ) = P ( r ) Z i ( r , θ ) exp ( j 2 π k r ) d r ,
U i ( k , ϕ ) = ( 1 ) n 2 + m n + 1 J n + 1 ( 2 π k ) k Θ m ( ϕ ) .
cos ( z cos θ ) = J 0 ( z ) + 2 i = 1 ( 1 ) i J 2 i ( z ) cos ( 2 i θ ) ,
sin ( z cos θ ) = 2 i = 0 ( 1 ) i J 2 i + 1 ( z ) cos [ ( 2 i + 1 ) θ ] ,
0 1 R n m ( r ) J m ( k r ) r d r = ( 1 ) ( n m ) 2 n + 1 J n + 1 ( k ) k
0 J v + 2 p + 1 ( t ) J v + 2 q + 1 ( t ) t 1 d t = δ p q 2 ( v + 2 p + 1 ) ,
1 π U i ( k , ϕ ) U i * ( k , ϕ ) k d k d ϕ = δ i i .
W ( R r , θ ) = i = 0 N 2 1 a i ( k , ϕ ) exp ( j 2 π N k r ) ,
W ( R r , θ ) = a ( k , ϕ ) exp ( j 2 π k r ) d k ,
a ( k , ϕ ) = W ( R r , θ ) exp ( j 2 π k r ) d r .
c i = 1 π P ( r ) W ( R r , θ ) Z i ( r , θ ) d r = 1 π a ( k , ϕ ) d k P ( r ) Z i ( r , θ ) exp ( j 2 π k r ) d r = 1 π a ( k , ϕ ) U i * ( k , ϕ ) d k .
c i = 1 π l = 0 N 2 1 a l ( k , ϕ ) U i * ( k , ϕ ) .
a ( k , ϕ ) = i = 0 c i Z i ( r , θ ) exp ( j 2 π k r ) d r = i = 0 c i P ( r ) Z i ( r , θ ) exp ( j 2 π k r ) d r = i = 0 c i U i ( k , ϕ ) .
W ( x , y ) x = j 2 π u a ( u , v ) exp [ j 2 π ( u x + v y ) ] d u d v ,
W ( x , y ) y = j 2 π v a ( u , v ) exp [ j 2 π ( u x + v y ) ] d u d v .
W ( x , y ) x = b u exp [ j 2 π ( u x + v y ) ] d x d y ,
W ( x , y ) y = b v exp [ j 2 π ( u x + v y ) ] d x d y .
b u ( u , v ) = j 2 π u a ( u , v ) ,
b v ( u , v ) = j 2 π v a ( u , v ) .
u b u ( u , v ) + v b v ( u , v ) = j 2 π ( u 2 + v 2 ) a ( u , v ) .
a ( u , v ) = j u b u ( u , v ) + v b v ( u , v ) 2 π ( u 2 + v 2 ) .

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