Abstract

A novel method of reconstructing wavefront aberrations by use of Zernike polynomials for radial shearing interferometers is discussed. This method uses matrix formalism to calculate the Zernike coefficients of a wavefront under test and shows the validity of reconstructing an arbitrary wavefront aberration from an interferogram taken by a radial shearing interferometer. We also propose a new interferometer setup to determine the shape and the direction (concave or convex) of wavefront aberration in a single measurement.

© 2007 Optical Society of America

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References

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  1. R. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
    [CrossRef]
  2. D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992), Chap. 5.
  3. D. Malacara, Appl. Opt. 13, 1781 (1974).
    [CrossRef] [PubMed]
  4. D. Li, P. Wang, X. Li, H. Yang, and H. Chen, Opt. Lett. 30, 492 (2005).
    [CrossRef] [PubMed]
  5. T. M. Jeong, D.-K. Ko, and J. Lee, Opt. Lett. 30, 3009 (2005).
    [CrossRef] [PubMed]

2005 (2)

1974 (1)

1961 (1)

R. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[CrossRef]

Appl. Opt. (1)

J. Sci. Instrum. (1)

R. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[CrossRef]

Opt. Lett. (2)

Other (1)

D. Malacara, Optical Shop Testing (Wiley-Interscience, 1992), Chap. 5.

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Figures (2)

Fig. 1
Fig. 1

(a) Wavefront under test, W 1 ( ρ , ϕ ) . (b) Expanded wavefront under test, W 2 ( ρ , ϕ ) . (c) Wavefront profile calculated from the interferogram in the RSI, W ( ρ , ϕ ) . (d) Reconstructed wavefront from the wavefront profile of W ( ρ , ϕ ) .

Fig. 2
Fig. 2

Conceptual setup for a RSI to measure the magnitude and the direction of the wavefront aberration in a single measurement.

Tables (1)

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Table 1 Mathematical Expressions for the Radial Polynominals and the Reduced Radial Polynomials in the Zernike Polynomials

Equations (13)

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E 1 ( ρ , ϕ ) = E 10 ( ρ , ϕ ) exp { i [ 2 π W 1 ( ρ , ϕ ) λ ] } ,
E 2 ( ρ , ϕ ) = E 20 ( ρ , ϕ ) exp { i [ 2 π W 2 ( ρ , ϕ ) λ ] } .
I ( ρ , ϕ ) = E 10 2 ( ρ , ϕ ) + E 20 2 ( ρ , ϕ ) + 2 E 10 ( ρ , ϕ ) E 20 ( ρ , ϕ ) cos [ θ ( ρ , ϕ ) ] ,
θ ( ρ , ϕ ) = 2 π λ [ W 1 ( ρ , ϕ ) W 2 ( ρ , ϕ ) ] = 2 π λ W ( ρ , ϕ ) .
W 1 ( ρ , ϕ ) = n , m μ n m Z n m ( ρ , ϕ ) ,
W 2 ( ρ , ϕ ) = n , m v n m Z n m ( ρ , ϕ ) ,
W ( ρ , ϕ ) = n , m u n m Z n m ( ρ , ϕ ) .
n , m ( μ n m ν n m ) Z n m ( ρ , ϕ ) = n , m u n m Z n m ( ρ , ϕ ) .
W 2 ( ρ , ϕ ) = n , m ν n m Z n m ( ρ , ϕ ) = n , m ν n m R n m ( ρ ) P n m ( ϕ ) = W 1 ( a ρ , ϕ ) = n , m μ n m R n m ( a ρ ) P n m ( ϕ ) .
n , m μ n m R n m ( a ρ ) P n m ( ϕ ) = { μ 0 0 + 3 ( a 2 1 ) μ 2 0 + } R 0 0 ( ρ ) P 0 0 ( ϕ ) + { a μ 1 1 + 8 a ( a 2 1 ) μ 3 1 + } R 1 1 ( ρ ) P 1 1 ( ϕ ) + { a μ 1 1 + 8 a ( a 2 1 ) μ 3 1 + } R 1 1 ( ρ ) P 1 1 ( ϕ ) + = ν 0 0 R 0 0 ( ρ ) P 0 0 ( ϕ ) + ν 1 1 R 1 1 ( ρ ) P 1 1 ( ϕ ) + ν 1 1 R 1 1 ( ρ ) P 1 1 ( ϕ ) + .
[ ν 2 2 ν 2 0 ν 2 2 ] = [ M 44 M 45 M 46 M 54 M 55 M 56 M 64 M 65 M 66 ] [ μ 2 2 μ 2 0 μ 2 2 ] .
[ μ 2 2 μ 2 0 μ 2 2 ] = [ 1 M 44 M 45 M 46 M 54 1 M 55 M 56 M 64 M 65 1 M 66 ] 1 [ u 2 2 u 2 0 u 2 2 ] .
μ 2 0 = μ 2 0 + 0.2 + 15 a 2 ( a 2 1 ) μ 4 0 ( 1 a 2 ) .

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