Abstract

The aim of this paper is to describe and demonstrate what should be done to the measured Zernike coefficients when conjugating the pupil and wavefront sensor planes with a ${4}f$ system. I provide theoretical and experimental evidence. The experimental setup consisted of two ${4}f$ systems of magnifications 1 and 1/3 with their corresponding wavefront sensors at their ends. Spherical and cylindrical trial lenses were measured. In addition, I measured a phase plate with high-order aberrations. I show that the Zernike coefficients of the wavefront expansion at two planes conjugated by a ${4}f$ system are related independently of the magnification of the ${4}f$ system by the following equation: ${b_i} = {(- 1)^{n}}{a_i}$, with $n$ being the order of the radial Zernike polynomial.

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References

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  1. S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
    [Crossref]
  2. J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
    [Crossref]
  3. S. Bará, E. Pailos, J. Arines, N. López-Gil, and L. Thibos, “Estimating the eye aberration coefficients in resized pupils: is it better to refit or to rescale?” J. Opt. Soc. Am. A 31, 114–123 (2014).
    [Crossref]
  4. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
    [Crossref]
  5. G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539–543 (2006).
    [Crossref]
  6. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

2014 (1)

2008 (1)

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[Crossref]

2007 (1)

2006 (2)

Acosta, E.

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[Crossref]

Ares, J.

Arines, J.

Bara, S.

Bará, S.

S. Bará, E. Pailos, J. Arines, N. López-Gil, and L. Thibos, “Estimating the eye aberration coefficients in resized pupils: is it better to refit or to rescale?” J. Opt. Soc. Am. A 31, 114–123 (2014).
[Crossref]

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[Crossref]

Dai, G. M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

López-Gil, N.

Lundström, L.

Pailos, E.

Prado, P.

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[Crossref]

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated, and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[Crossref]

Thibos, L.

Unsbo, P.

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Figures (2)

Fig. 1.
Fig. 1. ${4}f$ relay system conjugating planes ${\Pi _1}$ and ${\Pi _2}$.
Fig. 2.
Fig. 2. Experimental setup.

Tables (3)

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Table 1. Measured Zernike Coefficients

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Table 2. Estimated Radius of Curvature ( m 1 )

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Table 3. Zernike Coefficients Measured at Planes Π 1 and Π 3

Equations (3)

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W i ( x , y ) = i = 1 a i Z i ( x r 1 , y r 1 ) ; W o ( r , s ) = i = 1 b i Z i ( r r 2 , s r 2 ) .
W i ( x , y ) = i = 1 a i Z i ( x r 1 , y r 1 ) = i = 1 a i Z i ( f 1 f 2 r r 1 , f 1 f 2 s r 1 ) = i = 1 a i Z i ( r r 2 , s r 2 ) .
W i ( x , y ) = i = 1 a i Z i ( x r 1 , y r 1 ) = i = 1 a i ( 1 ) n Z i ( r r 2 , s r 2 ) = i = 1 b i Z i ( r r 2 , s r 2 ) = W o ( r , s ) ,

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