Abstract

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, Ll(x)Lm(y), where l and m are positive integers (including zero) and Ll(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial Ll(x)Lm(y), there is a corresponding orthonormal polynomial Ll(y)Lm(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram–Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.

© 2012 Optical Society of America

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References

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  1. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  3. V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 2004).
  4. V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils,” Appl. Opt. 49, 6924–6929 (2010).
    [CrossRef]
  5. J. C. Burfoot, “Third-order aberrations of ‘doubly symmetric’ systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
    [CrossRef]
  6. C. G. Wynne, “The primary aberrations of anamorphotic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
    [CrossRef]
  7. V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils,” presented at the annual meeting of OSA, San Jose, Calif., 16Oct.2011.
  8. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
  9. V. N. Mahajan and G.-M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
    [CrossRef]

2010 (1)

2007 (1)

1954 (2)

J. C. Burfoot, “Third-order aberrations of ‘doubly symmetric’ systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
[CrossRef]

C. G. Wynne, “The primary aberrations of anamorphotic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Burfoot, J. C.

J. C. Burfoot, “Third-order aberrations of ‘doubly symmetric’ systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
[CrossRef]

Dai, G.-M.

Korn, A.

A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Mahajan, V. N.

V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils,” Appl. Opt. 49, 6924–6929 (2010).
[CrossRef]

V. N. Mahajan and G.-M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 2004).

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).

V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils,” presented at the annual meeting of OSA, San Jose, Calif., 16Oct.2011.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Wynne, C. G.

C. G. Wynne, “The primary aberrations of anamorphotic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
[CrossRef]

Appl. Opt. (1)

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

Proc. Phys. Soc. B (2)

J. C. Burfoot, “Third-order aberrations of ‘doubly symmetric’ systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
[CrossRef]

C. G. Wynne, “The primary aberrations of anamorphotic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
[CrossRef]

Other (5)

V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils,” presented at the annual meeting of OSA, San Jose, Calif., 16Oct.2011.

A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 2004).

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Tables (4)

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Table 1. Orthonormal Polynomials Fj(x,y) for an Anamorphic System with a Circular Exit Pupil, where x=ρcosθ and y=ρsinθ and 0ρ1

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Table 2. Comparison of Defocus, Coma, and Spherical Aberration Orthonormal Polynomials for Anamorphic Systems with Square and Circular Pupils

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Table 3. Appropriate Polynomials for an Anamorphic System with Different Pupil Shapes

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Table 4. Appropriate Polynomials for a Rotationally Symmetric System with Different Pupil Shapes

Equations (15)

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Wcx(x,y)=x3.
σcx2=[Wcx(x,y)]2Wcx(x,y)2,
g(x,y)=1π0102πg(x,y)ρdρdθ,
σcx2=[Wcx(x,y)]2,=1π0102πρ6cos6θρdρdθ,=5/64.
Wbcx(x,y)=x3+bx.
σbcx2=564+b4+b24.
Wbcx(x,y)=x3(1/2)x.
G1=Q1=1,
Gj+1=Qj+1+k=1jcj+1,kFk,
Fj+1=Gj+1Gj+1=Gj+1[1πx2+y21Gj+12dxdy]1/2,
cj+1,k=1πx2+y21Qj+1Fkdxdy.
Qj+1Fk.
Fj+1=Nj+1[Qj+1k=1jQj+1FkFk],
1πx2+y21Fj(x,y)Fj(x,y)dxdy=δjj.
1π0102πFj(x,y)Fj(x,y)ρdρdθ=δjj.

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