Abstract

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.

© 2010 Optical Society of America

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References

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  1. W. T. Welford, Aberrations of the Symmetrical Optical System (Hilger, 1986).
  2. J. C. Burfoot, “Third-order aberrations of ‘doubly symmetric’ systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
    [CrossRef]
  3. C. G. Wynne, “The primary aberrations of anamorphotic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
    [CrossRef]
  4. H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970).
  5. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. I: The first-order foundation of and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, 2574–2584 (2009).
    [CrossRef] [PubMed]
  6. R. Barakat and L. Riseberg, “Diffraction theory of the aberrations of a slit aperture,” J. Opt. Soc. Am. 55, 878–881 (1965). There is an error in their polynomial S2, which should read as x2−1/3.
    [CrossRef]
  7. V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
    [CrossRef]
  8. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. II. Primary aberration theory for cylindrical anamorphic systems,” Appl. Opt. 48, 2836–2842 (2009).
    [CrossRef] [PubMed]
  9. V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE Press, 2001), second printing.
  10. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  11. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE Press, 2004), second printing.
  12. V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–17 (2003).
    [CrossRef]
  13. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
  14. J. L. Rayces, “Least-squares fitting of orthogonal polynomials to the wave-aberration function,” Appl. Opt. 31, 2223–2228(1992).
    [CrossRef] [PubMed]
  15. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
    [CrossRef]

2009 (2)

2007 (1)

2003 (1)

V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–17 (2003).
[CrossRef]

1992 (1)

1965 (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]

Barakat, R.

Born, M.

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

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970).

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

V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–17 (2003).
[CrossRef]

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE Press, 2001), second printing.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE Press, 2004), second printing.

Rayces, J. L.

Riseberg, L.

Sasian, J.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Hilger, 1986).

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]

Yuan, S.

Appl. Opt. (3)

J. Opt. Soc. Am. (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]

Proc. SPIE (1)

V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. SPIE 5173, 1–17 (2003).
[CrossRef]

Other (7)

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

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D.Malacara, ed. (Wiley, 2007), pp. 498–546.
[CrossRef]

W. T. Welford, Aberrations of the Symmetrical Optical System (Hilger, 1986).

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970).

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE Press, 2001), second printing.

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

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics (SPIE Press, 2004), second printing.

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Figures (2)

Fig. 1
Fig. 1

Schematic of an anamorphic imaging system consisting of orthogonal cylindrical lenses in a configuration called crossed cylinders. The system is symmetric about the y z and z x planes whose intersection defines the optical axis z. A fan of rays in the z x plane is shown originating at a point P in the center of a square object. The cylindrical lens L 1 acts as a plane-parallel plate on these rays and transmits them without any bending. When the transmitted rays are incident on the cylindrical lens L 2 , they are refracted by it just like a spherical lens and focused at the image point P .

Fig. 2
Fig. 2

Legendre polynomials P n ( x ) as a function of x. (a) Even n and (b) odd n.

Tables (2)

Tables Icon

Table 1 Legendre Polynomials L n ( x ) = 2 n + 1 P n ( x ) for a Unit Slit Pupil Orthonormal over the Interval [ 1 , 1 ]

Tables Icon

Table 2 Orthonormal Aberration Polynomials Q j ( x , y ) for an Anamorphic System with a Rectangular Pupil, where the ( x , y ) Coordinates of a Pupil Point have been Normalized by the Half Widths ( a , b ) of the Pupil

Equations (28)

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p 2 , x 2 , p x , q 2 , y 2 , and q y .
W ( p , q ; x , y ) = i , j , k , l , m , n C i , j , k , l , m , n ( p 2 ) i ( q 2 ) j ( x 2 ) k ( y 2 ) l ( p x ) m ( q y ) n ,
degree = 2 ( i + j + k + l + m + n ) .
W ( p , q ; x , y ) = ( C 1 p 3 + C 2 p q 2 ) x + ( C 3 p 2 q + C 4 q 3 ) y + ( C 5 p 2 + C 6 q 2 ) x 2 + C 7 p q x y + ( C 8 p 2 + C 9 q 2 ) y 2 + C 10 p x y 2 + C 11 q y x 2 + C 12 p x 3 + C 13 q y 3 + C 14 x 2 y 2 + C 15 x 4 + C 16 y 4 ,
W c x ( x , y ) = x 3 .
σ c x 2 = [ W c x ( x , y ) ] 2 W c x ( x , y ) 2 ,
g ( x , y ) = 1 1 1 1 g ( x , y ) d x d y 1 1 1 1 d x d y = ( 1 / 4 ) 1 1 1 1 g ( x , y ) d x d y .
W b c x ( x , y ) = x 3 + b x .
σ b c x 2 = 1 7 + 2 b 5 + b 2 3 .
W b c x ( x , y ) = x 3 ( 3 / 5 ) x .
1 2 1 1 P n ( x ) P n ( x ) d x = 1 2 n + 1 δ n n ,
P n ( x ) = ( 1 ) n P n ( x ) .
P n ( 1 ) = 1 ,
P n ( 1 ) = { 1 for even n 1 for odd n ,
P n ( 0 ) = 0 for odd n . P n ( 0 ) is positive or negative depending on whether n / 2 is even or odd .
( n + 1 ) P n + 1 ( x ) = ( 2 n + 1 ) x P n ( x ) n P n 1 ( x ) .
L n ( x ) = 2 n + 1 P n ( x ) .
1 2 1 1 L n ( x ) L n ( x ) d x = δ n n .
Q j ( x , y ) = L l ( x ) L m ( y ) ,
Q 1 ( x , y ) = L 0 ( x ) L 0 ( y ) = 1 .
1 4 1 1 1 1 Q j ( x , y ) Q j ( x , y ) d x d y = δ j j .
N n = ( 1 / 2 ) ( n + 1 ) ( n + 2 ) .
Q 32 ( x , y ) = L 4 ( x ) L 3 ( y ) .
W ( x , y ) = j A j Q j ( x , y ) ,
A j = ( 1 / 4 ) 1 1 1 1 W ( x , y ) Q j ( x , y ) d x d y .
W ( x , y ) = A 1 .
[ W ( x , y ) ] 2 = j A j 2 .
σ W 2 = [ W ( x , y ) ] 2 W ( x , y ) 2 = j 1 A j 2 .

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