Abstract

Aberrations of optical systems are usually analyzed by decomposing the wavefront aberration at a pupil into a series of terms. The Seidel and Zernike series are two common examples, each with its own strengths and weaknesses. A new aberration series is proposed that combines some of the strengths of each. The new series is consistent with the traditional system of aberration types but enables the straightforward characterization and comparison of different optical systems, independent of pupil size or shape. Expressions for the conversion between the new and common aberration series are given, and the physical interpretations of the different aberrations are discussed.

© 2009 Optical Society of America

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References

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2007 (1)

2006 (2)

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr., Microfabr., Microsyst. 5, 030501 (2006).
[CrossRef]

G. Ming Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539-543 (2006).
[CrossRef]

2005 (1)

L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vision Sci. 82, 358-369 (2005).
[CrossRef]

2004 (2)

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

J. D. Marsack, L. N. Thibos, and R. A. Applegate, “Metrics of optical quality derived from wave aberrations predict visual performance,” J. Vision 4, 322-328 (2004).
[CrossRef]

2003 (1)

2002 (1)

1997 (1)

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description of statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

1996 (1)

1994 (1)

1983 (1)

1982 (1)

1976 (2)

1964 (1)

1956 (1)

H. H. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London, Sect. B 69, 562-576 (1956).
[CrossRef]

1950 (1)

Applegate, R. A.

J. D. Marsack, L. N. Thibos, and R. A. Applegate, “Metrics of optical quality derived from wave aberrations predict visual performance,” J. Vision 4, 322-328 (2004).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and its Applications, Vol. 35 of OSA Trends in Optics and Photonics Series, V.Lakshminarayanan ed. (Optical Society of America, 2000), pp. 232-244.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

Bradley, A.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

Brown, J. W.

R. V. Churchill and J. W. Brown, Complex Variables and Applications, 4th ed. (McGraw-Hill, 1984).

Campbell, C. E.

Chen, L.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vision Sci. 82, 358-369 (2005).
[CrossRef]

Cheng, X.

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

Churchill, R. V.

R. V. Churchill and J. W. Brown, Complex Variables and Applications, 4th ed. (McGraw-Hill, 1984).

Conforti, G.

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr., Microfabr., Microsyst. 5, 030501 (2006).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1, 2nd ed. (Cambridge U. Press, 1992).

Guirao, A.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vision Sci. 82, 358-369 (2005).
[CrossRef]

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis, 2nd ed. (Dover, 1987).

Hopkins, H. H.

H. H. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London, Sect. B 69, 562-576 (1956).
[CrossRef]

Horner, D.

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description of statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

Janssen, A. J. E. M.

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr., Microfabr., Microsyst. 5, 030501 (2006).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981).

Keller, H. B.

Keller, J. B.

Kintner, E. C.

E. C. Kintner, “On the mathematics of the Zernike polynomials,” Opt. Acta 23, 679-680 (1976).
[CrossRef]

Kneisly, J. A.

Landgrave, J. E. A.

Lundström, L.

Mahajan, V. N.

Marsack, J. D.

J. D. Marsack, L. N. Thibos, and R. A. Applegate, “Metrics of optical quality derived from wave aberrations predict visual performance,” J. Vision 4, 322-328 (2004).
[CrossRef]

Ming Dai, G.

Moya-Cessa, J. R.

Porter, J.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vision Sci. 82, 358-369 (2005).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1, 2nd ed. (Cambridge U. Press, 1992).

Schwiegerling, J.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and its Applications, Vol. 35 of OSA Trends in Optics and Photonics Series, V.Lakshminarayanan ed. (Optical Society of America, 2000), pp. 232-244.

Singer, B.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vision Sci. 82, 358-369 (2005).
[CrossRef]

Stavroudis, O. N.

O. N. Stavroudis, “Simpler derivation of the formulas for generalized ray tracing,” J. Opt. Soc. Am. 66, 1330-1333 (1976).
[CrossRef]

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics, Vol. 38 of Pure and Applied Physics (Academic, 1972).

Struik, D. J.

D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Dover, 1988).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1, 2nd ed. (Cambridge U. Press, 1992).

Thibos, L. N.

J. D. Marsack, L. N. Thibos, and R. A. Applegate, “Metrics of optical quality derived from wave aberrations predict visual performance,” J. Vision 4, 322-328 (2004).
[CrossRef]

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description of statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and its Applications, Vol. 35 of OSA Trends in Optics and Photonics Series, V.Lakshminarayanan ed. (Optical Society of America, 2000), pp. 232-244.

Tyson, R. K.

Unsbo, P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1, 2nd ed. (Cambridge U. Press, 1992).

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and its Applications, Vol. 35 of OSA Trends in Optics and Photonics Series, V.Lakshminarayanan ed. (Optical Society of America, 2000), pp. 232-244.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems, Adam Hilger Series on Optics and Optoelectronics (Institute of Physics, 1986).

Wheeler, W.

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description of statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981).

Williams, D. R.

L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vision Sci. 82, 358-369 (2005).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

Appl. Opt. (1)

J. Microlithogr., Microfabr., Microsyst. (1)

A. J. E. M. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr., Microfabr., Microsyst. 5, 030501 (2006).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

J. Vision (2)

X. Cheng, A. Bradley, and L. N. Thibos, “Predicting subjective judgment of best focus with objective image quality metrics,” J. Vision 4, 310-321 (2004).
[CrossRef]

J. D. Marsack, L. N. Thibos, and R. A. Applegate, “Metrics of optical quality derived from wave aberrations predict visual performance,” J. Vision 4, 322-328 (2004).
[CrossRef]

Opt. Acta (1)

E. C. Kintner, “On the mathematics of the Zernike polynomials,” Opt. Acta 23, 679-680 (1976).
[CrossRef]

Opt. Lett. (2)

Optom. Vision Sci. (2)

L. N. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description of statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

L. Chen, B. Singer, A. Guirao, J. Porter, and D. R. Williams, “Image metrics for predicting subjective image quality,” Optom. Vision Sci. 82, 358-369 (2005).
[CrossRef]

Proc. Phys. Soc. London, Sect. B (1)

H. H. Hopkins, “The frequency response of optical systems,” Proc. Phys. Soc. London, Sect. B 69, 562-576 (1956).
[CrossRef]

Other (9)

D. J. Struik, Lectures on Classical Differential Geometry, 2nd ed. (Dover, 1988).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics, Vol. 38 of Pure and Applied Physics (Academic, 1972).

F. B. Hildebrand, Introduction to Numerical Analysis, 2nd ed. (Dover, 1987).

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and its Applications, Vol. 35 of OSA Trends in Optics and Photonics Series, V.Lakshminarayanan ed. (Optical Society of America, 2000), pp. 232-244.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Vol. 1, 2nd ed. (Cambridge U. Press, 1992).

R. V. Churchill and J. W. Brown, Complex Variables and Applications, 4th ed. (McGraw-Hill, 1984).

W. T. Welford, Aberrations of Optical Systems, Adam Hilger Series on Optics and Optoelectronics (Institute of Physics, 1986).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999).
[PubMed]

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1981).

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Figures (2)

Fig. 1
Fig. 1

Diagram of a general optical system showing the wavefront at the entrance- and exit-pupil planes and the imaginary reference surface from which the wavefront aberration Φ is defined.

Fig. 2
Fig. 2

Artificial set of Zernike coefficients to represent a wavefront aberration over a circular pupil of 7.0 mm diameter, and the corresponding set of Zernike coefficients for the same wavefront aberration over a scaled concentric pupil of 6.0 mm diameter. Also shown are the new-series coefficients of the wavefront aberration. These are independent of pupil size.

Tables (1)

Tables Icon

Table 1 First Fifteen Terms of Each of the Extended-Seidel, Zernike, and New Expansions a

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

Φ ( x , y ) = n , m = 0 β n m x n y m ,
β n m = 1 n ! m ! n m Φ x n y m ( 0 , 0 ) .
Φ ( x , y ) = n , m = 0 N , M β n m x n y m ,
Φ ( r , ϕ ) = k = 0 N l = 0 k R k { C k l ρ k cos l ϕ + S k l ρ k sin l ϕ } .
Φ ( ρ , ϕ ) = n = 0 N m = n n a n m Z n m ( ρ , ϕ ) ,
Z n m ( ρ , ϕ ) = N n m R n m ( ρ ) { cos ( m ϕ ) , if m 0 sin ( m ϕ ) , otherwise } ,
R n m ( ρ ) = s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! s ! ( n m 2 s ) ! ( n + m 2 s ) ! ρ n 2 s .
N n m = ( 2 δ m , 0 ) ( n + 1 ) ,
0 1 0 2 π Z n m ( ρ , ϕ ) Z n m ( ρ , ϕ ) ρ d ϕ d ρ = π δ n , n δ m , m .
( rms ) 2 = 1 π 0 1 0 2 π Φ 2 ( ρ , ϕ ) ρ d ϕ d ρ ,
rms = ( n = 0 N m = n n a n m 2 ) 1 2 ,
Φ ( r , ϕ ) = k = 0 N l = 0 k R k [ C k l ρ k cos ( l ϕ ) + S k l ρ k sin ( l ϕ ) ] ,
k = [ 1 + 8 j 1 2 ] ,
l = 2 j k ( k + 2 ) ,
j 0 = N ( N + 3 ) 2 .
a n m = 2 ( 2 δ m , 0 ) π 0 1 0 2 π Φ ( ρ , ϕ ) Z n m ( ρ , ϕ ) ρ d ϕ d ρ ,
0 1 ρ k + 1 + n 2 s d ρ = 1 k + n 2 s + 2 ,
a n m = 2 2 δ m , 0 k = 0 N s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! N n m s ! ( n m 2 s ) ! ( n + m 2 s ) ! ( k + n 2 s + 2 ) R k C ¯ k , m ,
C ¯ k , m = { C k , m if m 0 s k , m otherwise } ,
a n m = 2 2 δ m , 0 k = 0 N R k C ¯ k , m { [ ( G F 0 + G ) F 1 + G ] F 2 + + G } F ( n m ) 2 1 + G ( k + m + 2 ) ,
F s = ( n s ) ( s + 1 ) ( k + n 2 s ) ( n m 2 s ) ( n + m 2 s ) ( k + n 2 s + 2 ) ,
G = ( 1 ) ( n m ) 2 ( n + m 2 m ) .
( a b ) = exp [ ln Γ ( a ) ln Γ ( a b ) ln Γ ( b ) ] ,
C ¯ k , m = 1 R k s = 0 ( N k ) 2 N k + 2 s , m ( 1 ) s ( k + s ) ! s ! ( k m 2 ) ! ( k + m 2 ) ! a k + 2 s , m ,
( k + s ) ! s ! ( k m 2 ) ! ( k + m 2 ) ! = exp [ ln Γ ( k + s + 1 ) ln Γ ( s + 1 ) ln Γ ( k m 2 + 1 ) ln Γ ( k + m 2 + 1 ) ] .
n = 0 N m = n n a n m Z n m ( ρ , ϕ ) = n = 0 N m = n n a n m Z n m ( ϵ ρ , ϕ ) .
N n m ( 2 δ m , 0 ) ( n + 1 ) a n m = k = m N N k m a k m 0 1 R k m ( ϵ ρ ) R n m ( ρ ) ρ d ρ ,
0 1 R k m ( ϵ ρ ) R n m ( ρ ) ρ d ρ = { R k n ( ϵ ) R k n + 2 ( ϵ ) 2 ( n + 1 ) if k n 0 otherwise } .
a n m = 2 δ m , 0 2 N n m k = n N a k m N k m [ R k n ( ϵ ) R k n + 2 ( ϵ ) ] ,
Φ ( ρ , ϕ ) = n = 0 N m = 0 n a n , m Z n m ( ρ , ϕ )
cos m ϕ = ( 1 + δ m , 0 ) 2 m 1 cos m ϕ + m t = 1 m 1 ( 1 ) t ( m t 1 ) ! 2 m 2 t 1 t ( m 2 t ) ! ( t 1 ) ! cos m 2 t ϕ
C k , l = n = 0 N m = 0 n α k l n m a n , m ,
α k l n m = ( 1 ) ( n k ) 2 ( n + k 2 ) ! N n , m ( n k 2 ) ! ( k + m 2 ) ! ( k m 2 ) ! { ( 1 + δ m , 0 ) 2 l 1 if m = l ( 1 ) ( m l ) 2 m 2 l ( m + l 2 1 ) ! ( m l ) l ! ( m l 2 1 ) ! if m l } ,
cos l ϕ = 1 2 l t = 0 l l ! t ! ( l t ) ! cos ( l 2 t ) ϕ
a n , m = k = 0 N l = m k β n m k l C k , l ,
β n m k l = N n , m l ! 2 l 1 ( l + m 2 ) ! ( l m 2 ) ! s = 0 ( n m ) 2 ( 1 ) s ( n s ) ! s ! ( n m 2 s ) ! ( n + m 2 s ) ! ( n 2 s + k + 2 )
M = Φ , x x + Φ , y y 2 ,
M = 2 C 20 = 2 C 20 .
M = 2 R 2 s = 0 ( N 2 ) 2 N 2 + 2 s , 0 ( 1 ) s ( 2 + s ) ! s ! a 2 + 2 s , 0 = 4 R 2 [ 3 a 2 , 0 3 5 a 4 , 0 + 6 7 a 6 , 0 ] .
J 0 = Φ , x x Φ , y y 2 ,
J 45 = Φ , x y .
J 0 = 2 C 2 , 2 = 2 ( C 2 , 2 S 2 , 2 ) = 2 R 2 s = 0 ( N 2 ) 2 N 2 + 2 s , 2 ( 1 ) s ( 2 + s ) ! s ! 2 ! a 2 + 2 s , 2 = 2 R 2 [ 6 a 2 , 2 3 10 a 4 , 2 + 6 14 a 6 , 2 ] ,
J 45 = 2 S 2 , 2 = 2 R 2 s = 0 ( N 2 ) 2 N 2 + 2 s , 2 ( 1 ) s ( 2 + s ) ! s ! 2 ! a 2 + 2 s , 2 = 2 R 2 [ 6 a 2 , 2 3 10 a 4 , 2 + 6 14 a 6 , 2 ] .
a 1 , ± 1 = k = 1 k odd N 2 R k k + 3 C ¯ k , ± 1 ,
F 0 = C 3 , 1 , F 90 = S 3 , 1 , T 0 = C 3 , 3 , T 90 = S 3 , 3
F 0 = 6 R 3 ( 2 a 3 , 1 4 3 a 5 , 1 + 20 a 7 , 1 ) ,
F 90 = 6 R 3 ( 2 a 3 , 1 4 3 a 5 , 1 + 20 a 7 , 1 ) ,
T 0 = 2 R 3 ( 2 a 3 , 3 4 3 a 5 , 3 + 20 a 7 , 3 ) ,
T 90 = 2 R 3 ( 2 a 3 , 3 4 3 a 5 , 3 + 20 a 7 , 3 ) .
F 0 = Φ , x x x + Φ , x y y 8 ,
F 90 = Φ , x x y + Φ , y y y 8 ,
T 0 = Φ , x x x 3 Φ , x y y 24 ,
T 90 = 3 Φ , x x y Φ , y y y 24 .
M = ( M , x ) 2 + ( M , y ) 2 ,
M = ( 4 F 0 ) 2 + ( 4 F 90 ) 2 = 4 F ,
T 0 = J 0 , x J 45 , y 12 ,
T 90 = J 0 , y J 45 , x 12 .
S = Φ , x x x x + 2 Φ , x x y y + Φ , y y y y 64 ,
A 0 = Φ , x x x x Φ , y y y y 48 ,
A 90 = Φ , x x x y + Φ , x y y y 24 ,
Q 0 = Φ , x x x x 6 Φ , x x y y + Φ , y y y y 192 ,
Q 90 = Φ , x x x y Φ , x y y y 48 .
S = 6 R 4 ( 5 a 4 , 0 5 7 a 6 , 0 + 45 a 8 , 0 ) ,
A 0 = 4 2 R 4 ( 5 a 4 , 2 5 7 a 6 , 2 + 45 a 8 , 2 ) ,
A 90 = 4 2 R 4 ( 5 a 4 , 2 5 7 a 6 , 2 + 45 a 8 , 2 ) ,
Q 0 = 2 R 4 ( 5 a 4 , 4 5 7 a 6 , 4 + 45 a 8 , 4 ) ,
Q 90 = 2 R 4 ( 5 a 4 , 4 5 7 a 6 , 4 + 45 a 8 , 4 ) .

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