Abstract

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.

© 2014 Optical Society of America

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References

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  1. F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastme-thode,” Physica (Utrecht) 1(7-12), 689–704 (1934).
    [Crossref]
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    [Crossref] [PubMed]
  4. W. Lukosz, “Der Einfluss der Aberrationen aud die optische Ubertragungsfunktion bei kleinen Orts-Frequenzen,” Opt. Acta (Lond.) 10(1), 1–19 (1963).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2014 (1)

2013 (1)

2011 (1)

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011).
[Crossref]

2010 (1)

2007 (2)

2004 (1)

2003 (2)

L. Giraud and J. Langou, “Robust selective Gram-Schmidt reorthogonalization,” SIAM J. Sci. Comput. 25, 417–441 (2003).
[Crossref]

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003).
[Crossref]

2001 (1)

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

1995 (1)

1994 (2)

1992 (1)

1989 (1)

W. Hoffmann, “Iterative algorithms for Gram–Schmidt orthogonalization,” Computing 41(4), 335 (1989).
[Crossref]

1981 (1)

1976 (1)

1963 (1)

W. Lukosz, “Der Einfluss der Aberrationen aud die optische Ubertragungsfunktion bei kleinen Orts-Frequenzen,” Opt. Acta (Lond.) 10(1), 1–19 (1963).
[Crossref]

1934 (1)

F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastme-thode,” Physica (Utrecht) 1(7-12), 689–704 (1934).
[Crossref]

Chow, W. W.

Collins, M. J.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

Dai, G. M.

Davis, B.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

Díaz, J. A.

DiVittorio, M.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003).
[Crossref]

Ellerbroek, B.

Geary, J. M.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011).
[Crossref]

Gilbreath, C.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003).
[Crossref]

Giraud, L.

L. Giraud and J. Langou, “Robust selective Gram-Schmidt reorthogonalization,” SIAM J. Sci. Comput. 25, 417–441 (2003).
[Crossref]

Greivenkamp, J. E.

Hoffmann, W.

W. Hoffmann, “Iterative algorithms for Gram–Schmidt orthogonalization,” Computing 41(4), 335 (1989).
[Crossref]

Iskander, D. R.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

Langou, J.

L. Giraud and J. Langou, “Robust selective Gram-Schmidt reorthogonalization,” SIAM J. Sci. Comput. 25, 417–441 (2003).
[Crossref]

Liu, F.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011).
[Crossref]

Lukosz, W.

W. Lukosz, “Der Einfluss der Aberrationen aud die optische Ubertragungsfunktion bei kleinen Orts-Frequenzen,” Opt. Acta (Lond.) 10(1), 1–19 (1963).
[Crossref]

Mahajan, V. N.

Miller, J. M.

Mozurkewich, D.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003).
[Crossref]

Navarro, R.

Noll, R. J.

Rayces, J. L.

Reardon, P. J.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011).
[Crossref]

Restaino, S. R.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003).
[Crossref]

Robinson, B. M.

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011).
[Crossref]

Schwiegerling, J.

Swantner, W.

Teare, S. W.

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003).
[Crossref]

Upton, R.

Zernike, F.

F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastme-thode,” Physica (Utrecht) 1(7-12), 689–704 (1934).
[Crossref]

Appl. Opt. (6)

Computing (1)

W. Hoffmann, “Iterative algorithms for Gram–Schmidt orthogonalization,” Computing 41(4), 335 (1989).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (Lond.) (1)

W. Lukosz, “Der Einfluss der Aberrationen aud die optische Ubertragungsfunktion bei kleinen Orts-Frequenzen,” Opt. Acta (Lond.) 10(1), 1–19 (1963).
[Crossref]

Opt. Eng. (2)

F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011).
[Crossref]

S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003).
[Crossref]

Opt. Lett. (2)

Physica (Utrecht) (1)

F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastme-thode,” Physica (Utrecht) 1(7-12), 689–704 (1934).
[Crossref]

SIAM J. Sci. Comput. (1)

L. Giraud and J. Langou, “Robust selective Gram-Schmidt reorthogonalization,” SIAM J. Sci. Comput. 25, 417–441 (2003).
[Crossref]

Other (5)

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).

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

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, Hoboken, NY, USA, 2007).

B. R. A. Nijboer, The diffraction theory of aberrations, Ph.D. thesis, University of Groningen, 1942.

B. O'Neill, Elementary Differential Geometry, Rev. 2nd ed. (Academic Press, New York, NY, 2006).

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Figures (5)

Fig. 1
Fig. 1 Mapping of the unit circle D onto a connected set M through diffeomorphism φ(u,v).
Fig. 2
Fig. 2 General angular sector [ θ 1 , θ 2 ] of an elliptical (semiaxes A, B) annulus (h = a/A = b/B) with orientation α.
Fig. 3
Fig. 3 Basis functions obtained for the particular case of the Zernike circle polynomial Z 3 3 (trefoil) corresponding to the different cases listed in Table 1. The fringes in the interferograms correspond to 1 wavelength of optical path difference.
Fig. 4
Fig. 4 Orthogonal elliptical polynomials corresponding to Zernike wavefront aberrations: tilt E 1 1 , defocus E 2 0 , astigmatism E 2 2 , coma E 3 1 , trefoil E 3 3 and spherical aberration E 4 0 . The fringes in the interferograms correspond to 1 wavelength of optical path difference.
Fig. 5
Fig. 5 Orthogonal annular polynomials corresponding to equivalent Zernike wavefront aberrations.

Tables (3)

Tables Icon

Table 1 Range or constant values for the parameters corresponding to the different mappings of the unit circle. All parameters are positive.

Tables Icon

Table 2 Expressions of the orthogonal elliptical polynomials up to order n = 4 where A is the major semi axis; e= 1 B 2 / A 2 is the eccentricity; and α is the orientation of the ellipse.

Tables Icon

Table 3 Expressions of the orthogonal annular basis functions (polynomial quotients) up to order n = 4. A and a = hA are the outer and inner radii.

Equations (29)

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

R=A1
u :=(u,v)= φ 1 (x,y):=(u(x,y),v(x,y)),
δ i,j = 1 π D Z i (u,v) Z j (u,v)dudv= 1 π M Z i ( φ 1 (x,y)) Z j ( φ 1 (x,y))| J(x,y) |dxdy
ds:=dxdy=rdrdθ
K ¯ j (x,y):=Q(x,y) Z j ( φ 1 (x,y)).
f(u,v):= Q 1 (φ(u,v)) f ¯ (φ(u,v))= j c j Z j (u,v),
c j := 1 π D f (u,v) Z j (u,v)dudv.
Z j (u,v)= Q 1 (φ(u,v)) K ¯ j (φ(u,v));
f ¯ (x,y):= j c j K ¯ j (x,y)
c j := 1 π M f ¯ (x,y) K ¯ j (x,y) Q 2 (x,y)| J(x,y) |dxdy.
(x,y)= φ 2 (X,Y)=(XcosαYsinα,Xsinα+Ycosα)
(X,Y)= φ 2 1 (x,y)=(xcosα+ysinα,xsinα+ycosα)
(X,Y)= φ 1 ( u,v )=(A[h+(1h)ρ]cos[ θ 1 + θ 2 θ 1 2π ϕ],B[h+(1h)ρ]sin[ θ 1 + θ 2 θ 1 2π ϕ])
(u,v)= φ 1 1 (X,Y)=( r (X,Y)h 1h cos[ 2π( θ (X,Y) θ 1 ) θ 2 θ 1 ], r (X,Y)h 1h sin[ 2π( θ (X,Y) θ 1 ) θ 2 θ 1 ] )
(u,v)= φ 1 (x,y)=( r ( x,y )h 1h cos[ 2π( θ ( x,y ) θ 1 ) θ 2 θ 1 ], r ( x,y )h 1h sin[ 2π( θ ( x,y ) θ 1 ) θ 2 θ 1 ] ),
h 2 ( X A ) 2 + ( Y B ) 2 = [h+(1h)ρ] 2 = r 2 1
θ 1 θ =arctan( AY BX )= θ 1 + θ 2 θ 1 2π ϕ θ 2 .
J(x,y)= 2π( r h) AB r (1h) 2 ( θ 2 θ 1 ) .
G j (x,y):= K j (x,y)= K j ( XcosαYsinα,Xsinα+Ycosα ) = K j ( A r cos θ cosαB r sin θ sinα,A r cos θ sinα+B r sin θ cosα ) := Z j ( r h 1h cos[ 2π( θ θ 1 ) θ 2 θ 1 ], r h 1h sin[ 2π( θ θ 1 ) θ 2 θ 1 ] ).
Z n m (ρ,ϕ):={ N n m R n |m| (ρ)cos(mϕ), m0, N n m R n |m| (ρ)sin(mϕ), m<0,
N n m = 2(n+1) ( 1+ δ m0 ) and R n | m | (ρ)= s=0 ( n| m | ) /2 (1) s (ns)! s!( n|m| 2 s )!( n+|m| 2 s )! ρ n2s
Z n m ( r h 1h , 2π( θ θ 1 ) θ 2 θ 1 )={ N n m R n |m| ( r h 1h )cos(m 2π( θ θ 1 ) θ 2 θ 1 ), m0, N n m R n |m| ( r h 1h )sin(m 2π( θ θ 1 ) θ 2 θ 1 ), m<0,
E ¯ n m (x,y):= 1 AB E n m (x,y)= 1 AB Z n m ((xcosα+ysinα)/A,(xsinα+ycosα)/B)
E ¯ n m ( r , θ )={ 1 AB N n m R n |m| ( r )cos(m θ ), m0, 1 AB N n m R n |m| ( r )sin(m θ ), m<0,
(x,y)=φ(u,v)=(A[h+(1h)ρ]cosϕ,A[h+(1h)ρ]sinϕ)
(u,v)= φ 1 (x,y)=( rhA A( 1h ) cosθ, rhA A( 1h ) sinθ).
O n m (r,θ):= Z n m ( rhA A( 1h ) cosθ, rhA A( 1h ) sinθ)
O ¯ n m (r,θ):= rhA r A( 1h ) Z n m ( rhA A( 1h ) cosθ, rhA A( 1h ) sinθ)
O n m (r,θ)={ 2(n+1) ( 1+ δ m0 ) s=0 ( n| m | ) /2 (1) s (ns)! s!( n+|m| 2 s )!( n|m| 2 s )! ( rhA A( 1h ) ) n2s cos(mθ), m0, 2(n+1) ( 1+ δ m0 ) s=0 ( n| m | ) /2 (1) s (ns)! s!( n+|m| 2 s )!( n|m| 2 s )! ( rhA A( 1h ) ) n2s sin(mθ), m<0.

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