Abstract

A recurrence relation for the first-order Cartesian derivatives of the Zernike polynomials is derived. This relation is used with the Clenshaw method to determine an efficient method for calculating the derivatives of any linear series of Zernike polynomials.

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  1. V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
    [CrossRef]
  2. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [CrossRef]
  3. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13861 (2010).
    [CrossRef]
  4. A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt. 51, 3054–3058 (2012).
    [CrossRef]
  5. S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).
  6. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  7. G. Rousset, Adaptive Optics in Astronomy (Cambridge University, 1999), Chap. 5.
  8. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  9. G. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
  10. P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014). [Note that the “+” signs in Eqs. (6) and (13) should be “−” signs.]
    [CrossRef]
  11. C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Comput. 9, 118–120 (1955).
    [CrossRef]
  12. E. C. Kintner, “On the mathematics of the Zernike polynomials,” Opt. Acta 23, 679–680 (1976).
    [CrossRef]
  13. A. Prata and W. V. T. Rusch, “Algorithm for computation of Zernike polynomials expansion coefficients,” Appl. Opt. 28, 749–754 (1989).
    [CrossRef]
  14. J. C. Mason and D. C. Handscomb, Chebyshev Polynomials (Chapman & Hall, 2003).
  15. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  16. Wolfram Research Inc., Mathematica (Wolfram Research, 2010), Version 8.0.
  17. F. J. Smith, “An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation,” Math. Comput. 19, 33–36 (1965).
    [CrossRef]

2014 (1)

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014). [Note that the “+” signs in Eqs. (6) and (13) should be “−” signs.]
[CrossRef]

2012 (2)

A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt. 51, 3054–3058 (2012).
[CrossRef]

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

2011 (1)

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

2010 (1)

1994 (1)

1989 (1)

1980 (1)

1976 (2)

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

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

1965 (1)

F. J. Smith, “An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation,” Math. Comput. 19, 33–36 (1965).
[CrossRef]

1955 (1)

C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Comput. 9, 118–120 (1955).
[CrossRef]

Born, M.

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

Clenshaw, C. W.

C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Comput. 9, 118–120 (1955).
[CrossRef]

Cuby, J.-G.

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Dai, G.

G. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).

Dohlen, K.

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Ferrari, M.

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Fleck, A.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

Forbes, G. W.

Gray, M.

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Handscomb, D. C.

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials (Chapman & Hall, 2003).

Kintner, E. C.

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

Lakshminarayanan, V.

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

Le Mignant, D.

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Mahajan, V. N.

Mason, J. C.

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials (Chapman & Hall, 2003).

Mikš, A.

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014). [Note that the “+” signs in Eqs. (6) and (13) should be “−” signs.]
[CrossRef]

Noll, R. J.

Novák, J.

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014). [Note that the “+” signs in Eqs. (6) and (13) should be “−” signs.]
[CrossRef]

Novák, P.

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014). [Note that the “+” signs in Eqs. (6) and (13) should be “−” signs.]
[CrossRef]

Pascal, S.

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Prata, A.

Rousset, G.

G. Rousset, Adaptive Optics in Astronomy (Cambridge University, 1999), Chap. 5.

Rusch, W. V. T.

Smith, F. J.

F. J. Smith, “An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation,” Math. Comput. 19, 33–36 (1965).
[CrossRef]

Southwell, W. H.

Vives, S.

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Wolf, E.

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

Yabe, A.

Appl. Opt. (3)

J. Mod. Opt. (1)

V. Lakshminarayanan and A. Fleck, “Zernike polynomials: a guide,” J. Mod. Opt. 58, 545–561 (2011).
[CrossRef]

J. Opt. Soc. Am. (2)

Math. Comput. (2)

C. W. Clenshaw, “A note on the summation of Chebyshev series,” Math. Comput. 9, 118–120 (1955).
[CrossRef]

F. J. Smith, “An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation,” Math. Comput. 19, 33–36 (1965).
[CrossRef]

Opt. Acta (1)

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

Opt. Express (1)

Opt. Lasers Eng. (1)

P. Novák, J. Novák, and A. Mikš, “Fast and robust computation of Cartesian derivatives of Zernike polynomials,” Opt. Lasers Eng. 52, 7–12 (2014). [Note that the “+” signs in Eqs. (6) and (13) should be “−” signs.]
[CrossRef]

Proc. SPIE (1)

S. Pascal, M. Gray, S. Vives, D. Le Mignant, M. Ferrari, J.-G. Cuby, and K. Dohlen, “New modelling of freeform surfaces for optical design of astronomical instruments,” Proc. SPIE 8450, 845053 (2012).

Other (5)

G. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).

G. Rousset, Adaptive Optics in Astronomy (Cambridge University, 1999), Chap. 5.

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials (Chapman & Hall, 2003).

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

Wolfram Research Inc., Mathematica (Wolfram Research, 2010), Version 8.0.

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Figures (2)

Fig. 1.
Fig. 1.

Computation times of some recurrence relations for (/x)Znm(x,y) for a point (x,y) and for values of nN and all possible values of m. The three methods are: the method of the current work [Eq. (26)], the recurrence relation from Noll [Eqs. (11) and (14)], and the preferred method of Novák et al. [Eqs. (13) and (14)].

Fig. 2.
Fig. 2.

Error, ϵ, of a 64-bit (double-precision) floating-point computation of the Zernike derivative (/x)Z260(x,y) plotted as a function of radial distance, ρ, along a radial line at angle θ=π/4 to the x axis. Shown are the errors for the direct recurrence relation of the present work (in black circles) and the Recurrence 1 method of Novák et al. (in blue open diamonds). The respective envelopes of these two sets of points are almost indistinguishable from one another. Note that the scale of the vertical axis is ×1014. Also shown are the corresponding Zernike derivative values themselves (dashed line, right-hand axis).

Tables (1)

Tables Icon

Table 1. Expressions for the Zernike Polynomials Znm(x,y) up to Sixth Order and Their First-Order Cartesian Derivativesa

Equations (70)

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z(x,y)=n=0m=nnanmZnm(x,y),
Znm(ρ,θ)=NnmRnm(ρ)Θm(θ),
Θm(θ)={cos(mθ)ifm0,sin(mθ)ifm<0,
n0,(n|m|)is even,|m|n,
Rnm(ρ)=s=0n|m|2(1)s(ns)!s!(n|m|2s)!(n+|m|2s)!ρn2s,
01Rnm(ρ)Rnm(ρ)ρdρ=12(n+1)δmn,
Nnm=(2δm0)(n+1),
k1Rn+2m(ρ)=(k2ρ2+k3)Rnm(ρ)+k4Rn2m(ρ),
k1=n(n+|m|+2)(n|m|+2),k2=4n(n+1)(n+2),k3=2|m|2(n+1)2n(n+1)(n+2),k4=(n+|m|)(n|m|)(n+2),
Rn0(0)=(1)n/2ifnis even,Rn1(0)=0ifnis odd,Rnm(1)=1.
ρRnm(ρ)=n[Rn1|m1|(ρ)+Rn1|m+1|(ρ)]+ρRn2m(ρ),
ρRnm(ρ)=nρn1,
k1ddρRn+2m(ρ)=2k2ρRnm(ρ)+(k2ρ2+k3)ddρRnm(ρ)+k4ddρRn2m(ρ).
[x,y]=J[ρ,θ],
J=(cosθsinθρsinθcosθρ),
Vnm(ρ,θ)=NnmRnm(ρ)exp(imθ),
Vnm=Zn|m|+isgn(m)Zn|m|,
sgn(m)={+1ifm>0,0ifm=0,1ifm<0.
ρVnm=n[eiθbn1,m1nmVn1m1+eiθbn1,m+1nmVn1m+1]+bn2,mnmρVn2Vm,
bnmnm=NnmNnm.
ρZn|m|=ncosθ[bn1,m1nmZn1|m1|+bn1,m+1nmZn1|m+1|]+nsinθ[sgn(m1)bn1,m1nmZn1|m1|+sgn(m+1)bn1,m+1nmZn1|m+1|]+bn2,mnmρZn2|m|,
sgn(m)ρZn|m|=ncosθ[sgn(m1)bn1,m1nmZn1|m1|+sgn(m+1)bn1,m+1nmZn1|m+1|]+nsinθ[bn1,m1nmZn1|m1|bn1,m+1nmZn1|m+1|]+sgn(m)bn2,mnmρZn2|m|.
ρZnm=ncosθ[bn1,m1nmZn1αm|m1|+αmsgn(m+1)bn1,m+1nmZn1αm|m+1|]+nsinθ[αmsgn(m1)bn1,m1nmZn1αm|m1|+bn1,m+1nmZn1αm|m+1|]+bn2,mnmρZn2m,
αm={+1ifm0,1ifm<0.
ρZnm(x,y)=cosθxZnm(x,y)+sinθyZnm(x,y).
xZnm=n[bn1,m1nmZn1αm|m1|+αmsgn(m+1)bn1,m+1nmZn1αm|m+1|]+bn2,mnmxZn2m,
yZnm=n[αmsgn(m1)bn1,m1nmZn1αm|m1|+bn1,m+1nmZn1αm|m+1|]+bn2,mnmyZn2m.
xZn±n=Nnnnρn1Θ±(n1)(θ)δn,1,
yZn±n=Nnnnρn1Θ(n1)(θ)δ±n,1,
xZnm=n=0n1[an,m1nmZnαm|m1|+αmsgn(m+1)an,m+1nmZnαm|m+1|],
yZnm=n=0n1[αmsgn(m1)an,m1nmZnαm|m1|+an,m+1nmZnαm|m+1|],
anmnm={2δm02δm0(n+1)(n+1)ifn|m|,(n|m|)is even,n|m|,and(n|m|)is even,0otherwise.
z(x,y)=m=NNSm,
Sm=n=|m|NanmNnmΘm(θ)Rnm(ρ)=j=0ηcjR2j+|m|m(ρ),
Sm=g0ρ|m|,
gj+ujgj+1+vj+1gj+2=cj,
uj=k2ρ2+k3k1,
vj=k4k1,
ξz(x,y)=m=NNSm(ξ),
Sm(ξ)=n=|m|NanmξZnm(x,y).
Sm(ξ)=c¯Tp¯(ξ),
p¯j(ξ)+μjp¯j1(ξ)=wj(ξ),
p¯0(x)=N|m|m|m|ρ|m|1Θαm(|m|1)(θ),p¯0(y)=αmN|m|m|m|ρ|m|1Θαm(|m|1)(θ),
μj=bn2,mnm,
wj(x)=n[bn1,m1nmZn1αm|m1|+αmsgn(m+1)bn1,m+1nmZn1αm|m+1|],wj(y)=n[αmsgn(m1)bn1,m1nmZn1αm|m1|+bn1,m+1nmZn1αm|m+1|],
A¯p¯(ξ)=d¯,
A¯=(1μ11μ21μη1),
d¯j={p¯0(ξ)ifj=0,wj(ξ)otherwise.
hTA¯=c¯T,
Sm(ξ)=hTd=h0p¯0(ξ)+j=1ηhjwj(ξ),
hj+μj+1hj+1=c¯j,
pj=R2j+|m|1|m1|(ρ),
qj=R2j+|m|1|m+1|(ρ),
Sm(ξ)=h0p¯0(ξ)+cTp+cTq,
cj(x)=hjnNnmΘαm|m1|,cj(x)=αmsgn(m+1)hjnNnmΘαm|m+1|,
cj(y)=αmsgn(m1)hjnNnmΘαm|m1|,cj(y)=hjnNnmΘαm|m+1|.
pj+1+μjpj+νjpj1=0,
qj+1+μjqj+νjqj1=0,
p0=R|m|1|m1|,p1=R|m|+1|m1|,
q0=R|m|1|m+1|,q1=R|m|+1|m+1|.
Ap=d,
A=(1μ11ν2μ21ν3μ31νη1μη11),
d1=p1={(m+1)ρm+1mρm1form>0,ρ|m|+1otherwise,d2=ν1p0={m(m+3)2(m+1)2ρm1form>0,0otherwise,dj=0forηj3.
hTA=cT,
Sm(ξ,p)=cTp=hTd=h1d1+h2d2.
hj+μjhj+1+νj+1hj+2=cj,
Sm(ξ,q)=cTq=h1d1+h2d2,
d1=q1={ρm+1form0,(|m|+1)ρ|m|+1|m|ρ|m|1otherwise,d2=ν1q0={0form0,m(m3)2(m1)2ρ|m|1otherwise,
hj+μjhj+1+νj+1hj+2=cj,
Sm(ξ)=h0p¯0(ξ)+(h1d1+h2d2)+(h1d1+h2d2),

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