Abstract

The partial derivatives and Laplacians of the Zernike circle polynomials occur in various places in the literature on computational optics. In a number of cases, the expansion of these derivatives and Laplacians in the circle polynomials are required. For the first-order partial derivatives, analytic results are scattered in the literature. Results start as early as 1942 in Nijboer’s thesis and continue until present day, with some emphasis on recursive computation schemes. A brief historic account of these results is given in the present paper. By choosing the unnormalized version of the circle polynomials, with exponential rather than trigonometric azimuthal dependence, and by a proper combination of the two partial derivatives, a concise form of the expressions emerges. This form is appropriate for the formulation and solution of a model wavefront sensing problem of reconstructing a wavefront on the level of its expansion coefficients from (measurements of the expansion coefficients of) the partial derivatives. It turns out that the least-squares estimation problem arising here decouples per azimuthal order m, and per m the generalized inverse solution assumes a concise analytic form so that singular value decompositions are avoided. The preferred version of the circle polynomials, with proper combination of the partial derivatives, also leads to a concise analytic result for the Zernike expansion of the Laplacian of the circle polynomials. From these expansions, the properties of the Laplacian as a mapping from the space of circle polynomials of maximal degree N, as required in the study of the Neumann problem associated with the transport-of-intensity equation, can be read off within a single glance. Furthermore, the inverse of the Laplacian on this space is shown to have a concise analytic form.

© 2014 Optical Society of America

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References

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    [CrossRef]

2014

2001

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

1999

A. Capozzoli, “On a new recursive formula for Zernike polynomials first order derivatives,” Atti della Fondazione Giorgio Ronchi 54, 767–774 (1999).

1996

1995

1993

1988

1987

1976

1964

H. H. Hopkins, “Canonical pupil coordinates in geometrical and diffraction image theory,” Jpn. J. Appl. Phys. 3, 31 (1964).
[CrossRef]

1963

W. Lukosz, “Der Einfluss der Aberrationen auf die optische Übertragungsfunktion bei Kleinen Orts-Frequenzen,” Opt. Acta 10, 1–19 (1963).
[CrossRef]

Acosta, E.

Bará, S.

Boisvert, R. F.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (Cambridge University, 2010).

Born, M.

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

Braat, J.

Braat, J. J. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

Capozzoli, A.

A. Capozzoli, “On a new recursive formula for Zernike polynomials first order derivatives,” Atti della Fondazione Giorgio Ronchi 54, 767–774 (1999).

Clark, C. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (Cambridge University, 2010).

Dirksen, P.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

Gureyev, T. E.

Hopkins, H. H.

H. H. Hopkins, “Canonical pupil coordinates in geometrical and diffraction image theory,” Jpn. J. Appl. Phys. 3, 31 (1964).
[CrossRef]

Janssen, A. J. E. M.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

Lozier, D. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (Cambridge University, 2010).

Lukosz, W.

W. Lukosz, “Der Einfluss der Aberrationen auf die optische Übertragungsfunktion bei Kleinen Orts-Frequenzen,” Opt. Acta 10, 1–19 (1963).
[CrossRef]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (Rijksuniversiteit Groningen, 1942).

Noll, R. J.

Nugent, K. A.

Olver, F. W. J.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (Cambridge University, 2010).

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Ríos, S.

Roberts, A.

Roddier, C.

Roddier, F.

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Stephenson, P. C. L.

van Haver, S.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

Wolf, E.

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

Appl. Opt.

Atti della Fondazione Giorgio Ronchi

A. Capozzoli, “On a new recursive formula for Zernike polynomials first order derivatives,” Atti della Fondazione Giorgio Ronchi 54, 767–774 (1999).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Refract. Surg.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Jpn. J. Appl. Phys.

H. H. Hopkins, “Canonical pupil coordinates in geometrical and diffraction image theory,” Jpn. J. Appl. Phys. 3, 31 (1964).
[CrossRef]

Opt. Acta

W. Lukosz, “Der Einfluss der Aberrationen auf die optische Übertragungsfunktion bei Kleinen Orts-Frequenzen,” Opt. Acta 10, 1–19 (1963).
[CrossRef]

Other

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

ANSI Z80.28–2009, “Ophthalmics: methods of reporting optical aberrations of eyes,” American National Standards Institute, 2009.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.

B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (Rijksuniversiteit Groningen, 1942).

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Handbook of Mathematical Functions (Cambridge University, 2010).

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Tables (1)

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Table 1. Equations (54) and (68) for Znm with m0, n6

Equations (106)

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δx1=W(X1,Y1)X1,δy1=W(X1,Y1)Y1.
t2Q+2ikQz=0,
kIz=t·[It(kW)]=It2(kW)+tI·t(kW),
Znm(ν,μ)=Znm(ρ,ϑ)=Rn|m|(ρ)eimϑ,
ν+iμ=ρexp(iϑ);ν=ρcosϑ,μ=ρsinϑ,
Rn|m|(ρ)=Pn|m|2(0,|m|)(2ρ21)
(ν±iμ)Znm(ν,μ)=(ν±iμ)[Rn|m|((ν2+μ2)1/2)exp[imarctan(μ/ν)]]=((Rn|m|)(ρ)mρRn|m|(ρ))exp[i(m±1)ϑ],
(ddρ±mρ)(Rn|m|Rn2|m|)=2nRn1|m1|,
(ddρ±mρ)Rn2(n1|m1|)|m|=(ddρ±mρ)R|m1|1|m|=0,
(ddρ±mρ)Rn|m|=2l=012(n1|m1|)(n2l)Rn12l|m1|.
2n=|m1|(2)(n1)(n+1)Rn|m1|,
k=i(2)jmeansk=i,i+2,,j.
(ν±iμ)Znm(ν,μ)=2l=012(n|m|)(n2l)Zn12lm±1.
Znmν=l=012(n|m|)(n2l)Zn12lm1+l=012(n|m|)(n2l)Zn12lm+1,
Znmμ=il=012(n|m|)(n2l)Zn12lm1il=012(n|m|)(n2l)Zn12lm+1.
(ν±iμ)Znm=(ν±iμ)Zn2m+2nZn1m±1,
Znmν=Zn2mν+n(Zn1m1+Zn1m+1)
Znmμ=Zn2mμ+in(Zn1m1Zn1m+1)
Rn|m|(ρ)=s=012(n|m|)(ns12(n|m|))(12(n|m|)s)ρn2s
ddρ(Rn|m|Rn2|m|)=n(Rn1|m1|+Rn2|m+1|),
Rn|m|(ρ)=(1)nm20Jn+1(t)Jm(ρt)dt,0ρ<1,
W(ν,μ)=m=n=|m|(2)αnmZnm(ν,μ),
α=(αnm)m=,,,n=|m|(2),
β¯±=((β¯±)nm)m=,,n=|m|(2),
β±=A±α=(2(n+1)n=n(2)αn+1m1)m=,,,n=|m|(2).
γZC2=(γ,γ)ZC=m=n=|m|(2)|γnm|22(n+1).
ν2+μ21|Znm(ν,μ)|2dνdμ=π(n+1)
Aαβ¯ZC22=A+αβ¯+ZC2+Aαβ¯ZC2
A=[A+A];Aα=[A+αAα]ZC2,β¯=[β¯+β¯]ZC2
α^=(AHA)1AHβ¯,
AHδ=A+Hδ++AHδ,δ=[δ+δ]ZC2
A±Hγ=(2(n+1)n=|m|(2)nγn1m±1)m=,,,n=|m|(2),γZC.
AHAγ=(4(n+1)n=|m|(2)Bnnmγnm)m=,,,n=|m|(2),γZC,
nn=min(n,n)
Bnm=|m|+12(n|m|)(n+|m|+2).
AHAα=AHβ¯.
(AHβ¯)nm=2(n+1)n=|m|(2)n((β¯+)n1m+1+(β¯)n1m1),
n=|m|(2)Bnnmαnm=ψnm,n=|m|(2),
ψnm=n=|m|(2)n12((β¯+)n1m+1+(β¯)n1m1),
j=0IMijxj=ci,i=0,1,,I,
M=(bmin(i,j))i,j=0,1,,I.
c=L1d,
d=(dk)k=0,1,,I=(φ|m|+2km)k=0,1,,I,
φnm=12(β¯+)n1m+1+12(β¯)n1m1,
x=M1L1d=(LM)1d.
(LM)1=[a0a1000a1a2aI1aI00aI]
aj=1bjbj1,j=0,1,,I,
xj=ajdjaj+1dj+1,j=0,1,,I1;xI=aIdI.
bj=B|m|+2jm=|m|+2j(|m|+j+1),j=0,1,,I,
a0=1|m|;aj=12(|m|+2j),j=1,2,,I.
α^nm=CnmφnmCn+2mφn+2m,n=|m|(2)(|m|+I2),
α^|m|+2Im=C|m|+2Imφ|m|+2Im,
C|m|m=1|m|,n=|m|;Cnm=12n,n=(|m|+2)(2)(|m|+2I).
(2ν2+2μ2)Znm=4t=012(n|m|)1(n12t)(nt)(t+1)Zn22tm.
2ν2+2μ2=(ν+iμ)(νiμ),
(2ν2+2μ2)Znm=2l=0n1|m1|2(n2l)(ν+iμ)Zn12lm1=2l=0n1|m1|2k=1n12l1|m1+1|22(n12l2k)Zn12l12km1+1=4l=0n1|m1|2k=0n2l2|m|2(n2l)(n12l2k)Zn22l2km.
n1|m1|2={12(n|m|),m1,12(n|m|)1,m0.
(2ν2+2μ2)Znm=4l=0p1k=0p1l(n2l)(n12l2k)Zn22l2km,
4t=0p1l,k0;l+k=t(n2l)(n12t)Zn22tm=4t=0p1(n12t)Zn22tmSnt,
Snt=l,k0;l+k=t(n2l)=l=0t(n2l)=(nt)(t+1),
ΔZnm=s=|m|(2)(n2)(s+1)(n+s+2)(ns)Zsm.
Δφ=f,nφ=ψ,
Bnsm={(s+1)(n+s+2)(ns),s=|m|(2)(n2),0,s=n(2)Nm,
Nm=|m|+212(N|m|)
f=m=NNn=|m|(2)NmαnmZnm.
Δ(n=(|m|+2)(2)(Nm+2)βnmZnm)=Znm.
n=s(2)NmBn+2,smβn+2m=δsn,s=|m|(2)Nm.
Znm=Δ[14(n+2)(n+1)Zn+2m12n(n+2)Znm+14n(n+1)Zn2m],
n[Znm(ρ,ϑ)]=(Rn|m|)(1)exp[imϑ]=12(n(n+2)m2)exp[imϑ],
Wν±iWμ=2m=n=|m|(2)(n+1)(n=n(2)αn+1m1)Znm.
2n=|m1±1|(2)(n=n(2)(n+1)αn+1m1)Znm1±1.
n=n(2)αn+1m1={1,n+1n1,0,otherwise.
2n=|m1±1|(2)(n11)(n+1)Znm1±1=2n=(|m1±1|+1)(2)n1nZn1m1±1=2l=012(n11|m1±1|)(n12l)Zn112lm1±1=(ν±iμ)Zn1m1
γ=[γ+γ],δ=[δ+δ],
(Aγ,δ)ZC2=(A+γ+,δ+)ZC+(Aγ,δ)ZC=(γ+,A+Hδ+)ZC+(γ,AHδ)ZC=(γ,AHδ)ZC2,
AHδ=A+Hδ++AHδ.
A+α=(2(n+1)n=n(2)αn+1m1)m=,,,n=|m|(2).
(A+α,β)ZC=m=n=|m|(2)(A+α)nm(βnm)*2(n+1)=m=(n=|m|(2)(n=n(2)αn+1m1)(βnm)*)=m=(n=|m|(2)n=|m|(2)nαn+1m1(βnm)*)=m=(n=(|m+1|+1)(2)n=|m+1|(2)(n1)αnm(βnm+1)*).
(A+α,β)ZC=m=0(n=(m+2)(2)n=(m+1)(2)(n1)αnm(βnm+1)*)+m=1(n=|m|(2)n=(|m|1)(2)(n1)αnm(βnm+1)*)=m=0(n=(m+2)(2)n=(m+2)(2)nαnm(βn1m+1)*)+m=1(n=|m|(2)n=|m|(2)nαnm(βn1m+1)*).
(A+α,β)ZC=m=0(n=m(2)n=m(2)nαnm(βn1m+1)*)+m=1(n=|m|(2)n=|m|(2)nαnm(βn1m+1)*)=m=(n=|m|(2)n=|m|(2)nαnm(βn1m+1)*).
(A+α,β)ZC=m=(n=|m|(2)αnm(2(n+1)n=|m|(2)nβn+1m+1)*2(n+1))=(α,A+Hβ)ZC,
A+Hβ=(2(n+1)n=|m|(2)nβn1m+1)m=,,n=|m|(2).
(Aα,β)ZC=(α,AHβ)ZC,
AHβ=(2(n+1)n=|m|(2)βn1m1)m=,,,n=|m|(2).
(A+HA+γ)nm=2(n+1)n=|m|(2)n(A+γ)n1m+1,
(A+γ)n1m+1=2(n+11)n=(n1)(2)γn+1(m+1)1=2nn=n(2)γnm.
(A+HA+γ)nm=2(n+1)n=|m|(2)n,n1|m+1|2nn=n(2)γnm.
(AHAγ)nm=2(n+1)n=|m|(2)nn1|m1|2nn=n(2)γnm.
|m+1|={|m|+1,m0|m|1,m<0,|m1|={|m|1,m>0|m|+1,m0.
(AHAγ)nm=(A+HA+γ)nm+(AHAγ)nm=4(n+1)n=|m|(2)nn=n(2)nεn|m|γnm,
(AHAγ)nm=4(n+1)n=|m|(2)γnmn=|m|(2)(nn)nεn|m|,
n=|m|(2)nnεn|m|=|m|+12(n|m|)(n+|m|+2),
L=[10001100011]
M=(bmin(i,j))i,j=0,1,,I=[b0b0b0b0b0b1b1b1b0b1b2b2b0b1b2bI].
LM=[b0b0b0b00b1b0b1b0b1b0bI1bI2bI1bI200bIbI1]
U=[1c00001c11cI1001],ci=bibi1bi+1bi,
ULM=[b0b1000b1b0000bIbI1]=D
(Bn+2,sm)s,n=|m|(2)Nm
s=|m|+2u,n=|m|+2k,u,k=0,1,,K,
C=(Cuk)u,k=0,1,,K,
Cuk={4(|m|+2u+1)(|m|+k+u+2)(k+1u),ku,0,k<u,
Euk=12(|m|+k+u+2)(k+1u)=j=uk(12|m|+j+1)
[110001111001],[1d0/d10001d1/d21dK1/dK001]
U2(U1E)=U2[d0d0d00d1d100dK]=[d0000d1000dK]=D2.
C1=D21U2U1D11.
(C1)kk=(U2U1)kk4(|m|+2k+2)(|m|+2k+1)={14(|m|+2k+2)(|m|+2k+1),k=k,12(|m|+2k+2)(|m|+2k+4),k=k1,14(|m|+2k+4)(|m|+2k+5),k=k2.

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