Abstract

New Bessel-series representations for the calculation of the diffraction integral are presented yielding the point-spread function of the optical system, as occurs in the Nijboer–Zernike theory of aberrations. In this analysis one can allow an arbitrary aberration and a defocus part. The representations are presented in full detail for the cases of coma and astigmatism. The analysis leads to stably converging results in the case of large aberration or defocus values, while the applicability of the original Nijboer–Zernike theory is limited mainly to wave-front deviations well below the value of one wavelength. Because of its intrinsic speed, the analysis is well suited to supplement or to replace numerical calculations that are currently used in the fields of (scanning) microscopy, lithography, and astronomy. In a companion paper [J. Opt. Soc. Am. A 19, 860 (2002)], physical interpretations and applications in a lithographic context are presented, a convergence analysis is given, and a comparison is made with results obtained by using a numerical package.

© 2002 Optical Society of America

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References

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  1. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1985).
  3. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).
  4. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  5. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  6. S. C. Biswas, J.-E. Villeneuve, “Diffraction of a laser beam by a circular aperture under the combined effect of three primary aberrations,” Appl. Opt. 25, 2221–2232 (1986).
    [CrossRef] [PubMed]
  7. M. A. A. Neil, M. J. Booth, T. Wilson, “New modal wave-front sensor: theoretical analysis,” J. Opt. Soc. Am. A 17, 1098–1107 (2000).
    [CrossRef]
  8. K. Nienhuis, “On the influence of diffraction on image formation in the presence of aberrations,” Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1948).
  9. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

2002 (1)

2000 (1)

1986 (1)

1980 (1)

1976 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Biswas, S. C.

Booth, M. J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1985).

Braat, J. J. M.

Dirksen, P.

Janssen, A. J. E. M.

Neil, M. A. A.

Nienhuis, K.

K. Nienhuis, “On the influence of diffraction on image formation in the presence of aberrations,” Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1948).

Nijboer, B. R. A.

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

Noll, R. J.

Silva, D. E.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Villeneuve, J.-E.

Wang, J. Y.

Wilson, T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1985).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1985).

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

K. Nienhuis, “On the influence of diffraction on image formation in the presence of aberrations,” Ph.D. thesis (University of Groningen, Groningen, The Netherlands, 1948).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

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Figures (2)

Fig. 1
Fig. 1

A point source at O emits a spherical wave toward the schematically represented optical system. In the image space an aberrated wave front W leaves the exit pupil (center at E) and comes to a focus close to the image plane through O. The spherical reference wave front is denoted by S, and the wave-front aberration is given by the perpendicular distance between S and W. The phase function Φ is derived from W through Φ=2πW/λ, where λ is the wavelength of the monochromatic radiation. The normalized Cartesian pupil coordinates are denoted by (ν, μ), and the coordinates (x, y) in the image plane have been normalized with respect to the diffraction unit λ/NA, where NA is the image-side numerical aperture of the optical system. Note that the analysis in this paper is not limited to on-axis object and image points.

Fig. 2
Fig. 2

Contour plot of the modulus of the point-spread function U(x, y) with aberration Φ(ρ, θ)=αρ3 cos θ (coma), where α=1 and f=0, π/4, π, 2π.

Equations (109)

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U(x, y)=1πn ν2+μ21 exp[i(ν2+μ2)f+iΦ(ν, μ)]×exp(2πiνx+2πiμy)dνdμ.
Φ(ν, μ)Φ(ρ, θ)=n,mαnmRnm(ρ)cos mθ,
U(x, y)=1π01ρ exp(ifρ2)×02π exp[iΦ(ρ, θ+ϕ)]×exp(2πiρr cos θ)dθdρ,
02π exp[iΦ(ρ, θ+ϕ)]exp(2πiρr cos θ)dθ
=k=0ikk!02πΦk(ρ, θ+ϕ)exp(2πiρr cos θ)dθ.
02πΦ(ρ, θ+ϕ)exp(2πiρr cos θ)dθ
=n,mαnmRnm(ρ)02π[cos m(θ+ϕ)]×exp(2πiρr cos θ)dθ=2πn,mαnmimRnm(ρ)Jm(2πρr)cos mϕ.
01ρ exp(ifρ2)Rnm(ρ)Jm(2πρr)dρ,
01ρRnm(ρ)Jm(2πρr)dρ=(-1)(n-m)/2 Jn+1(v)v,
v=2πr.
exp(ifρ2)=exp12ifπfs=0(2s+1)isJs+1/212fR2s0(ρ).
Rnm(ρ)R2s0(ρ)=pApRpm(ρ).
ρm+2k=p=0km+2p+1m+p+k+1kpm+k+ppRm+2pm(ρ),
02πΦk(ρ, θ+ϕ)exp(2πiρr cos θ)dθ
ρnJm(2πρr)cos mϕ
Tnm01ρn+1 exp(ifρ2)Jm(2πρr)dρ.
Tnm=exp(if)l=1(-2if)l-1j=0ptljJm+l+2j(v)vl,
v=2πr,p=12(n-m),q=12(n+m),
tlj=(-1)jm+l+2jq+1pj×m+j+l-1l-1q+l+jq+1,
j=0,1,,l=1,2,.
pk=p(p-1)(p-k+1)k!,
U(x, y)201ρ exp(ifρ2)J0(2πρr)dρ+2in,mαnmim cos mϕ×01ρ exp(ifρ2)Rnm(ρ)Jm(2πρr)dρ.
01ρRnm(ρ)exp(ifρ2)Jm(2πρr)dρ
=exp(if)l=1(-2if)l-1j=0pvljJm+l+2j(v)lvl,
vlj=(-1)p(m+l+2j)m+j+l-1l-1×j+l-1l-1l-1p-jq+l+jl,
j=0, 1,,l=1, 2,,
02πΦk(ρ, θ+ϕ)exp(2πiρr cos θ)dθ
=n1,m1,, nk,mkαn1m1αnkmkRn1m1(ρ)Rnkmk(ρ)
×02π[cos m1(θ+ϕ)][cos mk(θ+ϕ)]
×exp(2πiρr cos θ)dθ.
cos x cos y=12cos(x+y)+12cos(x-y)=14cos(x+y)+14cos(x-y)+14cos(-x+y)+14cos(-x-y),
cos m1xcos mkx=12kε1,,εk=±1 cosl=1kεlmlx.
02π[cos ν(θ+ϕ)]exp(2πiρr cos θ)dθ
=2πiν cos νϕ Jν(2πρr).
Rnm(ρ)=s=0p(-1)s(n-s)!ρn-2ss!(q-s)!(p-s)!,
p=12(n-m),q=12(n+m),
Rn1m1(ρ)Rnkmk(ρ)
m1++mktn1++nk
βnmρnJm(2πρr)cos mϕ
01ρ exp(ifρ2)02πΦk(ρ, θ+ϕ)exp(2πiρr cos θ)dθdρ,
Φ(ρ, θ)=α ρ3 cos θ,Φ(ρ, θ)=γρ2 cos 2θ
U(x, y)=1π01ρ exp(ifρ2)02π exp[iαρ3 cos(θ+ϕ)]×exp(2πiρr cos θ)dθdρ.
U(x, y)=201ρ exp(ifρ2)J0(αρ3)J0(2πρr)dρ+4j=1(-1)j01ρ exp(ifρ2)Jj(αρ3)×Jj(2πρr)dρ cos jϕ.
01ρ exp(ifρ2)Jj(αρ3)Jj(2πρr)dρ
=12αjk=0(-14α2)kk!(k+j)!T3j+6k,j.
U(x, y)=l=0Clαl,
C2m=2-14mk=0mεk(-1)kT6m,2k(m-k)!(m+k)!cos 2kϕ,
C2m+1=-2-14mk=0m(-1)kT6m+3,2k+1(m-k)!(m+k+1)!×cos(2k+1)ϕ.
U(x, y)=1π01ρ exp(ifρ2)×02π exp[iγρ2 cos 2(θ+ϕ)]exp(2πiρr cos θ)dθdρ.
02π exp[iγρ2 cos 2(θ+ϕ)]exp(2πiρr cos θ)dθ
=2πJ0(γρ2)J0(2πρr)+4πj=1(-i)jJj(γρ2)J2j(2πρr)cos 2jϕ.
U(x, y)=l=0Dlγl,
D2m=2-14mk=0mεkT4m,4k(m-k)!(m+k)!cos 4kϕ,
D2m+1=-2i-14mk=0mT4m+2,4k+2(m-k)!(m+k+1)!×cos 2(2k+1)ϕ,
Tnm=exp(if)l=1(-2if)l-1j=0ptljJm+l+2j(v)vl,
tlj=(-1) jm+l+2jq+1pj×m+j+l-1l-1q+l+jq+1,
j=0,1,,l=1,2,.
Tnm=12πrn+20vtn+1 exp(iβt2)Jm(t)dt
=:12πrn+2Hnm(v),
Fnm(0)(v)=vnJm(v),
Fnm(l)(v)=0vtFnm(l-1)(t)dt,l=1,2,.
Hnm(v)=exp(iβv2)l=1(-2iβ)l-1Fnm(l)(v),
Fnm(l)(v)=vn+lj=0m+2j+lq+1×(-1)jpjm+j+l-1l-1q+j+lq+1Jm+2j+l(v).
αk=α(α-1)(α-k+1)k!=Γ(α+1)Γ(α-k+1)Γ(k+1)
0vtn+1Jm(t)dt
=vn+1k=0(m+2k+1)Γ(-p+k)Γ(q+1)Γ(-p)Γ(q+k+2)
×Jm+2k+1(v).
(-1)jpj=-p(-p+1)(-p+j-1)j!=Γ(-p+j)Γ(-p)j!.
m+2j+lq+1(-1) jpjm+j+l-1l-1q+j+lq+1
=Γ(-p+j)Γ(-p)Γ(j+l)Γ(j+1)Γ(q+1)Γ(q+j+l+1)
×m+j+l-1l-1(m+2j+l).
Fnm(l+1)(v)=0vtFnm(l)(t)dt=j=0Γ(-p+j)Γ(-p)Γ(j+l)Γ(j+1)Γ(q+1)Γ(q+j+l+1)×m+j+l-1l-1(m+2j+l)×0vtm+l+1Jm+2j+l(t)dt.
0vtn+l+1Jm+2j+l(t)dt=vn+l+1k=0[m+2(j+k)+l+1]Γ(-p+j+k)Γ(q+j+l+1)Γ(-p+j)Γ(q+l+j+k+2)Jm+2(j+k)+l+1(v).
Fnm(l+1)(v)=vn+l+1j,k=0Γ(-p+j+k)Γ(-p)Γ(j+l)Γ(j+1)×Γ(q+1)Γ(q+j+k+l+2)m+j+l-1l-1×(m+2j+l)×[m+2(j+k)+l+1]Jm+2(j+k)+l+1(v).
Fnm(l+1)(v)=vn+l+1s=0Γ(-p+s)Γ(-p)Γ(q+1)Γ(q+s+l+2)×(m+2s+l+1)×Jm+2s+l+1(v)j=0sΓ(j+l)Γ(j+1)×m+j+l-1l-1(m+2j+l).
j=0sΓ(j+l)Γ(j+1)m+j+l-1l-1(m+2 j+l)
=Γ(s+l+1)Γ(s+1)m+s+ll.
j=0s(j+l-1)(j+1)(m+j+l-1)
(m+j+1)(m+2j+l)
=1l(s+l)(s+1)(m+s+l)(m+s+1),
Fnm(l+1)(v)=vn+l+1s=0Γ(-p+s)Γ(-p)Γ(q+1)Γ(q+s+l+2)×Γ(s+l+1)Γ(s+1)m+s+ll×(m+2s+l+1)Jm+2s+l+1(v),
01ρRnm(ρ)exp(ifρ2)Jm(2πρr)dρ
=exp(if)l=1(-2if)l-1j=0pvljJm+l+2j(v)lvl,
vlj=(-1)p(m+l+2j)m+j+l-1l-1×j+l-1l-1l-1p-jq+l+jl,
j=0, 1,,l=1,2,.
01ρRnm(ρ)exp(ifρ2)Jm(2πρr)dρ
=s=0p(-1)s(n-s)!s!(q-s)!(p-s)!Tn-2s,m,
01ρRnm(ρ)exp(ifρ2)Jm(2πρr)dρ
=exp(if)l=1(-2if)l-1
×s=0pj=0p-s(-1)s(n-s)!tlj(n-2s, m)s!(q-s)!(p-s)!Jm+l+2j(v)vl,
tlj(n-2s, m)
=(-1) jm+l+2 jq-s+1p-sj×m+j+l-1l-1j+q-s+lq-s+1
=(-1) j(m+l+2 j)m+j+l-1l-1×(p-s)!(q-s)!(p-j-s)!(j+q-s+l)!×(j+l-1)!j!.
[]=s=0pj=0p-s(-1)j(m+l+2j)×m+j+l-1l-1(j+l-1)!j!×Jm+l+2j(v)vl(-1)s(n-s)!s!(p-j-s)!(j+q-s+l).
[]=j=0p(-1)j(m+l+2j)m+j+l-1l-1×(j+l-1)!j!Jm+l+2j(v)vl×s=0p-j(-1)s(n-s)!s!(p-j-s)!(j+q-s+l)!.
S(k, n, i)=s=0k(-1)s(n-s)!s!(k-s)!(n-s-i)!=1k!s=0k(-1)sks(n-s)!(n-s-i)!.
S(k, n, i)
=0,0i<k(-1)kk-i-1k(n-k)!(n-i)!,i<0.
S(k, n, i)=1k!(Δkf)(n),
f(x)=Γ(x+1)Γ(x-i+1),x>min(-1, i-1),
(Δg)(x)=g(x)-g(x-1)
f(x)=1(x+1)(x+t)=(-1)t-1(t-1)!(Δt-1g)(x),
g(x)=1x+t.
S(k, n, i)=1k!Δk(-1)t-1(t-1)!Δt-1g(n)=(-1)t-1k!(t-1)!(Δk+t-1g)(n).
S(k, n, i)=(k+t-1)!k!(t-1)!(-1)k(n-k+1)(n+t)=(-1)kk+t-1k(n-k)!(n+t)!,
S(p-j, n, p-j-l)
=0,p-jl(-1)p-jl-1p-j(q+j)!(q+l+j)!,p-j<l.
[]=j=0pvljJm+l+2j(v)lvl,
vlj=(-1)p(m+l+2j)m+j+l-1l-1(j+l-1)!j!×l-1p-j(q+j)!(q+l+j)!l=(-1)p(m+l+2j)m+j+l-1l-1j+l-1l-1×l-1p-jq+l+jl,

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