Abstract

We assess the validity of an extended Nijboer–Zernike approach [J. Opt. Soc. Am. A 19, 849 (2002)], based on recently found Bessel-series representations of diffraction integrals comprising an arbitrary aberration and a defocus part, for the computation of optical point-spread functions of circular, aberrated optical systems. These new series representations yield a flexible means to compute optical point-spread functions, both accurately and efficiently, under defocus and aberration conditions that seem to cover almost all cases of practical interest. Because of the analytical nature of the formulas, there are no discretization effects limiting the accuracy, as opposed to the more commonly used numerical packages based on strictly numerical integration methods. Instead, we have an easily managed criterion, expressed in the number of terms to be included in the Bessel-series representations, guaranteeing the desired accuracy. For this reason, the analytical method can also serve as a calibration tool for the numerically based methods. The analysis is not limited to pointlike objects but can also be used for extended objects under various illumination conditions. The calculation schemes are simple and permit one to trace the relative strength of the various interfering complex-amplitude terms that contribute to the final image intensity function.

© 2002 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).
  2. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  3. B. R. A. Nijboer, “The diffraction theory of aberrations,” Ph.D. dissertation (University of Groningen, Groningen, The Netherlands, 1942).
  4. A. J. E. M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
    [CrossRef]
  5. SOLID-C, a software product (release 5.6.2) (SIGMA-C GmbH, Thomas-Dehlerstrasze 9, D-81737 Munich, Germany).
  6. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  7. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  8. V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33, 8121–8124 (1994).
    [CrossRef] [PubMed]
  9. C. J. Progler, A. K. Wong, “Zernike coefficients, are they really enough?” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 40–52 (2000).
    [CrossRef]
  10. G. Szegö, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).

2002 (1)

1994 (1)

1980 (1)

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Abramowitz, M.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Janssen, A. J. E. M.

Mahajan, V. N.

Nijboer, B. R. A.

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

Progler, C. J.

C. J. Progler, A. K. Wong, “Zernike coefficients, are they really enough?” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 40–52 (2000).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Silva, D. E.

Stegun, I. A.

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

Szegö, G.

G. Szegö, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).

Wang, J. Y.

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

Wong, A. K.

C. J. Progler, A. K. Wong, “Zernike coefficients, are they really enough?” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 40–52 (2000).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

Proc. R. Soc. London Ser. A (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other (6)

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

SOLID-C, a software product (release 5.6.2) (SIGMA-C GmbH, Thomas-Dehlerstrasze 9, D-81737 Munich, Germany).

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

C. J. Progler, A. K. Wong, “Zernike coefficients, are they really enough?” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 40–52 (2000).
[CrossRef]

G. Szegö, Orthogonal Polynomials, 4th ed. (American Mathematical Society, Providence, 1975).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the wave propagating from the exit pupil (center at E0) toward the image plane (center at P0). The diameter of the exit pupil is 2ρ0, and the distance from pupil to image plane is R. The real-space image-plane coordinates are (X, Y). In this paper, the exit pupil coordinates are normalized to unity by means of the value of ρ0 and denoted by (ν, μ); the image-plane coordinates are normalized with the aid of the diffraction unit λ/NA and denoted by (x, y). NA (=ρ0/R) is the image-side numerical aperture of the optical system.

Fig. 2
Fig. 2

The real part (left-hand column), the imaginary part (central column), and the squared modulus or intensity (right-hand column) of various radial functions Vnm have been displayed, each time for two values of the defocus parameter f (drawn curve, f=0; dashed curve, f=π). From top to bottom, the values of the indices (n, m) are (0, 0), (2, 2), (3, 1), and (4, 0).

Fig. 3
Fig. 3

Cross sections of the image intensity for a delta-function object (curves labeled “Extended Nijboer–Zernike” and “Numerical integration”) and for a rectangular contact hole in the object plane (curves labeled “Extended Nijboer–Zernike, 0.3 µm hole” and “SOLID-C, 0.3 µm hole”). The aberration term is fourth-order spherical aberration, and the magnitude is given by the value of α4,0 (2π/6), corresponding to the “just”-diffraction-limited case. The deviations between the analytical computation and the strictly numerical integration method (typically 5×10-5) are not visible in the figure; the deviations between the data obtained for the contact hole (typically 0.001–0.002) are especially visible for low values of the radial coordinate v.

Fig. 4
Fig. 4

Aerial image intensity calculation due to a mask with two contact holes (diameter 0.1 µm, spacing 0.2 µm). Solid curves: calculated by using Eqs. (9) and (10) for the functions Vnm; dashed curves: SOLID-C package, λ=0.248 µm, NA=0.6, in-focus situation. The typical difference in normalized intensity between the two resulting curves amounts to 0.01.

Fig. 5
Fig. 5

Image-plane intensity distribution in the case of a set of high-frequency Zernike coefficients that represent the result of, e.g., manufacturing errors in the wave front exiting from an optical system. A set of coefficients in the range q=6,,12 has been introduced (n ranges from 6 to 24, m from 0 to 12). The central maximum has been truncated in the plot. Note that the diffracted intensity is concentrated within a circle given by v=n+3, the extent of the highest-order Bessel function present in the image-plane intensity function. The numerals representing the intensity in the contour plot are a measure of the relative intensity and have to be multiplied by 10-2.

Fig. 6
Fig. 6

Wave-front deviation PQ in the case of defocusing. The spherical wave (radius R2, center of curvature M2) is projected onto a defocused image plane through M1 and perpendicular to the axis OM2. The aperture angle is denoted by p. The defocus distance z has been heavily exaggerated in the figure.

Tables (1)

Tables Icon

Table 1 Convergence of the Analytically Calculated Image Intensity I(x, y)=|U(x, y)|2 As a Function of the Number of Terms L Included in the Series Expansion for the Amplitude U (Interval |v|30)

Equations (63)

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U(x, y)=1πν2+μ21A(ν, μ)exp[iΦ(ν, μ)]×exp[i(ν2+μ2)f]exp[i2π(νx+μy)]dνdμ,=1π01ρ exp(ifρ2)×02πA(ρ, θ)exp[iΦ(ρ, θ)]×exp[i2πrρ cos(θ-ϕ)]dθdρ,
U(x, y)=1πk=0ikk!01ρ exp(ifρ2)02πΦk(ρ, θ+ϕ)×exp(i2πrρ cos θ)dθdρ.
Φ(ρ, θ)=n,mαnmRnm(ρ)cos mθ,
U(x, y)201ρ exp(ifρ2)J0(2πρr)dρ+2in,mimαnm01ρ exp(ifρ2)Rnm(ρ)×Jm(2πρr)dρ cos mϕ.
01ρRnm(ρ)Jm(2πρr)dρ=(-1)(n-m)/2Jn+1(2πr)2πr
A(ρ, θ)exp[iΦ(ρ, θ)]=n,mβnmRnm(ρ)cos mθ,
U(x, y)=2n,mβnmimVnm cos mϕ,
Vnm=01ρ exp(ifρ2)Rnm(ρ)Jm(2πρr)dρ
Vnm=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,
v=2πr,p=n-m2,q=n+m2.
|f|2π,v20,0pq6.
U(x, y)2V00+2in,mαnmimVnm cos mϕ,
Tnm cos mϕ,
Tnm=01ρn+1 exp(ifρ2)Jm(2πρr)dρ,
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,
Φ(ρ, θ)=αρ3 cos θ,γρ2 cos 2θ,δρ4,
U(x, y)=j=0Cjαj,j=0Djγj,
x=XNAλ,y=YNAλ,
v=2πx2+y2,f=2πλZ(1-1-NA2).
U(x, y)=2J1(v)v.
U(x, y)2J1(v)v+n,mim+1αnm(-1)(n-m)/2Jn+1(v)vcos mϕ.
U(x, y)2V00+n,mim+1αnmVnm cos mϕ,
A1,Φ(ρ)=α4,0R40(ρ)=2π6(6ρ4-6ρ2+1),
0ρ1,
max|v|30||V4,0(L)|2-|V4,0(40)|2|,
B(l; a, b, c, d)
a+l-1l-1b+l-1l-1l-1cd+ll
B(l+1; a, b, c, d)
(l+1)(l+a)(l+b)l(l-c)(l+d+1)B(l; a, b, c, d).
A(ρ, θ)exp[iΦ(ρ, θ)]=n,mβnmRnm(ρ)cos mθ.
U(x, y)=2 exp(iα4,0)k=0(6iα4,0)kk!T4k,0,
max|v|30||U|2-|UN|2|,
I(x, y)=μ(x0-x0, y0-y0)F(x0, y0)F*(x0, y0)×U(x-x0, y-y0)U*(x-x0, y-y0)dx0dx0dy0dy0,
μ(x0, y0)
=I(p, q)exp{i[px0+iqy0]}dpdqI(p, q)dpdq,
μ(v)=2J1(σv)σv.
I(x, y)=|F(x0, y0)U(x0, y0)|2.
I(x, y)=|F(x0, y0)|2|U(x0, y0)|2.
F(x0, y0)=nAnδ(x0-an, y0-bn).
I(x, y)=n,mAnAm*μ(an-am, bn-bm)×U(x-an, y-bn)U*(x-am, y-bm).
(Wf+R1)2+z2+2(Wf+R1)z cos p=R22.
Wf=z(1-cos p)=z[1-(1-sin2 p)1/2].
Wf=zsin2 p2+sin4 p8+sin6 p16+=12zs02ρ2+18zs04ρ4+116zs06ρ6+.
Wfz(bˆ0+bˆ1ρ2)
(1-sin2 p)1/2=-12(1-c0)n=012n-11-c01+c0n-1-12n+31-c01+c0n+1R2n0(ρ),
|Vnm|Mm(v)2n+1,|Tnm|Mm(v)2n+2,
Mm(v)=max0uv|Jm(u)|1.
01ρ|Rnm(ρ)|2 dρ=12(n+1),
V00=T00=exp(if)l=1(-2if)l-1Jl(v)vl.
Jl(v)vlv=0=2-ll!,
V00(v=0)=12exp(if)l=1(-if)l-1l!.
l-1vljtlj=(l-1)!q!p!(q+j)!(l-1-p+j)!,j=0,,p,
Qlj=l-1vljJl+m+2j(v)Jl(v),j=0,,p,
|l-1vlj|(l+m+j)!(l-1+j)!(l+q+j)!(l-1+j-p)!pjq+jpexp[-(q+1)p/l]pjq+jp.
0Js(v)Js-1(v)v/s1+[1-(v/s)2]1/2,0vs.
12-14-c1/2=c1-c1-c1-,0c14.
0Jl+m+2j(v)Jl(v)bm+2j,0vl,
b=v/(l+1)1+{1-[v/(l+1)]2}1/2.
j=0pQljexp[-(q+1)p/l]j=0ppjq+jpbq-p+2j.
exp[-(q+1)q/l]j=0qqjq+jqb2j
j=0qqjq+jqb2j=Pq(1+2b2)(b+1+b2)2q+12(πbq)1/2(1+b2)1/4,

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