Abstract

By analyzing aerial images, we characterize the lowest order coma aberration measurements for the projection optics of a microlithography exposure apparatus based on scalar diffraction theory. Our developed method for measuring the coma aberration exploits the intensity difference between the sidelobe peaks appearing near the boundaries of the bright field (“negative”) single-line or plural-line patterns. Our method further demonstrates linearity between the intensity difference of the sidelobe peaks and the amount of residual lowest order coma aberration. We analyze the coma aberration sensitivity formula and determine the duty ratio of the line-and-space pattern that realizes the highest aberration sensitivity.

© 2014 Optical Society of America

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References

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  1. J. Bruning, “Optical lithography: 40 years and holding,” Proc. SPIE 6520, 652004 (2007).
    [CrossRef]
  2. M. Born and E. Wolf, “The diffraction theory of aberrations,” in Principles of Optics (Cambridge University, 1999), pp. 517–553.
  3. R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20, 16436–16449 (2012).
    [CrossRef]
  4. J. P. Kirk, “Review of photoresist-based lens evaluation methods,” Proc. SPIE 4000, 2–8 (2000).
    [CrossRef]
  5. H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
    [CrossRef]
  6. N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, “In-situ measurement of lens aberrations,” Proc. SPIE 4000, 18–29 (2000).
    [CrossRef]
  7. R. Barakat and A. Houston, “Line spread and edge spread functions in the presence of off-axis aberrations,” J. Opt. Soc. Am. 55, 1132–1135 (1965).
    [CrossRef]
  8. R. Barakat and A. Houston, “Diffraction image of a single bar in the presence of off-axis aberrations,” J. Opt. Soc. Am. 56, 1402–1403 (1966).
    [CrossRef]
  9. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 208, 408–432 (1953).

2012

2007

J. Bruning, “Optical lithography: 40 years and holding,” Proc. SPIE 6520, 652004 (2007).
[CrossRef]

2001

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

2000

N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, “In-situ measurement of lens aberrations,” Proc. SPIE 4000, 18–29 (2000).
[CrossRef]

J. P. Kirk, “Review of photoresist-based lens evaluation methods,” Proc. SPIE 4000, 2–8 (2000).
[CrossRef]

1966

1965

1953

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 208, 408–432 (1953).

Barakat, R.

Born, M.

M. Born and E. Wolf, “The diffraction theory of aberrations,” in Principles of Optics (Cambridge University, 1999), pp. 517–553.

Bruning, J.

J. Bruning, “Optical lithography: 40 years and holding,” Proc. SPIE 6520, 652004 (2007).
[CrossRef]

Busath, D. R.

N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, “In-situ measurement of lens aberrations,” Proc. SPIE 4000, 18–29 (2000).
[CrossRef]

Dierichs, M.

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

Dunn, C.

Farrar, N. R.

N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, “In-situ measurement of lens aberrations,” Proc. SPIE 4000, 18–29 (2000).
[CrossRef]

Gray, R. W.

Greevenbroek, H. V.

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 208, 408–432 (1953).

Houston, A.

Kirk, J. P.

J. P. Kirk, “Review of photoresist-based lens evaluation methods,” Proc. SPIE 4000, 2–8 (2000).
[CrossRef]

Laan, H. V. D.

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

McCoo, E.

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

Pongers, R.

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

Rolland, J. P.

Smith, A. H.

N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, “In-situ measurement of lens aberrations,” Proc. SPIE 4000, 18–29 (2000).
[CrossRef]

Stoffels, F.

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

Taitano, D.

N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, “In-situ measurement of lens aberrations,” Proc. SPIE 4000, 18–29 (2000).
[CrossRef]

Thompson, K. P.

Willekers, R.

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, “The diffraction theory of aberrations,” in Principles of Optics (Cambridge University, 1999), pp. 517–553.

J. Opt. Soc. Am.

Opt. Express

Proc. R. Soc. Lond. Ser. A

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 208, 408–432 (1953).

Proc. SPIE

J. P. Kirk, “Review of photoresist-based lens evaluation methods,” Proc. SPIE 4000, 2–8 (2000).
[CrossRef]

H. V. D. Laan, M. Dierichs, H. V. Greevenbroek, E. McCoo, F. Stoffels, R. Pongers, and R. Willekers, “Aerial image measurement methods for fast aberration set-up and illumination pupil verification,” Proc. SPIE 4346, 394–407 (2001).
[CrossRef]

N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, “In-situ measurement of lens aberrations,” Proc. SPIE 4000, 18–29 (2000).
[CrossRef]

J. Bruning, “Optical lithography: 40 years and holding,” Proc. SPIE 6520, 652004 (2007).
[CrossRef]

Other

M. Born and E. Wolf, “The diffraction theory of aberrations,” in Principles of Optics (Cambridge University, 1999), pp. 517–553.

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Figures (9)

Fig. 1.
Fig. 1.

Asymmetry of sidelobe intensity with coma aberration, Z7=0.05λ, five plural L&S patterns, duty ratio of 0.5, and pitch P=0.8μm. The arrows point to the first sidelobe peaks. The simulation parameters were λ=0.365μm, NA=0.5, and circular illumination σ=0.3: (a) positive pattern and (b) negative pattern.

Fig. 2.
Fig. 2.

Intensity difference of the sidelobe defined in Eq. (2), plotted with respect to Z7 and σ variation: (a) positive pattern, and (b) negative pattern.

Fig. 3.
Fig. 3.

Coordinate diagram of the coherent image coordination corresponding to Eq. (5). The model is handled in one dimension, so that the directional cosine only uses the x axis.

Fig. 4.
Fig. 4.

Positive single line in Eq. (31) for amplitude U0(x)/(π/λ), ringing waves Si(kNAx) and Si(kNAx+), and D=4.0μm. The calculation parameters are λ=0.365μm and NA=0.5.

Fig. 5.
Fig. 5.

Negative single line in Eq. (43) for amplitude U0(x)/(π/λ), ringing waves Si(kNAx) and Si(kNAx+), and D=4.0μm. The calculation parameters are λ=0.365μm and NA=0.5.

Fig. 6.
Fig. 6.

Coherent image simulation results for the intensity difference ΔI in Eq. (2), showing the linear dependency of coma aberration Z7 for positive and negative single-line patterns, where D=4.0μm. The simulation parameters are λ=0.365μm, NA=0.5, and coherence factor σ=0.0.

Fig. 7.
Fig. 7.

Duty ratio dependency of the intensity difference ΔIposi under the coherent illumination σ=0.0. In addition to the numerical simulation result, the fitting curves using the parameters of Fitting No. 1 and Fitting No. 2 are plotted in (a). Similarly, the numerical simulation result and the fitting curves using Fitting No. 1 are plotted in (b). These imaging parameters are exactly the same as the coherent case (σ=0.0) in Fig. 9.

Fig. 8.
Fig. 8.

Integral sine function defined in Eq. (32).

Fig. 9.
Fig. 9.

Duty ratio dependence of the aberration sensitivity under a wide range of illumination coherency σ=0.00.7 for plural L&S patterns with pitch P=0.8μm and the duty ratio between 0.0 and 1.0. The simulation parameters are λ=0.365μm and NA=0.5.

Tables (2)

Tables Icon

Table 1. Result of Fitting Parameter for Positive L&S

Tables Icon

Table 2. Result of Fitting Parameter for Negative L&S

Equations (66)

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σ=NA(Illumination optics)NA(projection optics).
ΔI=I(xR)I(xL),
T˜(ξ)=dxT(x)eikξx,
W(ξ,η)=Z7[3ξ3+(3η22)ξ],
I(x)=|NANAdξT˜(ξ)G(ξ,0)eikξx|2=|NANAdξT˜(ξ)eikZ7(3ξ3)eik(x2Z7)ξ|2,
G(ξ,0)=m=0(ik)mm!{(3ξ3)Z7}m1+ik(3ξ3)Z7k22[(3ξ3)Z7]2.
I(x)|U(x)|2|U0(x)+U1(x)Z7+U2(x)(Z7)2|2,
U(x)=U0(x)+U1(x)Z7+U2(x)(Z7)2,
U0(x)=NANAdξT˜(ξ)eikξx,
U1(x)=ikNANAdξT˜(ξ)(3ξ3)eikξx,
U2(x)=k22NANAdξT˜(ξ)(3ξ3)2eikξx.
U0*(x)=NANAdξT˜*(ξ)eikξx=NANAd(ξ)T˜(ξ)eikξx=NANAdξT˜(ξ)eikξx=U0(x),
U1*(x)=ikNANAdξ(3ξ3)T˜(ξ)eikξx=ik+NANAd(ξ)(3ξ3)T˜(ξ)eikξx=U1(x),
U2*(x)=k22NANAdξ(3ξ3)2T˜(ξ)eikξx=k22NANAd(ξ)(3ξ3)2T˜(ξ)eikξx=U2(x).
U0(x)=NANAdξT˜(ξ)eikξx=NANAd(ξ)T˜(ξ)eikξx=NANAdξT˜(ξ)eikξx=U0(x),
U2(x)=k22NANAdξ(3ξ3)2T˜(ξ)eikξx=k22NANAd(ξ)(3ξ3)2T˜(ξ)eikξx=U2(x).
U1(x)=ikNANAdξ(3ξ3)T˜(ξ)eikξx=ik+NANAd(ξ)(3ξ3)T˜(ξ)eikξx=U1(x).
U0(x)|x=±x0=0,
0=2{U0(x)+U1(x)Z7}×{U0(x)+U1(x)Z7}|x=±x0+ζ1.
ζ1=U1(x0)2U0(x0)Z7,
0=2U(x)U(x)|x=±x0+ζ1+ζ2.
ζ2=±12U0(x0)[123U0(x0)(ζ1)2+2U1(x0)ζ1Z7+U2(x0)(Z7)2],
xL=x0+ζ1ζ2,
xR=x0+ζ1+ζ2.
ΔI|U(xR)|2|U(xL)|2|U0(x0)+U0(x0)(ζ1+ζ2)+122U0(x0)(ζ1+ζ2)2+U1(x0)Z7+U1(x0)(ζ1+ζ2)Z7+U2(x0)(Z7)2|2|U0(x0)+U0(x0)(ζ1ζ2)+122U0(x0)(ζ1ζ2)2+U1(x0)Z7+U1(x0)(ζ1ζ2)+U2(x0)(Z7)2|2=|U0(x0)+ε+U1(x0)Z7|2|U0(x0)+εU1(x0)Z7|2=2[U0(x0)+ε]×2U1(x0)Z74U0(x0)U1(x0)Z7,
εU0(x0)(ζ1+ζ2)+122U0(x0)(ζ1+ζ2)2+U1(x0)(ζ1+ζ2)Z7+U2(x0)(Z7)2122U0(x0)[(ζ1)2+2ζ1ζ2]+U1(x0)ζ1Z7+U2(x0)(Z7)2={12[U1(x0)]22U0(x0)+U2(x0)}(Z7)2.
(aberration sensitivity)=ΔIZ7|Z70.
(aberration sensitivity)=4U0(x0)U1(x0).
T(x)=rect[D2,D2],
T˜(ξ)=Dsinc(πDλξ),
U0(x)=NANAdξDsinc(πDλξ)eikξx=λπ0NAdξ1ξ{sin[k(x+D2)ξ]sin[k(xD2)ξ]}=λπ[Si(kNAx)Si(kNAx+)],
Si(x)=0xdtsintt,
x±=xD2.
U1(x)=ikNANAdξ3ξ3Dsinc(πDλξ)eikξx=3kλπ[0kNAx+dξξ2cosξ(kx+)30kNAxdξξ2cosξ(kx)3]=3(NA)3kλπ[u1(kNAx)u1(kNAx+)],
u1(x)=2xcos(x)+(x22)sin(x)x3.
0=|U0(x)|2=2U0(x)U0(x)U0(x)[Si(kNAx)Si(kNAx+)][Si()Si(kNAx+)]=Si(kNAx+)=sin(kNAx+)kNAx+.
kNAx+|x=x0=π,
x0=D2+πkNA.
kNAD=2πλ(NA)D1,
ΔIposi4U0(x0)U1(x0)Z74λπ[Si()Si(π)]×3(NA)3kλπ2(π)cos(π)π3Z7=24λπ[Si(π)π2](NA)3kλπ3Z7.
T(x)=1rect[D2,D2],
T˜(ξ)=λδ(ξ)Dsinc(πDλξ).
U0(x)=NANAdξ[λδ(ξ)Dsinc(πDλξ)]eikξx=λπ[πSi(kNAx)+Si(kNAx+)].
ΔInega4U0(x0)U1(x0)Z74λπ[πSi()+Si(π)]×3(NA)3kλπ2(π)cos(π)π3Z7=24λπ[π2+Si(π)](NA)3kλπ3Z7.
ΔInega/Z7ΔIposi/Z7|Z70=Si(π)+π2Si(π)π212.17.
T˜(ξ)=m=(1)(N+1)mND×sinc(πDPm)sinc[πNPλ(ξmλP)].
U0(x)=λDπP[C00th(x)+C01st(x)sinc(πDP)],
U1(x0)=λDπP[C10th(x)+C11st(x)sinc(πDP)],
ΔIposi=(λDπP)2[a0+a1sinc(πDP)+a2{sinc(πDP)}2]Z7,
a0=C00th(x0)C10th(x0),
a1=C00th(x0)C11st(x0)+C01st(x0)C10th(x0),
a2=C01st(x0)C11st(x0).
(λDπP)2{sinc(πDP)}2sin2(πDP).
T˜(ξ)=λδ(ξ)m=(1)(N+1)mND×sinc(πDPm)sinc[πNPλ(ξmλP)].
U0(x)=λλDπP{C00th(x)+C01st(x)sinc(πDP)},
U1(x)=λDπP[C10th(x)+C11st(x)sinc(πDP)].
ΔInega=λDπP[b0+b1sinc(πDP)]Z7+ΔIposi,
λDπPsinc(λDπP)sin(λDπP).
0={U0(x)+U1(x)Z7}|x=±x0+ζ1U0(x)|x=±x0+2U0(x)|x=±x0ζ1+U1(x)|x=±x0Z7=2U0(x)|x=±x0ζ1+U1(x)|x=±x0Z7.
0=2U0(x0)ζ1+U1(x0)Z7.
0=U0(x)|x=±x0+ζ1+ζ2+U1(x)|x=±x0+ζ1+ζ2Z7+U2(x)|x=±x0+ζ1+ζ2(Z7)2U0(x)|x=±x0+2U0(x)|x=±x0(ζ1+ζ2)+12!3U0(x)|x=±x0(ζ1+ζ2)2+U1(x)|x=±x0Z7+2U1(x)|x=±x0(ζ1+ζ2)Z7+U2(x)|x=±x0(Z7)22U0(x)|x=±x0ζ2+123U0(x)|x=±x0(ζ1)2+2U1(x)|x=±x0(ζ1+ζ2)Z7+U2(x)|x=±x0(Z7)2,
0=2U0(x0)ζ2±123U0(x0)(ζ1)2±2U1(x0)ζ1Z7±U2(x0)(Z7)2.
Si(x)=n=0(1)nx2n+1(2n+1)(2n+1)!,Si(x)=π2cosxn=0(1)n(2n)!x2n+1
sinxn=0(1)n(2n+1)!x2n+2.
ΔIposi(λDπP)2,
ΔInegaλDπP.

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