Abstract

Imaging properties of a birefringent lens, in which the fast (or the slow) axis is distributed in the radial direction whereas magnitude of birefringence varies as a quadratic function of the pupil radius, are investigated by calculating a point-spread function. It is found that the point image is analytically described by using the Lommel function as well as the zero-order Bessel function, and a localized intensity null surrounded by bright regions in all directions can be realized at a geometrical focus under certain conditions. The magnitude of birefringence that is tolerable in image formations is also discussed, assuming that the lens is applied to microlithography.

© 2002 Optical Society of America

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References

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2000 (4)

1998 (3)

1994 (1)

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

1993 (1)

X. Yang, C. Xiang, G. Zhang, “Birefringent common-path interferometer for testing large convex spherical surfaces,” Opt. Eng. 32, 1080–1083 (1993).
[CrossRef]

1992 (1)

1991 (1)

H. Fukuda, A. Imai, T. Terasawa, S. Okazaki, “New approach to resolution limit and advanced image formation techniques in optical lithography,” IEEE Trans. Electron Devices 38, 67–75 (1991).
[CrossRef]

1990 (2)

1989 (2)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).
[CrossRef]

1986 (1)

1985 (1)

1984 (1)

F. Ratajczyk, “A method of calculation of permissible birefringence in lenses of the optical instruments,” Optik (Stuttgart) 68, 61–68 (1984).

1957 (1)

1941 (1)

Andrews, L. C.

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (Oxford U. Press, Oxford, UK, 1998), Chap. 6.

Arlt, J.

Ashkin, A.

Bandyopadhyay, P.

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

Bjorkholm, J. E.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 9.

Chipman, L. J.

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).
[CrossRef]

Chipman, R. A.

J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I: Formulation and example,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
[CrossRef]

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).
[CrossRef]

Chu, S.

Condell, W. J.

Dayson, L.

Duncan, A. J.

Dziedzic, J. M.

Fukuda, H.

H. Fukuda, A. Imai, T. Terasawa, S. Okazaki, “New approach to resolution limit and advanced image formation techniques in optical lithography,” IEEE Trans. Electron Devices 38, 67–75 (1991).
[CrossRef]

Gahagan, K. T.

Ghosh, A.

S. Sanyal, A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000).
[CrossRef]

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 3.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4.

Harris, M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Hill, C. A.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Imai, A.

H. Fukuda, A. Imai, T. Terasawa, S. Okazaki, “New approach to resolution limit and advanced image formation techniques in optical lithography,” IEEE Trans. Electron Devices 38, 67–75 (1991).
[CrossRef]

Iwata, K.

Jones, C.

Kikuta, H.

Kinnstatter, K.

Lesso, J. P.

Maenhoudt, M.

M. Maenhoudt, S. Verhaegen, K. Ronse, P. Zandbergen, E. Muzio, “Limits of optical lithography,” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 373–387 (2000).
[CrossRef]

McGuire, J. P.

Muzio, E.

M. Maenhoudt, S. Verhaegen, K. Ronse, P. Zandbergen, E. Muzio, “Limits of optical lithography,” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 373–387 (2000).
[CrossRef]

Ojima, M.

Okazaki, S.

H. Fukuda, A. Imai, T. Terasawa, S. Okazaki, “New approach to resolution limit and advanced image formation techniques in optical lithography,” IEEE Trans. Electron Devices 38, 67–75 (1991).
[CrossRef]

Padgett, M. J.

Ratajczyk, F.

F. Ratajczyk, “A method of calculation of permissible birefringence in lenses of the optical instruments,” Optik (Stuttgart) 68, 61–68 (1984).

Ronse, K.

M. Maenhoudt, S. Verhaegen, K. Ronse, P. Zandbergen, E. Muzio, “Limits of optical lithography,” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 373–387 (2000).
[CrossRef]

Sanyal, S.

S. Sanyal, A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321–2325 (2000).
[CrossRef]

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

Shimomura, H.

Sibbett, W.

Suzuki, A.

Y. Unno, A. Suzuki, “Analyses of imaging performance degradation caused by birefringence residual in lens materials,” in Optical Microlithography XIV, C. J. Progler, ed., Proc. SPIE4346, 1306–1317 (2001).
[CrossRef]

Swartzlander, G. A.

Terasawa, T.

H. Fukuda, A. Imai, T. Terasawa, S. Okazaki, “New approach to resolution limit and advanced image formation techniques in optical lithography,” IEEE Trans. Electron Devices 38, 67–75 (1991).
[CrossRef]

Unno, Y.

Vaughan, J. M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Verhaegen, S.

M. Maenhoudt, S. Verhaegen, K. Ronse, P. Zandbergen, E. Muzio, “Limits of optical lithography,” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 373–387 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 9.

Xiang, C.

X. Yang, C. Xiang, G. Zhang, “Birefringent common-path interferometer for testing large convex spherical surfaces,” Opt. Eng. 32, 1080–1083 (1993).
[CrossRef]

Yang, X.

X. Yang, C. Xiang, G. Zhang, “Birefringent common-path interferometer for testing large convex spherical surfaces,” Opt. Eng. 32, 1080–1083 (1993).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Fort Worth, Texas, 1991), Chap. 1.

Yonezawa, S.

Zandbergen, P.

M. Maenhoudt, S. Verhaegen, K. Ronse, P. Zandbergen, E. Muzio, “Limits of optical lithography,” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 373–387 (2000).
[CrossRef]

Zhang, G.

X. Yang, C. Xiang, G. Zhang, “Birefringent common-path interferometer for testing large convex spherical surfaces,” Opt. Eng. 32, 1080–1083 (1993).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Electron Devices (1)

H. Fukuda, A. Imai, T. Terasawa, S. Okazaki, “New approach to resolution limit and advanced image formation techniques in optical lithography,” IEEE Trans. Electron Devices 38, 67–75 (1991).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

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

Opt. Commun. (1)

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Opt. Eng. (4)

X. Yang, C. Xiang, G. Zhang, “Birefringent common-path interferometer for testing large convex spherical surfaces,” Opt. Eng. 32, 1080–1083 (1993).
[CrossRef]

S. Sanyal, P. Bandyopadhyay, A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng. 37, 592–599 (1998).
[CrossRef]

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

R. A. Chipman, L. J. Chipman, “Polarization aberration diagrams,” Opt. Eng. 28, 100–106 (1989).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

F. Ratajczyk, “A method of calculation of permissible birefringence in lenses of the optical instruments,” Optik (Stuttgart) 68, 61–68 (1984).

Other (8)

Y. Unno, A. Suzuki, “Analyses of imaging performance degradation caused by birefringence residual in lens materials,” in Optical Microlithography XIV, C. J. Progler, ed., Proc. SPIE4346, 1306–1317 (2001).
[CrossRef]

M. Maenhoudt, S. Verhaegen, K. Ronse, P. Zandbergen, E. Muzio, “Limits of optical lithography,” in Optical Microlithography XIII, C. J. Progler, ed., Proc. SPIE4000, 373–387 (2000).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 3.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 9.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Fort Worth, Texas, 1991), Chap. 1.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4.

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (Oxford U. Press, Oxford, UK, 1998), Chap. 6.

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Figures (14)

Fig. 1
Fig. 1

Model of an optical system used for the analyses. It is assumed that L2 has birefringence that is rotationally symmetric around the optical axis, and the entire system is aberration free when the magnitude of birefringence is zero.

Fig. 2
Fig. 2

Pupil transmission characteristics of the birefringent lens shown in Fig. 1. If it is assumed that an input beam is linearly polarized in the y direction, Aγ(ρ, α) and Wγ(ρ, α) represent amplitude transmittance and phase distribution on the exit pupil, respectively, for the γ polarization component (γ=x, y).

Fig. 3
Fig. 3

Division of the focal region by lines of z/r=±1. Points in R-I are located in the geometrical shadow of a converging beam, whereas points in R-II are illuminated directly.

Fig. 4
Fig. 4

Radial distributions of C(r, z) and S(r, z) calculated for discrete values of z between 0 and 1. Distributions for z<0 are obtained by C(r, -z)=C(r, z) and S(r, -z)=-S(r, z).

Fig. 5
Fig. 5

Radial distribution of the point-spread function obtained in the case of zero birefringence.

Fig. 6
Fig. 6

Logarithmic intensity plots of I0(r, z; a) on a cross-sectional plane containing the x and z axes. The parameter a is varied between 0 and 1 with a step of 0.25, and the intensity is normalized by the maximum in each case.

Fig. 7
Fig. 7

Intensity distributions of I0(r, z; a) on the optical axis (r=0) plotted in a linear scale.

Fig. 8
Fig. 8

Logarithmic intensity plots of I2(r, z; a) on a cross-sectional plane containing the x and z axes. The parameter a is varied between 0.25 and 1 with a step of 0.25, and the intensity is normalized by the maximum in each case. The result for a=0 is omitted because it is zero in the entire region.

Fig. 9
Fig. 9

Intensity distributions of I2(r, z; a) on the z=0 plane plotted in a linear scale.

Fig. 10
Fig. 10

Helical wave front that contributes to the creation of U2(r, z). Corresponding to the change of α from 0 to 2π, a phase change of 4π is observed.

Fig. 11
Fig. 11

Logarithmic intensity plots of I(r, z; a) on a cross-sectional plane containing the x and z axes, obtained as the summation of I0(r, z; a) and I2(r, z; a). The intensity is again normalized by the maximum in each case.

Fig. 12
Fig. 12

Variation of the total energy assigned to the two image components. E0 and E2 are calculated by 0I0(r, z; a)r dr and 0I2(r, z; a)r dr, respectively, as a function of a.

Fig. 13
Fig. 13

Contour plot of the Strehl intensity calculated as functions of a and z. The hatching represents a region where the image formation is considered ideal, and the depth of focus is given by 2Δz(a), which is reduced with an increase of a.

Fig. 14
Fig. 14

Relationships between the total lens thickness (D) and the maximum magnitude of birefringence (B) that can be neglected in image formations. The condition becomes severer as the wavelength is shortened and/or the lens thickness is increased.

Tables (1)

Tables Icon

Table 1 Number K in Eq. (19) Needed to Keep Calculation Errors C(r, z) [Eq. (20)] and S(r, z) [Eq. (21)] under 10-4, 10-8, and 10-12a

Equations (60)

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px(ρ, α)py(ρ, α)=R(-α)exp(-iπaρ2)00exp(iπaρ2)×R(α)cos ωsin ω,
R(α)=cos αsin α-sin αcos α,
px(ρ, α)py(ρ, α)=f(ρ, α)g(ρ, α)g(ρ, α)f*(ρ, α)cos ωsin ω,
f(ρ, α)=sin2 α exp(iπaρ2)+cos2 α exp(-iπaρ2),
g(ρ, α)=(sin α cos α)[exp(-iπaρ2)-exp(iπaρ2)],
I[μ(ρ, α)]=0102πμ(ρ, α)×exp[-i2πrρ cos(α-θ)-iπzρ2]ρ dρ dα,
Ux(r, z, θ; ω)Uy(r, z, θ; ω)=I[f(ρ, α)]I[g(ρ, α)]I[g(ρ, α)]I[f*(ρ, α)]cos ωsin ω,
Iˆω(r, z, θ)=|Ux(r, z, θ; ω)|2+|Uy(r, z, θ; ω)|2
Iˆ(r, z, θ)=0πIˆω(r, z, θ)dω0πdω.
0π sin2 ω dω=0π cos2 ω dω=π/2,
0π sin ω cos ω dω=0,
Iˆ(r, z, θ)=12{|I[f(ρ, α)]|2+|I[f*(ρ, α)]|2}+|I[g(ρ, α)]|2,
Iˆ(r, z, θ)=0102π exp[-i2πrρ cos(α-θ)-iπzρ2]ρ dρdα2
Iˆ(r, z, θ)=|U0(r, z)|2,
U0(r, z)2π01J0(2πrρ)exp(-iπzρ2)ρ dρ,
U0(r, z)=C(r, z)-iS(r, z),
C(r, z)=2π01J0(2πrρ)cos(πzρ2)ρ dρ,
S(r, z)=2π01J0(2πrρ)sin(πzρ2)ρ dρ.
Ln(r, z)s=0K[(-1)s(z/r)n+2sJn+2s(2πr)]
C(r, z)=cos(πz)zL1(r, z)+sin(πz)zL2(r, z),
S(r, z)=sin(πz)zL1(r, z)-cos(πz)zL2(r, z),
Iˆ(r, z, θ)=[C(r, z)]2+[S(r, z)]2,
Iˆ(r, z, θ)=12{|F(r, z, θ)|2+|F(r,-z, θ+π)|2}+|G(r, z, θ)|2.
F(r, z, θ)=12 0102π exp[-i2πrρ cos(α-θ)]×{exp[-iπ(z+a)ρ2]+exp[-iπ(z-a)ρ2]}ρ dρdα+12 0102π cos(2α)exp[-i2πrρ cos(α-θ)]×{exp[-iπ(z+a)ρ2]-exp[-iπ(z-a)ρ2]}ρ dρdα,
G(r, z, θ)=12 0102π sin(2α)exp[-i2πrρ cos(α-θ)]×{exp[-iπ(z+a)ρ2]-exp[-iπ(z-a)ρ2]}ρ dρdα,
02π sin(2α)exp[-i2πrρ cos(α-θ)]dα=-2π sin(2θ)J2(2πrρ),
02π cos(2α)exp[-i2πrρ cos(α-θ)]dα=-2π cos(2θ)J2(2πrρ),
F(r, z, θ)=12[U0(r, z+a)+U0(r, z-a)]-cos(2θ)2[U2(r, z+a)-U2(r, z-a)],
G(r, z, θ)=-sin(2θ)2[U2(r, z+a)-U2(r, z-a)],
U2(r, z)2π01J2(2πrρ)exp(-iπzρ2)ρ dρ.
U0(r,-z)=U0*(r, z),
U2(r,-z)=U2*(r, z),
12{|F(r, z, θ)|2+|F(r,-z, θ+π)|2}=14|U0(r, z+a)+U0(r, z-a)|2+cos2(2θ)4|U2(r, z+a)-U2(r, z-a)|2
I(r, z; a)=I0(r, z; a)+I2(r, z; a),
I0(r, z; a)=14|U0(r, z+a)+U0(r, z-a)|2,
I2(r, z; a)=14|U2(r, z+a)-U2(r, z-a)|2.
I0(r, z; a)=14[C(r, z+a)+C(r, z-a)]2+14[S(r, z+a)+S(r, z-a)]2,
J2(v)=-J0(v)+2J1(v)v,
J1(v)=-dJ0(v)dv,
J2(2πrρ)=-J0(2πrρ)-12π2r2ρ dJ0(2πrρ)dρ.
U2(r, z)=-U0(r, z)-1πr2×[J0(2πr)exp(-iπz)-1+izU0(r, z)],
Re[U2(r, z)]=-C(r, z)-1πr2[zS(r, z)+J0(2πr)cos(πz)-1],
Im[U2(r, z)]=S(r, z)-1πr2[zC(r, z)-J0(2πr)sin(πz)],
I2(r, z; a)=14[WR(r, z; a)]2+14[WI(r, z; a)]2,
WR(r, z; a)Re[U2(r, z+a)-U2(r, z-a)]=-C(r, z+a)+C(r, z-a)-1πr2[(z+a)S(r, z+a)-(z-a)S(r, z-a)-2J0(2πr)sin(πz)sin(πa)],
WI(r, z; a)Im[U2(r, z+a)-U2(r, z-a)]=S(r, z+a)-S(r, z-a)-1πr2[(z+a)C(r, z+a)-(z-a)C(r, z-a)-2J0(2πr)cos(2πz)sin(πa)]
I0(r, z; a)=I0(r, z;-a)=I0(r,-z; a),
I2(r, z; a)=I2(r, z;-a)=I2(r,-z; a),
J2(2πrρ)=-exp(-i2θ)2π 02π exp(i2α)×exp[-i2πrρ cos(α-θ)] dα,
U2(r, z)=-exp(-i2θ)I[exp(i2α)],
0[I0(r, z; a)+I2(r, z; a)]r dr=0.5,
IS(z, a)I(0, z; a)/I(0, 0; 0),
IS(r, a)=14[sinc(z+a)+sinc(z-a)]2+14 sinπ z+a2sincz+a2+sinπ z-a2sincz-a22,
IS(0, a)=[sinc(a)]2=1-π2a23+2π4a445- .
02π sin(2α)exp[-i2πrρ cos(α-θ)]dα=12i exp(i2θ)02π exp(-i2πrρ cos u)exp(i2u)du-exp(-i2θ)02π exp(-i2πrρ cos u)exp(-i2u)du,
02π exp(iv cos u)exp(inu)du=in2πJn(v),
02π exp(iv cos u)exp(i2u)du=-2πJ2(v)
02π exp(-iv cos u)exp(i2u)du=-2πJ2(v)
02π exp(-iv cos u)exp(-i2u)du=-2πJ2(v).
02π sin(2α)exp[-i2πrρ cos(α-θ)]dα=-2π sin(2θ)J2(2πrρ).

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