Abstract

Analysis and optimization of diffraction effects in nanolithography through multilayered media with a fast and accurate field-theoretical approach is presented. The scattered field through an arbitrary two-dimensional (2D) mask pattern in multilayered media illuminated by a TM-polarized incident wave is determined by using an electric field integral equation formulation. In this formulation the electric field is represented in terms of complex images Green’s functions. The method of moments is then employed to solve the resulting integral equation. In this way an accurate and computationally efficient approximate method is achieved. The accuracy of the proposed method is vindicated through comparison with direct numerical integration results. Moreover, the comparison is made between the results obtained by the proposed method and those obtained by the full-wave finite-element method. The ray tracing method is combined with the proposed method to describe the imaging process in the lithography. The simulated annealing algorithm is then employed to solve the inverse problem, i.e., to design an optimized mask pattern to improve the resolution. Two binary mask patterns under normal incident coherent illumination are designed by this method, where it is shown that the subresolution features improve the critical dimension significantly.

© 2012 Optical Society of America

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References

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  1. D. Shao and S. Chen, “Surface plasmon assisted contact scheme nanoscale photolithography using an UV lamp,” J. Vac. Sci. Technol. B 26, 227–231 (2008).
    [CrossRef]
  2. A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
    [CrossRef]
  3. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
    [CrossRef]
  4. R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
    [CrossRef]
  5. A. Grbic and R. Merlin, “Near-field focusing plates and their design,” IEEE Trans. Antennas Propag. 56, 3159–3165 (2008).
    [CrossRef]
  6. A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99, 1806–1815 (2011).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
  8. J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
    [CrossRef]
  9. K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlith. Microfab. Microsys. 1, 253 (2002).
    [CrossRef]
  10. F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
    [CrossRef]
  11. M. S. Yeung, “Three-dimensional mask transmission simulation using a single integral equation method,” Proc. SPIE 3334, 704–713 (1998).
    [CrossRef]
  12. Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theor. Tech. 39, 588–592 (1991).
    [CrossRef]
  13. J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microwave Theor. Tech. 40, 595–600 (1992).
    [CrossRef]
  14. M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag. 53, 3644–3653 (2005).
    [CrossRef]
  15. H. Alaeian and R. Faraji-Dana, “A fast and accurate analysis of 2-D periodic devices using complex images Green’s functions,” J. Lightwave Technol. 27, 2216–2223 (2009).
    [CrossRef]
  16. Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag. 37, 229–234 (1989).
    [CrossRef]
  17. A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microwave Theor. Tech. 58, 602–613 (2010).
    [CrossRef]
  18. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).
  19. T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography.” Ph.D. thesis (University of California at Berkeley, 2001).
  20. A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the ’simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13, 262–280 (1987).
    [CrossRef]
  21. E. D. Palik, ed., Handbook of Optical Constants of Solids 11 (Academic, 1991), pp. 374–385.

2011

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99, 1806–1815 (2011).
[CrossRef]

2010

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microwave Theor. Tech. 58, 602–613 (2010).
[CrossRef]

2009

2008

D. Shao and S. Chen, “Surface plasmon assisted contact scheme nanoscale photolithography using an UV lamp,” J. Vac. Sci. Technol. B 26, 227–231 (2008).
[CrossRef]

A. Grbic and R. Merlin, “Near-field focusing plates and their design,” IEEE Trans. Antennas Propag. 56, 3159–3165 (2008).
[CrossRef]

2007

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
[CrossRef]

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

2006

F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
[CrossRef]

2005

M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag. 53, 3644–3653 (2005).
[CrossRef]

2003

J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[CrossRef]

2002

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlith. Microfab. Microsys. 1, 253 (2002).
[CrossRef]

1998

M. S. Yeung, “Three-dimensional mask transmission simulation using a single integral equation method,” Proc. SPIE 3334, 704–713 (1998).
[CrossRef]

1992

J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microwave Theor. Tech. 40, 595–600 (1992).
[CrossRef]

1991

Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theor. Tech. 39, 588–592 (1991).
[CrossRef]

1989

Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag. 37, 229–234 (1989).
[CrossRef]

1987

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the ’simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13, 262–280 (1987).
[CrossRef]

1982

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[CrossRef]

Adam, K.

F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
[CrossRef]

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlith. Microfab. Microsys. 1, 253 (2002).
[CrossRef]

Aksun, M. I.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microwave Theor. Tech. 58, 602–613 (2010).
[CrossRef]

M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag. 53, 3644–3653 (2005).
[CrossRef]

Alaeian, H.

Alparslan, A.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microwave Theor. Tech. 58, 602–613 (2010).
[CrossRef]

Azpiroz, J. T.

J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[CrossRef]

Burchard, P.

J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[CrossRef]

Chen, S.

D. Shao and S. Chen, “Surface plasmon assisted contact scheme nanoscale photolithography using an UV lamp,” J. Vac. Sci. Technol. B 26, 227–231 (2008).
[CrossRef]

Chow, Y. L.

J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microwave Theor. Tech. 40, 595–600 (1992).
[CrossRef]

Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theor. Tech. 39, 588–592 (1991).
[CrossRef]

Corana, A.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the ’simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13, 262–280 (1987).
[CrossRef]

Dural, G.

M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag. 53, 3644–3653 (2005).
[CrossRef]

Fang, D. G.

J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microwave Theor. Tech. 40, 595–600 (1992).
[CrossRef]

Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theor. Tech. 39, 588–592 (1991).
[CrossRef]

Faraji-Dana, R.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Grbic, A.

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99, 1806–1815 (2011).
[CrossRef]

A. Grbic and R. Merlin, “Near-field focusing plates and their design,” IEEE Trans. Antennas Propag. 56, 3159–3165 (2008).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

Hesselink, L.

F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
[CrossRef]

Howard, G. E.

J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microwave Theor. Tech. 40, 595–600 (1992).
[CrossRef]

Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theor. Tech. 39, 588–592 (1991).
[CrossRef]

Hua, Y.

Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag. 37, 229–234 (1989).
[CrossRef]

Imani, M. F.

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99, 1806–1815 (2011).
[CrossRef]

Levenson, M. D.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[CrossRef]

Marchesi, M.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the ’simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13, 262–280 (1987).
[CrossRef]

Martini, C.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the ’simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13, 262–280 (1987).
[CrossRef]

Matteo, J.

F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
[CrossRef]

Merlin, R.

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99, 1806–1815 (2011).
[CrossRef]

A. Grbic and R. Merlin, “Near-field focusing plates and their design,” IEEE Trans. Antennas Propag. 56, 3159–3165 (2008).
[CrossRef]

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
[CrossRef]

Michalski, K. A.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microwave Theor. Tech. 58, 602–613 (2010).
[CrossRef]

Milanfar, P.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

Neureuther, A. R.

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlith. Microfab. Microsys. 1, 253 (2002).
[CrossRef]

Pistor, T. V.

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography.” Ph.D. thesis (University of California at Berkeley, 2001).

Poonawala, A.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

Ridella, S.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the ’simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13, 262–280 (1987).
[CrossRef]

Sarkar, T. K.

Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag. 37, 229–234 (1989).
[CrossRef]

Schellenberg, F.

F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
[CrossRef]

Shao, D.

D. Shao and S. Chen, “Surface plasmon assisted contact scheme nanoscale photolithography using an UV lamp,” J. Vac. Sci. Technol. B 26, 227–231 (2008).
[CrossRef]

Simpson, R. A.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[CrossRef]

Sun, L.

F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
[CrossRef]

Thomas, E. M.

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99, 1806–1815 (2011).
[CrossRef]

Viswanathan, N. S.

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[CrossRef]

Yablonovitch, E.

J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[CrossRef]

Yang, J. J.

J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microwave Theor. Tech. 40, 595–600 (1992).
[CrossRef]

Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theor. Tech. 39, 588–592 (1991).
[CrossRef]

Yeung, M. S.

M. S. Yeung, “Three-dimensional mask transmission simulation using a single integral equation method,” Proc. SPIE 3334, 704–713 (1998).
[CrossRef]

ACM Trans. Math. Softw.

A. Corana, M. Marchesi, C. Martini, and S. Ridella, “Minimizing multimodal functions of continuous variables with the ’simulated annealing’ algorithm,” ACM Trans. Math. Softw. 13, 262–280 (1987).
[CrossRef]

IEEE Trans. Antennas Propag.

M. I. Aksun and G. Dural, “Clarification of issues on the closed-form Green’s functions in stratified media,” IEEE Trans. Antennas Propag. 53, 3644–3653 (2005).
[CrossRef]

Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag. 37, 229–234 (1989).
[CrossRef]

A. Grbic and R. Merlin, “Near-field focusing plates and their design,” IEEE Trans. Antennas Propag. 56, 3159–3165 (2008).
[CrossRef]

IEEE Trans. Electron Devices

M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices ED-29, 1828–1836 (1982).
[CrossRef]

IEEE Trans. Image Process.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography—an inverse imaging problem,” IEEE Trans. Image Process. 16, 774–788 (2007).
[CrossRef]

IEEE Trans. Microwave Theor. Tech.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microwave Theor. Tech. 58, 602–613 (2010).
[CrossRef]

Y. L. Chow, J. J. Yang, D. G. Fang, and G. E. Howard, “A closed form spatial Green’s function for the thick microstrip substrate,” IEEE Trans. Microwave Theor. Tech. 39, 588–592 (1991).
[CrossRef]

J. J. Yang, Y. L. Chow, G. E. Howard, and D. G. Fang, “Complex images of an electric dipole in homogenous and layered dielectrics between two ground planes,” IEEE Trans. Microwave Theor. Tech. 40, 595–600 (1992).
[CrossRef]

J. Lightwave Technol.

J. Microlith. Microfab. Microsys.

K. Adam and A. R. Neureuther, “Domain decomposition methods for the rapid electromagnetic simulation of photomask scattering,” J. Microlith. Microfab. Microsys. 1, 253 (2002).
[CrossRef]

J. Vac. Sci. Technol. B

D. Shao and S. Chen, “Surface plasmon assisted contact scheme nanoscale photolithography using an UV lamp,” J. Vac. Sci. Technol. B 26, 227–231 (2008).
[CrossRef]

Microelectron. Eng.

F. Schellenberg, K. Adam, L. Sun, J. Matteo, and L. Hesselink, “Polarization effects in plasmonic masks,” Microelectron. Eng. 83, 919–922 (2006).
[CrossRef]

Proc. IEEE

A. Grbic, R. Merlin, E. M. Thomas, and M. F. Imani, “Near-field plates: metamaterial surfaces/arrays for subwavelength focusing and probing,” Proc. IEEE 99, 1806–1815 (2011).
[CrossRef]

Proc. SPIE

M. S. Yeung, “Three-dimensional mask transmission simulation using a single integral equation method,” Proc. SPIE 3334, 704–713 (1998).
[CrossRef]

J. T. Azpiroz, P. Burchard, and E. Yablonovitch, “Boundary layer model to account for thick mask effects in photolithography,” Proc. SPIE 5040, 1611–1619 (2003).
[CrossRef]

Science

R. Merlin, “Radiationless electromagnetic interference: evanescent-field lenses and perfect focusing,” Science 317, 927–929 (2007).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

E. D. Palik, ed., Handbook of Optical Constants of Solids 11 (Academic, 1991), pp. 374–385.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

T. V. Pistor, “Electromagnetic simulation and modeling with applications in lithography.” Ph.D. thesis (University of California at Berkeley, 2001).

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Figures (9)

Fig. 1.
Fig. 1.

Geometry of the problem.

Fig. 2.
Fig. 2.

Approximation paths in different planes.

Fig. 3.
Fig. 3.

Complex images Green’s functions versus numerical integration; d1=0, d2=80nm, n1=1.563, n2=0.84j1.65, n3=1, and λ=193nm; source is located at z=λ, x=0, and (a) z=0, (b) z=80nm; source at z=40nm, x=0, and (c) z=0, (d) z=80nm.

Fig. 4.
Fig. 4.

The structure of two lines with equal widths W placed at a distance S.

Fig. 5.
Fig. 5.

Instantaneous electric field in etched areas for S=2λ, (a) our method and (b) COMSOL, and for S=λ/2, (c) our method and (d) COMSOL.

Fig. 6.
Fig. 6.

Geometry of the first pattern consisting of three lines.

Fig. 7.
Fig. 7.

Image intensities of the first pattern (a) below the mask at z=d2λ/5 and (b) in the image plane, both the aerial and final images.

Fig. 8.
Fig. 8.

Geometry of the second pattern consisting of five lines.

Fig. 9.
Fig. 9.

Image intensities of the second pattern (a) below the mask at z=d2λ/5 and (b) in the image plane, both the aerial and final images.

Tables (2)

Tables Icon

Table 1. Parameters Used in Approximating the Green’s Functions

Tables Icon

Table 2. Parameters of the Simulated Annealing Algorithma

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

Ey=jωAy+Ey,inc,B=×A,
2Ay+kn2Ay=μJeffn,kn2=ω2μϵ0ϵrn,n=1,2,3,
Jeffn=ω2ϵ0(ϵrϵrn)Ay,
Eym=jωμJeffn(x,z)Gmn(x,z;x,z)dxdz+Ey,inc,
Gmn(x,z,x,z)=12π+G˜mn(kx,z;z)2jkznexp(jkx(xx))dkx,
kzn=kn2kx2,n=1,2,3,
G˜11=exp(jkz1|zz|)+Aexp(jkz1(2d1+z)),
G˜21=Bexp(jkz2(d1+z))+Cexp(jkz2(d1+2d2+z)),
G˜31=Dexp(jkz1(d1+z)),
G˜12=Bexp(jkz2(d1+z))+Cexp(jkz2(d1+2d2+z)),
G˜22=exp(jkz2|zz|)+Eexp(jkz2(2d1+z+z))+F(exp(jkz2(2d12d2+zz))+exp(jkz2(2d1+2d2+zz)))+Hexp(jkz2(2d2+z+z)),
G˜32=Iexp(jkz3(d2+z)),
A=(kz1kz2)(kz2+kz3)+(kz1+kz2)(kz2kz3)exp(2jkz2(d2d1))(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
B=2kz2(kz2+kz3)exp(jkz1(d1+z))(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
C=2kz2(kz2kz3)exp(jkz1(d1+z))(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
D=4kz2kz3exp(jkz3(d2+z))exp(jkz2(d2d1))(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
E=(kz2+kz3)(kz2kz1)(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
F=(kz2kz3)(kz2kz1)(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
H=(kz2kz3)(kz2+kz1)(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
I=2kz3((kz1+kz2)exp(jkz2(d2+z))+(kz2kz1)exp(jkz2(2d1+d2z)))(kz1+kz2)(kz2+kz3)+(kz1kz2)(kz2kz3)exp(2jkz2(d2d1)),
kzn=k3[jt+knk3(1tT)],0tT,n=1,2,3,
F˜(kzi)|Cn=1NF˜anF˜exp(kzibnF˜),i=1,2,3,
12π+exp(jkz|zz|)2jkzexp(jkz(xx))dkx=14jH02(k|rr|),
Ey(x,z)=m=1MαmPm(x,z).
αn=Ey,inc(xn,zn)+m=1MαnZnm,n=1toM,
Znm=μω2ϵ0Pm(x,z)(ϵrϵr(x,z))G(xn,zn;x,z)dxdz,
[IZ][α]=[Zmask][α]=[Ey,inc],
[Zmask]=[ZBackground][Pattern],
Ey|below the mask=Ey,inc|below the mask+μω2ϵ0etched areas(ϵrϵr(x,z))E(x,z)G(x,zbelow the mask;x,z)dxdz.
Ey,scattered|below the mask=l=1LβlPl(x,zbelow the mask),
[Ey|below the mask]=[Ey,inc|below the mask]+[Zbelow the mask][α],
|kx|2πNA(1+σ)λR=kc
Ey,scattered|below the mask=jωμJequivalent(x,zbelow the mask)Gfree-space(x,z;x,z)dx,
[ZJtoE][Jequivalent]=[Ey,scattered|below the mask],
ZJtoE,mn=ωμ4H02(k3|xnxm|).
H02(k3|rnrm|)=1π+exp(jkx(xnxm))kz3exp(jkz3|znzm|)dkx
kout,x=Rkin,x,kout,z=k32kout,x2.
|Ey,entrance pupil(kin,x)|2rdθ=|Ey,exit pupil(kout,x)|2ρdθ,
Ey,line source image(xn,zn;xm,zm)=ωμ4πkckcf(kx)(exp(jRkx(xnxm))kin,z)exp((jkout,z|znzm|))dkx,
f(kx)=(kout,zRkin,z)1/2
[Zlens][ZJtoE]1[Ey,scattered|below the mask]+[Ey,inc|image plane]κ=[Zlens][ZJtoE]1[Zbelow the mask][Zmask]1([Ey,inc|etched areas])+[Ey,inc|image plane]κ=[Ey,aerial image],
U(I(x)tr)={0I(x)tr1I(x)>tr,

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