Abstract

We consider the design problem of creating coatings that are either highly reflective or highly transparent. The goal is to create an optical element, consisting of planar dielectric layers, that reflects (or transmits) energy over a given range of wavelengths and angles of incidence. The approach that we take is to formulate the problem as a minimax optimization problem. We demonstrate that the approach can be effective in producing coatings of a few layers with desirable properties.

© 2004 Optical Society of America

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References

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  1. W. W. Hager, R. Rostamian, “Optimal coatings, bang–bang controls, and gradient techniques,” Opt. Control Appl. Methods 8, 1–20 (1987).
    [CrossRef]
  2. H. Konstanty, F. Santosa, “Optimal design of minimally reflective coatings,” Wave Motion 21, 291–309 (1995).
    [CrossRef]
  3. W. W. Hager, R. Rostamian, D. Wang, “The wave annihilation technique and the design of nonreflective coatings,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 1388–1424 (2000).
    [CrossRef]
  4. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
    [CrossRef] [PubMed]
  5. D. Felbacq, B. Guizal, F. Zolla, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604–4607 (1998).
    [CrossRef]
  6. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1990).
  7. R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).
  8. C. Charalambous, A. R. Conn, “An efficient method to solve the minimax problem directly,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 162–187 (1978).
    [CrossRef]
  9. A. R. Conn, “An efficient second order method to solve the (constrained) minimax problem,” (Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1979).
  10. Y. Li, “An efficient algorithm for nonlinear minimax problems,” Ph.D. dissertation (Computer Science Department, University of Waterloo, Waterloo, Ontario, Canada, 1988).
  11. A. R. Conn, Y. Li, “A structure-exploiting algorithm for nonlinear minimax problems,” SIAM J. Optim. 2, 242–263 (1992).
    [CrossRef]
  12. A. R. Conn, N. I. M. Gould, D. Orban, Ph. L. Toint, “A primal-dual trust-region algorithm for non-convex nonlinear programming,” Math. Program. 87, 215–249 (2000).
    [CrossRef]
  13. G. Di Pillo, L. Grippo, S. Lucidi, “A smooth method for the finite minimax problem,” Math. Program. 60, 187–214 (1993).
    [CrossRef]
  14. W. W. Hager, D. L. Presler, “Dual techniques for minimax,” SIAM J. Control Optim. 25, 660–685 (1987).
    [CrossRef]
  15. T. Plantenga, “A trust region method for nonlinear programming based on primal interior-point techniques,” SIAM J. Sci. Comput. (USA) 20, 282–305 (1998).
    [CrossRef]
  16. G. D. Erdmann, “A new minimax algorithm and its application to optics problems,” Ph.D. dissertation (University of Minnesota, Minneapolis, Minn., 2003).
  17. J. Nocedal, S. J. Wright, Numerical Optimization (Springer, New York, 1999).
  18. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
    [CrossRef]

2002 (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
[CrossRef]

2000 (2)

W. W. Hager, R. Rostamian, D. Wang, “The wave annihilation technique and the design of nonreflective coatings,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 1388–1424 (2000).
[CrossRef]

A. R. Conn, N. I. M. Gould, D. Orban, Ph. L. Toint, “A primal-dual trust-region algorithm for non-convex nonlinear programming,” Math. Program. 87, 215–249 (2000).
[CrossRef]

1998 (3)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

D. Felbacq, B. Guizal, F. Zolla, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604–4607 (1998).
[CrossRef]

T. Plantenga, “A trust region method for nonlinear programming based on primal interior-point techniques,” SIAM J. Sci. Comput. (USA) 20, 282–305 (1998).
[CrossRef]

1995 (1)

H. Konstanty, F. Santosa, “Optimal design of minimally reflective coatings,” Wave Motion 21, 291–309 (1995).
[CrossRef]

1993 (1)

G. Di Pillo, L. Grippo, S. Lucidi, “A smooth method for the finite minimax problem,” Math. Program. 60, 187–214 (1993).
[CrossRef]

1992 (1)

A. R. Conn, Y. Li, “A structure-exploiting algorithm for nonlinear minimax problems,” SIAM J. Optim. 2, 242–263 (1992).
[CrossRef]

1987 (2)

W. W. Hager, D. L. Presler, “Dual techniques for minimax,” SIAM J. Control Optim. 25, 660–685 (1987).
[CrossRef]

W. W. Hager, R. Rostamian, “Optimal coatings, bang–bang controls, and gradient techniques,” Opt. Control Appl. Methods 8, 1–20 (1987).
[CrossRef]

1978 (1)

C. Charalambous, A. R. Conn, “An efficient method to solve the minimax problem directly,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 162–187 (1978).
[CrossRef]

Benoit, G.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
[CrossRef]

Charalambous, C.

C. Charalambous, A. R. Conn, “An efficient method to solve the minimax problem directly,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 162–187 (1978).
[CrossRef]

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1990).

Conn, A. R.

A. R. Conn, N. I. M. Gould, D. Orban, Ph. L. Toint, “A primal-dual trust-region algorithm for non-convex nonlinear programming,” Math. Program. 87, 215–249 (2000).
[CrossRef]

A. R. Conn, Y. Li, “A structure-exploiting algorithm for nonlinear minimax problems,” SIAM J. Optim. 2, 242–263 (1992).
[CrossRef]

C. Charalambous, A. R. Conn, “An efficient method to solve the minimax problem directly,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 162–187 (1978).
[CrossRef]

A. R. Conn, “An efficient second order method to solve the (constrained) minimax problem,” (Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1979).

Di Pillo, G.

G. Di Pillo, L. Grippo, S. Lucidi, “A smooth method for the finite minimax problem,” Math. Program. 60, 187–214 (1993).
[CrossRef]

Erdmann, G. D.

G. D. Erdmann, “A new minimax algorithm and its application to optics problems,” Ph.D. dissertation (University of Minnesota, Minneapolis, Minn., 2003).

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Felbacq, D.

D. Felbacq, B. Guizal, F. Zolla, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604–4607 (1998).
[CrossRef]

Fink, Y.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Fletcher, R.

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

Gould, N. I. M.

A. R. Conn, N. I. M. Gould, D. Orban, Ph. L. Toint, “A primal-dual trust-region algorithm for non-convex nonlinear programming,” Math. Program. 87, 215–249 (2000).
[CrossRef]

Grippo, L.

G. Di Pillo, L. Grippo, S. Lucidi, “A smooth method for the finite minimax problem,” Math. Program. 60, 187–214 (1993).
[CrossRef]

Guizal, B.

D. Felbacq, B. Guizal, F. Zolla, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604–4607 (1998).
[CrossRef]

Hager, W. W.

W. W. Hager, R. Rostamian, D. Wang, “The wave annihilation technique and the design of nonreflective coatings,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 1388–1424 (2000).
[CrossRef]

W. W. Hager, R. Rostamian, “Optimal coatings, bang–bang controls, and gradient techniques,” Opt. Control Appl. Methods 8, 1–20 (1987).
[CrossRef]

W. W. Hager, D. L. Presler, “Dual techniques for minimax,” SIAM J. Control Optim. 25, 660–685 (1987).
[CrossRef]

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
[CrossRef]

Joannopoulos, J. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Konstanty, H.

H. Konstanty, F. Santosa, “Optimal design of minimally reflective coatings,” Wave Motion 21, 291–309 (1995).
[CrossRef]

Li, Y.

A. R. Conn, Y. Li, “A structure-exploiting algorithm for nonlinear minimax problems,” SIAM J. Optim. 2, 242–263 (1992).
[CrossRef]

Y. Li, “An efficient algorithm for nonlinear minimax problems,” Ph.D. dissertation (Computer Science Department, University of Waterloo, Waterloo, Ontario, Canada, 1988).

Lucidi, S.

G. Di Pillo, L. Grippo, S. Lucidi, “A smooth method for the finite minimax problem,” Math. Program. 60, 187–214 (1993).
[CrossRef]

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Nocedal, J.

J. Nocedal, S. J. Wright, Numerical Optimization (Springer, New York, 1999).

Orban, D.

A. R. Conn, N. I. M. Gould, D. Orban, Ph. L. Toint, “A primal-dual trust-region algorithm for non-convex nonlinear programming,” Math. Program. 87, 215–249 (2000).
[CrossRef]

Plantenga, T.

T. Plantenga, “A trust region method for nonlinear programming based on primal interior-point techniques,” SIAM J. Sci. Comput. (USA) 20, 282–305 (1998).
[CrossRef]

Presler, D. L.

W. W. Hager, D. L. Presler, “Dual techniques for minimax,” SIAM J. Control Optim. 25, 660–685 (1987).
[CrossRef]

Rostamian, R.

W. W. Hager, R. Rostamian, D. Wang, “The wave annihilation technique and the design of nonreflective coatings,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 1388–1424 (2000).
[CrossRef]

W. W. Hager, R. Rostamian, “Optimal coatings, bang–bang controls, and gradient techniques,” Opt. Control Appl. Methods 8, 1–20 (1987).
[CrossRef]

Santosa, F.

H. Konstanty, F. Santosa, “Optimal design of minimally reflective coatings,” Wave Motion 21, 291–309 (1995).
[CrossRef]

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
[CrossRef]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Toint, Ph. L.

A. R. Conn, N. I. M. Gould, D. Orban, Ph. L. Toint, “A primal-dual trust-region algorithm for non-convex nonlinear programming,” Math. Program. 87, 215–249 (2000).
[CrossRef]

Wang, D.

W. W. Hager, R. Rostamian, D. Wang, “The wave annihilation technique and the design of nonreflective coatings,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 1388–1424 (2000).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Wright, S. J.

J. Nocedal, S. J. Wright, Numerical Optimization (Springer, New York, 1999).

Zolla, F.

D. Felbacq, B. Guizal, F. Zolla, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604–4607 (1998).
[CrossRef]

J. Math. Phys. (1)

D. Felbacq, B. Guizal, F. Zolla, “Limit analysis of the diffraction of a plane wave by a one-dimensional periodic medium,” J. Math. Phys. 39, 4604–4607 (1998).
[CrossRef]

Math. Program. (2)

A. R. Conn, N. I. M. Gould, D. Orban, Ph. L. Toint, “A primal-dual trust-region algorithm for non-convex nonlinear programming,” Math. Program. 87, 215–249 (2000).
[CrossRef]

G. Di Pillo, L. Grippo, S. Lucidi, “A smooth method for the finite minimax problem,” Math. Program. 60, 187–214 (1993).
[CrossRef]

Nature (London) (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature (London) 420, 650–653 (2002).
[CrossRef]

Opt. Control Appl. Methods (1)

W. W. Hager, R. Rostamian, “Optimal coatings, bang–bang controls, and gradient techniques,” Opt. Control Appl. Methods 8, 1–20 (1987).
[CrossRef]

Science (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

W. W. Hager, R. Rostamian, D. Wang, “The wave annihilation technique and the design of nonreflective coatings,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 60, 1388–1424 (2000).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

C. Charalambous, A. R. Conn, “An efficient method to solve the minimax problem directly,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 15, 162–187 (1978).
[CrossRef]

SIAM J. Control Optim. (1)

W. W. Hager, D. L. Presler, “Dual techniques for minimax,” SIAM J. Control Optim. 25, 660–685 (1987).
[CrossRef]

SIAM J. Optim. (1)

A. R. Conn, Y. Li, “A structure-exploiting algorithm for nonlinear minimax problems,” SIAM J. Optim. 2, 242–263 (1992).
[CrossRef]

SIAM J. Sci. Comput. (USA) (1)

T. Plantenga, “A trust region method for nonlinear programming based on primal interior-point techniques,” SIAM J. Sci. Comput. (USA) 20, 282–305 (1998).
[CrossRef]

Wave Motion (1)

H. Konstanty, F. Santosa, “Optimal design of minimally reflective coatings,” Wave Motion 21, 291–309 (1995).
[CrossRef]

Other (6)

A. R. Conn, “An efficient second order method to solve the (constrained) minimax problem,” (Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1979).

Y. Li, “An efficient algorithm for nonlinear minimax problems,” Ph.D. dissertation (Computer Science Department, University of Waterloo, Waterloo, Ontario, Canada, 1988).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1990).

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).

G. D. Erdmann, “A new minimax algorithm and its application to optics problems,” Ph.D. dissertation (University of Minnesota, Minneapolis, Minn., 2003).

J. Nocedal, S. J. Wright, Numerical Optimization (Springer, New York, 1999).

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Figures (9)

Fig. 1
Fig. 1

Problem geometry.

Fig. 2
Fig. 2

Optimal design of an 18-layer transparent film. The initial design (top) has a maximum reflection of 0.0379 for the TE or TM mode over the wavelength range 9.8–14.6 μm and the angle range 0°–50°. The optimized design (bottom) has a maximum reflection of 0.00205.

Fig. 3
Fig. 3

Optimal design of a 27-layer nondissipative reflective film. Here the layer thickness is fixed, and the indices of the layers are optimized. The initial design (top) has a maximum transmission of 0.00526 for the TE or TM mode over the wavelength range 9.8–14.6 μm and the angle range 0°–85°. The optimized design (bottom) has a maximum transmission of 0.000443.

Fig. 4
Fig. 4

Optimal design of an 18-layer nondissipative reflective film. Here the indices of refraction are fixed, and the layer thicknesses are optimized. The initial design (top) has a maximum transmission of 0.00526 for the TE or TM mode over the wavelength range 9.8–14.6 μm and the angle range 0°–85°. The optimized design (bottom) has a maximum transmission of 0.000351. Note that the structure is no longer periodic and that, in the interior, the ratio of layer thicknesses has changed. Moreover, the first high-index layer is now somewhat thinner.

Fig. 5
Fig. 5

Optimal design of a 12-layer dissipative reflective film. Here the indices of refraction are fixed, and the layer thicknesses are optimized. The initial design (top) has a maximum transmission of 0.175 for the TE or TM mode over the wavelength range 9.8–14.6 μm and the angle range 0°–85°. The optimized design (bottom), which is no longer periodic, has a maximum transmission of 0.117. The improvement is less dramatic in comparison with that for the nondissipative cases.

Fig. 6
Fig. 6

Optimal design of a 27-layer nondissipative low-contrast reflective film. Here the layer thickness is fixed, and the indices of the layers are optimized. The initial design (top) has a maximum transmission of 0.0224 for the TE or TM mode over the wavelength range 11.8–12.6 μm and the angle range 0°–85°. The optimized design (bottom) has a maximum transmission of 0.00244.

Fig. 7
Fig. 7

Optimal design of an 18-layer non-dissipative low-contrast reflective film. Here the indices of refraction are fixed, and the layer thicknesses are optimized. The initial design (top) has a maximum transmission of 0.0224 for the TE or TM mode over the wavelength range 11.8–12.6 μm and the angle range 0°–85°. The optimized design (bottom) has a maximum transmission of 0.00313. Note that the structure is no longer periodic and that, in the interior, the ratio of layer thicknesses has changed. Moreover, the first high-index layer is now somewhat thinner.

Fig. 8
Fig. 8

Optimal design of an 18-layer dissipative low-contrast reflective film. Here the indices of refraction are fixed, and the layer thicknesses are optimized. The initial design (top) has a maximum transmission of 0.270 for the TE or TM mode over the wavelength range 11.8–12.6 μm and the angle range 0°–85°. The optimized design (bottom), which is no longer periodic, has a maximum transmission of 0.121. The improvement is less dramatic in comparison with that for the nondissipative cases.

Fig. 9
Fig. 9

Optimal design of a 27-layer nondissipative reflective film. Here the layer thickness is fixed, and the indices of the layers are optimized. The initial design (top) has a maximum transmission of 1.00 for the TE or TM mode over the wavelength range 9.8–14.6 μm and the angle range 0°–85° (at some angle and wavelength, we have near-perfect transmission). The optimized design (bottom) has a maximum transmission of 0.129.

Tables (4)

Tables Icon

Table 1 Transparent Film Results

Tables Icon

Table 2 High-Contrast Mirror Results

Tables Icon

Table 3 Low-Contrast Bandgap Mirror Results

Tables Icon

Table 4 Low-Contrast Nonbandgap Mirror Results

Equations (48)

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

ejy=Aj[exp(-iαjz)+R˜jexp(2iαjdj+iαjz)].
RjTE=αj-αj+1αj+αj+1,RjTM=j+1αj-jαj+1j+1αj+jαj+1.
R˜N=RN
R˜j=Rj+R˜j+1exp(2iαj+1hj)1+RjR˜j+1exp(2iαj+1hj)for0j<N
n=[n1, n2,, nN]
h=[h1, h2,, hN]
R(n, h, θ, ω)=|R˜0|2.
minnNmaxωΩ,θΘR(n, h=h0, θ, ω),
minnNmaxωΩ,θΘT(n, h=h0, θ, ω).
minhHmaxωΩ,θΘR(n=n0, h, θ, ω)
minhHmaxωΩ,θΘT(n=n0, h, θ, ω)
minnNωΩ,θΘR(n, h=h0, θ, ω).
fj(n, h)=R(n, h, θj, ωj)
fj(n, h)=T(n, h, θj, ωj)
minxχmax1jm fj(x).
min(z,x)R×χ zsubjecttozfj(x)
forallj{1, 2,, m}.
cj(z, x)=sgn(xi-li)|xi-li|ν.
cj(z, x)=sgn(ui-xi)|ui-xi|ν.
min(z,x)RN+1 zsubjecttocj(z, x)0
forallj{1, 2,,m+p}.
ϕ(z, x, μ)=z-μj=1m+plog[cj(z, x)],
min(z,x)RN ϕ(z, x, μ),
m(z+t, x+s)=ϕ(z, x, μ)+g, (t, s)+12(t, s), H(t, s).
min(t,x)Δ m(z+t, x+s).
ϕz=1-μj=1m1cj(z, x)=0,
ϕxk=-μj=1m+p1cj(z, x)cj(z, x)xk=0.
J˜(x)=c1/x1c1/x2c1/xNcm+p/x1cm+p/x2cm+p/xNT
yT=μc1, μc2,,μcm+p,
1-j=1myj=0,
J˜y=0.
yj>0forallj=1, 2,, m+p,
cjyj=μforallj=1, 2,, m+p.
cj>0forallj=1, 2,, m+p.
J(z, x)=c1/zc1/x1c1/xNcm+p/zcm+p/x1cm+p/xNT,
H=JC-1YJT-i=1m+pyi(z,x)(z,x)ci.
mintk,j,sk,j mk,j(zk,j+tk,j, xk,j+sk,j)subjectto(tk,j, sk,j)k,jΔk,j.
ϕ(zk,j+tk,j, xk,j+sk,j)<ϕ(zk,j, xk,j)
hRjTE=hRjTM=0,hhRjTE=hhRjTM=0.
hR˜N=0,hhR˜N=0.
R˜j=Rj+R˜j+1exp(2iαj+1hj)1+RjR˜j+1exp(2iαj+1hj)fjgj.
hfj=(2iαj+1R˜j+1ej+hR˜j+1)exp(2iαj+1hj),
hhfj={(2iαj+1)2R˜j+1ejejT+2iαj+1[hR˜j+1ejT+ej(hR˜j+1)T]+hhR˜j+1}exp(2iαj+1hj),
hgj=Rjhfj.
hR˜j=(1-Rj2)hfjgj2,
hhR˜j=1-Rj2gj3 [gjhhfj-2Rjhfj(hfj)T].
hR(n, h, θ, ω)=2 Re(R˜0¯hR˜0),
hhR(n, h, θ, ω)=2 Re[R˜0¯hhR˜0+hR˜0¯(hR˜0)T].

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