Abstract

A simulation tool based on the finite-difference time-domain (FDTD) technique is developed to model the electromagnetic interaction of a focused optical Gaussian beam in two dimensions incident on a simple model of a corrugated dielectric surface plated with a thin film of realistic metal. The technique is a hybrid approach that combines an intensive numerical method near the surface of the grating, which takes into account the optical properties of metals, with a free-space transform to obtain the radiated fields. A description of this technique is presented along with numerical examples comparing gratings made with realistic and perfect conductors. In particular, a demonstration is given of an obliquely incident beam focused on a uniform grating and a normally incident beam focused on a nonuniform grating. The gratings in these two cases are coated with a negative-permittivity thin film, and the scattered radiation patterns for these structures are studied. Both TE and TM polarizations are investigated. Using this hybrid FDTD technique results in a complete and accurate simulation of the total electromagnetic field in the near field as well as in the far field of the grating. It is shown that there are significant differences in the performances of the realistic metal and the perfect metal gratings.

© 1995 Optical Society of America

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References

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  1. E. Kriezis, P. Pandelakis, A. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
    [Crossref]
  2. T. Park, H. Eom, K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905–911 (1993).
    [Crossref]
  3. J. Aguilar, E. Méndez, “Imaging optically thick objects in scanning microscopy: perfectly conducting surfaces,” J. Opt. Soc. Am. A 11, 155–167 (1994).
    [Crossref]
  4. J. Kann, T. Milster, F. Froehlich, R. Ziolkowski, J. Judkins, “Near-field optical detection of asperities in dielectric surfaces,” J. Opt. Soc. Am. A 12, 501–512 (1995).
    [Crossref]
  5. M. J. Barth, R. R. McLeod, R. W. Ziolkowski, “A near and far-field projection algorithm for finite-difference time-domain codes,” J. Electromagn. Waves Appl. 6, 5–18 (1992).
  6. J. G. Maloney, G. S. Smith, W. R. Scott, “Accurate computation of the radiation from simple antennas using the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 38, 1059–1068 (1990).
    [Crossref]
  7. K. S. Yee, D. Ingham, K. Shlager, “Time-domain extrapolation to the far field based on FDTD calculations,” IEEE Trans. Antennas Propag. 39, 410–413 (1991).
    [Crossref]
  8. R. Luebbers, K. S. Kunz, M. Schneider, F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propag. 39, 429–433 (1991).
    [Crossref]
  9. R. Luebbers, D. Ryan, J. Beggs, “A two-dimensional time-domain near-zone to far-zone transformation,” IEEE Trans. Antennas Propag. 40, 848–851 (1992).
    [Crossref]
  10. J. Marvin, CRC Handbook of Laser Science and Technology (CRC, Cleveland, Ohio, 1986), Vol. 3, pp. 193–196.
  11. R. Ziolkowski, J. Judkins, “Optical nonlinear wakefield vortices: results from full-wave vector Maxwell equation simulations in two spatial dimensions and time,” in Integrated Photonics Research, Post-Deadline Papers, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper Pd14, pp. 50–51.
  12. T. Kashiwa, I. Fukal, “A treatment by the FD-TD method of the dispersive characteristics associated with electronic polarization,” Microwave Opt. Technol. Lett. 3, 1326–1328 (1990).
    [Crossref]
  13. R. M. Joseph, S. C. Hagness, A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
    [Crossref] [PubMed]
  14. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
    [Crossref]
  15. R. W. Ziolkowski, J. B. Judkins, “NL-FDTD modeling of linear and nonlinear corrugated waveguides,” J. Opt. Soc. Am. B 11, 1565–1575 (1994).
    [Crossref]
  16. K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
    [Crossref]
  17. G. Mur, “Absorbing boundary conditions for the finite-difference approximation to the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
    [Crossref]
  18. J. Kong, Electromagnetic Waves (Wiley, New York, 1986), pp. 495–499.

1995 (1)

1994 (3)

1993 (1)

1992 (2)

M. J. Barth, R. R. McLeod, R. W. Ziolkowski, “A near and far-field projection algorithm for finite-difference time-domain codes,” J. Electromagn. Waves Appl. 6, 5–18 (1992).

R. Luebbers, D. Ryan, J. Beggs, “A two-dimensional time-domain near-zone to far-zone transformation,” IEEE Trans. Antennas Propag. 40, 848–851 (1992).
[Crossref]

1991 (3)

K. S. Yee, D. Ingham, K. Shlager, “Time-domain extrapolation to the far field based on FDTD calculations,” IEEE Trans. Antennas Propag. 39, 410–413 (1991).
[Crossref]

R. Luebbers, K. S. Kunz, M. Schneider, F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propag. 39, 429–433 (1991).
[Crossref]

R. M. Joseph, S. C. Hagness, A. Taflove, “Direct time integration of Maxwell’s equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses,” Opt. Lett. 16, 1412–1414 (1991).
[Crossref] [PubMed]

1990 (3)

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[Crossref]

T. Kashiwa, I. Fukal, “A treatment by the FD-TD method of the dispersive characteristics associated with electronic polarization,” Microwave Opt. Technol. Lett. 3, 1326–1328 (1990).
[Crossref]

J. G. Maloney, G. S. Smith, W. R. Scott, “Accurate computation of the radiation from simple antennas using the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 38, 1059–1068 (1990).
[Crossref]

1982 (1)

K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[Crossref]

1981 (1)

G. Mur, “Absorbing boundary conditions for the finite-difference approximation to the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[Crossref]

Aguilar, J.

Barth, M. J.

M. J. Barth, R. R. McLeod, R. W. Ziolkowski, “A near and far-field projection algorithm for finite-difference time-domain codes,” J. Electromagn. Waves Appl. 6, 5–18 (1992).

Beggs, J.

R. Luebbers, D. Ryan, J. Beggs, “A two-dimensional time-domain near-zone to far-zone transformation,” IEEE Trans. Antennas Propag. 40, 848–851 (1992).
[Crossref]

Eom, H.

Froehlich, F.

Fukal, I.

T. Kashiwa, I. Fukal, “A treatment by the FD-TD method of the dispersive characteristics associated with electronic polarization,” Microwave Opt. Technol. Lett. 3, 1326–1328 (1990).
[Crossref]

Hagness, S. C.

Hunsberger, F.

R. Luebbers, K. S. Kunz, M. Schneider, F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propag. 39, 429–433 (1991).
[Crossref]

Hunsberger, F. P.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[Crossref]

Ingham, D.

K. S. Yee, D. Ingham, K. Shlager, “Time-domain extrapolation to the far field based on FDTD calculations,” IEEE Trans. Antennas Propag. 39, 410–413 (1991).
[Crossref]

Joseph, R. M.

Judkins, J.

J. Kann, T. Milster, F. Froehlich, R. Ziolkowski, J. Judkins, “Near-field optical detection of asperities in dielectric surfaces,” J. Opt. Soc. Am. A 12, 501–512 (1995).
[Crossref]

R. Ziolkowski, J. Judkins, “Optical nonlinear wakefield vortices: results from full-wave vector Maxwell equation simulations in two spatial dimensions and time,” in Integrated Photonics Research, Post-Deadline Papers, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper Pd14, pp. 50–51.

Judkins, J. B.

Kann, J.

Kashiwa, T.

T. Kashiwa, I. Fukal, “A treatment by the FD-TD method of the dispersive characteristics associated with electronic polarization,” Microwave Opt. Technol. Lett. 3, 1326–1328 (1990).
[Crossref]

Kong, J.

J. Kong, Electromagnetic Waves (Wiley, New York, 1986), pp. 495–499.

Kriezis, E.

Kunz, K. S.

R. Luebbers, K. S. Kunz, M. Schneider, F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propag. 39, 429–433 (1991).
[Crossref]

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[Crossref]

Luebbers, R.

R. Luebbers, D. Ryan, J. Beggs, “A two-dimensional time-domain near-zone to far-zone transformation,” IEEE Trans. Antennas Propag. 40, 848–851 (1992).
[Crossref]

R. Luebbers, K. S. Kunz, M. Schneider, F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propag. 39, 429–433 (1991).
[Crossref]

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[Crossref]

Maloney, J. G.

J. G. Maloney, G. S. Smith, W. R. Scott, “Accurate computation of the radiation from simple antennas using the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 38, 1059–1068 (1990).
[Crossref]

Marvin, J.

J. Marvin, CRC Handbook of Laser Science and Technology (CRC, Cleveland, Ohio, 1986), Vol. 3, pp. 193–196.

McLeod, R. R.

M. J. Barth, R. R. McLeod, R. W. Ziolkowski, “A near and far-field projection algorithm for finite-difference time-domain codes,” J. Electromagn. Waves Appl. 6, 5–18 (1992).

Méndez, E.

Milster, T.

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite-difference approximation to the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[Crossref]

Pandelakis, P.

Papagiannakis, A.

Park, T.

Ryan, D.

R. Luebbers, D. Ryan, J. Beggs, “A two-dimensional time-domain near-zone to far-zone transformation,” IEEE Trans. Antennas Propag. 40, 848–851 (1992).
[Crossref]

Schneider, M.

R. Luebbers, K. S. Kunz, M. Schneider, F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propag. 39, 429–433 (1991).
[Crossref]

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[Crossref]

Scott, W. R.

J. G. Maloney, G. S. Smith, W. R. Scott, “Accurate computation of the radiation from simple antennas using the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 38, 1059–1068 (1990).
[Crossref]

Shlager, K.

K. S. Yee, D. Ingham, K. Shlager, “Time-domain extrapolation to the far field based on FDTD calculations,” IEEE Trans. Antennas Propag. 39, 410–413 (1991).
[Crossref]

Smith, G. S.

J. G. Maloney, G. S. Smith, W. R. Scott, “Accurate computation of the radiation from simple antennas using the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 38, 1059–1068 (1990).
[Crossref]

Standler, R. B.

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[Crossref]

Taflove, A.

Umashankar, K. R.

K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[Crossref]

Yee, K. S.

K. S. Yee, D. Ingham, K. Shlager, “Time-domain extrapolation to the far field based on FDTD calculations,” IEEE Trans. Antennas Propag. 39, 410–413 (1991).
[Crossref]

Yoshitomi, K.

Ziolkowski, R.

J. Kann, T. Milster, F. Froehlich, R. Ziolkowski, J. Judkins, “Near-field optical detection of asperities in dielectric surfaces,” J. Opt. Soc. Am. A 12, 501–512 (1995).
[Crossref]

R. Ziolkowski, J. Judkins, “Optical nonlinear wakefield vortices: results from full-wave vector Maxwell equation simulations in two spatial dimensions and time,” in Integrated Photonics Research, Post-Deadline Papers, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper Pd14, pp. 50–51.

Ziolkowski, R. W.

R. W. Ziolkowski, J. B. Judkins, “NL-FDTD modeling of linear and nonlinear corrugated waveguides,” J. Opt. Soc. Am. B 11, 1565–1575 (1994).
[Crossref]

M. J. Barth, R. R. McLeod, R. W. Ziolkowski, “A near and far-field projection algorithm for finite-difference time-domain codes,” J. Electromagn. Waves Appl. 6, 5–18 (1992).

IEEE Trans. Antennas Propag. (4)

J. G. Maloney, G. S. Smith, W. R. Scott, “Accurate computation of the radiation from simple antennas using the finite-difference time-domain method,” IEEE Trans. Antennas Propag. 38, 1059–1068 (1990).
[Crossref]

K. S. Yee, D. Ingham, K. Shlager, “Time-domain extrapolation to the far field based on FDTD calculations,” IEEE Trans. Antennas Propag. 39, 410–413 (1991).
[Crossref]

R. Luebbers, K. S. Kunz, M. Schneider, F. Hunsberger, “A finite-difference time-domain near zone to far zone transformation,” IEEE Trans. Antennas Propag. 39, 429–433 (1991).
[Crossref]

R. Luebbers, D. Ryan, J. Beggs, “A two-dimensional time-domain near-zone to far-zone transformation,” IEEE Trans. Antennas Propag. 40, 848–851 (1992).
[Crossref]

IEEE Trans. Electromagn. Compat. (3)

R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat. 32, 222–227 (1990).
[Crossref]

K. R. Umashankar, A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat. EMC-24, 397–405 (1982).
[Crossref]

G. Mur, “Absorbing boundary conditions for the finite-difference approximation to the time-domain electromagnetic-field equations,” IEEE Trans. Electromagn. Compat. EMC-23, 377–382 (1981).
[Crossref]

J. Electromagn. Waves Appl. (1)

M. J. Barth, R. R. McLeod, R. W. Ziolkowski, “A near and far-field projection algorithm for finite-difference time-domain codes,” J. Electromagn. Waves Appl. 6, 5–18 (1992).

J. Opt. Soc. Am. A (4)

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

Microwave Opt. Technol. Lett. (1)

T. Kashiwa, I. Fukal, “A treatment by the FD-TD method of the dispersive characteristics associated with electronic polarization,” Microwave Opt. Technol. Lett. 3, 1326–1328 (1990).
[Crossref]

Opt. Lett. (1)

Other (3)

J. Kong, Electromagnetic Waves (Wiley, New York, 1986), pp. 495–499.

J. Marvin, CRC Handbook of Laser Science and Technology (CRC, Cleveland, Ohio, 1986), Vol. 3, pp. 193–196.

R. Ziolkowski, J. Judkins, “Optical nonlinear wakefield vortices: results from full-wave vector Maxwell equation simulations in two spatial dimensions and time,” in Integrated Photonics Research, Post-Deadline Papers, Vol. 10 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper Pd14, pp. 50–51.

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Figures (8)

Fig. 1
Fig. 1

FDTD stencil for TM fields. Nodes for the y field components are located at the cell center, and the x and z field components are located at the edges of the FDTD cell. Vectors PL and J are considered volume quantities, and hence their nodes are located at the centers of the cells also. The fields E and PL are staggered in time from H and J.

Fig. 2
Fig. 2

Near and far zones are treated separately. In the near zone, Maxwell’s equations and the Lorentz equation are solved on a rectangular-grid FDTD mesh. The mesh is truncated by a second-order Mur condition to reduce unwanted reflections. The mesh is subdivided into two regions: a total-field region encompassing the total field and the geometry, and a scattered-field region in which the incident field is absent.

Fig. 3
Fig. 3

Problem geometry, a uniform rectangular toothed grating. The grating is constructed from a homogeneous dielectric covered by a thin layer of conductor. The grating period is Λ = 1.17 μm, and the thickness of the layer is tf = 48.75 nm.

Fig. 4
Fig. 4

Beam incident on the corrugation at angle θi and diffracted into multiple reflection angles. For λ = 0.78 μm and θi = 19.5° two scattered beams are present: the zeroth order, θR = −19.5°, and the first order, θB = 19.5°.

Fig. 5
Fig. 5

In the far zone a transform technique is used to calculate the scattered fields, based on the near-zone results. Two idealized power detectors, D1 and D2, are placed in the far zone, to the left and to the right, respectively, of the center of the grating.

Fig. 6
Fig. 6

Power in the reflected beam (m = 0 order) and in the diffracted beam (m = 1 order) is measured in the far zone by the idealized detectors D1 and D2, respectively, as a′ varies from 0.1 to 1.0. These signals are generated by integration of the radiation pattern derived from the near field by Eq. (6a). The differential signal S is calculated and plotted versus a′ for a grating with a gold film and a P.E.C. grating. (a) TE-polarized incident beam, (b) TM-polarized incident beam. Additionally, in the TM case (b) S is calculated for a wide (w0 = 7.02 μm) beam, and the results are compared with results derived from a mode-matching formulation for a TM plane wave and an infinite P.E.C. grating.

Fig. 7
Fig. 7

Power in the diffracted beam (m = 1 order) is measured by the idealized detector D2 in the far zone for a TE-polarized incident beam as h varies from 0 to 0.50 μm. (a) The power at D2 is plotted versus h for four gratings with different conducting films and for a P.E.C. grating. (b) The remaining power, detected at D1, is plotted versus h for the same five gratings.

Fig. 8
Fig. 8

On-axis intensities are produced with the FDTD simulation and the near-field transform [Eq. (14)] for a w0 = 4.68 μm, normally incident Gaussian beam reflecting off a Fresnel grating. The horizontal axis is the distance along the beam axis. Results for the four metallic films and a P.E.C. grating are shown.

Equations (28)

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n L ( ω ) = R ( ω ) = ( χ 0 ω 0 2 ω 0 2 - ω 2 + j Γ ω + 0 ) 1 / 2 .
ω 0 = ω { 1 - χ 0 ( r - / 0 ) / [ ( r - / 0 ) 2 + i 2 ] } 1 / 2 ,
Γ = ω χ 0 i ( r - / 0 ) 2 + i 2 - χ 0 ( r - / 0 ) .
f ( x , t ) = A ( x ) cos [ ω t + ϕ ( x ) ] = ½ [ A w ( x ) exp ( j ω t ) + A w * ( x ) exp ( - j ω t ) ] .
A w ( x ) = 2 T 0 T f ( x , t ) exp ( - j ω t ) d t .
E y scat ( R , θ ) 1 2 λ R sampling plane d x exp [ j k sin ( θ ) x ] × [ E y ( x , z t ) cos ( θ ) + η H x ( x , z t ) ] ,
E y scat ( R , θ ) = δ 2 λ R FFT [ E y ( x , z t ) cos ( θ ) + η H x ( x , z t ) ] ,
θ = sin - 1 ( n λ / N δ )
E y ( x , z = z t + d ) = 1 2 π - E ˜ ( k x , z t ) × exp [ - j ( k x x + k z d ) ] d k x ,
E y ( n ) ( x = n δ , z = z t + d ) = 1 N FFT - 1 { FFT [ E 0 ( x , z t ) ] T z ( d ) } ,
t H = - 1 μ 0 × E ,
t E = 1 × H - 1 t P L ,
2 t 2 P L + Γ t P L + ω 0 2 P L = 0 χ 0 ω 0 2 E ,
t J + Γ J = ω 0 2 ( 0 χ 0 E - P L ) ,
t P L = J .
E 0 ( x ) = [ ( 8 π ) 1 / 2 η w 0 P 0 ] 1 / 2 exp [ - ( x w 0 ) 2 ] .
H x inc ( x , z s ) = 1 η Im { Ψ H x ( x , z s ) exp [ j ω ( i Δ t ) ] } ,
E y inc ( x , z s ) = Im { Ψ E y ( x , z s ) exp [ j ω ( i + 1 / 2 ) Δ t ] } ,
Ψ E y ( x , z = z s ) = 1 2 π - E ˜ 0 ( k x , 0 ) × exp [ - j ( k x x + k z z ) ] d k x ,
Ψ H x ( x , z = z s ) = 1 2 π k - E ˜ 0 ( k x , 0 ) [ k z cos ( θ i ) - k x sin ( θ i ) ] exp [ - j ( k x x + k z z ) ] d k x ,
Ψ E y ( n ) ( x = n δ , z = z s ) = 1 N FFT - 1 { FFT [ E 0 ( x , 0 ) ] exp ( - j k z z ) } ,
Ψ H x ( n ) ( x = n δ , z = z s ) = 1 N k FFT - 1 { FFT [ E 0 ( x , 0 ) ] exp ( - j k z z ) × [ k z cos ( θ i ) - k x sin ( θ i ) ] } ,
t H = - 1 μ 0 × E - 1 μ 0 M source ,
t E = 1 × H - 1 t P L - 1 J source ,
H x scat ( x , z s ) = H x total ( x , z s ) - Δ t δ μ 0 E y inc ( x , z s ) ,
E y total ( x , z s ) = E y scat ( x , z s ) + Δ t δ H x inc ( x , z s ) .
sin θ m = sin θ i - m λ n Λ ,
h = h 0 [ 1 - ( x / 2 w 0 ) 2 ] ,

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