Abstract

The ponderomotive interaction of high-power laser beams with collisional plasma is modeled in the nonrelativistic regime and is simulated using the powerful finite-difference time-domain (FDTD) method for the first time in literature. The nonlinear and dissipative dielectric constant function of the collisional plasma is deduced that takes the ponderomotive effect into account and is implemented in the discrete framework of FDTD algorithms. Maclaurin series expansion approach is applied for implementing the obtained physical model and the time average of the square of light field is extracted by numerically evaluating an integral identity based on the composite trapezoidal rule for numerical integration. Two numerical examples corresponding to two different types of laser beams, Gaussian beam and vortex Laguerre-Gaussian beam, propagating in collisional plasma, are presented for specified laser and plasma parameters to verify the validity of the proposed FDTD-based approach. Simulation results show the anticipated self-focusing and attenuation phenomena of laser beams and the deformation of the spatial density distributions of electron plasma along the beam propagation path. Due to the flexibility of FDTD method in light beam excitation and accurate complex material modeling, the proposed approach has a wide application prospect in the study of the complex laser-plasma interactions in a small scale.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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  9. M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
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  21. T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antenn. Propag. 58(8), 2641–2648 (2010).
    [Crossref]
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    [Crossref]
  29. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
    [Crossref] [PubMed]
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2013 (1)

A. R. Niknam and N. Akhlaghipour, “Microwave ponderomotive action on the inhomogeneous collisioless and collisional plasma,” Waves Random Complex Media 23(2), 183–199 (2013).
[Crossref]

2012 (1)

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beams in plasma under plasma density ramp,” Opt. Commun. 285(21-22), 4483–4487 (2012).
[Crossref]

2011 (3)

N. Kant, S. Saralch, and H. Singh, “Ponderomotive self-focusing of a short laser pulse under a plasma density ramp,” Nukleonika 56(2), 149–153 (2011).

P. Ding and J. Pu, “Progagation of Laguerre-Gaussian vortex beam,” Wuli Xuebao 60(9), 094204 (2011).

Y. Wang, C. Yuan, Z. Zhou, L. Li, and Y. Du, “Propagation of Gaussian laser Beam in cold plasma of Drude model,” Phys. Plasmas 18(11), 113105 (2011).
[Crossref]

2010 (3)

2009 (2)

M. S. Sodha, S. K. Mishra, and S. Misra, “Focusing of dark hollow Gaussian electro-magnetic beams in a plasma,” Laser Part. Beams 27(1), 57–68 (2009).
[Crossref]

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

2006 (1)

N. S. Saini and T. S. Gill, “Self-focusing and self-phase modulation of an elliptic Gaussian laser beam in collisionless magneto plasma,” Laser Part. Beams 24(3), 447–453 (2006).
[Crossref]

2005 (1)

T. Y. Chien, C. L. Chang, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, “Spatially localized self-injection of electrons in a self-modulated laser wake-field accelerator by using a laser induced transient density ramp,” Phys. Rev. Lett. 94(11), 115003 (2005).
[Crossref] [PubMed]

2004 (2)

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref] [PubMed]

2000 (1)

J. Roden and S. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000).
[Crossref]

1997 (1)

M. Okoniewaki, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guided Wave Lett. 7(5), 121–123 (1997).
[Crossref]

1996 (2)

D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antenn. Propag. 44(6), 792–797 (1996).
[Crossref]

E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Overview of plasma based accelerator concepts,” IEEE Trans. Plasma Sci. 24(2), 252–288 (1996).
[Crossref]

1995 (1)

J. Lindl, “Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain,” Phys. Plasmas 2(11), 3933–4024 (1995).
[Crossref]

1994 (1)

1992 (1)

D. M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antenn. Propag. 40(10), 1223–1230 (1992).
[Crossref]

1990 (1)

S. Suckewer and C. H. Skinner, “Soft x-ray lasers and their applications,” Science 247(4950), 1553–1557 (1990).
[Crossref] [PubMed]

1989 (1)

1980 (1)

D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. 27(6), 1829–1833 (1980).
[Crossref]

1900 (1)

P. Drude, “Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte,” Ann. Phys. 308(11), 369–402 (1900).
[Crossref]

Akhlaghipour, N.

A. R. Niknam and N. Akhlaghipour, “Microwave ponderomotive action on the inhomogeneous collisioless and collisional plasma,” Waves Random Complex Media 23(2), 183–199 (2013).
[Crossref]

Amendt, P.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Berakdar, J.

Berger, R. I.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Burnett, N. H.

Chang, C. L.

T. Y. Chien, C. L. Chang, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, “Spatially localized self-injection of electrons in a self-modulated laser wake-field accelerator by using a laser induced transient density ramp,” Phys. Rev. Lett. 94(11), 115003 (2005).
[Crossref] [PubMed]

Chen, S. Y.

T. Y. Chien, C. L. Chang, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, “Spatially localized self-injection of electrons in a self-modulated laser wake-field accelerator by using a laser induced transient density ramp,” Phys. Rev. Lett. 94(11), 115003 (2005).
[Crossref] [PubMed]

Chien, T. Y.

T. Y. Chien, C. L. Chang, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, “Spatially localized self-injection of electrons in a self-modulated laser wake-field accelerator by using a laser induced transient density ramp,” Phys. Rev. Lett. 94(11), 115003 (2005).
[Crossref] [PubMed]

Corkum, P. B.

Ding, P.

P. Ding and J. Pu, “Progagation of Laguerre-Gaussian vortex beam,” Wuli Xuebao 60(9), 094204 (2011).

Dongare, M. B.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Drude, P.

P. Drude, “Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte,” Ann. Phys. 308(11), 369–402 (1900).
[Crossref]

Du, Y.

Y. Wang, C. Yuan, Z. Zhou, L. Li, and Y. Du, “Propagation of Gaussian laser Beam in cold plasma of Drude model,” Phys. Plasmas 18(11), 113105 (2011).
[Crossref]

Esarey, E.

E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Overview of plasma based accelerator concepts,” IEEE Trans. Plasma Sci. 24(2), 252–288 (1996).
[Crossref]

Feng, L.

Fisher, R.

D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. 27(6), 1829–1833 (1980).
[Crossref]

Fulari, V. J.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Gedney, S.

J. Roden and S. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000).
[Crossref]

Gill, T. S.

N. S. Saini and T. S. Gill, “Self-focusing and self-phase modulation of an elliptic Gaussian laser beam in collisionless magneto plasma,” Laser Part. Beams 24(3), 447–453 (2006).
[Crossref]

Glendinning, S. G.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Glenzer, S. H.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Haan, S. W.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Hulse, C.

Jia, Y.

Kant, N.

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beams in plasma under plasma density ramp,” Opt. Commun. 285(21-22), 4483–4487 (2012).
[Crossref]

N. Kant, S. Saralch, and H. Singh, “Ponderomotive self-focusing of a short laser pulse under a plasma density ramp,” Nukleonika 56(2), 149–153 (2011).

Kauffman, R. L.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Kelley, D. F.

D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antenn. Propag. 44(6), 792–797 (1996).
[Crossref]

Knoesen, A.

Krall, J.

E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Overview of plasma based accelerator concepts,” IEEE Trans. Plasma Sci. 24(2), 252–288 (1996).
[Crossref]

Kumar, A.

N. Kant, M. A. Wani, and A. Kumar, “Self-focusing of Hermite-Gaussian laser beams in plasma under plasma density ramp,” Opt. Commun. 285(21-22), 4483–4487 (2012).
[Crossref]

Landen, O. L.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Lee, C. H.

T. Y. Chien, C. L. Chang, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, “Spatially localized self-injection of electrons in a self-modulated laser wake-field accelerator by using a laser induced transient density ramp,” Phys. Rev. Lett. 94(11), 115003 (2005).
[Crossref] [PubMed]

Li, L.

Y. Wang, C. Yuan, Z. Zhou, L. Li, and Y. Du, “Propagation of Gaussian laser Beam in cold plasma of Drude model,” Phys. Plasmas 18(11), 113105 (2011).
[Crossref]

Lin, J. Y.

T. Y. Chien, C. L. Chang, C. H. Lee, J. Y. Lin, J. Wang, and S. Y. Chen, “Spatially localized self-injection of electrons in a self-modulated laser wake-field accelerator by using a laser induced transient density ramp,” Phys. Rev. Lett. 94(11), 115003 (2005).
[Crossref] [PubMed]

Lin, Z.

Lindl, J.

J. Lindl, “Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain,” Phys. Plasmas 2(11), 3933–4024 (1995).
[Crossref]

Lindl, J. D.

J. D. Lindl, P. Amendt, R. I. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, “The physics basis for ignition using indirect-drive targets on the National Ignition Facility,” Phys. Plasmas 11(2), 339–491 (2004).
[Crossref]

Luebbers, R. J.

D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antenn. Propag. 44(6), 792–797 (1996).
[Crossref]

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref] [PubMed]

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref] [PubMed]

Merewether, D. E.

D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. 27(6), 1829–1833 (1980).
[Crossref]

Mishra, S. K.

M. S. Sodha, S. K. Mishra, and S. Misra, “Focusing of dark hollow Gaussian electro-magnetic beams in a plasma,” Laser Part. Beams 27(1), 57–68 (2009).
[Crossref]

Misra, S.

M. S. Sodha, S. K. Mishra, and S. Misra, “Focusing of dark hollow Gaussian electro-magnetic beams in a plasma,” Laser Part. Beams 27(1), 57–68 (2009).
[Crossref]

Mrozowski, M.

M. Okoniewaki, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guided Wave Lett. 7(5), 121–123 (1997).
[Crossref]

Navare, S. T.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Niknam, A. R.

A. R. Niknam and N. Akhlaghipour, “Microwave ponderomotive action on the inhomogeneous collisioless and collisional plasma,” Waves Random Complex Media 23(2), 183–199 (2013).
[Crossref]

Okoniewaki, M.

M. Okoniewaki, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guided Wave Lett. 7(5), 121–123 (1997).
[Crossref]

Ou, P.

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref] [PubMed]

Patil, S. D.

M. V. Takale, S. T. Navare, S. D. Patil, V. J. Fulari, and M. B. Dongare, “Self-focusing and defocusing of TEM0p Hermite-Gaussian laser beams in collisionless plasma,” Opt. Commun. 282(15), 3157–3162 (2009).
[Crossref]

Potter, M.

T. Tan and M. Potter, “FDTD discrete planewave (FDTD-DPW) formulation for a perfectly matched source in TFSF simulations,” IEEE Trans. Antenn. Propag. 58(8), 2641–2648 (2010).
[Crossref]

Pu, J.

P. Ding and J. Pu, “Progagation of Laguerre-Gaussian vortex beam,” Wuli Xuebao 60(9), 094204 (2011).

Roden, J.

J. Roden and S. Gedney, “Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000).
[Crossref]

Saini, N. S.

N. S. Saini and T. S. Gill, “Self-focusing and self-phase modulation of an elliptic Gaussian laser beam in collisionless magneto plasma,” Laser Part. Beams 24(3), 447–453 (2006).
[Crossref]

Saralch, S.

N. Kant, S. Saralch, and H. Singh, “Ponderomotive self-focusing of a short laser pulse under a plasma density ramp,” Nukleonika 56(2), 149–153 (2011).

Singh, H.

N. Kant, S. Saralch, and H. Singh, “Ponderomotive self-focusing of a short laser pulse under a plasma density ramp,” Nukleonika 56(2), 149–153 (2011).

Skinner, C. H.

S. Suckewer and C. H. Skinner, “Soft x-ray lasers and their applications,” Science 247(4950), 1553–1557 (1990).
[Crossref] [PubMed]

Smith, F. W.

D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci. 27(6), 1829–1833 (1980).
[Crossref]

Sodha, M. S.

M. S. Sodha, S. K. Mishra, and S. Misra, “Focusing of dark hollow Gaussian electro-magnetic beams in a plasma,” Laser Part. Beams 27(1), 57–68 (2009).
[Crossref]

Sprangle, P.

E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Overview of plasma based accelerator concepts,” IEEE Trans. Plasma Sci. 24(2), 252–288 (1996).
[Crossref]

Stuchly, M. A.

M. Okoniewaki, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guided Wave Lett. 7(5), 121–123 (1997).
[Crossref]

Suckewer, S.

S. Suckewer and C. H. Skinner, “Soft x-ray lasers and their applications,” Science 247(4950), 1553–1557 (1990).
[Crossref] [PubMed]

Sullivan, D. M.

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Supplementary Material (6)

NameDescription
» Visualization 1: MP4 (231 KB)      Gaussian beam propagating in free space
» Visualization 2: MP4 (572 KB)      Gaussian beam propagating in plasma
» Visualization 3: MP4 (294 KB)      Gaussian beam propagating in plasma-cross section views
» Visualization 4: MP4 (182 KB)      vortex Laguerre-Gaussian beam propagating in free space
» Visualization 5: MP4 (449 KB)      vortex Laguerre-Gaussian beam propagating in plasma
» Visualization 6: MP4 (312 KB)      vortex Laguerre-Gaussian beam propagating in plasma-cross section views

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Figures (4)

Fig. 1
Fig. 1 The longitudinal-section views of the simulation results of Example 1 (see Visualization 1 and Visualization 2). (a) The transient distribution of field component E x of Gaussian beam in free space (a) and in plasma (b) over the x O z plane; the normalized distribution of light field magnitude in free space (b) and in plasma (e) over the x O z plane; (c) free space without plasma; (f) the normalized density profile of plasma deformed by the ponderomotive force over the x O z plane.
Fig. 2
Fig. 2 The spatial distribution of light field magnitude of Gaussian laser beam in plasma (see Visualization 3) on several cross-section planes specified by the different z values where λ L = 20 Δ z .
Fig. 3
Fig. 3 The longitudinal-section views of the simulation results of Example 2 (see Visualization 4 and Visualization 5). (a) The transient distribution of field component E x of vLG laser beam in free space (a) and in plasma (b) over the x O z plane; the normalized distribution of light field magnitude in free space (b) and in plasma (e) over the x O z plane; (c) free space without plasma; (f) the normalized density profile of plasma deformed by the ponderomotive force over the x O z plane.
Fig. 4
Fig. 4 The spatial distribution of light field magnitude of vLG laser beam in plasma (see Visualization 6) on several cross-section planes specified by the different z values where λ L = 20 Δ z .

Equations (20)

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f p = n ( r ) e 2 2 m e ω L 2 E 2 ( r , t ) ,
f pg = k B T e n ( r ) ,
n ( r ) e 2 2 m e ω L 2 E 2 ( r , t ) = k B T e n ( r ) .
n ( r ) = n 0 ( r ) exp ( e 2 E 2 ( r , t ) 2 m e k B T e ω L 2 ) ,
ε ( r , ω ) = ε 0 ( 1 ω p 2 ( r ) ω 2 i ν c ω ) ,
ε ( r , ω ) = ε 0 [ 1 ω p 0 2 ( r ) ω 2 i ν c ω exp ( α E 2 ( r , t ) ) ]
D ( r ) = ε ( r , ω ) E ( r ) ,
E n + 1 = b 0 D n + 1 + b 1 D n + b 2 D n 1 ε 0 ( a 0 ω ¯ p 2 + b 0 ) a 1 ω ¯ p 2 + b 1 a 0 ω ¯ p 2 + b 0 E n a 2 ω ¯ p 2 + b 2 a 0 ω ¯ p 2 + b 0 E n 1 ,
a 0 = 6 + 3 ν ¯ c ν ¯ c 2 , a 1 = 60 4 ν ¯ c 2 , a 2 = 6 3 ν ¯ c ν ¯ c 2 ,
b 0 = 72 + 36 ν ¯ c 3 ν ¯ c 3 , b 1 = 144 , b 2 = 72 36 ν ¯ c + 3 ν ¯ c 3
E 2 ( r , t ) E 2 ( r ˜ , t ) = E ¯ x 2 ( r ˜ , t ) + E ¯ y 2 ( r ˜ , t ) + E ¯ z 2 ( r ˜ , t ),
r ˜ = e x ( i + 1 2 ) Δ x + e y ( j + 1 2 ) Δ y + e z ( k + 1 2 ) Δ z ,
E ¯ x ( r ˜ , t ) = 1 4 { E x [( i + 1 2 ) Δ x , j Δ y , k Δ z , t ] + E x [( i + 1 2 ) Δ x , ( j + 1 ) Δ y , k Δ z , t ] + E x [( i + 1 2 ) Δ x , j Δ y , ( k + 1 ) Δ z , t ] + E x [( i + 1 2 ) Δ x , ( j + 1 ) Δ y , ( k + 1 ) Δ z , t ] } ,
E ¯ y ( r ˜ , t ) = 1 4 { E y [ i Δ x , ( j + 1 2 ) Δ y , k Δ z , t ] + E y [( i + 1 ) Δ x , ( j + 1 2 ) Δ y , k Δ z , t ] + E y [ i Δ x , ( j + 1 2 ) Δ y , ( k + 1 ) Δ z , t ] + E y [( i + 1 ) Δ x , ( j + 1 2 ) Δ y , ( k + 1 ) Δ z , t ] } ,
E ¯ z ( r ˜ , t ) = 1 4 { E z [ i Δ x , j Δ y , ( k + 1 2 ) Δ z , t ] + E z [( i + 1 ) Δ x , j Δ y , ( k + 1 2 ) Δ z , t ] + E z [ i Δ x , ( j + 1 ) Δ y , ( k + 1 2 ) Δ z , t ] + E z [( i + 1 ) Δ x , ( j + 1 ) Δ y , ( k + 1 2 ) Δ z , t ] } .
E 2 ( r , t ) = 1 T L t 0 t 0 + T L E 2 ( r , t )d t = 2 T L t 0 t 0 + T L / 2 E 2 ( r , t )d t ,
E 2 ( r ˜ , t ) 1 N T { E 2 [ r ˜ , ( n N T 2 ) Δ t ] + E 2 ( r ˜ , n Δ t ) + 2 m = n N T / 2 + 1 n 1 E 2 ( r ˜ , m Δ t ) }
R e l a t i v e E r r o r = 4 T L { Δ t 3 12 m = n N T / 2 + 1 n d 2 [ cos 2 ( ω L t + φ 0 α ) ] d t 2 | t = m Δ t } = 0.
E ( r , z = 0 , t ) = e x E 0 exp [ ( r w 0 ) 2 ] sin ( ω L t )
E ( ρ , z = 0 ) = e x E 0 ( 2 ρ w 0 ) m exp ( ρ 2 w 0 2 ) e x p ( i m φ ) ,

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