Abstract

Diffraction of a three-dimensional (3D) spatiotemporal optical pulse by a phase-shifted Bragg grating (PSBG) is considered. The pulse diffraction is described in terms of signal transmission through a linear system with a transfer function determined by the reflection or transmission coefficient of the PSBG. Resonant approximations of the reflection and transmission coefficients of the PSBG as functions of the angular frequency and the in-plane component of the wave vector are obtained. Using these approximations, a hyperbolic partial differential equation (Klein–Gordon equation) describing a general class of transformations of the incident 3D pulse envelope is derived. A solution to this equation is found in the form of a convolution integral. The presented rigorous simulation results fully confirm the proposed theoretical description. The obtained results may find application in the design of new devices for spatiotemporal pulse shaping and for optical information processing and analog optical computing.

© 2016 Optical Society of America

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References

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2016 (1)

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

2015 (5)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

D. A. Bykov and L. L. Doskolovich, “ω–kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92(1), 013845 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

2014 (6)

2013 (1)

2012 (1)

2011 (1)

2009 (2)

2008 (1)

2007 (3)

2006 (2)

S. M. Sepke and D. P. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31(10), 1447–1449 (2006).
[Crossref] [PubMed]

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

2002 (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

1995 (1)

Alù, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Azaña, J.

Berger, N. K.

Bezus, E. A.

Boudreau, S.

Bykov, D. A.

Cao, P.

Carballar, A.

Castaldi, G.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Chen, J.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Chu, S. T.

Coldren, L. A.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Dong, J.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Doskolovich, L. L.

Engheta, N.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Ferrera, M.

Fischer, B.

Galdi, V.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Gaylord, T. K.

Gippius, N. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Golovastikov, N. V.

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015).
[Crossref]

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

Grann, E. B.

Guzzon, R. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Hu, X.

Ishihara, T.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Jiang, X.

Kazanskiy, N. L.

Khonina, S. N.

Kulishov, M.

Larochelle, S.

Levit, B.

Li, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Little, B. E.

Liu, W.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Lomakin, V.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

Lu, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Lu, M.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Michielssen, E.

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

Moharam, M. G.

Monticone, F.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Morandotti, R.

Moss, D. J.

Muljarov, E. A.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Norberg, E. J.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Pan, T.

Park, Y.

Parker, J. S.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Plant, D. V.

Pommet, D. A.

Qiu, C.

Quoc Ngo, N.

Razzari, L.

Rivas, L. M.

Ruan, Z.

Sepke, S. M.

Serafimovich, P. G.

Silva, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Slavík, R.

Soifer, V. A.

Su, Y.

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Tremblay, C.

Umstadter, D. P.

Wu, J.

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Yang, T.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Yang, Y.

Yao, J.

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Zhang, X.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Zheng, A.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Zhou, G.

Zhou, L.

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

IEEE Trans. Antenn. Propag. (1)

V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. 54(3), 970–984 (2006).
[Crossref]

J. Exp. Theor. Phys. (1)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating,” J. Exp. Theor. Phys. 121(5), 785–792 (2015).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (1)

W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10(3), 190–195 (2016).
[Crossref]

Opt. Commun. (1)

N. V. Golovastikov, D. A. Bykov, L. L. Doskolovich, and E. A. Bezus, “Spatial optical integrator based on phase-shifted Bragg gratings,” Opt. Commun. 338, 457–460 (2015).
[Crossref]

Opt. Express (7)

Opt. Lett. (8)

N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40(15), 3492–3495 (2015).
[Crossref] [PubMed]

S. M. Sepke and D. P. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31(10), 1447–1449 (2006).
[Crossref] [PubMed]

N. L. Kazanskiy, P. G. Serafimovich, and S. N. Khonina, “Use of photonic crystal cavities for temporal differentiation of optical signals,” Opt. Lett. 38(7), 1149–1151 (2013).
[Crossref] [PubMed]

Z. Ruan, “Spatial mode control of surface plasmon polariton excitation with gain medium: from spatial differentiator to integrator,” Opt. Lett. 40(4), 601–604 (2015).
[Crossref] [PubMed]

N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009).
[Crossref] [PubMed]

L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. Larochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009).
[Crossref] [PubMed]

Phys. Rev. A (1)

D. A. Bykov and L. L. Doskolovich, “ω–kx Fano line shape in photonic crystal slabs,” Phys. Rev. A 92(1), 013845 (2015).
[Crossref]

Phys. Rev. B (1)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).
[Crossref]

Sci. Rep. (1)

T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, and J. Chen, “All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator,” Sci. Rep. 4, 5581 (2014).
[PubMed]

Science (1)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Other (3)

T. Myint-U and L. Debnath, Linear Partial Differential Equations for Scientists and Engineers (Birkhäuser, 2007).

A. Erdélyi, Tables of Integral Transforms (McGraw-Hill, 1954), Vol. I.

G. Baym, Lectures on Quantum Mechanics (Benjamin, 1969).

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Figures (6)

Fig. 1
Fig. 1 Incident pulse P inc ( x,y,z,t ) and reflected pulse P ref ( x,y,z,t ) upon diffraction by a multilayer structure.
Fig. 2
Fig. 2 (a) Transmission spectra | T( k x ,ω ) | of the PSBG at N l =29 calculated using RCWA (left half, k x <0 ) and using the resonant approximation (12) (right half, k x >0 ) in the case of TM-polarization. (b) Reflection spectra | R( k x ,ω ) | of the PSBG at N l =9 calculated using RCWA (left half, k x <0 ) and using the resonant approximation (12) (right half, k x >0 ) in the case of TM-polarization.
Fig. 3
Fig. 3 (a) Envelope of the pulse incident on a PSBG with N l =29 layers. (b) Absolute value of the envelope of the transmitted pulse calculated using RCWA and Eqs. (5), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). The insets show the corresponding isosurfaces of the incident and transmitted pulses in ( x,y,t ) space.
Fig. 4
Fig. 4 (a) Envelope of the pulse incident on a PSBG with N l =9 layers. (b) Absolute value of the envelope of the reflected pulse calculated using RCWA and Eqs. (4), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). (c) Absolute value of the temporal derivative of the incident pulse. The insets show the corresponding isosurfaces of the incident and reflected pulses in ( x,y,t ) space.
Fig. 5
Fig. 5 (a) Envelope of the pulse incident of a PSBG with N l =9 layers. (b) Absolute value of the reflected pulse envelope calculated using RCWA and Eqs. (4), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). (c) Laplacian of the envelope of the incident pulse. The insets show the corresponding isosurfaces of the incident and reflected pulses in ( x,y,t ) space.
Fig. 6
Fig. 6 (a) Envelope of the pulse incident on a PSBG with N l =9 layers. (b) Absolute value of the envelope of the reflected pulse calculated using RCWA and Eqs. (4), (6) (left half, x<0 ) and using the proposed analytical model (12), (21), (22) (right half, x>0 ). (c) Estimate of the envelope of the reflected pulse obtained using Eq. (20). The insets show the corresponding isosurfaces of the incident and reflected pulses in ( x,y,t ) space.

Equations (34)

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E( x,y,z,t )=P( x,y,z,t )exp{ i ω 0 c n sup zi ω 0 t } = V( k x , k y ,ω )exp{ i k x x+i k y yi k z zi( ω+ ω 0 )t }d k x d k y dω ,
E( x,y,0,t )=v( x,y,t )exp{ i ω 0 t },
v( x,y,t )=P( x,y,0,t )= V( k x , k y ,ω )exp{ i k x x+i k y yiωt }d k x d k y dω
u R ( x,y,t )= V( k x , k y ,ω ) R ˜ ( k x , k y ,ω+ ω 0 )exp{ i k x x+i k y yiωt }d k x d k y dω ,
u T ( x,y,t )= V( k x , k y ,ω ) T ˜ ( k x , k y ,ω+ ω 0 )exp{ i k x x+i k y yiωt }d k x d k y dω ,
R ˜ ( k x , k y ,ω+ ω 0 )= R ТE ( k x , k y ,ω+ ω 0 )+ k x 2 k x 2 + k y 2 [ R ТM ( k x , k y ,ω+ ω 0 ) R ТE ( k x , k y ,ω+ ω 0 ) ], T ˜ ( k x , k y ,ω+ ω 0 )= T ТE ( k x , k y ,ω+ ω 0 )+ k x 2 k x 2 + k y 2 [ T ТM ( k x , k y ,ω+ ω 0 ) T ТE ( k x , k y ,ω+ ω 0 ) ].
H R ( k x , k y ,ω )= R ˜ ( k x , k y ,ω+ ω 0 ), H T ( k x , k y ,ω )= T ˜ ( k x , k y ,ω+ ω 0 ).
R( k x , k y ,ω )=R( k x 2 + k y 2 ,0,ω ),T( k x , k y ,ω )=T( k x 2 + k y 2 ,0,ω ).
n ˜ 1 h 1 = n ˜ 2 h 2 = λ B /4,
R( k x ,ω )= γ R v g 2 k x 2 ( ω ω z R )( ω ω z2 R ) v g 2 k x 2 ( ω ω p )( ω ω p2 ) , T( k x ,ω )= γ T v g 2 k x 2 ( ω ω z T )( ω ω z2 T ) v g 2 k x 2 ( ω ω p )( ω ω p2 ) ,
R( ω * )=R ( ω ) * ,T( ω * )=T ( ω ) * .
R( k x , k y ,ω )=R( k x 2 + k y 2 ,ω )= γ R v g 2 ( k x 2 + k y 2 )( ω ω z R )( ω+ ω z R ) v g 2 ( k x 2 + k y 2 )( ω ω p )( ω+ ω p * ) , T( k x , k y ,ω )=T( k x 2 + k y 2 ,ω )= γ T v g 2 ( k x 2 + k y 2 )( ω ω z T )( ω+ ω z T* ) v g 2 ( k x 2 + k y 2 )( ω ω p )( ω+ ω p * ) .
R TM ( 0,ω )= R TE ( 0,ω ), T TM ( 0,ω )= T TE ( 0,ω ).
H R,T ( k x , k y ,ω )= H R,T 1 ( k x , k y ,ω ) H R,T 2 ( k x , k y ,ω ),
H R,T 1 ( k x , k y ,ω )= γ R,T [ v g 2 ( k x 2 + k y 2 )( ω+ ω 0 ω z R,T )( ω+ ω 0 + ω z R,T* ) ],
H R,T 2 ( k x , k y ,ω )= [ v g 2 ( k x 2 + k y 2 )( ω+ ω 0 ω p )( ω+ ω 0 + ω p * ) ] 1 ,
U R,T ( k x , k y ,ω )= H R,T ( k x , k y ,ω )V( k x , k y ,ω ).
[ H R,T 2 ( k x , k y ,ω ) ] 1 U R,T ( k x , k y ,ω )= H R,T 1 ( k x , k y ,ω )V( k x , k y ,ω ).
2 u R,T t 2 v g 2 Δ u R,T 2i( ω 0 iIm ω p ) u R,T t ( ω 0 ω p )( ω 0 + ω p * ) u R,T = f R,T ,
f R,T ( x,y,t )= γ R,T ( 2 v t 2 v g 2 Δv2i( ω 0 iIm ω z R,T ) v t ( ω 0 ω z R,T )( ω 0 + ω z R,T* )v ),
u R,T ( x,y,t )= + + + f R,T ( ξ,η,τ )h( xξ,yη,tτ )dξdη dτ ,
h( x,y,t )= exp{ i ω 0 t+Im ω p t }cos( t 2 ( x 2 + y 2 ) / v g 2 Re ω p ) 2π v g 2 t 2 ( x 2 + y 2 ) / v g 2 , x 2 + y 2 v g t.
f R ( 0,0,0 )= 4 v g 2 σ 2 + | ω z | 2 2 σ t 2 ω 0 2 .
2 g( x,y,t ) t 2 v g 2 Δg( x,y,t )+ ( Re ω p ) 2 g( x,y,t )=f( x,y,t ),
f( x,y,t )= γ R,T ( 2 p t 2 v g 2 Δp+ ( Re ω z R,T ) 2 p )exp{ Im ω z R,T tIm ω p t },
u R,T ( x,y,t )=g( x,y,t )exp{ i ω 0 t+Im ω p t }, v( x,y,t )=p( x,y,t )exp{ i ω 0 t+Im ω z R,T t }.
G( k x , k y ,t )= 1 2π g( ξ,η,t )exp{ i k x ξ+i k y η }dξdη , F( k x , k y ,t )= 1 2π f( ξ,η,t )exp{ i k x ξ+i k y η }dξdη .
2 G( k x , k y ,t ) t 2 +( v g 2 ( k x 2 + k y 2 )+ ( Re ω p ) 2 )G( k x , k y ,t )=F( k x , k y ,t )
G( k x , k y ,t )= 0 t w( k x , k y ,tτ )F( k x , k y ,τ )dτ ,
g( x,y,t )= 1 2π G( k x , k y ,t )exp{ i k x xi k y y }d k x d k y = 1 ( 2π ) 2 0 t [ f( ξ,η,τ ) [ w( k x , k y ,tτ )exp{ i k x ( ξx )+i k y ( ηy ) }d k x d k y ]dξdη ]dτ .
1 2π + sin( y k 2 + q 2 ) k 2 + q 2 exp( ikx )dk ={ 1 2 J 0 ( q y 2 q 2 ), | x |<y, 0, | x |>y,
g( x,y,t )= 1 4π v g 0 t dτ × f( ξ,η,τ )[ J 0 ( v g 2 ( tτ ) 2 ( ξx ) 2 ( Re ω p ) v g 2 2 + k y 2 )exp{ i k y ( ηy ) }d k y ]dξdη.
1 2π + J 0 ( d k 2 + q 2 )exp( iky )dk ={ 1 π cos( q d 2 y 2 ) d 2 y 2 , | y |<d, 0, | y |>d,
g( x,y,t )= 1 2π v g 2 0 t dτ cos( Re[ ω p ] ( tτ ) 2 ( ( ξx ) 2 + ( ηy ) 2 ) / v g 2 ) ( tτ ) 2 ( ( ξx ) 2 + ( ηy ) 2 ) / v g 2 f( ξ,η,τ )dξdη .

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