Abstract

The Laguerre–Gaussian (LG) beam expansion is described as a numerical and physical model of paraxial ultrashort pulse diffraction in the time domain. An overview of the dynamics of higher-order ultrashort planar LG modes is given through numerical simulations, and the finite width of these beams is shown to induce a dispersive-like axial broadening of the fields, which creates related variations in the on-axis amplitude of such pulses. The propagation of a pulsed plane wave scattered at an aperture is then illustrated as a finite weighted sum of individual planar LG pulses, which allows for intuitive illustration of the convergence of this expansion technique. By applying such an expansion to diffraction at a hard aperture, the planar pulsed LG beams are described as the paraxial analogs of the Bessel boundary waves typically observed in such situations, with both exhibiting superluminal group velocities along the optical axis. Numerical results of pulse diffraction at an aperture highlight the suitability of the LG expansion method for efficient and practical simulation of ultrashort fields in the paraxial regime.

© 2013 Optical Society of America

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2012

2011

R. J. Mahon and J. A. Murphy, “Gaussian beam mode analysis of optical pulses,” Proc. SPIE 8171, 81710H (2011).
[CrossRef]

M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
[CrossRef]

2010

2009

M. V. Vasnetsov and V. A. Pas’ko, “Group velocity of Gaussian beams,” Ukr. J. Phys. 54, 50–52 (2009).

2008

M. F. Kling and M. J. J. Vrakking, “Attosecond electron dynamics,” Annu. Rev. Phys. Chem. 59, 463–492 (2008).
[CrossRef]

A. M. Nugrowati, S. F. Pereira, and A. S. van de Nes, “Near and intermediate fields of an ultrashort pulse transmitted through Young’s double-slit experiment,” Phys. Rev. A 77, 053810 (2008).
[CrossRef]

Z. Yang, Z. Yang, and S. Zhang, “Carrier-envelope phase of ultrashort pulsed Laguerre–Gaussian beam,” Chin. Opt. Lett. 6, 189–191 (2008).
[CrossRef]

D. P. Kelly, B. M. Hennelly, A. Grün, and K. Unterrainer, “Numerical sampling rules for paraxial regime pulse diffraction calculations,” J. Opt. Soc. Am. A 25, 2299–2308 (2008).
[CrossRef]

2007

2006

Q. Zou and B. Lü, “Anomalous spectral behaviour near phase singularities in diffraction of pulsed Laguerre–Gaussian beams,” J. Opt. A 8, 531–536 (2006).

Y. Liu and B. Lü, “Truncated Hermite–Gauss series expansion and its application,” Optik 117, 437–442 (2006).
[CrossRef]

C. J. Zapata-Rodríguez, “Temporal effects in ultrashort pulsed beams focused by planar diffracting elements,” J. Opt. Soc. Am. A 23, 2335–2341 (2006).
[CrossRef]

2005

D. Deng, H. Guo, D. Han, M. Liu, and C. Li, “Effects of dispersion and longitudinal chromatic aberration on the focusing of isodiffracting pulsed Gaussian light beam,” Phys. Lett. A 334, 73–80 (2005).
[CrossRef]

2004

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004).
[CrossRef]

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

2003

N. Trappe, J. A. Murphy, and S. Withington, “The Gaussian beam mode analysis of classical phase aberrations in diffraction-limited optical systems,” Eur. J. Phys. 24, 403–412 (2003).
[CrossRef]

M. Lefrançois and S. F. Pereira, “Time evolution of the diffraction pattern of an ultrashort laser pulse,” Opt. Express 11, 1114–1122 (2003).
[CrossRef]

2002

M. A. Porras, R. Borghi, and M. Santarsiero, “Superluminality in Gaussian beams,” Opt. Commun. 203, 183–189 (2002).
[CrossRef]

W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A 19, 49–53 (2002).
[CrossRef]

2001

S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001).
[CrossRef]

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
[CrossRef]

S. Feng and H. G. Winful, “Higher-order transverse modes of ultrashort isodiffracting pulses,” Phys. Rev. E 63, 046602 (2001).
[CrossRef]

2000

1999

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single cycle terahertz pulses,,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[CrossRef]

1998

1997

1996

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197–203(1996).
[CrossRef]

M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771–778 (1996).
[CrossRef]

1993

J. A. Murphy and A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

1992

1991

1988

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1987

J. Durnin, J. J. Miceli, and J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

1983

1980

E. Cavanagh and B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
[CrossRef]

Ait-Ameur, K.

Andrews, L. C.

Balma, M.

Beracha, I.

Bokor, J.

Bor, Z.

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[CrossRef]

Borghi, R.

M. A. Porras, R. Borghi, and M. Santarsiero, “Superluminality in Gaussian beams,” Opt. Commun. 203, 183–189 (2002).
[CrossRef]

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197–203(1996).
[CrossRef]

Bowlan, P.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2007), pp. 379–380.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Budiarto, E.

Cagniot, E.

Cavanagh, E.

E. Cavanagh and B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
[CrossRef]

Citrin, D. S.

S. Nekkanti, D. Sullivan, and D. S. Citrin, “Simulation of spatiotemporal terahertz pulse shaping in 3-D using conductive apertures of finite thickness,” IEEE J. Quantum Electron. 37, 1226–1231 (2001).
[CrossRef]

Climent, V.

Cook, B. D.

E. Cavanagh and B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
[CrossRef]

Dartora, C. A.

Deng, D.

D. Deng, H. Guo, D. Han, M. Liu, and C. Li, “Effects of dispersion and longitudinal chromatic aberration on the focusing of isodiffracting pulsed Gaussian light beam,” Phys. Lett. A 334, 73–80 (2005).
[CrossRef]

Ding, D.

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberhardt, W.

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Egan, A.

J. A. Murphy and A. Egan, “Examples of Fresnel diffraction using Gaussian modes,” Eur. J. Phys. 14, 121–127 (1993).
[CrossRef]

Feng, S.

S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001).
[CrossRef]

S. Feng and H. G. Winful, “Higher-order transverse modes of ultrashort isodiffracting pulses,” Phys. Rev. E 63, 046602 (2001).
[CrossRef]

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, “Direct observation of the Gouy phase shift with single cycle terahertz pulses,,” Phys. Rev. Lett. 83, 3410–3413 (1999).
[CrossRef]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

Fernández-Alonso, M.

Fromager, M.

Gagnon, J.

M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
[CrossRef]

Gan, X. S.

Gao, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Goorjian, P. M.

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197–203(1996).
[CrossRef]

Goulielmakis, E.

M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
[CrossRef]

Gradstehyn, I. S.

I. S. Gradstehyn and I. M. Rhyzik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007), p. 811.

Grguras, I.

M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
[CrossRef]

Grün, A.

Gu, M.

Guo, H.

D. Deng, H. Guo, D. Han, M. Liu, and C. Li, “Effects of dispersion and longitudinal chromatic aberration on the focusing of isodiffracting pulsed Gaussian light beam,” Phys. Lett. A 334, 73–80 (2005).
[CrossRef]

W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A 19, 49–53 (2002).
[CrossRef]

Gürtler, A.

Hagness, S. C.

Han, D.

D. Deng, H. Guo, D. Han, M. Liu, and C. Li, “Effects of dispersion and longitudinal chromatic aberration on the focusing of isodiffracting pulsed Gaussian light beam,” Phys. Lett. A 334, 73–80 (2005).
[CrossRef]

Hellwarth, R. W.

Helm, H.

Hennelly, B. M.

Heyman, E.

Hofstetter, M.

M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
[CrossRef]

Horváth, Z. L.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[CrossRef]

Z. L. Horváth and Z. Bor, “Reshaping of femtosecond pulses by the Gouy phase shift,” Phys. Rev. E 60, 2337–2346 (1999).
[CrossRef]

Hu, W.

Jacquemin, R.

Jeong, S.

Jepsen, P. U.

Jiang, Z.

Joseph, R. M.

Judkins, J. B.

Kelly, D. P.

Kempe, M.

Klebniczki, J.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Kleineberg, U.

M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
[CrossRef]

Kling, M. F.

M. F. Kling and M. J. J. Vrakking, “Attosecond electron dynamics,” Annu. Rev. Phys. Chem. 59, 463–492 (2008).
[CrossRef]

Kovács, A. P.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Krausz, F.

M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
[CrossRef]

Kurdi, G.

Z. L. Horváth, J. Klebniczki, G. Kurdi, and A. P. Kovács, “Experimental investigation of the boundary wave pulse,” Opt. Commun. 239, 243–250 (2004).
[CrossRef]

Lancis, J.

Lefrançois, M.

Li, C.

D. Deng, H. Guo, D. Han, M. Liu, and C. Li, “Effects of dispersion and longitudinal chromatic aberration on the focusing of isodiffracting pulsed Gaussian light beam,” Phys. Lett. A 334, 73–80 (2005).
[CrossRef]

Liu, M.

D. Deng, H. Guo, D. Han, M. Liu, and C. Li, “Effects of dispersion and longitudinal chromatic aberration on the focusing of isodiffracting pulsed Gaussian light beam,” Phys. Lett. A 334, 73–80 (2005).
[CrossRef]

Liu, Y.

Y. Liu and B. Lü, “Truncated Hermite–Gauss series expansion and its application,” Optik 117, 437–442 (2006).
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[CrossRef]

Q. Zou and B. Lü, “Anomalous spectral behaviour near phase singularities in diffraction of pulsed Laguerre–Gaussian beams,” J. Opt. A 8, 531–536 (2006).

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M. Schultze, A. Wirth, I. Grguras, M. Uiberacker, T. Uphues, A. J. Verhoef, J. Gagnon, M. Hofstetter, U. Kleineberg, E. Goulielmakis, and F. Krausz, “State-of-the-art attosecond metrology,” J. Electron Spectrosc. Relat. Phenom. 184, 68–77 (2011).
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P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt. 14, 015701 (2012).
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Laser Phys.

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Opt. Express

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Optik

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Phys. Rev. E

Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
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Figures (13)

Fig. 1.
Fig. 1.

FDTD simulation of the general behavior of a 1D ultrashort plane wave pulse, propagating from left to right, scattered at a narrow aperture. The propagation of the generated boundary waves (BWs) and geometric wave (GW) is indicated. The field is shown as the absolute value of the electric field and clipped to enhance the BWs.

Fig. 2.
Fig. 2.

Structure of the cylindrically symmetric LG function ψ10(r0), from Eq. (2) without the normalization factor, compared with the Bessel function J0(κ10r0). The correlation between the Bessel and LG functions increases with larger values of m.

Fig. 3.
Fig. 3.

Expansion coefficients for syntheses of a circular aperture, an annular aperture with inner radius a/2, and a hemispherical function (a2r02)1/2. Calculated from Eq. (6) with M=49 and w0=a/M1/2.

Fig. 4.
Fig. 4.

Propagation of the m=15 planar Laguerre–Gaussian pulse |Ψ15(r,z,t)|2 from the numerical solution of Eq. (14). Calculated with w0=600λ0 and a large bandwidth of γ=0.6 at the planes z= (a) 0, (b) z0/2, (c) z0, and (d) 2z0. The fields are displayed as a decibel scale representation of their normalized intensity between 35 and 0 dB.

Fig. 5.
Fig. 5.

Time-domain simulations for the on-axis temporal amplitude of the m=30 (gray) and m=100 pulsed LG modes Re{Ψm(r=0,z,t)} as simulated from Eq. (14). The fields are predicted at z= (a) 0, (b) z0/4, (c) z0/2, (d) z0, (e) 2z0, (f) 8z0, with w0=300λ0 and γ=0.3.

Fig. 6.
Fig. 6.

Group velocity, vm(z), of the planar Laguerre–Gaussian pulses of different indices, with γ=0.1 and w0=300λ0.

Fig. 7.
Fig. 7.

Axial time delay, Tm(z), of the temporal amplitude envelope of the pulsed Laguerre–Gaussian modes |Ψm(r=0,z,t)| in relation to the half-duration of the pulse τ0, with w0=300λ0 and γ=0.1.

Fig. 8.
Fig. 8.

Variation in the HWHM τ(z) of the on-axis temporal field |Ψm(r=0,z,t)| at each plane z related to the half-duration τ0 of the source signal defined at z=0. w0=300λ0 and γ=0.1.

Fig. 9.
Fig. 9.

Maximum value of |Ψm(r=0,z,t)| at each plane z, with γ=0.3 and w0=300λ0. The on-axis amplitude of the m=0 pulsed LG mode varies negligibly on this axis scale from w0/w(ω0,z).

Fig. 10.
Fig. 10.

(a)–(c) Axial convergence of an LG pulse expansion (solid curve) toward the result obtained with the conventional FDI (dashed curve) for a pulse diffracted at a circular aperture with M= (a) 2, (b) 50, (c) 100. (d)–(f) A two-dimensional (1+1D) intensity representation |EM(r,z,t)|2 of the respective data in (a)–(c) within 40 and 0 dB. The fields are predicted with a fractional bandwidth of γ=0.3 at z=a2/(25λ0).

Fig. 11.
Fig. 11.

Normalized mean square error (NMSE) between the LG expansion Re{EM(r=0,z,t)} and the corresponding FDI calculation of the on-axis field of a γ=0.3 plane wave pulse diffracted by a circular aperture with various values of the Fresnel number N0. For the case of M=0, we assumed a value of w0=a.

Fig. 12.
Fig. 12.

LG expansion coefficients for the grating structure defined by Eq. (28), with M=499. Recall that the mode indices have discrete integer values only.

Fig. 13.
Fig. 13.

Paraxial LG expansion calculation, with M=499, of a plane wave pulse with a γ=0.4 Gaussian spectrum diffracted by the annular aperture structure defined by Eq. (28). The fields are calculated with the Fresnel number N0= (a) 1, (b) 5, (c) 10, (d) 20. The 30dB contours of the m=0 (dotted curve), m=9 (dash-dotted curve), m=90 (dashed curve), and m=249 (solid curve) component LG pulselets are superimposed and are shown on alternating sides of r=0 for clarity of the total field. The intensity data is shown on a normalized logarithmic dB scale between 40 and 0 dB, except in (a), which has a lower limit of 50dB.

Equations (30)

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

E(r,z,t)=12πS(ω)ECW(ω,r,z)exp(iωt)dω,
ψm(r0)=2πw02Lm(2r02w02)exp(r02w02),
ρm=w02m+1.
κmK2m+1,
E0(r0)m=0MAmψm(r0),
Am=2πw020aE0(r0)Lm(2r02w02)exp(r02w02)2πr0dr0.
w0=a/M.
ψm(ω,r,z)=2πw2(ω,z)Lm(2r2w2(ω,z))exp(r2w2(ω,z))exp(iωc(z+r22R(ω,z)))exp(i(2m+1)tan1(2czωw02)),
w(ω,z)=w01+(2czωw02)2
R(ω,z)=z(1+(ωw022cz)2),
EM(ω,r,z)=m=0MAmψm(ω,r,z),
EM(r,z,t)=12πS(ω){m=0MAm(ω)ψm(ω,r,z)}exp(iωt)dω.
EM(r,z,t)=m=0MAmΨm(ω,r,z,t),
Ψm(r,z,t)=12πS(ω)ψm(ω,r,z)exp(iωt)dω.
s(t)=exp(ln2(tt0)2τ02)exp(iω0t),
S(ω)=τ02ln2exp(τ02(ωω0)24ln2).
ϕ(ω,z)=ωcz(2m+1)tan1(2czωw02),
kz(ω,z)=ϕ(ω,z)z=ωc2c(2m+1)ωw02(1+4c2z2ω2w04).
nG(ω,z)=12c2(2m+1)ω2w02(1+4c2z2ω2w04).
v(z)=cnG(ω0,z)+ω0nG(ω,z)ω|ω=ω0,
vm(z)=c(1(m+12)w02(z2z02)(z2+z02)2)1,
Tm(z)=[0zdzvm(z)]zc=(2m+1)2czw024c2z2+w04ω02.
Tm(z0)=(m+1/2)/ω0.
z=z0/3.
NMSE=100×t|Re{EF(0,z,t)}Re{EM(0,z,t)}|2dtt|Re{EF(0,z,t)}|2dt.
Ts(z)=a2+z2czcandTp(z)=a22cz,
vs(z)=ca2+z2zandvp(z)=2cz2a22z2,
krs(z)=k0aa2+z2andkrp(z)=k0az,
κμ(ω0,z)=k0w0z2(M12)+1=k0az,forzz0
E0(r0)={1+sgn[cos(2.5πr0/a)]}/2,

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