Energy flow characteristics of vector X-Waves

Mohamed A. Salem; Hakan Bağcı

doi:10.1364/OE.19.008526

Energy flow characteristics of vector X-Waves

Mohamed A. Salem and Hakan Bağcı

Optics Express
Vol. 19,
Issue 9,
pp. 8526-8532
(2011)
•https://doi.org/10.1364/OE.19.008526

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Mohamed A. Salem and Hakan Bağcı, "Energy flow characteristics of vector X-Waves," Opt. Express 19, 8526-8532 (2011)

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Related Topics
Table of Contents Category
- Physical Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Propagation (350.5500)
- Waves (350.7420)
- Optical tweezers or optical manipulation (350.4855)
- Electromagnetic optics (260.2110)

About this Article
History
- Original Manuscript: February 10, 2011
- Revised Manuscript: March 26, 2011
- Manuscript Accepted: March 29, 2011
- Published: April 18, 2011
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 6, Iss. 5

Accessible

Open Access

Abstract

The vector form of X-Waves is obtained as a superposition of transverse electric and transverse magnetic polarized field components. It is shown that the signs of all components of the Poynting vector can be locally changed using carefully chosen complex amplitudes of the transverse electric and transverse magnetic polarization components. Negative energy flux density in the longitudinal direction can be observed in a bounded region around the centroid; in this region the local behavior of the wave field is similar to that of wave field with negative energy flow. This peculiar energy flux phenomenon is of essential importance for electromagnetic and optical traps and tweezers, where the location and momenta of micro-and nanoparticles are manipulated by changing the Poynting vector, and in detection of invisibility cloaks.

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References

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Year
|
Author
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Publication

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (J. Wiley & Sons, 2008).
[Crossref]
R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
[Crossref] [PubMed]
R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]
E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]
A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]
A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]
E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).
Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]
L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
[Crossref] [PubMed]
B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009).
[Crossref]
H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]
A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
[Crossref]
E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[Crossref]
A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.
J. D. Jackson, Classical Electrodynamics3rd ed. (Wiley, 1999).
J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]
M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[Crossref] [PubMed]
M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
[Crossref]

2011 (2)

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

2010 (2)

L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
[Crossref] [PubMed]

M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[Crossref] [PubMed]

2009 (1)

B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009).
[Crossref]

2007 (1)

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
[Crossref]

2006 (1)

M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
[Crossref]

2004 (2)

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Scattering of X-waves from a circular disk using a time domain incremental theory of diffraction,” Prog. Electromagn. Res. 44, 103–129 (2004).
[Crossref]

2003 (1)

E. Recami, M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. Hernández-Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003).
[Crossref]

1998 (1)

E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).

1997 (1)

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

1992 (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting x waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

1989 (2)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
[Crossref] [PubMed]

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.

Ambrosio, L. A.

Attiya, A. M.

Bagci, H.

M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[Crossref] [PubMed]

Besieris, I. M.

Chen, H.

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Chen, M.

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Ciattoni, A.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Conti, C.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Dartora, C. A.

Ding, J.

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Donnelly, R.

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

El-Diwany, E.

Gradshtein, I.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Greenleaf, J. F.

Hernández-Figueroa, H. E.

Heyman, E.

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics3rd ed. (Wiley, 1999).

Jeffery, A.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Lu, J.-Y.

Nóbrega, K. Z.

Novitsky, A. V.

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
[Crossref]

Novitsky, D. V.

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
[Crossref]

Porto, P. D.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Power, D.

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

Recami, E.

E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).

Ryzhik, I.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Salem, M. A.

M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[Crossref] [PubMed]

Shaarawi, A. M.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.

Wang, H.-T.

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Wu, B. I.

B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009).
[Crossref]

Zamboni-Rached, M.

M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
[Crossref]

Zhang, B.

B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009).
[Crossref]

Zhang, B.-F.

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Zheng, Zh.

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
[Crossref] [PubMed]

Zwillinger, D.

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

Appl. Opt. (1)

Zh. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

IEEE Trans. Antennas Propag. (2)

R. Donnelly and D. Power, “The behavior of electromagnetic localized waves at a planar interface,” IEEE Trans. Antennas Propag. 45, 580–591 (1997).
[Crossref]

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007).
[Crossref]

M. Zamboni-Rached, “Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures,” J. Opt. Soc. Am. A 23, 2166–2176 (2006).
[Crossref]

Materials Today (1)

H. Chen and M. Chen, “Flipping photons backward: reversed Cherenkov radiation,” Materials Today 14, 24–41 (2011).
[Crossref]

Opt. Express (2)

M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[Crossref] [PubMed]

Phys. Rev. A (2)

E. Recami, “On localized “X-shaped” superluminal solutions to maxwell equations,” Phys. Rev. A 252, 586–610 (1998).

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2032 (1989).
[Crossref] [PubMed]

Phys. Rev. E (1)

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[Crossref]

Phys. Rev. Lett. (1)

B. Zhang and B. I. Wu, “Electromagnetic detection of a perfect invisibility cloak”, Phys. Rev. Lett. 103, 243901 (2009).
[Crossref]

Prog. Electromagn. Res. (1)

Other (4)

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (J. Wiley & Sons, 2008).
[Crossref]

A. Jeffery, I. Gradshteǐn, D. Zwillinger, and I. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. (Dover, 1964), chap. 15.

J. D. Jackson, Classical Electrodynamics3rd ed. (Wiley, 1999).

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Fig. 1

A comparison between the z-component of the Poynting vector at the centroid plane of the zero-order vector X-Wave with n = 1, a = 2×10⁻¹⁶ s and V = 1.5c for two different configurations. Configuration 1 (a) is for the amplitudes $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = 1 / \sqrt{μ_{0}}$ ; and Configuration 2 (b) is for the amplitudes $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = i / \sqrt{μ_{0}}$ .

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Fig. 2

A comparison between the net energy flux in the z-direction in a finite circular cross section with radius ρ ₀ of the zero-order vector X-Wave with n = 1, a = 2 × 10⁻¹⁶ s and V = 1.5c. Configuration 1 (dashed line) is for the amplitudes $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = 1 / \sqrt{μ_{0}}$ ; and Configuration 2 (solid line) is for the amplitudes $A_{e} = 1 / \sqrt{ɛ_{0}}$ and $A_{h} = i / \sqrt{μ_{0}}$ .

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Equations (20)

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{\tilde{Ψ}}_{n} (k_{ρ}, k_{z}, ω; m, α) = 2^{n} {(k_{z} + \frac{α}{V})}^{m} e^{- a V k_{z}} e^{- a α} δ (k_{ρ}^{2} - [\frac{ω^{2}}{c^{2}} - k_{z}^{2}]) δ (ω - [V k_{z} + α]),

Ψ_{n} (ρ, ϕ, z, t) = \int_{0}^{\infty} d k_{z} \int_{- \infty}^{\infty} d ω \int_{0}^{\infty} d k_{ρ} {\tilde{Ψ}}_{n} (k_{ρ}, k_{z}, ω; m, α) J_{n} (k_{ρ} ρ) e^{i n ϕ} e^{i (k_{z} z - ω t)} .

Ψ_{n} (ρ, ϕ, ζ) = e^{i n ϕ} \frac{{(γ ρ)}^{n}}{τ^{1 + m + n}} \frac{(m + n)!}{n!}_{2} F_{1} (\frac{(1 + m + n)}{2}, \frac{(2 + m + n)}{2}; 1 + n; - η),

E = \nabla (\nabla \cdot Π_{e}) - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} Π_{e} - μ_{0} \nabla \times (\frac{\partial}{\partial t} Π_{h}),

H = ɛ_{0} \nabla \times (\frac{\partial}{\partial t} Π_{e}) + \nabla (\nabla \cdot Π_{h}) - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} Π_{h},

Ψ_{1} (ρ, ϕ, ζ) = 2 e^{i ϕ} \frac{\sqrt{1 + η - 1}}{γ ρ \sqrt{1 + η}} .

E_{ρ} (ρ, ϕ, ζ) = 2 ℜ {\frac{i γ e^{i ϕ}}{X^{(5 / 2)}} [Ξ A_{e} + i μ_{0} V X A_{h}]},

E_{ϕ} (ρ, ϕ, ζ) = 2 ℜ {\frac{i γ e^{i ϕ}}{X^{(5 / 2)}} [i X A_{e} - V μ_{0} Ξ A_{h}]},

E_{z} (ρ, ϕ, ζ) = 6 ℜ {\frac{γ^{3} ρ e^{i ϕ}}{X^{2} \sqrt{1 + η}} A_{e}},

H_{ρ} (ρ, ϕ, ζ) = 2 ℜ {\frac{e^{i ϕ} γ}{X^{(5 / 2)}} [i Ξ A_{h} + V ɛ_{0} X A_{e}]},

H_{ϕ} (ρ, ϕ, ζ) = 2 ℜ {\frac{i γ e^{i ϕ}}{X^{(5 / 2)}} [i X A_{h} + V ɛ_{0} Ξ A_{e}]},

H_{z} (ρ, ϕ, ζ) = 6 ℜ {\frac{γ^{3} ρ e^{i ϕ}}{X^{2} \sqrt{1 + η}} A_{h}},

\begin{array}{l} S_{ρ} (ρ, ϕ) = & 6 \frac{γ^{4} ρ a V^{2} ξ}{χ^{5}} {sin (2 ϕ) [ɛ_{0} ℜ {A_{e}^{2}} + μ_{0} ℜ {A_{h}^{2}}] \\ + 2 cos (2 ϕ) [ɛ_{0} A_{e}^{R} A_{e}^{I} + μ_{0} A_{h}^{R} A_{h}^{I}]}, \end{array}

\begin{array}{l} S_{ϕ} (ρ, ϕ) & = 6 \frac{γ^{4} ρ a V}{χ^{2}} {2 ξ [A_{h}^{R} A_{e}^{I} - A_{h}^{I} A_{e}^{R}] \\ + V χ [ɛ_{0} | A_{e} |^{2} + μ_{0} | A_{h} |^{2}] \\ + V χ cos (2 ϕ) [ɛ_{0} ℜ {A_{e}^{2}} + μ_{0} ℜ {A_{h}^{2}}] \\ - V χ sin (2 ϕ) [ɛ_{0} ℑ {A_{e}^{2}} + μ_{0} ℑ {A_{h}^{2}}]}, \end{array}

\begin{array}{l} S_{z} (ρ, ϕ) & = 2 \frac{γ^{2}}{χ^{5}} {2 (γ^{2} + 2) χ ξ [A_{e}^{I} A_{h}^{R} - A_{e}^{R} A_{h}^{I}] \\ + V (χ^{2} + ξ^{2}) [ɛ_{0} | A_{e} |^{2} + μ_{0} | A_{h} |^{2}] \\ - V (ξ^{2} - χ^{2}) cos (2 ϕ) [ɛ_{0} ℜ {A_{e}^{2}} + μ_{0} ℜ {A_{h}^{2}}] \\ + V (ξ^{2} - χ^{2}) sin (2 ϕ) [ɛ_{0} ℑ {A_{e}^{2}} + μ_{0} ℑ {A_{h}^{2}}]} . \end{array}

P (ρ_{0}) = \int_{0}^{ρ_{0}} d ρ \int_{- π}^{π} d ϕ ρ S (ρ, ϕ),

P_{ρ} = 0,

P_{ϕ} = \frac{3 π^{2} γ}{32 {(a V)}^{4}} {4 V [ɛ_{0} | A_{e} |^{2} + μ_{0} | A_{h} |^{2}] + (A_{h}^{I} A_{e}^{R} - A_{e}^{I} A_{h}^{R})},

P_{z} = \frac{3 π}{2 a^{4} V^{3}} [ɛ_{0} | A_{e} |^{2} + μ_{0} | A_{h} |^{2}]

ρ_{0_{max}} = \frac{a V}{\sqrt{3} γ} \sqrt{- 4 - \frac{4 \times 2^{1 / 3} (P - 2)}{Q} + 2^{2 / 3} Q},

Abstract

References

2011 (2)

2010 (2)

2009 (1)

2007 (1)

2006 (1)

2004 (2)

2003 (1)

1998 (1)

1997 (1)

1992 (1)

1989 (2)

Abramowitz, M.

Ambrosio, L. A.

Attiya, A. M.

Bagci, H.

Besieris, I. M.

Chen, H.

Chen, M.

Ciattoni, A.

Conti, C.

Dartora, C. A.

Ding, J.

Donnelly, R.

El-Diwany, E.

Gradshtein, I.

Greenleaf, J. F.

Hernández-Figueroa, H. E.

Heyman, E.

Jackson, J. D.

Jeffery, A.

Lu, J.-Y.

Nóbrega, K. Z.

Novitsky, A. V.

Novitsky, D. V.

Porto, P. D.

Power, D.

Recami, E.

Ryzhik, I.

Salem, M. A.

Shaarawi, A. M.

Stegun, I. A.

Wang, H.-T.

Wu, B. I.

Zamboni-Rached, M.

Zhang, B.

Zhang, B.-F.

Zheng, Zh.

Ziolkowski, R. W.

Zwillinger, D.

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

IEEE Trans. Antennas Propag. (2)

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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

Materials Today (1)

Opt. Express (2)

Phys. Rev. A (2)

Phys. Rev. E (1)

Phys. Rev. Lett. (1)

Prog. Electromagn. Res. (1)

Other (4)

Cited By

Figures (2)

Equations (20)

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