- Email: [email protected]
Comment on: “A theoretical model to explain the mechanism of light wave propagation through non-metallic nanowires” [Opt. Commun. 283 (2010) 4085]
Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Q1 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63Q3 Q2 64 65 66
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Discussion
Comment on: “A theoretical model to explain the mechanism of light wave propagation through non-metallic nanowires” [Opt. Commun. 283 (2010) 4085] Afshin Moradi a,b,n a b
Department of Basic Sciences, Kermanshah University of Technology, Kermanshah, Iran Department of Nano Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
art ic l e i nf o Article history: Received 25 June 2012 Received in revised form 1 October 2013 Accepted 4 October 2013
a b s t r a c t In a recent article [R. P. Dwivedi, H.-I. Lee, O. Beom-Hoan, E.-H. Lee, Opt. Commun, 283 (2010) 4085], E.-H. Lee et al. studied the mechanism of light wave propagation through non-metallic nanowires of sub-diffraction dimension and presented the general expressions of the dispersions relations of TE and TM modes, respectively. Here we show that the system generally disallows the separation of the TE and TM modes, Q4 except for the case of modes with no angular dependence. & 2013 Published by Elsevier B.V.
Following Lee et al. calculations [1], the electric current density flowing on the cylinder surface may be written as ^ J ðr; tÞ; Jðr; tÞ ¼ en0 2D vðr; tÞ ¼ sE
ð1Þ
where E J ðr; tÞ ¼ Ez e^ z þ Eθ e^θ is the tangential component of the electromagnetic field and s^ is the conductivity tensor of the unbound electron fluid. We define the Fourier–Bessel (FB) transform Am(q) of an arbitrary function Aðθ; z; tÞ by Z þ1 þ1 dqAm ðqÞ exp½ iðmθ þ qz ωtÞ; ð2Þ Aðθ; z; tÞ ¼ ∑ m ¼ 1
1
By eliminating the induced density n1 ðr; tÞ, from Eq. (1) in Ref. [1] and applying Eqs. (1) and (2) in the present work, we find ! szz szθ s^ ¼ ; ð3Þ sθz sθθ in0 2D e2 ω2 αm2 =a2 ; szz ¼ me ω ω2 αq2m szθ ¼ sθz ¼
sθθ ¼
in0 2D e2 αqm=a ; me ω ω2 αq2m
in0 2D e2 ω2 αq2 ; me ω ω2 αq2m
n
Correspondence address: Department of Basic Sciences, Kermanshah University of Technology, Kermanshah, Iran. Tel.: þ98 918 331 2692. E-mail address: [email protected]
where q2m ¼ q2 þ m2 =a2 , and the terms with α ¼ v2F =3 and vF ¼ ð2EF =me Þ1=2 in the above equations, come from the internal interaction force in the electron gas [1–4]. The field components for r 4a and r o a can be expressed in terms of Ez and Hz, and it is readily shown that these satisfy 2
d Ez dr
2
þ
1 dEz m2 κ 2i þ 2 Ez ¼ 0; r dr r
ð4Þ
where κ 2i ¼ q2 ɛ i ω2 =c2 (with i¼ 1,2) and c is the light speed. The same equation holds for Hz. The relevant solution of Eq. (4) and the corresponding solution for Hz are ( Ez ¼
Am I m ðκ 1 rÞ
Bm K m ðκ 2 rÞ
ðr o aÞ ðr 4 aÞ;
( Hz ¼
C m I m ðκ 1 rÞ
Dm K m ðκ 2 rÞ
ðr o aÞ ðr 4 aÞ
ð5Þ
The field components have to satisfy the usual boundary conditions at r ¼ a, that is, continuity of Ez and Eθ and discontinuity of Hz and H θ , due to the conductivity s^ associated with the 2D layer of unbound electrons, as shown in Ref. [2]. After doing some algebra, we find that except in the special case m¼ 0 the boundary conditions on Hz and H θ cannot be satisfied by solutions in which either Ez ¼ 0 or H z ¼ 0. Thus electromagnetic wave in the system do not have pure TM mode or TE mode character. The boundary conditions on Hz, Ez, H θ and Eθ give four linear equations and solvability condition takes the form 2 ! qm2 4sθθ Ψ m Υ m 1 a iωμ0 κ 42 κ 41
1
κ 22
1
κ 21
!2 3 5
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.10.018
Please cite this article as: A. Moradi, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.10.018i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
A. Moradi / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
( " ! !# qm Ψm Υm 1 1 Ψ þ m a κ 22 κ 21 κ 22 κ 21 !) qm iωμ0 Υ m Ψ m szθ þ sθθ 2 aκ 1 " # qm ¼ iɛ 0 ω ɛ 1 Ψ m ɛ 2 Υ m szz szθ 2 iωμ0 Υ m Ψ m sθθ þ Ψ m Υ m ; aκ 1
þ szθ
ð6Þ
Ψm ¼
1 I ′m ðκ 1 aÞ ; κ 1 Im ðκ 1 aÞ
Υm ¼
1 K ′m ðκ 2 aÞ ; κ 2 K m ðκ 2 aÞ
where ɛ 0 and μ0 are the permittivity and permeability of free space, respectively. Eq. (6) is the dispersion relation of electromagnetic wave propagation through the non-metallic nanowires.
This equation describes modes in which the field amplitude are maximum at r ¼ a and decrease with distance from the nanowire's surface. In summary, by using the linearized hydrodynamic model in conjunction with Maxwell equations we have examined the propagation of the electromagnetic wave through the nonmetallic nanowires. It has been found that the system generally disallows the separation of the TE and TM modes, except for the case of modes with no angular dependence. In this way, we have obtained the general expression of dispersion relation for the electromagnetic wave with mixed TE and TM modes in system. References [1] [2] [3] [4]
R.P. Dwivedi, H.-I. Lee, B.-h. O, E.-H. Lee, Opt. Commun. 283 (2010) 4085. A. Moradi, Photon Nanostruct. Fundam. Appl. 11 (2013) 85. A. Moradi, Appl. Phys. A 113 (2013) 97. A. Moradi, Appl. Phys. B 111 (2013) 127.
Please cite this article as: A. Moradi, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.10.018i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Q1 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63Q3 Q2 64 65 66
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Discussion
Comment on: “A theoretical model to explain the mechanism of light wave propagation through non-metallic nanowires” [Opt. Commun. 283 (2010) 4085] Afshin Moradi a,b,n a b
Department of Basic Sciences, Kermanshah University of Technology, Kermanshah, Iran Department of Nano Science, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
art ic l e i nf o Article history: Received 25 June 2012 Received in revised form 1 October 2013 Accepted 4 October 2013
a b s t r a c t In a recent article [R. P. Dwivedi, H.-I. Lee, O. Beom-Hoan, E.-H. Lee, Opt. Commun, 283 (2010) 4085], E.-H. Lee et al. studied the mechanism of light wave propagation through non-metallic nanowires of sub-diffraction dimension and presented the general expressions of the dispersions relations of TE and TM modes, respectively. Here we show that the system generally disallows the separation of the TE and TM modes, Q4 except for the case of modes with no angular dependence. & 2013 Published by Elsevier B.V.
Following Lee et al. calculations [1], the electric current density flowing on the cylinder surface may be written as ^ J ðr; tÞ; Jðr; tÞ ¼ en0 2D vðr; tÞ ¼ sE
ð1Þ
where E J ðr; tÞ ¼ Ez e^ z þ Eθ e^θ is the tangential component of the electromagnetic field and s^ is the conductivity tensor of the unbound electron fluid. We define the Fourier–Bessel (FB) transform Am(q) of an arbitrary function Aðθ; z; tÞ by Z þ1 þ1 dqAm ðqÞ exp½ iðmθ þ qz ωtÞ; ð2Þ Aðθ; z; tÞ ¼ ∑ m ¼ 1
1
By eliminating the induced density n1 ðr; tÞ, from Eq. (1) in Ref. [1] and applying Eqs. (1) and (2) in the present work, we find ! szz szθ s^ ¼ ; ð3Þ sθz sθθ in0 2D e2 ω2 αm2 =a2 ; szz ¼ me ω ω2 αq2m szθ ¼ sθz ¼
sθθ ¼
in0 2D e2 αqm=a ; me ω ω2 αq2m
in0 2D e2 ω2 αq2 ; me ω ω2 αq2m
n
Correspondence address: Department of Basic Sciences, Kermanshah University of Technology, Kermanshah, Iran. Tel.: þ98 918 331 2692. E-mail address: [email protected]
where q2m ¼ q2 þ m2 =a2 , and the terms with α ¼ v2F =3 and vF ¼ ð2EF =me Þ1=2 in the above equations, come from the internal interaction force in the electron gas [1–4]. The field components for r 4a and r o a can be expressed in terms of Ez and Hz, and it is readily shown that these satisfy 2
d Ez dr
2
þ
1 dEz m2 κ 2i þ 2 Ez ¼ 0; r dr r
ð4Þ
where κ 2i ¼ q2 ɛ i ω2 =c2 (with i¼ 1,2) and c is the light speed. The same equation holds for Hz. The relevant solution of Eq. (4) and the corresponding solution for Hz are ( Ez ¼
Am I m ðκ 1 rÞ
Bm K m ðκ 2 rÞ
ðr o aÞ ðr 4 aÞ;
( Hz ¼
C m I m ðκ 1 rÞ
Dm K m ðκ 2 rÞ
ðr o aÞ ðr 4 aÞ
ð5Þ
The field components have to satisfy the usual boundary conditions at r ¼ a, that is, continuity of Ez and Eθ and discontinuity of Hz and H θ , due to the conductivity s^ associated with the 2D layer of unbound electrons, as shown in Ref. [2]. After doing some algebra, we find that except in the special case m¼ 0 the boundary conditions on Hz and H θ cannot be satisfied by solutions in which either Ez ¼ 0 or H z ¼ 0. Thus electromagnetic wave in the system do not have pure TM mode or TE mode character. The boundary conditions on Hz, Ez, H θ and Eθ give four linear equations and solvability condition takes the form 2 ! qm2 4sθθ Ψ m Υ m 1 a iωμ0 κ 42 κ 41
1
κ 22
1
κ 21
!2 3 5
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.10.018
Please cite this article as: A. Moradi, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.10.018i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
A. Moradi / Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
( " ! !# qm Ψm Υm 1 1 Ψ þ m a κ 22 κ 21 κ 22 κ 21 !) qm iωμ0 Υ m Ψ m szθ þ sθθ 2 aκ 1 " # qm ¼ iɛ 0 ω ɛ 1 Ψ m ɛ 2 Υ m szz szθ 2 iωμ0 Υ m Ψ m sθθ þ Ψ m Υ m ; aκ 1
þ szθ
ð6Þ
Ψm ¼
1 I ′m ðκ 1 aÞ ; κ 1 Im ðκ 1 aÞ
Υm ¼
1 K ′m ðκ 2 aÞ ; κ 2 K m ðκ 2 aÞ
where ɛ 0 and μ0 are the permittivity and permeability of free space, respectively. Eq. (6) is the dispersion relation of electromagnetic wave propagation through the non-metallic nanowires.
This equation describes modes in which the field amplitude are maximum at r ¼ a and decrease with distance from the nanowire's surface. In summary, by using the linearized hydrodynamic model in conjunction with Maxwell equations we have examined the propagation of the electromagnetic wave through the nonmetallic nanowires. It has been found that the system generally disallows the separation of the TE and TM modes, except for the case of modes with no angular dependence. In this way, we have obtained the general expression of dispersion relation for the electromagnetic wave with mixed TE and TM modes in system. References [1] [2] [3] [4]
R.P. Dwivedi, H.-I. Lee, B.-h. O, E.-H. Lee, Opt. Commun. 283 (2010) 4085. A. Moradi, Photon Nanostruct. Fundam. Appl. 11 (2013) 85. A. Moradi, Appl. Phys. A 113 (2013) 97. A. Moradi, Appl. Phys. B 111 (2013) 127.
Please cite this article as: A. Moradi, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.10.018i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132