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Azimuthal electromagnetic surface waves on an annular magnetized plasma
Physics Letters A 318 (2003) 415–424 www.elsevier.com/locate/pla
Azimuthal electromagnetic surface waves on an annular magnetized plasma B. Shokri a,b,∗ , B. Jazi c a Physics Department and Laser Center of Shahid Beheshti University, Evin, Tehran, Iran b Institute for Studies in Theoretical Physics, 19395-3351 Tehran, Iran c Physics Department of Shahid Beheshti University, Evin, Tehran, Iran
Received 28 August 2003; accepted 17 September 2003 Communicated by F. Porcelli
Abstract The dispersion relation of azimuthal electromagnetic surface waves propagating perpendicular to the external magnetic field on an annular magnetized plasma rotating around the symmetric axis in a cylindrical metallic waveguide is obtained. The frequency region for E-modes and B-modes is investigated. Furthermore, the graphs of frequency spectra and radial dependency of the electric fields on the external magnetic field strength, geometric dimensions of the waveguide, and thickness of the annular plasma are presented. 2003 Published by Elsevier B.V. PACS: 52.25 Keywords: Surface wave; Magnetized plasma
1. Introduction It is well known that surface waves are qualitatively a type of electromagnetic oscillations of a bounded medium. They propagate on the interface of two media. Furthermore, surface wave characteristics essentially depend on the properties of medium surface, or more exactly, on the boundary conditions [1–4]. Surface waves propagating on the unmagnetized plasma–vacuum interface, are often E-type with field components Ez , Ex , By , if the interface of two media is the yz plane. When their phase velocity is small these waves are assumed to be potential waves [1,3]. On the other hand significant attention was paid to the development of plasma sources and investigations of properties of the produced plasma in relativistic plasma microwave electronics [5]. The operation of a plasma ˇ microwave oscillator is based on the Cerenkov interaction of a high current relativistic electron beam (REB) with slow eigenmode of a plasma waveguide placed in a magnetic field [6,7]. Then it seems that any plasma source * Corresponding author.
E-mail address: [email protected] (B. Shokri). 0375-9601/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.physleta.2003.09.033
416
B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
can be used in studies of microwaves oscillators based on REB–plasmas interaction. It is obvious that the problem of electron beam–plasma interaction is simplest if a plasma is placed inside a cylindrical metal waveguide with smooth walls and the plasma column is uniform over its length and azimuth [6–8]. On the other hand a great interest in the problem of gas discharges sustained on the surface waves is caused by their applications for coating by plasma-enhanced, chemical vapor deposition and/or etching for microelectronics. Parameters of such discharges depend on many factors, such as type of used electromagnetic surface waves [9]. There are plasma sources which can produce the completely ionized annular plasmas improving the interaction with REB through its coupling with slow surface waves on the annular plasma [9]. In this case, the study of azimuthal surface waves and their excitation on the cylindrical surfaces of an annular magnetized plasma has become important [10–12]. The waves propagating along the azimuthal angle, across an external axial steady magnetic field, refers as the azimuthal surface wave (ASW). On the other hand the cylindrical presentation of Maxwell’s equations for collisionless cold magnetized plasmas is complicated and strongly coupled [1–4]. For azimuthal modes kz = 0 we have shown that it can be reduced and separated into two independent modes, if the external magnetic field being along the Z-axis is directed to the symmetric axis of the waveguide. The problem is studied using the model of a magnetized cold collisionless dielectric tensor in which the ion response has been neglected. In the present Letter, we investigate the azimuthal electromagnetic surface waves (ASW) propagating perpendicular to the external magnetic field on a thin annular cold neutral magnetized plasma column placed in a cylindrical metallic waveguide. Dividing surface waves into the B-modes and E-modes respect to the direction of the propagation of these waves [1,2] with field components (Ez , Br , Bϕ ) and (Bz , Er , Eϕ ), respectively, we obtain the dispersion relations and the frequency spectra of these waves, separately. It will be shown that there is a frequency band in which the azimuthal surface E-modes propagating on an annular magnetized plasma is converted to the azimuthal volume waves. In this frequency band all waves in the both vacuum and plasma regions of the waveguide are azimuthal volume waves. It will be obtained that surface E-modes on an annular magnetized plasma can be considered as potential waves if the electron cyclotron frequency greatly exceeds the electron plasma frequency. It will be shown that the high frequency (HF) region of surface E-modes lies between the upper hybrid frequency and higher cutoff frequency of extraordinary HF waves [1]. This work is presented in four sections. In Section 2 the general formulations and configuration of waves in on an annular magnetized plasma is given. In Section 3 the dispersion relation and its graphs for surface E-mode and B-mode on an annular magnetized plasma are investigated. Finally, a summary and conclusion is presented.
2. Configuration and general equations We consider a long annular column of a magnetized plasma with external and internal radii b and a, respectively, surrounded by a cylindrical loss-free metal waveguide with radius R > b, as shown in Fig. 1. The external axial magnetic field is B0 = B0 zˆ where zˆ is a unit vector along the symmetry axis of the plasma column. In the high frequency region the plasma is assumed to be cold and collisionless. To study the dispersion relations of waves in this configuration, we make use Maxwell’s equations: B = 1 ∂ (˜εE), ∇× c ∂t E = − 1 ∂ B, ∇× c ∂t B = 0, ∇. E = 4πe(ni − ne ), ∇.
(1) (2) (3) (4)
where ε˜ is the dielectric tensor and ni , ne is the perturbed value of ions and electrons densities, respectively; E and B are the perturbed values of the electric and magnetic fields, respectively. In a magneto-active cold plasma the
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Fig. 1. Sketch of an annular magnetized plasma in a metallic cylindrical waveguide.
dielectric tensor has the following form [1,3]: ε⊥ ig 0 ε˜ = −ig ε⊥ 0 , 0 0 ε
where ε⊥ = 1 − ε = 1 − g=
2 ωpe
ω2 − Ωe2
,
2 ωpe
, ω2 2 Ω −ωpe e
ω(ω2 − Ωe2 )
(5) (6)
.
(7)
Here ωpe is the electron plasma frequency and Ωe is the electron cyclotron frequency. In the high frequency region, ion response can be neglected. In the linear approximation the perturbed fields B and E can be written as a superposition of monochromatic plane waves [1,3]: ϕ, z, t) = B(r,
3
eˆi Bi (r) exp −i(ωt − kz z − mϕ) ,
(8)
eˆi Ei (r) exp −i(ωt − kz z − mϕ) .
(9)
i=1
ϕ, z, t) = E(r,
3 i=1
Here eˆi is a unit vector in cylindrical coordinates and m is an integer. Substituting Eqs. (8), (9) into Eqs. (1), (2), we find the system of equations describing general behavior of electric and magnetic fields in this geometry as follows: dEz ω 2 m ω2 m ω3 dBz + χ Bz − g 3 + igkz 2 Ez , Er = ξ −1 −ikz χ 2 (10) dr c r c dr c r ω3 m ω 2 dBz ω2 dEz −1 2m Eϕ = ξ (11) + kz χ Ez − gkz 2 − ig 3 Bz , i χ c dr r c dr c r
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ω ω2 m g 2 ω2 m 2 dBz 2 2 ω dEz − ε⊥ Ez + gkz + igkz 2 Bz , Br = ξ −ikz χ χ + dr c ε⊥ c 2 r c dr c r
2 2 2 ω ω dBz g ω dEz 2m 2m ω + k B + igk E χ − gk Bϕ = ξ −1 −iε⊥ χ2 + z z z 2 z , z c ε⊥ c2 dr r c dr r c −1
(12) (13)
where ω2 ω4 , ξ = χ 4 − g2 4 . 2 c c In the above equations the field components Ez , Bz are strongly coupled with each other as follows:
ω g 2 g 2 ω2 ε
2 ∇⊥ χ2 + Ez − ξ Ez = ikz ∇ Bz , ε⊥ c 2 ε⊥ c ε⊥ ⊥ ω 2 2 Bz − ξ Bz = −ikz g∇⊥ Ez , χ 2 ∇⊥ c where χ 2 = kz2 − ε⊥
(14)
(15) (16)
m2 1 d d (17) r − 2, r dr dr r is the transverse Laplasian operator. Eqs. (15), (16) can be separated into two independent equations of Bz and Ez for the azimuthal modes (kz = 0). In this case B-modes and E-modes with components (Ez , Br , Bϕ ) and (Bz , Er , Eϕ ) appears, respectively. These modes will be investigated separately, in the next section. 2 ∇⊥ =
3. Waves separation and dispersion relations 3.1. Azimuthal B-mode (kz = 0, Bz = 0, Ez = 0) In this case combining Eqs. (12), (13) and (15) show that a B-mode can be obtained from the equations: 2
1 d m2 ω d (18) ε − 2 Ez = 0, r Ez + r dr dr c r ic dEz cm , Br = Ez . Bϕ = (19) ω dr ω r These modes propagate perpendicular to the external magnetic field, rotating around the symmetry axis with wavenumber and phase velocity k⊥ = m/r and vph = rω/m, respectively. It is obvious that by substituting ε = 1 into the above equations they can be generalized to the vacuum region. We should solve Eq. (18) separately, in three regions and match their solutions at the interface surfaces r = a, b by making use of boundary conditions {Ez }r=a,b = 0,
{Bϕ }r=a,b = 0,
Ez(r=R) = 0,
where the notation {A}r=a,b means a jump of quantity A in crossing the plasma-vacuum interface at r = a, b. Since surface wave fields should decrease from the boundaries toward the plasma region [1,3,4], the magnitude of ε in the Eq. (18) must be negative which is satisfied in the frequency region ω < ωpe . Therefore, we can find the solution of Eq. (18) as follows: r < a, A1 Jm (γ0 r) a < r < b, Ez = A2 Km (γ r) + A3 Im (γ r), (20) A4 Jm (γ0 r) + A5 Nm (γ0 r), b < r < R,
B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
Fig. 2. Graph of dispersion relation given by Eq. (22) for dominate modes (m = ±1) versus to ratio of radii of the annular plasma (α = b/a) for two arbitrary electron plasma frequencies ωpe .
where ω γ= c
ωpe ω
2 − 1,
γ0 =
ω , c
419
Fig. 3. Graph of solutions number of Eq. (22) versus to ωpe for α = b/a = 1.4.
(21)
are real constants in the frequency region ω < ωpe ; Jm , Nm are Bessel and Numan functions and Km , Im are modified Bessel functions, respectively. Here (A1 , A2 , A3 , A4 , A5 ) are real constants. Finally from Eqs. (19), (20) and making use of boundary conditions, we obtain the dispersion relation of azimuthal surface B-mode propagating on an annular magnetized plasma perpendicular to the external magnetic field rotating around the symmetric axis in a metallic cylindrical waveguide as follows: N (γ b) Jm (γ0 b) 0 −Km (γ b) −Im (γ b) m 0 0 0 Jm (γ0 a) −Km (γ a) −Im (γ a) (γ a) −γ I (γ a) = 0. (22) 0 0 γ0 Jm (γ0 a) −γ Km m γ N (γ b) γ J (γ b) 0 −γ Km (γ b) −γ Im (γ b) 0 m 0 0 m 0 Nm (γ0 R) Jm (γ0 R) 0 0 0 In the above equation the derivative of each function is referred to its whole argument. The pointing vector associated with these modes is s|B -mode =
4π Ez {Br ϕˆ0 − Bϕ rˆ0 } cos2 (ωt − mϕ), c
(23)
where ϕˆ0 and rˆ0 are unit vectors in the cylindrical coordinates. It must be noted that for an azimuthal propagation (perpendicular to the external magnetic field), radial flux of energy must be existed [9–11]. To proceed the investigation of Eq. (22), we present its graphs in different cases. As shown in Fig. 2 the frequency spectra of
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Fig. 4. Graph of dispersion relation given by Eq. (22) for dominate modes (m = ±1) versus to ratio of radii of the annular plasma (α = b/a) in two arbitrary electron plasma frequencies ωpe for R = 6 cm.
Fig. 5. Graph of solutions number of Eq. (22) versus to R for two arbitrary electron plasma frequencies for the surface B-mode.
dominate modes (m = ±1) versus to the ratio of radii of annular plasma (α = b/a) in two arbitrary plasma frequencies ωpe , are presented. In this figure we have considered a = 1 cm and R = 3 cm. It shows that, the number of spectra, increases by the increase of the plasma frequency. Here, we can find three values ω1 = 0.24 ωpe , ω2 = 0.72 ωpe , ω3 = 0.88 ωpe for an arbitrary value α = 1.4 where ωpe = 1 × 1011 Hz. The graph of solutions number of Eq. (22) versus to ωpe , is depicted in Fig. 3. For example three solutions of Eq. (22) are illustrated in Fig. 3. In Fig. 4 the frequency spectra of dominate modes (m = ±1) versus to the ratio of radii of annular plasma (α = b/a) in two arbitrary plasma frequencies ωpe , are presented for a = 1 cm and R = 6 cm. Comparing Figs. 2 and 4 shows that the number of spectra essentially depends on waveguide radius (R). The graph of solutions number of Eq. (22) versus to R can be obtained, from Fig. 5. Furthermore, the graph of Ez , versus to r in three frequencies ω1 , ω2 , ω3 obtained in Fig. 2, is presented in Fig. 6 for α = 1.4, a = 1 cm, R = 3 cm and ωpe = 1 × 1011 Hz. 3.2. Azimuthal E-mode (kz = 0, Ez = 0, Bz = 0) In this case Eqs. (10), (11) and (16) show that a E-mode (extraordinary wave) [1,3] can be obtained from the equations:
2
ε ⊥ − g 2 ω 2 m2 1 d dBz − 2 Bz = 0, r + r dr dr ε⊥ c r
dBz g dBz g −ic/ω c/ω + Bz , + Bz . ε⊥ Er = 2 ε⊥ Eϕ = 2 dr r dr r ε⊥ − g 2 ε⊥ − g 2
(24) (25)
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Fig. 6. Graph of Ez , versus to r in three frequencies ω1 , ω2 , ω3 obtained from Fig. 2 for a surface B-mode.
421
2 )/ε versus to ω/ω Fig. 7. Plot of (g 2 − ε⊥ pe for determination of ⊥ forbidden frequency band.
The latter equations can be generalized to the vacuum regions by substitution ε⊥ = 1 and g = 0. As done in the previous case (B-mode) the solutions of Eq. (24) are: A1 Jm (γ01 r) Bz = A2 Km (γ1 r) + A3 Im (γ1 r), A4 Jm (γ01 r) + A5 Nm (γ01 r),
r < a, a < r < b, b < r < R,
(26)
where ω γ1 = c
2 g 2 − ε⊥ , ε⊥
γ01 =
ω , c
(27)
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B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
Fig. 9. Graph of solutions number of Eq. (29) versus to ωpe for α = b/a = 1.4 in two arbitrary electron cyclotron frequencies Ωe for the surface E-mode.
Fig. 8. Graph of dispersion relation given by Eq. (29) for dominate modes (m = ±1) versus to ratio of radii of the annular plasma (α = b/a) for two arbitrary electron plasma frequencies ωpe and electron cyclotron frequencies Ωe for the surface E-mode.
Fig. 10. Graph of Eϕ , versus to r in the frequency ω = 0.08 ωpe obtained from Fig. 8 for the surface E-mode.
B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
423
are real constants in the frequency regions
ω ωpe
<
1+
1 2
− ωΩpee
Ωe2 2 ωpe
<
+4 , Ω2 < 12 ωΩpee + ω2e + 4 ,
+
ω ωpe
Ωe2 2 ωpe
(28)
pe
and (A1 , A2 , A3 , A4 , A5 ) are real constants. In the frequency regions (28) the azimuthal surface electromagnetic waves on the plasma column is matched with the azimuthal volume electromagnetic waves in the vacuum region determined by r < a or b < r < R regions. In the high frequency range the lower expression of the frequency region (28) associated with upper hybrid frequency and higher cutoff frequency for extraordinary HF waves [1,3, 10]. As shown in Fig. 7 there is a forbidden frequency band between the lower cutoff frequency of extraordinary HF waves and upper hybrid frequency [1,3] in which any azimuthal surface E-mode cannot exist on the annular magnetized plasma in a cylindrical waveguide. In the other word, in this frequency band only azimuthal volume electromagnetic waves appear in the plasma region. Fig. 7 shows that the forbidden frequency band width increases when Ωe /ωpe grows. In the unmagnetized case the frequency region ω < ωpe determines the region in which the azimuthal surface E-mode can appear. From Fig. 7, it is clear that when Ωe /ωpe 1 the surface E-modes can be considered as potential waves. Finally from Eqs. (24), (26), making use of the boundary conditions {Bz }r=a,b = 0, {Eϕ }r=a,b = 0, and Eϕ|r=R = 0, we obtain the dispersion relation of azimuthal surface electromagnetic E-mode on the annular magnetized plasma in the cylindrical waveguide as follows: Jm (γ01 b) 0 −Km (γ1 b) −Im (γ1 b) Nm (γ01b) −Km (γ1 a) −Im (γ1 a) 0 0 Jm (γ01 a) (γ a) −ψ K (γ a) − g K (γ a) −ψ I (γ a) − g I (γ a) 0 0 ψγ J m m 01 01 0 1 1 0 1 1 m m m a a = 0, ψγ N (γ b) ψγ J (γ b) (γ b) − g K (γ b) −ψ I (γ b) − g I (γ b) 0 −ψ K 01 m 01 0 m 1 0 m 1 01 m 01 b m b m 1 N (γ R) Jm (γ01 R) 0 0 0 m 01 (29) where ψ0 = γ1 ε⊥ ,
2 ψ = ε⊥ − g2 .
The pointing vector associated with these modes is s|E -mode =
4π Bz {−Er ϕˆ0 + Eϕ rˆ0 } cos2 (ωt − mϕ). c
(30)
Hear, we present the graphs of Eq. (29), in different cases. As shown in Fig. 8, the frequency spectra of dominate modes (m = ±1) versus to the ratio of radii of the annular plasma (α = b/a) for two arbitraryelectron plasma 2 − frequencies ωpe , and electron cyclotron frequency Ωe , in the low frequency region 2ω/ωpe < ( 4 + Ωe2 /ωpe Ωe /ωpe ) are illustrated. In this figure we have considered the signs ✷ for ωpe = Ωe = 3 × 1010 Hz and ◦ for ωpe = Ωe = 1 × 1011 Hz and the filled circle and filled box for Ωe = 1 × 1011 Hz = (10/3)ωpe and Ωe = 3 × 1010 Hz = (3/10)ωpe , respectively. The graph of solutions number of Eq. (29) versus to ωpe , is depicted in Fig. 9. As shown, the number of spectra is more sensitive to the plasma frequency than to the cyclotron frequency. We can find the number of spectra with respect to the plasma frequency for two arbitrary cyclotron frequencies in Fig. 9. In addition, variation of Eϕ /Eϕ(r=0) versus to r for frequency ω = 0.08 ωpe is presented in Fig. 10, where Eϕ( r=0) means the magnitude of Eϕ at r = 0.
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4. Summary and conclusion In this work starting with the general formulation of electromagnetic waves propagating in a cold collisionless annular magnetized plasma we showed that there are two independent modes of azimuthal electromagnetic surface waves propagating on an annular plasma column in a cylindrical metallic waveguide and perpendicular to the external magnetic field. These waves rotate around the symmetry axis with azimuthally and radially flux of energy. The dispersion relations and frequency spectra of the azimuthal surface B-modes and E-modes propagating on the annular magnetized plasma with field components (Ez , Br , Bϕ ) and (Bz , Er , Eϕ ) respectively, were obtained. It was shown that the number of solutions of the dispersion equation essentially depends on the external magnetic field strength, geometric dimensions of the waveguide, thickness of the annular plasma, and electron plasma frequency in both E-mode and B-mode cases. It was shown that in the HF frequency region the azimuthal surface E-modes on an annular magnetized plasma can appear in the frequency region between the upper hybrid frequency and higher cutoff frequency of extraordinary HF waves. Finally, we found that surface E-modes on an annular magnetized plasma can be considered as potential waves under the condition Ωe /ωpe 1.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
N.A. Krall, A.W. Trivelpiece, Principles of Plasma Physics, 1973. A.W. Trivelpiece, R.W. Gould, J. Appl. Phys. 31 (1959) 1784. A.F. Alexandrov, L.S. Bogdankevich, A.A. Rukhadze, Principles of Plasma Electrodynamics, Springer-Verlag, Heidelberg, 1984. D.M. Pozar, Microwave Engineering, 1990. V.L. Granatstein, I. Alexeff, High-Power Microwave Sources, Artech, Boston, MA, 1987. J.Q. Wu, Phys. Plasmas 4 (1997) 3064. B. Shokri, H. Ghomi, H. Latifi, Phys. Plasmas 7 (2000) 2671. A.V. Timofeev, B.N. Shvilkin, Usp. Fiz. Nauk 118 (1976) 273. C.M. Ferreira, M. Moisan, in: Microwave Discharges: Fundamentals and Applications, in: NATO Advanced Science Institutes Series B: Physics, Vol. 302, Plenum, New York, 1993. [10] V.O. Girka, A.E. Sporov, in: Proceedings of the 25th EPS Conference on Controlled Fusion and Plasma Physics, in: Europhysics Conference Abstracts, Vol. 22C, European Physical Society, 1998, p. 2837. [11] N.A. Azarenkov, V.O. Girka, I.V. Povlenko, in: Proceedings of the 25th EPS Conference on Controlled Fusion and Plasma Physics, in: Europhysics Conference Abstracts, Vol. 22C, European Physical Society, 1998, p. 2603. [12] O.T. Loza, Plasma Phys. Rep. 23 (1997) 201.
Azimuthal electromagnetic surface waves on an annular magnetized plasma B. Shokri a,b,∗ , B. Jazi c a Physics Department and Laser Center of Shahid Beheshti University, Evin, Tehran, Iran b Institute for Studies in Theoretical Physics, 19395-3351 Tehran, Iran c Physics Department of Shahid Beheshti University, Evin, Tehran, Iran
Received 28 August 2003; accepted 17 September 2003 Communicated by F. Porcelli
Abstract The dispersion relation of azimuthal electromagnetic surface waves propagating perpendicular to the external magnetic field on an annular magnetized plasma rotating around the symmetric axis in a cylindrical metallic waveguide is obtained. The frequency region for E-modes and B-modes is investigated. Furthermore, the graphs of frequency spectra and radial dependency of the electric fields on the external magnetic field strength, geometric dimensions of the waveguide, and thickness of the annular plasma are presented. 2003 Published by Elsevier B.V. PACS: 52.25 Keywords: Surface wave; Magnetized plasma
1. Introduction It is well known that surface waves are qualitatively a type of electromagnetic oscillations of a bounded medium. They propagate on the interface of two media. Furthermore, surface wave characteristics essentially depend on the properties of medium surface, or more exactly, on the boundary conditions [1–4]. Surface waves propagating on the unmagnetized plasma–vacuum interface, are often E-type with field components Ez , Ex , By , if the interface of two media is the yz plane. When their phase velocity is small these waves are assumed to be potential waves [1,3]. On the other hand significant attention was paid to the development of plasma sources and investigations of properties of the produced plasma in relativistic plasma microwave electronics [5]. The operation of a plasma ˇ microwave oscillator is based on the Cerenkov interaction of a high current relativistic electron beam (REB) with slow eigenmode of a plasma waveguide placed in a magnetic field [6,7]. Then it seems that any plasma source * Corresponding author.
E-mail address: [email protected] (B. Shokri). 0375-9601/$ – see front matter 2003 Published by Elsevier B.V. doi:10.1016/j.physleta.2003.09.033
416
B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
can be used in studies of microwaves oscillators based on REB–plasmas interaction. It is obvious that the problem of electron beam–plasma interaction is simplest if a plasma is placed inside a cylindrical metal waveguide with smooth walls and the plasma column is uniform over its length and azimuth [6–8]. On the other hand a great interest in the problem of gas discharges sustained on the surface waves is caused by their applications for coating by plasma-enhanced, chemical vapor deposition and/or etching for microelectronics. Parameters of such discharges depend on many factors, such as type of used electromagnetic surface waves [9]. There are plasma sources which can produce the completely ionized annular plasmas improving the interaction with REB through its coupling with slow surface waves on the annular plasma [9]. In this case, the study of azimuthal surface waves and their excitation on the cylindrical surfaces of an annular magnetized plasma has become important [10–12]. The waves propagating along the azimuthal angle, across an external axial steady magnetic field, refers as the azimuthal surface wave (ASW). On the other hand the cylindrical presentation of Maxwell’s equations for collisionless cold magnetized plasmas is complicated and strongly coupled [1–4]. For azimuthal modes kz = 0 we have shown that it can be reduced and separated into two independent modes, if the external magnetic field being along the Z-axis is directed to the symmetric axis of the waveguide. The problem is studied using the model of a magnetized cold collisionless dielectric tensor in which the ion response has been neglected. In the present Letter, we investigate the azimuthal electromagnetic surface waves (ASW) propagating perpendicular to the external magnetic field on a thin annular cold neutral magnetized plasma column placed in a cylindrical metallic waveguide. Dividing surface waves into the B-modes and E-modes respect to the direction of the propagation of these waves [1,2] with field components (Ez , Br , Bϕ ) and (Bz , Er , Eϕ ), respectively, we obtain the dispersion relations and the frequency spectra of these waves, separately. It will be shown that there is a frequency band in which the azimuthal surface E-modes propagating on an annular magnetized plasma is converted to the azimuthal volume waves. In this frequency band all waves in the both vacuum and plasma regions of the waveguide are azimuthal volume waves. It will be obtained that surface E-modes on an annular magnetized plasma can be considered as potential waves if the electron cyclotron frequency greatly exceeds the electron plasma frequency. It will be shown that the high frequency (HF) region of surface E-modes lies between the upper hybrid frequency and higher cutoff frequency of extraordinary HF waves [1]. This work is presented in four sections. In Section 2 the general formulations and configuration of waves in on an annular magnetized plasma is given. In Section 3 the dispersion relation and its graphs for surface E-mode and B-mode on an annular magnetized plasma are investigated. Finally, a summary and conclusion is presented.
2. Configuration and general equations We consider a long annular column of a magnetized plasma with external and internal radii b and a, respectively, surrounded by a cylindrical loss-free metal waveguide with radius R > b, as shown in Fig. 1. The external axial magnetic field is B0 = B0 zˆ where zˆ is a unit vector along the symmetry axis of the plasma column. In the high frequency region the plasma is assumed to be cold and collisionless. To study the dispersion relations of waves in this configuration, we make use Maxwell’s equations: B = 1 ∂ (˜εE), ∇× c ∂t E = − 1 ∂ B, ∇× c ∂t B = 0, ∇. E = 4πe(ni − ne ), ∇.
(1) (2) (3) (4)
where ε˜ is the dielectric tensor and ni , ne is the perturbed value of ions and electrons densities, respectively; E and B are the perturbed values of the electric and magnetic fields, respectively. In a magneto-active cold plasma the
B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
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Fig. 1. Sketch of an annular magnetized plasma in a metallic cylindrical waveguide.
dielectric tensor has the following form [1,3]: ε⊥ ig 0 ε˜ = −ig ε⊥ 0 , 0 0 ε
where ε⊥ = 1 − ε = 1 − g=
2 ωpe
ω2 − Ωe2
,
2 ωpe
, ω2 2 Ω −ωpe e
ω(ω2 − Ωe2 )
(5) (6)
.
(7)
Here ωpe is the electron plasma frequency and Ωe is the electron cyclotron frequency. In the high frequency region, ion response can be neglected. In the linear approximation the perturbed fields B and E can be written as a superposition of monochromatic plane waves [1,3]: ϕ, z, t) = B(r,
3
eˆi Bi (r) exp −i(ωt − kz z − mϕ) ,
(8)
eˆi Ei (r) exp −i(ωt − kz z − mϕ) .
(9)
i=1
ϕ, z, t) = E(r,
3 i=1
Here eˆi is a unit vector in cylindrical coordinates and m is an integer. Substituting Eqs. (8), (9) into Eqs. (1), (2), we find the system of equations describing general behavior of electric and magnetic fields in this geometry as follows: dEz ω 2 m ω2 m ω3 dBz + χ Bz − g 3 + igkz 2 Ez , Er = ξ −1 −ikz χ 2 (10) dr c r c dr c r ω3 m ω 2 dBz ω2 dEz −1 2m Eϕ = ξ (11) + kz χ Ez − gkz 2 − ig 3 Bz , i χ c dr r c dr c r
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ω ω2 m g 2 ω2 m 2 dBz 2 2 ω dEz − ε⊥ Ez + gkz + igkz 2 Bz , Br = ξ −ikz χ χ + dr c ε⊥ c 2 r c dr c r
2 2 2 ω ω dBz g ω dEz 2m 2m ω + k B + igk E χ − gk Bϕ = ξ −1 −iε⊥ χ2 + z z z 2 z , z c ε⊥ c2 dr r c dr r c −1
(12) (13)
where ω2 ω4 , ξ = χ 4 − g2 4 . 2 c c In the above equations the field components Ez , Bz are strongly coupled with each other as follows:
ω g 2 g 2 ω2 ε
2 ∇⊥ χ2 + Ez − ξ Ez = ikz ∇ Bz , ε⊥ c 2 ε⊥ c ε⊥ ⊥ ω 2 2 Bz − ξ Bz = −ikz g∇⊥ Ez , χ 2 ∇⊥ c where χ 2 = kz2 − ε⊥
(14)
(15) (16)
m2 1 d d (17) r − 2, r dr dr r is the transverse Laplasian operator. Eqs. (15), (16) can be separated into two independent equations of Bz and Ez for the azimuthal modes (kz = 0). In this case B-modes and E-modes with components (Ez , Br , Bϕ ) and (Bz , Er , Eϕ ) appears, respectively. These modes will be investigated separately, in the next section. 2 ∇⊥ =
3. Waves separation and dispersion relations 3.1. Azimuthal B-mode (kz = 0, Bz = 0, Ez = 0) In this case combining Eqs. (12), (13) and (15) show that a B-mode can be obtained from the equations: 2
1 d m2 ω d (18) ε − 2 Ez = 0, r Ez + r dr dr c r ic dEz cm , Br = Ez . Bϕ = (19) ω dr ω r These modes propagate perpendicular to the external magnetic field, rotating around the symmetry axis with wavenumber and phase velocity k⊥ = m/r and vph = rω/m, respectively. It is obvious that by substituting ε = 1 into the above equations they can be generalized to the vacuum region. We should solve Eq. (18) separately, in three regions and match their solutions at the interface surfaces r = a, b by making use of boundary conditions {Ez }r=a,b = 0,
{Bϕ }r=a,b = 0,
Ez(r=R) = 0,
where the notation {A}r=a,b means a jump of quantity A in crossing the plasma-vacuum interface at r = a, b. Since surface wave fields should decrease from the boundaries toward the plasma region [1,3,4], the magnitude of ε in the Eq. (18) must be negative which is satisfied in the frequency region ω < ωpe . Therefore, we can find the solution of Eq. (18) as follows: r < a, A1 Jm (γ0 r) a < r < b, Ez = A2 Km (γ r) + A3 Im (γ r), (20) A4 Jm (γ0 r) + A5 Nm (γ0 r), b < r < R,
B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
Fig. 2. Graph of dispersion relation given by Eq. (22) for dominate modes (m = ±1) versus to ratio of radii of the annular plasma (α = b/a) for two arbitrary electron plasma frequencies ωpe .
where ω γ= c
ωpe ω
2 − 1,
γ0 =
ω , c
419
Fig. 3. Graph of solutions number of Eq. (22) versus to ωpe for α = b/a = 1.4.
(21)
are real constants in the frequency region ω < ωpe ; Jm , Nm are Bessel and Numan functions and Km , Im are modified Bessel functions, respectively. Here (A1 , A2 , A3 , A4 , A5 ) are real constants. Finally from Eqs. (19), (20) and making use of boundary conditions, we obtain the dispersion relation of azimuthal surface B-mode propagating on an annular magnetized plasma perpendicular to the external magnetic field rotating around the symmetric axis in a metallic cylindrical waveguide as follows: N (γ b) Jm (γ0 b) 0 −Km (γ b) −Im (γ b) m 0 0 0 Jm (γ0 a) −Km (γ a) −Im (γ a) (γ a) −γ I (γ a) = 0. (22) 0 0 γ0 Jm (γ0 a) −γ Km m γ N (γ b) γ J (γ b) 0 −γ Km (γ b) −γ Im (γ b) 0 m 0 0 m 0 Nm (γ0 R) Jm (γ0 R) 0 0 0 In the above equation the derivative of each function is referred to its whole argument. The pointing vector associated with these modes is s|B -mode =
4π Ez {Br ϕˆ0 − Bϕ rˆ0 } cos2 (ωt − mϕ), c
(23)
where ϕˆ0 and rˆ0 are unit vectors in the cylindrical coordinates. It must be noted that for an azimuthal propagation (perpendicular to the external magnetic field), radial flux of energy must be existed [9–11]. To proceed the investigation of Eq. (22), we present its graphs in different cases. As shown in Fig. 2 the frequency spectra of
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Fig. 4. Graph of dispersion relation given by Eq. (22) for dominate modes (m = ±1) versus to ratio of radii of the annular plasma (α = b/a) in two arbitrary electron plasma frequencies ωpe for R = 6 cm.
Fig. 5. Graph of solutions number of Eq. (22) versus to R for two arbitrary electron plasma frequencies for the surface B-mode.
dominate modes (m = ±1) versus to the ratio of radii of annular plasma (α = b/a) in two arbitrary plasma frequencies ωpe , are presented. In this figure we have considered a = 1 cm and R = 3 cm. It shows that, the number of spectra, increases by the increase of the plasma frequency. Here, we can find three values ω1 = 0.24 ωpe , ω2 = 0.72 ωpe , ω3 = 0.88 ωpe for an arbitrary value α = 1.4 where ωpe = 1 × 1011 Hz. The graph of solutions number of Eq. (22) versus to ωpe , is depicted in Fig. 3. For example three solutions of Eq. (22) are illustrated in Fig. 3. In Fig. 4 the frequency spectra of dominate modes (m = ±1) versus to the ratio of radii of annular plasma (α = b/a) in two arbitrary plasma frequencies ωpe , are presented for a = 1 cm and R = 6 cm. Comparing Figs. 2 and 4 shows that the number of spectra essentially depends on waveguide radius (R). The graph of solutions number of Eq. (22) versus to R can be obtained, from Fig. 5. Furthermore, the graph of Ez , versus to r in three frequencies ω1 , ω2 , ω3 obtained in Fig. 2, is presented in Fig. 6 for α = 1.4, a = 1 cm, R = 3 cm and ωpe = 1 × 1011 Hz. 3.2. Azimuthal E-mode (kz = 0, Ez = 0, Bz = 0) In this case Eqs. (10), (11) and (16) show that a E-mode (extraordinary wave) [1,3] can be obtained from the equations:
2
ε ⊥ − g 2 ω 2 m2 1 d dBz − 2 Bz = 0, r + r dr dr ε⊥ c r
dBz g dBz g −ic/ω c/ω + Bz , + Bz . ε⊥ Er = 2 ε⊥ Eϕ = 2 dr r dr r ε⊥ − g 2 ε⊥ − g 2
(24) (25)
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Fig. 6. Graph of Ez , versus to r in three frequencies ω1 , ω2 , ω3 obtained from Fig. 2 for a surface B-mode.
421
2 )/ε versus to ω/ω Fig. 7. Plot of (g 2 − ε⊥ pe for determination of ⊥ forbidden frequency band.
The latter equations can be generalized to the vacuum regions by substitution ε⊥ = 1 and g = 0. As done in the previous case (B-mode) the solutions of Eq. (24) are: A1 Jm (γ01 r) Bz = A2 Km (γ1 r) + A3 Im (γ1 r), A4 Jm (γ01 r) + A5 Nm (γ01 r),
r < a, a < r < b, b < r < R,
(26)
where ω γ1 = c
2 g 2 − ε⊥ , ε⊥
γ01 =
ω , c
(27)
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B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
Fig. 9. Graph of solutions number of Eq. (29) versus to ωpe for α = b/a = 1.4 in two arbitrary electron cyclotron frequencies Ωe for the surface E-mode.
Fig. 8. Graph of dispersion relation given by Eq. (29) for dominate modes (m = ±1) versus to ratio of radii of the annular plasma (α = b/a) for two arbitrary electron plasma frequencies ωpe and electron cyclotron frequencies Ωe for the surface E-mode.
Fig. 10. Graph of Eϕ , versus to r in the frequency ω = 0.08 ωpe obtained from Fig. 8 for the surface E-mode.
B. Shokri, B. Jazi / Physics Letters A 318 (2003) 415–424
423
are real constants in the frequency regions
ω ωpe
<
1+
1 2
− ωΩpee
Ωe2 2 ωpe
<
+4 , Ω2 < 12 ωΩpee + ω2e + 4 ,
+
ω ωpe
Ωe2 2 ωpe
(28)
pe
and (A1 , A2 , A3 , A4 , A5 ) are real constants. In the frequency regions (28) the azimuthal surface electromagnetic waves on the plasma column is matched with the azimuthal volume electromagnetic waves in the vacuum region determined by r < a or b < r < R regions. In the high frequency range the lower expression of the frequency region (28) associated with upper hybrid frequency and higher cutoff frequency for extraordinary HF waves [1,3, 10]. As shown in Fig. 7 there is a forbidden frequency band between the lower cutoff frequency of extraordinary HF waves and upper hybrid frequency [1,3] in which any azimuthal surface E-mode cannot exist on the annular magnetized plasma in a cylindrical waveguide. In the other word, in this frequency band only azimuthal volume electromagnetic waves appear in the plasma region. Fig. 7 shows that the forbidden frequency band width increases when Ωe /ωpe grows. In the unmagnetized case the frequency region ω < ωpe determines the region in which the azimuthal surface E-mode can appear. From Fig. 7, it is clear that when Ωe /ωpe 1 the surface E-modes can be considered as potential waves. Finally from Eqs. (24), (26), making use of the boundary conditions {Bz }r=a,b = 0, {Eϕ }r=a,b = 0, and Eϕ|r=R = 0, we obtain the dispersion relation of azimuthal surface electromagnetic E-mode on the annular magnetized plasma in the cylindrical waveguide as follows: Jm (γ01 b) 0 −Km (γ1 b) −Im (γ1 b) Nm (γ01b) −Km (γ1 a) −Im (γ1 a) 0 0 Jm (γ01 a) (γ a) −ψ K (γ a) − g K (γ a) −ψ I (γ a) − g I (γ a) 0 0 ψγ J m m 01 01 0 1 1 0 1 1 m m m a a = 0, ψγ N (γ b) ψγ J (γ b) (γ b) − g K (γ b) −ψ I (γ b) − g I (γ b) 0 −ψ K 01 m 01 0 m 1 0 m 1 01 m 01 b m b m 1 N (γ R) Jm (γ01 R) 0 0 0 m 01 (29) where ψ0 = γ1 ε⊥ ,
2 ψ = ε⊥ − g2 .
The pointing vector associated with these modes is s|E -mode =
4π Bz {−Er ϕˆ0 + Eϕ rˆ0 } cos2 (ωt − mϕ). c
(30)
Hear, we present the graphs of Eq. (29), in different cases. As shown in Fig. 8, the frequency spectra of dominate modes (m = ±1) versus to the ratio of radii of the annular plasma (α = b/a) for two arbitraryelectron plasma 2 − frequencies ωpe , and electron cyclotron frequency Ωe , in the low frequency region 2ω/ωpe < ( 4 + Ωe2 /ωpe Ωe /ωpe ) are illustrated. In this figure we have considered the signs ✷ for ωpe = Ωe = 3 × 1010 Hz and ◦ for ωpe = Ωe = 1 × 1011 Hz and the filled circle and filled box for Ωe = 1 × 1011 Hz = (10/3)ωpe and Ωe = 3 × 1010 Hz = (3/10)ωpe , respectively. The graph of solutions number of Eq. (29) versus to ωpe , is depicted in Fig. 9. As shown, the number of spectra is more sensitive to the plasma frequency than to the cyclotron frequency. We can find the number of spectra with respect to the plasma frequency for two arbitrary cyclotron frequencies in Fig. 9. In addition, variation of Eϕ /Eϕ(r=0) versus to r for frequency ω = 0.08 ωpe is presented in Fig. 10, where Eϕ( r=0) means the magnitude of Eϕ at r = 0.
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4. Summary and conclusion In this work starting with the general formulation of electromagnetic waves propagating in a cold collisionless annular magnetized plasma we showed that there are two independent modes of azimuthal electromagnetic surface waves propagating on an annular plasma column in a cylindrical metallic waveguide and perpendicular to the external magnetic field. These waves rotate around the symmetry axis with azimuthally and radially flux of energy. The dispersion relations and frequency spectra of the azimuthal surface B-modes and E-modes propagating on the annular magnetized plasma with field components (Ez , Br , Bϕ ) and (Bz , Er , Eϕ ) respectively, were obtained. It was shown that the number of solutions of the dispersion equation essentially depends on the external magnetic field strength, geometric dimensions of the waveguide, thickness of the annular plasma, and electron plasma frequency in both E-mode and B-mode cases. It was shown that in the HF frequency region the azimuthal surface E-modes on an annular magnetized plasma can appear in the frequency region between the upper hybrid frequency and higher cutoff frequency of extraordinary HF waves. Finally, we found that surface E-modes on an annular magnetized plasma can be considered as potential waves under the condition Ωe /ωpe 1.
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