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Transmission of microwaves through thin disks
Volume 70A, number 5,6
PHYSICS LETTERS
2 April 1979
TRANSMISSIONOFMICROWAVESTHROUGHTHINDISK!3 M. Ya. AZBEL' Department of Physics and Astronomy, Tel-AvivUniversity, Tel-Aviv,Israel Received 25 January 1979
Anomalous transmission of microwaves through thin metal disks is considered. It is related to the attenuation of micro. waves in a semi-infinite metal.
It is well known that electromagnetic waves attenuate inside a metal. In the situation of a normal skin effect (when the skin depth 6 is much larger than the mean free path l) in the absence of a magnetic field this attenuation is monotonical, exponential and similar for e.g., an electric field E and a current j. But an anomalous skin effect (8 Q 1) and/or a strong magnetic field (r < 1, r is the Larmor radius) provide anomalous wave penetration into the metal. A fraction of the current penetrates [l-4] to a depth of order 1; a strong magnetic field in the “static skin effect” situation (r < 6 Q I) implies a different attenuation of electric field and current [5], while in the situation of cyclotron resonance it provides [6] maxima of electric field and current inside the metal and selective transparency of thin films; in the situation of electron paramagnetic resonance a thin film is also anomalously transparent to microwaves [3]. Recent experiments (see, e.g., refs. [7-lo]) demonstrated the anomalous transparency of thin films to microwaves and suggested a quantitative comparison of experimental results and theoretical calculations. Meanwhile, some of the theoretical results refer to a semi-infinite metal. This paper allows one to apply these results to the determination of the microwave transparency of thin films, accounting for the microwave reflection from both surfaces. (For simplicity, we shall not account for the anisotropy.) Suppose c#J(Z)is the solution of the Maxwell equa-
tioninafIlmO
(1)
Then, obviously, $(d - Z) is also the solution of eq. (l), and thus we can present the electric field E(Z) as E(Z) = A@(Z) t B$(d - Z) ,
A = [E(O) -
E@#@1/ [1 - @2@>l = E(O),
B = VW) - E(O)@(d)] / [l - #2(d)]
.
(3) (4)
According to eq. (I), @(O)= 1. Eqs. (2)-(4) represent the electric field inside the film, 0 < Z < d; outside the film, E(Z) = Ehc exp(-iwZ/c)
+ Er exp(iwZ/c)
,
whenZ
,
Now the continuity E(-0)
= E(+O) ,
of E and E’ at Z = 0, d:
4II4llC
=
- cZ,/2n),
W,/‘W
[W)
(6)
determine E,, E, through
together with eqs. (2)-(5) Einc: ErlEinc = -(l
(5)
when Z > d .
E(d + 0) = E(d - 0) ,
9’(O) = -4nio/c2Zs ’ This work was initiated while staying at the Department of Physics of the University of Pennsylvania.
(2)
where A and B can be expressed through E(0) and E(d):
(7) +
@‘(4/4’@>1 ,
,
where Z, is the surface impedance
(8) and cZ, < 1. When 455
Volume 70A, number 5, 6
PHYSICS LETTERS
the thickness of the film is much larger than the skin depth, d than in the leading order ~/d approximation Z~and 0(Z) are equal to their values for a semi-infinite metal (where On E(Z)/E(0)). The latter were explicitly evaluated in the quoted papers ~ The described procedure is immediately generalized to an anisotropic situation. ~‘
~,
.
In the “static skin effect” situation, ref.
[51provides the
solution for a finite film.
References [11 G.E. Reuter and E.H.
Sondheimer, Proc. Roy. Soc. London A195 (1948) 336.
456
2 April 1979
[2] M. Ya. Azbel’ and E.A. Kaner, J. Phys. Chem. Solids 6 (1958) 113. (31 M. Ya. Azbel’ and I.M. Lifshitz, in: Progress in low temperature physics. Vol. Ill, ed. C.J. Gorter (Amsterdam, 1961) Ch. VII. [4] M.Ya. Azbel’ and M.I. Kagarov, Uchenyi Zapiski Kharkov Univ. 6 (1955) 59. [5] M.Ya. Azbel’ and S. Ya. Rakhmanov, Soy. Phys. Solid State 11(1970)2580. [6] Azbel’, Soy. Phys. 12 (1961) 283.Lett. 54A [7] M.Ya. J. Lebech, M. Surma and JETP K. Saermark, Phys. (1975) 211. [8) M. Surma, J. Lebech and K. Saermark, Solid State Commun. 17 (1975) 1359. [9] M. Surma, J. Lebech and K. Saermark, Solid State Commun. 20 (1976) 493. [101 M. Surma, Program and Abstracts of Intern. Conf. on Solids and plasmas in high magnetic fields (MIT, 1978) p. 20.
PHYSICS LETTERS
2 April 1979
TRANSMISSIONOFMICROWAVESTHROUGHTHINDISK!3 M. Ya. AZBEL' Department of Physics and Astronomy, Tel-AvivUniversity, Tel-Aviv,Israel Received 25 January 1979
Anomalous transmission of microwaves through thin metal disks is considered. It is related to the attenuation of micro. waves in a semi-infinite metal.
It is well known that electromagnetic waves attenuate inside a metal. In the situation of a normal skin effect (when the skin depth 6 is much larger than the mean free path l) in the absence of a magnetic field this attenuation is monotonical, exponential and similar for e.g., an electric field E and a current j. But an anomalous skin effect (8 Q 1) and/or a strong magnetic field (r < 1, r is the Larmor radius) provide anomalous wave penetration into the metal. A fraction of the current penetrates [l-4] to a depth of order 1; a strong magnetic field in the “static skin effect” situation (r < 6 Q I) implies a different attenuation of electric field and current [5], while in the situation of cyclotron resonance it provides [6] maxima of electric field and current inside the metal and selective transparency of thin films; in the situation of electron paramagnetic resonance a thin film is also anomalously transparent to microwaves [3]. Recent experiments (see, e.g., refs. [7-lo]) demonstrated the anomalous transparency of thin films to microwaves and suggested a quantitative comparison of experimental results and theoretical calculations. Meanwhile, some of the theoretical results refer to a semi-infinite metal. This paper allows one to apply these results to the determination of the microwave transparency of thin films, accounting for the microwave reflection from both surfaces. (For simplicity, we shall not account for the anisotropy.) Suppose c#J(Z)is the solution of the Maxwell equa-
tioninafIlmO
(1)
Then, obviously, $(d - Z) is also the solution of eq. (l), and thus we can present the electric field E(Z) as E(Z) = A@(Z) t B$(d - Z) ,
A = [E(O) -
E@#@1/ [1 - @2@>l = E(O),
B = VW) - E(O)@(d)] / [l - #2(d)]
.
(3) (4)
According to eq. (I), @(O)= 1. Eqs. (2)-(4) represent the electric field inside the film, 0 < Z < d; outside the film, E(Z) = Ehc exp(-iwZ/c)
+ Er exp(iwZ/c)
,
whenZ
,
Now the continuity E(-0)
= E(+O) ,
of E and E’ at Z = 0, d:
4II4llC
=
- cZ,/2n),
W,/‘W
[W)
(6)
determine E,, E, through
together with eqs. (2)-(5) Einc: ErlEinc = -(l
(5)
when Z > d .
E(d + 0) = E(d - 0) ,
9’(O) = -4nio/c2Zs ’ This work was initiated while staying at the Department of Physics of the University of Pennsylvania.
(2)
where A and B can be expressed through E(0) and E(d):
(7) +
@‘(4/4’@>1 ,
,
where Z, is the surface impedance
(8) and cZ, < 1. When 455
Volume 70A, number 5, 6
PHYSICS LETTERS
the thickness of the film is much larger than the skin depth, d than in the leading order ~/d approximation Z~and 0(Z) are equal to their values for a semi-infinite metal (where On E(Z)/E(0)). The latter were explicitly evaluated in the quoted papers ~ The described procedure is immediately generalized to an anisotropic situation. ~‘
~,
.
In the “static skin effect” situation, ref.
[51provides the
solution for a finite film.
References [11 G.E. Reuter and E.H.
Sondheimer, Proc. Roy. Soc. London A195 (1948) 336.
456
2 April 1979
[2] M. Ya. Azbel’ and E.A. Kaner, J. Phys. Chem. Solids 6 (1958) 113. (31 M. Ya. Azbel’ and I.M. Lifshitz, in: Progress in low temperature physics. Vol. Ill, ed. C.J. Gorter (Amsterdam, 1961) Ch. VII. [4] M.Ya. Azbel’ and M.I. Kagarov, Uchenyi Zapiski Kharkov Univ. 6 (1955) 59. [5] M.Ya. Azbel’ and S. Ya. Rakhmanov, Soy. Phys. Solid State 11(1970)2580. [6] Azbel’, Soy. Phys. 12 (1961) 283.Lett. 54A [7] M.Ya. J. Lebech, M. Surma and JETP K. Saermark, Phys. (1975) 211. [8) M. Surma, J. Lebech and K. Saermark, Solid State Commun. 17 (1975) 1359. [9] M. Surma, J. Lebech and K. Saermark, Solid State Commun. 20 (1976) 493. [101 M. Surma, Program and Abstracts of Intern. Conf. on Solids and plasmas in high magnetic fields (MIT, 1978) p. 20.