Dynamic Modes in Open Gyromagnetic WaveguidesPietro de Santis and Giorgio Franceschetti Citation: Journal of Applied Physics 43, 2012 (1972); doi: 10.1063/1.1661442 View online: http://dx.doi.org/10.1063/1.1661442 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/43/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Robust one-way modes in gyromagnetic photonic crystal waveguides with different interfaces Appl. Phys. Lett. 97, 041112 (2010); 10.1063/1.3470873 On the use of leaky modes in open waveguides for the sound propagation modeling in street canyons J. Acoust. Soc. Am. 126, 2864 (2009); 10.1121/1.3259845 Microwaves in gyromagnetic waveguides J. Appl. Phys. 89, 535 (2001); 10.1063/1.1323748 Concept of TransverseMode Pattern in Gyromagnetic Waveguides J. Appl. Phys. 41, 2867 (1970); 10.1063/1.1659329 NONMAGNETOSTATIC VOLUME AND SURFACE WAVE MODES ON GYROMAGNETIC YIG RODWAVEGUIDES Appl. Phys. Lett. 16, 194 (1970); 10.1063/1.1653158

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6V. T. Buchwald, Proc. Roy. Soc. (London) 253, 563 (1959). 7M.J. Lighthill, Phil. Trans. Roy. Soc. (London) 252A, 397 (1960).

BE.P. Papadakis, J. Acoust. Soc. Am. 40, 863 (1966). 9M. G. Cohen, J. Appl. Phys. 38, 3821 (1967).

1"E. Gates, J. Acoust. Soc. Am. 41, 118 (1967). lIM.S. Kharusi and G. W. Farnell, J. Acoust. Soc. Am. 48,

665 (1970). 12See, for example, R. Courant, Differential and Integral Cal­

culus (Interscience, New York, 1959), Vol. I, p. 281.

Dynamic Modes in Open Gyromagnetic Waveguides*

Pietro de Santis and Giorgio Franceschetti Istituto Elettrotecnico, Universitd di Napoli, Napoli, Italy and Istituto Universitario Navale, Napoli, Italy

(Received 6 December 1971)

It is shown that the "dynamic modes" of a ferrite rod in vacuum are dielectric modes modi­fied by the presence of the induced magnetic anisotropy. A number of inconsistencies existing in the literature are pointed out.

The exact numerical analyses by Schott, Tao, and Frei­brunI on nonmagnetostatic volume and surface wave modes in open gyromagnetic waveguides (e. g., ferrite rods in free space) showed the existence of a new family of modes. These modes called "dynamic modes,,1 exist for (w~ +wowm)I/2 <w <00 (w, wo, and wm are the opera­tion, ferromagnetic resonance, and saturation magneti­zation angular frequency, respectively), and their dis­persion curves on a Brillouin diagram always display positive slope without any resonant frequency.

An experimental verification of above results was re­ported by Tao, Tully, and Schott2 using both polycrystal­line and single -crystal rods.

Since (i) these modes are not predicted by a quasistatic analysis, (ii) no other structure contiguous to the fer­rite rode exists to support them, (iii) no dual modes ex­ist in a gyroelectric open waveguide such as a gaseous plasma column in free space (see, for example, the numerical results by Likuski3), one is left somewhat puzzled as to their phYSical origin.

In the present work we will show that (a) they are the dielectric waveguide modes supported by an isotropic cylinder of dielectric constant Ef (Ef being the dielectric constant of the ferrite material) modified by the pres­ence of the induced magnetic anisotropy, (b) no gyro­electric counterpart exists in a gaseous plasma column because no dielectric waveguide mode exists for W - 00, and (c) a gyroelectric counterpart does exist in a semi­conductor rod because, for w - 00, it reduces to an iso­tropic dielectric cylinder of dielectric constant EL (EL being the semiconductor's lattice dielectric constant). Furthermore, on the basis of these results, we will show that some of the dispersion curves existing in the literature display in incorrect behavior.

Let us consider a generalized gyrotropic circular wave­guide of radius R referred to a coordinate system of

TABLE I. Dielectric and permittivity constants for considered materials as w _00.

Ferrite J.l2 = 0; J.ll =J.l3 =1-'0 E2 = 0; El=E3= Ef

Gaseous plasma J.l2 = 0; 1-'1 =1-'3=1-'0 E2 = 0; "1 = "3= EO

Solid-state plasma 1-'2 = 0; 1-'1 =1-'3=1-'0 E2=0; El=E3=EL

J. Appl. Phys., Vol. 43, No.4, April 1972

cylindrical coordinates (p, e, z), the z axis being coin­cident with the waveguide axis.

Assuming the z axis to be the gyrotropy axis, the dielec­tric and magnetic tensors have the form

11-1 -j11-2 0

W= jl1-2 11-1 0' (1)

o 0 11-3

where Ef and I1-f are real quantities independent of coor­dinates. Let the surrounding medium be vacuum (dielec­tric permittivity Eo, magnetic permeability 11-0)'

Assuming an exp(j(ne - ,8z)] space dependence for the field components, and following Van Trier's notations, 4

the characteristic equation for the structure under con­sideration is

MODIfIED DIELECTRIC MODES

c

--7~----------l-""'-__ MODIFIED MAGNETOSTATIC MODES

f3R

FIG. 1. DisperSion diagram for a ferrite rod in vacuum.

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J"(O'IR) In(O',fi) -Kn(I1 IR) 0

(c1;;a) In(I1IR) (~ ; aj I n(I1,fi) 0 -Kn(I1IR)

nf3 ) - jw MQ K'(11 R) =0,

Sl(R) S2(R) 2jfKn(O'/R 0'1 0'1 n I

(2 )

S3(R) S4(R) :l!:!!§ K~ (0' IR) nf3 cf.R Kn(I1IR)

111 1

where In(x) are the Bessel's functions of the first kind and Kn(x) the modified Bessel's functions of the second kind. 111 ,2 and al are the transversal wave numbers in the gyrotropic medium and in vacuo, respectively. SI(R), in terms of Van Trier's notations (see formulas 2.103 in Ref. 4) read

S (R) jn ( !.a....2) ( ) jWMg +s<fg '( ) l =bR -WM3

E1 -rll2 I n aIR + b 111Jn 111R , S (R) P!( !a ...2) ( ) jW/ls +s~ '( )

2 = bR - WM3€I - rlTi In 11,fi + b 0'2Jn a2R ,

S3(R) = ~~ (-j{3~ -p~) In(a1R) + t ~alJ~ (0'1R ) , SiR)=~~ (-jf3~ -p~) J,,(aaR) +*~a1J~(O'aR), (3)

where the prime indicates differentiation with respect to the argument and quantities a, b, r, p, q, s are defined in (2.11) and (2.15) or Ref. 4.

When W - 00, the situation depicted in Table I occurs and Eq. (2) reduces to:

In(aR) 0

0 I n(I1R)

n~ ) u2R In(uR -jw/lo J'(I1R)

u n

jW(€t or EL ) ~(I1R) nf3 u2R I n(I1R)

0'

where

0'= 0'11 w~., = 0'21 w~ .. == (W 2 Mo(Ef or E L) - )32)1/2.

Equation (4) is the characteristic equation for a dielec­tric rod of dielectric constant either Ef or EL'

It is then demonstrated that for W - 00, the dielectric modes are the normal modes for both the ferrite and the semiconductor plasma rod in vacuum.

As a consequence, the follOwing conclusions can be drawn:

(i) The (dual) similarity introduced in Ref. 2 between the "dynamic modes" in ferrite rods and the surface wave modes in magnetogaseous plasma columns studied by Granatstein and Schlesinger5 cannot be established, be­cause in the latter structure no modified dielectric modes exist. In Ref. (5), Granatstein and SchleSinger label as "dynamic modes" a class of resonating modes, which disappear for frequencies higher than (w; +w~)1/2 (WI> and we are the electron plasma and cyclotron fre­quency, respectively). In our opinion, it is somewhat misleading to adopt the same denomination of "dynamic modes" for two completely different classes of modes. The "dynamic modes" in a ferrite rod could be more properly named "modified dielectric modes" (analogous to "modified waveguide modes" in gyrotropic metal waveguides) .

(ii) For a ferrite rod in vacuum the "dynamic modes" must exhibit dispersion curves on a Brillouin diagram

-Kn(O',R) 0

0 -Kn(O'IR )

nf3 - jw./lo K' (0' R) 0, (4)

TRKn(uIR) u n 1 1 I

jW€o K'(O' R) n{3 RKn(O'IR ) 0' "I 0'1 1

which are comprise between the limit straight lines w/f3==c and w/f3=C/(Ef )1I2, c being the velocity of light in vacuum. In particular, the latter line must represent an oblique asymptote for all the "dynamic mode" dis­perSion curves. Therefore, in order to comply with this requirement, some of the dispersion curves in Fig. 4 and 6 of Ref. 1 must be corrected.

To continue with our analysis of the dispersion curves for a ferrite rod in vacuum, we notice that as W is low­er.ed toward the hybrid frequency line W == WI == (w~ + Wo

XW",)l f2 , the dielectric modes become more and more modified by the presence of the magnetic anistropy, in the same manner as the waveguide modes do in a metal waveguide filled with ferrite. At W = WI' the "dynamic mode" dispersion curves terminate at discrete cutoff points, where it can be rigorously demonstrated that CJw/CJf3==O (horizontal tangent). 6

Figure 1 shows what the correct dispersion diagrams look like. Note how the addition of the w/f,=C/(f)1I2 line to the diagram adds phYSical inSight to the propaga­tion phenomenon, inasmuch as it evidences the modified dielectric mode region for W > WI and the modified mag­netostatic mode region for W <Wi'

*Work sponsored by the Italian Consiglio Nazionale delle Ricerche.

IF.W. Schott, T.F. Tao, and R.A. Freibrun, J. Appl. Phys. 38, 3015 (1967).

J. App!. Phys., Vol. 43, No.4, April 1972

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2T.F. Tao, J.W. Tully, andF.W. Schott, Appl. Phys. Letters 14, 106 (1969).

3Likuski R., University of Illinois, Electrical Engineering Research Laboratory, Technical Documentary Report No. AL-TDR-64-157, (1964) (unpublished).

4For the sake of brevity we shall, henceforth, follow Van Trier's notations, and, for a more detailed definition of the

symbols, we refer the reader to his work "Guided electromag­netic waves in anisotropic media" [Appl. Sci. Res. 3 B, 305, (1953)].

5V. L. Granatstein and S. P. Schlesinger, J. Appl. Phys. 35, 2846 (1964).

6p. De Santis and G. Franceschetti, IEEE Trans. Microwave Theory and Techniques (to be published).

Behavior of a Pulsed-Discharge Laser with an Intracavity Absorber* Paul L. Houston, t David G. Sutton, f and Jeffrey I. Steinfeld§

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 3 May 1971; in final form 13 December 1971)

The asymmetrical pulse usually obtained from a transversely discharged medium-pressure laser may be transformed into a narrower pulse with a higher peak by the incorporation of an absorber into the laser cavity. The experimental results are well predicted by a pre­viously described model for passive Q switching.

Several authors have obtained high peak power from transversely discharged, high-pressure CO2 lasers .1-5

Many of these authors3- 5 report that the laser emits an asymmetrical pulse consisting of a sharp spike fol­lowed by a longer tail, particularly at lower pressures (0.1 atm or so). When a pressure of 1 atm or above is used in the laser discharge, this "afterpulse" is es­sentially eliminated. In this communication, we show that a much narrower pulse, with 2-3 times the peak intensity, is produced by the incorporation of an ab­sorber into the cavity. Furthermore, this phenomenon can be predicted adequately by a simple model used previously to describe passive Q switching in cw CO2

lasers.6

The transverse-discharge laser consisted of a dc power

12 f- -

10f- -

-

6f-( b)

-

4f- (0 ) -

2f- -

FIG. 1. (a) Experimental laser pulse with no SFs; (b) experi­mental laser pulse with 289 mTorr of SFs in cavity absorption cell. Time base, approximately 30 ,",sec full scale.

J. Appl. Phys., Vol. 43, No.4, April 1972

supply charging a 0.033-IlF capacitor to 20 kV. A spark gap discharged this voltage transversely along the laser tube. The cathode was formed by the pins of 130560-0 resistors spaced 1 cm apart, while the anode was made of a long copper bar. A totally reflecting mirror and a grating formed the cavity. With 25 Torr each of N2 and CO2 , and 100 Torr of helium, oscillation was obtained on any of the lines between P(6) and P(34) of the 00°1-1000 transition. The radiation passed through a Bausch & Lomb O. 5-m monochromator onto a Santa Barbara gold-doped germanium detector. The electrical signal was RC coupled to a Tektronix 553A oscilloscope with a type lA7A plug-in amplifier.

Sulfur hexafluoride was used as the intracavity absorb­er. The ll-cm absorption cell was filled to the desired

18

16

! 14

Q )( 1 2

>-+-

If)

10 (b) c: .,

0

c: 8 0 +-0 .c a.. 6

4

2

FIG. 2. (a) Calculated laser pulse in the absence of a saturable absorber; (b) same, with absorber density corresponding to 380 m Torr SFs. See text for discharge pumping furtction, and Ref. 6 for relaxation parameters used in the calculation. Time base as in Fig. 1.

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