Microwave Open Resonator Techniques 40884

8/10/2019 Microwave Open Resonator Techniques 40884

1/28

Microwave Open Resonator Techniques

Part I: Theory

Giuseppe Di Massa

Additional information is available at the end of the chapter

http://dx.doi.org/10.5!/5151"

1.Introduction

This chapter is devoted to the theory of open resonators. It is well known that lasers usesopenresonators as an oscillatory system. In the simplest case, this consists of two mirrors

facingeach other. This is the first but not the only application of open resonators, whosesalientfeatures consist in the fact that their dimensions are much larger than the wavelength

and thespectrum of their eigenvalues is much less dense than that of close cavity.

The origin of Open Resonators can be dated to the beginning of the twentieth century whentwo French physicists developed the classical Fabry-Perot interferometer or Etalon [1]. Thisnovel form of interference device was based on multiple reflection of waves between twoclosely spaced and highly reflecting mirrors.

In [2] and [3] a theory was developed for resonators with spherical mirrors andapproximatedthe modes by wave beams. The concept of electromagnetic wave beams was

also introducedin [5, 12] where was investigated the sequence of lens for the guidedtransmission ofelectromagnetic waves.

The use of open resonators either in the microwave region, or at higher frequencies (optical

region) has taken place over a number of decades. The related theory and its applications

have found a widespread use in several branches of optical physics and today isincorporatedin many scientific instruments [6].

In microwave region open resonators have also been proposed as cavities for quasi-opticalgyrotrons[16] and as an open cavity in a plasma beat wave accelerator experiment [9].

The use of Open Resonators as microwave Gaussian Beam Antennas [10, 11, 18] provides a

very interesting solution as they can provide very low sidelobes level. They are based on theresult that the field map at the mid section of an open resonator shows a gaussian

distributionthat can be used to illuminate a metallic grid or a dielectric sheet.

#!01! Di Massa$ licensee %n&ech. &his is an open access chapter distributed underthe terms of the'reative 'ommons Attribution (icense)http://creativecommons.org/licenses/b*/".0+$ ,hich permitsunrestricted use$distribution$ and reproduction in an* medium$ provided the original ,or- is properl*cited.

Chapter 1Chapter 0
http://dx.doi.org/10.5772/51513http://creativecommons.org/licenses/by/3.0)%2Cwhichhttp://creativecommons.org/licenses/by/3.0)%2Cwhichhttp://creativecommons.org/licenses/by/3.0)%2Cwhichhttp://dx.doi.org/10.5772/51513


2/28

Micro,aveMaterials2

For microwave applications a reliable description of the coupling between the cavity and thefeeding waveguide is necessary. Several papers deal with the coupling through a small holeora rectangular waveguide taking into account only for the fundamental cavity andwaveguidemode [7, 8, 17].

In [4] a complete analysis of the coupling between a rectangular waveguide and an opencavity has been developed taking into account for all the relevant eigenfunctions in the

waveguide and in the cavity.

In this paper we review the general theory of Open Resonators and propose to study thecoupling between a feeding coupling aperture given by a rectangular or circular waveguide.

Starting from the paraxial approximation of the wave equation, we derive the modal

expansion of the field into the cavity taking into account for the proper coordinate system.

The computation of the modal coefficients takes into account for the characteristics of themirrors, the ohmic and diffraction losses and coupling.

2.Open resonator theory

2.1.Parabolic approximation to wave equation

A parabolic equation was first introduced into the analysis of electromagnetic wavepropagation in [13] and [14]. Since then, it has been used in diffraction theory toobtain approximate (asymptotic) solutions when the wavelength is small compared to allcharacteristic dimensions. As open resonators usually satisfy this condition, the parabolicequation finds wide application in developing a theory of open resonators.

A rectangular field component of a coherent wave satisfies the scalar wave equation:2 2

u/ k u0 0 (1)

wherek2/is the propagation constant in the medium.

For a wave traveling in the zeta forward direction, assuming anejttime dependance, we

put:

u0 )x,y,z+ejkz (2)

whereis a slowly varying function which represents the deviation from a plane wave. By

inserting (2) into (1) and assuming thatvaries so slowly withzthat its second derivativecan

be neglected with respect to1k1

1, one obtains the well know parabolic approximation to the1

wave equation: 2 2

x2/

y22jk

z0 0 (3)

The differential equation (3), similar to the Schrodinger equation, has solution of the type:

)k

2

where: 0 ej P/

2q

r

(4)

r20 x2/y2 (5)

2 Will-be-set-by-IN-TECH

1z


3/28

The parameterP)z+ represents a complex phase shift associated to the propagation of thebeam along the z axis,q)z+ is the complex parameter which describe the Gaussian beamintensity with the distancerfrom the z axis.

The insertion of (4) in (3) gives the relations:

dq0 1 (6)

dzdP j

dz0

q(7)

The integration of (6)

yields:

q)z2+ q)z1+z (8)

which relates the intensity in the planez2with the intensity in the planez1.

A wave with a Gaussian intensity profile, as (4), is one the most important solutions ofequation (3) and is often calledfundamental mode.

Eo1

0.8

0.6

0.4

Eo

W e

0.2

0

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5

r

Figure 1.Amplitude distribution of cavity fundamental mode

Two real beam parameters, R and w, are introduced in relation to the above complex

parameters q by1 1

q0R

j

w2(9)

Introducing (9) in the solution (4), we obtain:)

r2 2

0 ej P/

R

r

e w2 (10)

Now the physical meaning of these two parameters becomes clear:

Microwave!"e#$eso#atorTec%#i&'es()artI*T%eory +

E


4/28

R(z) is the curvature radius of the wavefront that intersects the axis at z;

w(z) is the decrease of the field amplitude with the distance r from the axis.

Parameter w is calledbeam radiusand the term 2wbeam diameter. The Gaussian beamcontractsto a minimum diameter 2w0at beam waist where the phase is plane. The beam

parameter qat waist is given by:

q00j

and, using (8), at distance z from the waist:

w0(11)

w0

Combining (12) and (10), we have:

q0 q0/z0j

/z (12)

R)z+z1(z

R2l

z

(13)

an

dw2)z+

w2 z

1/2

(14)

wherezRis the Rayleigh distance:

02

zR0

w2

w23

(15)

The beam contour is an hyperbola with asymptotes inclined to the axis at anangle:

0w0

(16)

In (14)wis the beam radius,w0is the minimum beam radius (called beam waist) where onehas a plane phase front atz0 0 and R is the curvature radius of of the phase front at z. Itshould be noted that the phase front is not exactly spherical; therefore, its curvature radius isexactly equal to R only on the z-axis. The parameter of the Gaussian beam are illustrated inFig. 2.

Figure .Parameters of Gaussian beam


0

0


5/28

Dividing (14) by (13), the useful relation is obtained:

w2 w2

(17)R

The expression (17) is used to expressw0andzin terms ofwand R:

w2

00 (18))

w22

R

Rz0)

R2

w2

(19)

Inserting (11) in (7) we obtain the complex phase shift at distance z from the waist:

dP

dz0 jz/j

w0

(20)

Integration of (20) yields

jP)z+

lg

1j

z

w2

0 lg 1/

z2

w2

jarctan

z

w2

(21)

The real part of P represent the phase shift difference4between the Gaussian beam and anideal plane wave, while the imaginary part produces an amplitude factor

w0which gives the

decrease of intensity due to the expansion of the beam. Now we can write the fundamentalGaussian beam:

w0 j)kz4+r2 1/

jk

where:

u)r,z+ew

w2 2R

z

(22)

4 0 arctanw2

(23)

2.2.Stability of open resonator

A resonator with spherical mirrors of unequal curvature is representable as a periodicsequence of lens which can be stable or unstable. The stability condition assumes the form:

)2l5) 2l

5

0< 11

12

< 1 (24)

The above expression was previously derived in [3] from geometrical optics approach based

0

w2

1

1

2

0 5

0

0

w

0

R R


6/28

on equivalence of the resonator and a periodic sequence of parallel lens and independently

in[5] solving the integral equation for the field distribution of the resonant modes in thelimit ofinfinite Fresnel numbers.

Microwave!"e#$eso#atorTec%#i&'es()artI*T%eory 5


7/28

To show graphically which type of resonator is stable and which is unstable, it is useful toplota stability diagram on which each type of resonator type is represented by a point (Fig.3).

Note that both thempandmpfunctions are orthonormal on the transverse planesz=cost.

When we assume that the mirrors are sufficiently large to permit the total reflection of thegaussian beams of any relevant order, we can put:

6mpq0 u)/+ )+

mpq/ umpq (27)

1To be consistent with the parabolic approximation the condition|m/ n/ 1|


8/28

whereu)/+andu

)+represent Hermite Gauss beams propagating from left to right and from

mpq mpq

right to left,respectively.

Resonance occurs when the phase shift from one mirror to the other is a multiple of. Using

(2), (4) and (9) this condition leads to:

kmpq2l2)m/p/ 1+

tan1

)l

5

zR

0 )q/ 1+ (28)

whereqis the number of nodes of the axial standing wave pattern and 2l>>zRthe distance

between the mirrors (Fig.4).

The fundamental beat frequency7f0, i.e. the frequency spacing between successivelongitudinal resonances, is given by:c

7f004l

(29)

where c is the velocity of light. From (10) the resonant frequencyfof a mode can be expressedas:

fmpq

7f00 q/ 1/

1)mp1+cos1(1

2l5

R

(30)

The combined use of eqs. (2),(4) and (9) yields:

6mpq)x,y,z+0 mp)x,y,z+cos kmpqz )m/ s/ 1+ tan1z

/

r q

l

(31)

zR R)z+ 2

Figure ".Spherical Open Cavity

In the paraxial approximation the eigenfunctions6spqsatisfy the normalization relation

rrr

cavity

6mpq68

nstdxdydz

0

fl,mpqnst;

0,mpqI0 nst.(32)

Microwave!"e#$eso#atorTec%#i&'es()artI*T%eory ,

2


9/28

1Micro,ave9penesonator&echni;ues Vnen (33)n

H > Inhn (34)n

According to the results reported in sect. 2, the expressions for they polarized modes are:

en0 mp)x,y,z+cos

kmpqz )m/p/ 1+tan

1z

/ r

ql

/ y

zR R)z+ 2

hn0 mp)x,y,z+sin kmpqz )m/p/ 1+

tan1

z/

rq

l

/ x

z

R

and similarly for thex polarized ones.

In (33) the indexnsummarizes the indexes (mpq).

From Maxwell equations we get for the mode vectors [15]:

R)z+ 2

and for the coefficientsVn,In

In0

j0

knhn0 en (35)

1rrn Eh8ndS (36)

k2k2l S

Vn kn 1

rr

n E

h8

ndS0 j

0

n

In (37)

k2k2lS

whereSis the cavity surface,8 denotes the complex conjugate and0is the free spaceimpedance. Note explicitly that the tangential electric field appearing in expression (36) and(37) is the actual field over S. This is not given by expression (33), which, at variance withexpression (34), does not provide a representation for the tangential components uniformlyvalid up to the cavity boundaries. Let us divide the surfaceSinto three parts: the couplingaperture A, the mirrorsMand the (ideal) cavity boundary external to the the mirrors,M (see

Fig. 4). Hence:

In0j0 1frr

rr

n Eh8ndS/

rr

n Eh8ndS/

n Eh8ndS

(38)

k2k2l A M M

Strictly speaking, in one of the integrals over the mirrors surfaces the coupling aperture


n

n

n

2

2


10/28

Micro,aveMaterials1

shouldbe deleted. However, the error that we make in extending the integration to the

whole mirroris negligible provided that the waveguide dimension is much smaller than thatof the mirrors.


11/28

The Leontovic boundary condition:

n E01/

j

n

H n

(39)

whereinis the electric conductivity of the mirrors and I 2is the penetration depth,

can be applied to express the (tangential) electric field over the mirrors in terms of magnetic

one, given by (16). Hence:rr

n Eh8ndS0 > Im1/jrr

hmh8

ndS021/j

nmIm (40)

M

being:

m

M

1rr

>

nm02

h8

nMhmdS (41)

Outside the mirrors we can assume that the field is an outgoing locally plane wave(Fresnel-Kirchhoff approximation), so that onM :

n E0

on H n

Accordingly, we have for the last integral in (38):

(42)

rr rr

n Eh8ndS0 o> ImM m

M

hmh8

ndS2o> Im)nm nm+ (43)m

The last equality in (43) follows from the fact that:

rr

MMhmh

8

ndSrr

z0lhmh

8

ndxdy

/

rr

z0

l

hmh8

ndxdy0 2nm (44)

according to the orthogonality condition satisfied by the

functions.The cavity quality factorQnfor the n-th mode is defined

as:

Qn0 nWn

Pn

(45)

whereWnis the mean electromagnetic energy stored in the cavity andP

nis the power lost

when only the nth mode is excited at the resonance pulsationn. The power is lost due todiffraction and ohmic losses.

By taking (32) into account, we can express the diagonal term in (40) as a function of the

quality factor for the ohmic losses,Qrn,:rr

m


12/28

2nn0 |hn|2dS0

n0nl (46)

M rn

beingnthe skin depth at the resonant frequency.

The diffraction losses of a cavity can be calculated by taking into account for the diffractioneffects produced by the finite size of the mirrors. Under the simplifying assumption of

Microwave!"e#$eso#atorTec%#i&'es()artI*T%eory

Q


13/28

Micro,aveMaterials

quasi-optic nature of the problem (dimensions of the resonator large compared towavelengthand quasi-transverse electromagnetic fields) the Fresnel-Kirchhoff formulationcan be invokedfor the diffracted field from the mirrors. Hence we have for the diffractionloss of the nthmode:

Pdn0 Re)rr 1 5EnH8n n

dS 0

rr

0|In| |hn|2ds (47)

M2 2 M

and for the corresponding quality factor for the diffraction losses,Qd,:

orr

2 n0

l M|hn| dS0 Qd

n

(48)

Byusing(40,41,44,45)andtakingintoaccountthat0>>1andn/ 1forallrelevantmodes,equation(38)becomes:

In ) j01

)rr

Ehn ndS2

0>,Im

nm (49)

k2k2/kkn

jkkn l A m

wherein>, >nI0m

n

and:

Qr

n

QTn

1)1 1

5

0 /

(50)

QTn Qdn

Qrn

A metallic waveguide is assumed to feed the cavity. The waveguide field on the couplingapertureAis represented as:

Eg

> Vgeg

(51)

n nn

Hg

> Ighg

(52)n n

n

whereegandhgare TE electromagnetic modes of the waveguide:

Assuming that the mirror curvature can be neglected over the extension of the couplingaperture,fields (51, 52) verify the following orthonormality relation:

rr eg gnhm zdS0 nm (53)

A

Expressing the field over the coupling aperture A by means of expression (52) we obtain from(3):

where:

,

In2Fn> nmImm

Fn

0>

nmVg

(54)

Fn )jk/

lk2k2/

kkn jkkn

(55)

and:n Qrn

rr

QTn

rrnm0 e

gh8 n dS

0 h 8 hdS. (56)

8

mm

g

1 2


14/28

Micro,aveMaterialsm n n m

A A

By introducing the matricesAandB, whose elements are:

f1

anm0n=m

2nmn

I

0 m.

(57)


F


15/28

andnmrespectively2, and the vectorsI {In} andV

g {Vg

}, relation (3) can be written

in a compact form as:0A I B V

g B

)V V

+(58)

whereinV

/

andV

are the vectors of the incident and reflected waveguide modeamplitudes

respectively. By enforcing the continuity of the magnetic field tangential component over thecoupling aperture, we get:

/I0 Ig

01

10

)V

V

+

(59)

whereinB/is the adjoint (i.e., the transpose, beingB a real matrix) ofB and is thediagonalmatrix whose elements are the modes characteristic impedances, normalized to0.

From (48)and (49) we immediately obtain:

(I A1B B2

I0 V (60)0)

I B A1 0 (I /

/

A1

/

(61)

whereinIis the unit matrix andA1the inverse of the matrixA.

Solution of eq. (60) and (61) provides the answer to our problem. In particular, from (61) weget the (formal) expression for the feeding waveguide scattering matrixS:

S (I B A1

1

B

)

I B A1

(62)

4.Field on the coupling aperture

In order to solve the system (60-61) a suitable description of the field on the aperture A is

necessary. Any device, able to support electromagnetic field matching the cavity field on themirror (33-34), can be used to feed the cavity. In the following, the case of metallic andcircularwaveguide will be treated in detail.

4.1.Modes in rectangular waveguide

A rectangular metallic waveguide, with transverse dimensionsa b, is assumed to feed thecavity. The waveguide TE electromagnetic modes of the metallic rectangular waveguide, onthe coupling apertureA, is represented as:

hn0 hpq0

k

1

tp

q

/4pq

ab (63)

f

pasin p

)

x

aa cos2

q)

y/

b

b5x/

2

qcos

b

p)

xa


n

A1

B

B

B

V

B

V B

B

ro,ave9penesonator&echni;ues< =art%:&heor* 1"


16/28

a sin2 q

)b

5y

/y b 2

2Note explicitly thatnmandnmare real quantities, as both the cavity and waveguide mode vectors are real.Moreover

nm 0 mnso that the matrixAis symmetric.


17/28

Micro,ave9penesonator&echni;ues

From (79,80) we immediately get:

2

n I00

01V/

0

(80)

2F0n0n

V

n0@1n/

F0>k

0kk

A V1

(81)

HenceV

2 2

B 01

1

1/ F0101F0>kI01k0k

1F0>kk2

(82)

From (82) we get for the equivalent terminal impedance relative to the fundamental mode:

1Vg

1

0

1

01

2

V

0

1

n

1


20/28

1 B 0)

0k 52

Z 011 B

2

0>

kI

1

01k (83)

Microwave!"e#$eso#atorTec%#i&'es()artI*T%eory 1+

01


21/28

Taking into account for the expression (55) forF0, we get the equivalent circuit representationof Fig. 5.

Figure #.Equivalent Circuit

where the explicit expression for its elements are collected under Table 1.

L00l)H+ Cl/)k0l+2(F)

L0 )H+ R, 2/ (C)

Ta$%e 1.Expressions for the circuit elements of Fig.5

The value ofLedepends on the feeding waveguide and is reported in the following

subsections for rectangular and circular waveguides.

5.1. Rectangular waveguide

The cavity is assumed to be excited by the incident fundamentalTE10mode. Fromexpressions(51-52) and (63) follows that in the cavity are excited only the (0,0,q) mode, in

the feedingwaveguide are excited the modesTEn0.

When the cavity is fed by a rectangular waveguide under the approximation 3), the expression(56) for0kcan be explicitly evaluated, leading to:

4 1/

ab

0n0 n w2

n0 1, 3, (84)

Hence, for the sum at the right hand side of (83) we have:

0 > ) 0

k 52

D 1 1kj0> 2/

k03,5, 01

)2a

5 1

k0)2k3+

1)

)2k3

2

2a 1

15 l

j00ja

k0)2k3+

3

18

)3+

1

0jLe (85)

wherein)+ denotes the Rieman zeta function. From equation (85), we have the value ofLe:

Le0 16.5 1030aEhenryF (86)


>


22/28

5.2.Circular waveguide

The cavity is assumed to be excited by the incident fundamentalTE11mode. From

expressions(51, 52) and (69) follows that in the cavity are excited the (0,0,q) mode, in

the feedingwaveguide are excited the modesTE1r.When the cavity is fed by a circular waveguide under the approximation 3), the expression

(56) for0kcan be evaluated (Appendix A), leading to:

20r

% 1

q,2

1J)q,

1

+ w)l+ EI1I2F (87)

prp2ppr

q,prr a 2 l 2

, , w2)l+ 1 l q

I10

a

J1)kt1r+e0

sin

2

l

k00qltan

/ /zR R)l+ 2

d (88)

r

aI20 J1

)k,+e

w2

)l+

sin k00qltan1 /

q

l d (89)

0t1r zR R)l+ 2

allowing the computation ofLe.

Author details

Giuseppe Di Massa

University of Calabria, Italy

Appendix

A.Hermite polynomials

The Hermite polynomials are defined as:

Hn)x+ )1+nex

The differential equation:

2 n

xn

ex2

(90)

2y y

x22x

x/ 2y0 0 (91)

admits as solution the Hermite polynomialHn)x+. For the the Hermite polynomials thefollowing orthogonal relation holds:

r

x2

f 2nn!, n=m;

e 2Hn)x+Hm)x+D 0, nI0

m.

(92)

In the following some particular values with the recursion relation are reported:

H0)x+ 1

H1)x+ 2x

H2)x+ 4x2

2


l

l 2


23/28

H3)x+ 8x3

12x

H4)x+ 16x4

48x212

Hn12xHn)x+ 2nHn1)x+

B.Coupling cavity - circular waveguide

The x component of the magnetic field, for TE modes, in the waveguide is:

/

xpr0 p 1%

q,

2

1J)q,

+ (93)

q,pr prp

2ppr

p

aJp

, )k, + coscos)p+

Jp)k, +sinsin)p+

The x component of the magnetic field in the cavityis:

z r2 ql

where hxn0 mp)x,y,z+

sin

kmpqz )m/p/ 1+ tan1

/R

R)z+

2

(94)

1

mp)x,y,z+

w)z+

2H

2m/pm!p|

)2 xH w

)2yw

ex

p r2

l

w2)z+

(95)

According to the position of sect.4 we consider the mode (0,0,q) in the cavity, so the equation(95) reduces to:

1

00)x,y,z+

w)z+and the equation (94) on the mirror (for z=l):

/

2exp

r2lw2)z+

(96)

1/2

r2 2l

w2)l+

1 l r qhxn0 w)l+

e

sink00qltan

/ /zR R)l+ 2

(97)

rr0m0

Ah8h

gdS (98)

2

%

1

prp2

1

Jp)q,pr

tp tp

m

0

q,2

z

p


24/28

1

+w)l+ (99)

rr

q,

prAa

pp)ktpr+ coscos)p+

Jp)ktpr+ sinsin)p+ (100)

J, , ,

2 l 2

ew2)l+

sinkl

tan1 l/

ql

/ dd (101)

00q zR R)l+ 2



25/28

where:

2

0r

%

1

prp2

1

Jp

)q,pr

1

+ w)l+ EI1I2F (102)

s

in

)2p

q,pr

2l

lq

l

I10p

p2

J,)k,

+ew2

)l+

sink

l tan1

//

z

d

(103)

1aptpr0

2

l

00qR R)l+2

I20 sin)2p+

r a

pp2

Jp)k,t

pr+e

w2

)l+

sin

k00ql

tan1

/

ql

d

(104)

1

0

z

R

R)l+

2

that are not equal to zero for p=1,giving:q,pr

r a 2 l 2,w

1 l

q

I

0

a

J

1)k

t

1

r+e

0

sink00ql

tan

//

zR

R)l+

2

d(105)

lr a

l

I20 J

1

)k,

+e

w2

)

l+

sin

k00ql

tan1

/

/q

d(106)

0t1r zR R)l+ 2

Microwave!"e#$eso#atorTec%#i&'es()artI*T%eory 1,

,

l

2

l

2

l 2


26/28

6

.

R

ef

e

r

e

n

c

e

s[1]C.Fa

bry

andA.Per

ot,(1899),Theo

rieetApplication

sduneNouv

lleMethodd

eSpectrosco

pieInt

erfrentielle,

AnnChim.P

hys.,Vol.

7,Vol.16

,115-

143.

[2]G

.

D.

B

oy

d

a

nd

J

.

P.

Go

r

do

n,

(

19

61

)

,

Con

f

oc

al

m

u

lti

m

od

e resonator for millimeter

through outical wavelengthmasers,Bell Sys. Tech. J.,Vol.

40, 489-508.

[3] G. D. Boyd and H. Kogelnik,

(1962), Generalized confocal

resonator theory,Bell Sys.Tech. J., Vol. 41, 1347-1369.

[4] O. Bucci, G. Di Massa, (1992),Open resonator powered byrectangular waveguide,IEEProceedings-H, Vol. 139, 323-329.

[5] P. O. Clark, (1964). A selfconsistent field analisys of

Spherical-Mirror Fabry PerotResonators, Proc. of theIEEE, Vol. 53, No. 1, 36 - 41.

[6] R. N. Clarke, C. B.Rosemberg, (1982). Fabry-Perot and Open Resonatorsat Microwaveand Millimeterwave frequencies, 2-200 GHz,J. Phys. E: Sci. Instrum.,Vol.15, 9-24.

[7] A. L. Cullen, P.R. Yu, (1979),

Complex source-point theoryof the electromagnetic openresonator,Proc. R. Soc Lond.A., Vol. 366, 155-171.

[8] G. Di Massa, D. Cuomo, A.Cutolo, G. Delle Cave (1989),Open resonator formicrowaveapplication,IEEProc. H,Vol. 136, 159-164.

[9] G. Di Massa, R. Fedele, G.Miano, 1990, C. Nappi(1990), A beat waveexperiment in openresonator, Physica Scripta,Vol. T30, 122-126.

[10] G. Di Massa,L. Boccia,G.Amendola, (2005), AGaussian beam antenna

based on an openresonator,The European Conference onWireless Technology, Paris,

France.[11] G. Di Massa,L. Boccia, G.

Amendola, (2005),Gaussianbeam antennas based onopenresonator structures,28th ESA Antenna Workshop


27/28

onSpaceAntenn

aSystemsa

ndTechnolo

gies.[

1

2

]

H.

K

o

g

elnik,

T. Li,

(1966)

,

Laserbeam

and

reson

ator,

Proc.

IEEE,

Vol.

54,

1312-

1329.


28/28

! Micro,aveMaterials

[13] Leontovich M.A.(1944), Statistical Physics, OGIZ, Moskow.

[14] Leontovich M.A., Fock V. A. (1948), Investigation of Radio Wave Propagation,ed. B.A.Vendeski, AN SSSR, Moskow, 13-39.

[15] K. Kurokawa, (1969),An introduction to theory of of microwave circuits, Academic

Press, New York.[16] A. Perrenoud, M.Q. Tran, B. Isac, (1986), On the design of open resonator forquasi-

optical gyratron,Int. J. of Infrared and Millimeter Waves, Vol. 7, 427-446.

[17] V. N. Rodionova, A. Ya. Slepyan, G. Ya. Slepian, (1991) Oliner model for quasiopticalresonator to rectangular waveguide coupling elements,Electronic Letters, Vol. 27,1427-1428.

[18] R. Sauleau, P. Coquet, D. Thouroude, J.P. Daniel, and T. Matsui, (2003), RadiationCharacteristics and Performance of Millimeter-Wave Horn-Fed Gaussian BeamAntennas,IEEE Trans. Antennas and Propagation, Vol. 51, 379-386.


Documents

Microwave Open Resonator Techniques 40884