BIBLIOTHEEK TU Delft P 1979 4358
DIFFRACTION THEORY OF OPEN RESONATORS
H. BLOK
ERRATA
In (8.2) replace "S " by
"V
".
In (8.11) omit " = 0" and replace "on M^" by "on Mg".
In (8.12) omit " = 0" and replace "on M^" by "on M^".
"(i+Oc/axg)^}^" " {MH/^s:^f}l "
In (8.13) replace
^-r
by r—r
{^+{^^/^x^r}^
{i+Oc/aa;^) }^
In (8.27) replace "
z^iz^)
= "
W'i^ + iH^/^x^frh^ix^)
= ".
K
^1
In (9.7) replace " /_^ " b y " / J ".
In (I0.lt) replace "+
i{ka^/d)
" by " + |(fea^/d)".
In Fig. 11, pp.
h9
replace
"v^"
by '\_/'.
In (10.15) replace "
a{k) "
by " {a(fe)}"'' ".
In (10.16) replace "
ex^iikd)
" by "
exj>{ikd - i-n/k)
".
On page 53 replace " Re[/c^°^) = (2n+l)ïï/<i " by " Re(fe^°^) = (2n-l)ir/d ".
On page 5^ replace " to be (f = 1 meter. " by " to be d = 1 meter and the
case $ = 0 (associated with the first boundary value
problem) is considered. " •
1
In (11.1) replace " Re(fe) ^
K{x^/x^)
" by " ( ^ f ^ ) ^
K{x^\x^)
".
2'nd
In Fig.22, pp. 65»the radius of curvature of the curved mirrors =
d.
In (13.5) replace " when
~b < z < b,
" by " when -Z? < 2 < i>, 0 « 9 <
2TT,",
In (13.13) omit " (-1)" ".
In (I3.2U) replace " r cos(e) " by " r cos(e) ".
In (17.3) replace " - 2 ff-^ " by " - 2 ^f^^ " and replace
" ^(2:21^1)^ " ^y " ^(1^21^)^ "•
In (17.6) replace " {1 + O c ^ a x ) ^ +
iHj^yf}'^ "
by
" (1 + O ? /9a
f +
(3^ /ay
f}'^
".
STELLINGEN
I
Bij het berekenen van de eigenschappen van een cilindrische monopool-antenne is het in de literatuur gebruikelijk de monopool-antenne te vervangen door een cilindrische buis. Het is echter zonder veel extra complica-ties mogelijk uit te gaan van de integraalvergelijking voor de stroom-verdeling op een massieve antenne.
CHANG, D.C., IEEE Trans. AP, vol.l6, 1968, pp.58-64.
I I
Bij de interpretatie van de gemeten Doppler verschuiving van elektro-magnetische golven die tegen de ionosfeer reflecteren, kan een "Epstein model" van de ionosfeer een goed hulpmiddel zijn.
DAVIES, K. and D.M. BAKER, Radio Science, vol.l, 1966, pp.545-556; BLOK, H., Appl.Sci.Res., vol.l7, 1967, pp.331-339.
I I I
Bij de berekening van de verstrooiing van golven door een obstakel kan toepassing van het reciprociteitstheorema tot een stelsel vergelijkin-gen leiden dat wat numerieke oplosmethoden betreft, mogelijk voordelen biedt boven de gebruikelijke integraalvergelijkingen.
IV
De door SCHENCK aangevoerde overwegingen ten aanzien van de niet-een-duidigheid van de integraalvergelijkingsfonnulering voor akoestische stralingsproblemen zijn nodeloos gecompliceerd.
S C H E N C K , H.A., J.Acoust.Soc.Am., vol.44, 1968, ppAlAi.
V
De door BATES opgestelde vergelijkingen voor de berekening van de af-snijfrequenties van modi in golfpijpen met willekeurige dwarsdoorsnede zijn op veel eenvoudiger wijze te verkrijgen door gebruik te maken van het reciprociteitstheorema.
BAJES, R.H.T., IEEE Trans. MTT, vol.lT, 1969, pp.294-301.
VI
De door MARCATILI gegeven theorie over de propagatie van modi in een systeem van periodiek opgestelde dikke, astigmatische, lensachtige, focuserende elementen is niet correct.
MARCATILI, E.A.J., Bell System Tech.J., vol.XLVI, 1967, pp.2887-2904.
VII
Teneinde tot berekeningen aan een model van een actieve Fabry-Perot resonator te komen, kan de in dit proefschrift gegeven theorie worden uitgebreid tot een configuratie waarbij tussen de spiegels een oneindig uitgebreide plak aktief materiaal wordt geplaatst.
V I I I
Het i s aan de hand van een goede t h e o r e t i s c h e formulering n i e t in t e z i e n , welke a n a l o g i e i n eigenschappen e r t u s s e n de open F a b r y - P e r o t r e s o n a t o r en de b u n d e l g o l f g e l e i d e r van h e t i r i s - t y p e zou b e s t a a n .
LOTSCH, H.K.V., Physics Letters, voLll, 1964, pp.221-222.
IX
De in dit proefschrift gegeven exacte scalaire theorie resulteert in het geval van een open resonator bestaande uit twee gelijke, parallel-le, vlakke spiegels in een homogene integraalvergelijking van de eerste soort. Toepassing van de momentenmethode op deze integraalvergelijking door de onbekende functie te ontwikkelen naar een compleet stelsel functies, die ieder apart aan de "Kantenbedingung" voldoen, leidt voor parameters in het optische gebied tot een numeriek zeer slecht conver-gerend proces.
OOMS, G., AppLScURes., voL19, 1968, pp.198-212.
X
Bij het beraad omtrent vernieuwing van het programma van het eerste studiejaar van de studierichting der Elektrotechniek kan het in de
"Verkenningen in het eerste studiejaar teahnisohe natuurkunde" gerap-porteerde van groot belang zijn.
VASTENHOUW, J., W.M. van WOERDEN en ,A.D. WOLFF-ALBERS, Verkenningen in hel eerste studiejaar
technische natuurkunde. Sectie onderzoek, onderwijskundige dienst van de Technische Hogeschool Delft,
DIFFRAQION THEORY OF
OPEN RESONATORS
DIFFRAQION THEORY OF
OPEN RESONATORS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE
HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR.IR. C.J.D.M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT
TE VERDEDIGEN OP WOENSDAG 13 MEI 1970 TE 16 UUR
/Q1Q 9iS-J>
DOORHANS BLOK
ELEKTROTECHNISCH INGENIEUR
GEBOREN TE ROTTERDAM
9'^?
Bl/LIOTHELK
TECHNlSCHi: HOGESCHOOL
D£LFr
1970 DRUKKERIJ BRONDER-OFFSET N.V. ROTTERDAMDIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR
Aan mijn Ouders Aan Aartje,
CONTENTS
CHAPTER I INTRODUCTION 1 1. Introduction and general review of the literature 1
CHAPTER II SCALAR DIFFRACTION THEORY OF OPEN RESONATORS 5 2. Formulation of the problem, energy considerations
and justification of the scalar description 5 3. Formulation of the exact problem in terms of
inte-gral equations 13 4. The modified Kirchhoff approximation 16
5. A survey of passive open resonators in free space 17
6. The Fabry-Perot type of open resonator 20
7. The Fresnel approximation 24
8. Strip resonators 27 9. Mirrors of rectangular or circular shape 37
10. Discussion of the numerical techniques employed 46
11. Numerical results 54 12. Experiments with a microwave model of an open
res-onator with plane mirrors of circular shape 77
13. The ring-shaped open resonator 85
14. Numerical results 94
CHAPTER III ELECTROMAGNETIC DIFFRACTION THEORY OF OPEN RESONATORS 98 15. Formulation of the exact electromagnetic problem
in terms of vector integral equations 98 16. The modified Kirchhoff approximation 101 17. The vector integral equations of an open resonator
of the Fabry-Perot type 103
LIST OF SYMBOLS 110
SUMMARY 116
SAMENVATTING 118
C h a p t e r I
INTRODUCTION
7.
Introduation and general re-oiew of the literature
The main object of the present thesis is to develop a diffraction theory of open resonators and to compute the resonant frequencies and field distributions of its resonant modes.
Open resonators play an important role in measuring and other de-vices in the very short microwave region; for instance they are applied in plasma diagnostics. Also, they are frequently employed in optically active components such as lasers. In the present thesis two types of open resonators receive attention: (a) the Fabry-Perot type of open resonator where two (partially or totally) reflecting mirrors succes-sively reflect the waves, (b) the ring-shaped resonator which consists of a single reflecting ring. In a laser an optically active medium (i.e. a medium with inverted population distribution) is present, usu-ally between the mirrors of an open resonator of the Fabry-Perot type. If a certain resonant mode of the resonator has a high enough quality factor, the power gained by the stimulated emission in the active me-dium (induced by the stored energy of the mode) exceeds the power losses due to dissipation in the medium and to diffraction of the opti-cal waves around the mirrors. After PROKHOROV' and SCHAWLOW and TOWNES^ proposed in 1958 the use of a conventional Fabry-Perot structure as a resonator for masers operating at infrared or optical frequencies, a number of theoretical studies on this type of open resonators have been put forward. The main emphasis of these studies was lying upon the problem of the determination of the diffraction losses. We mention in particular the first of these studies by Fox and Li, which can be con-sidered as almost classical in the field (FOX and L I ^ ) . The majority of later studies can be considered as based upon the same ideas. The cor-ner stone in this theory is the definition of a mode in an open resona-tor as a time-harmonic (scalar) wave which, being launched from a
2 Introduction Sect.1
certain part of the system of reflectors, reproduces itself up to a constant factor after having completed a cycle of reflections in the system. Using this point of departure Fox and Li and others (e.g. BOYD and GORDON"*, STREIFER^'^, BERGSTEIN and SCHACHTER'', B E R G S T E I N and
MAROM^, M C C U M B E R ^ , T O R A L D O D I F R A N C I A ^ " , C U L S H A W ^ ^ and SOOHOOl^) have
analyzed the modes in several resonator systems consisting of two mir-rors, while G O U B A U and SCHWERING^^ and KUMAGAI, MORI and YOSHIDA^'* have analyzed the ring-shaped resonator. A review of papers based upon this approach has been published by KOGELNIK and LI^^ and KOGELNIK^^. More recently, different approaches to the open resonator problem have been presented by e.g., LOTSCH 1'', OGURA and YOSHIDA^^, RISKEN^^ and
V A I N S H T E I N ^ " . The latter approaches have been reviewed in four exten-sive papers by LOTSCH^l•22.23,24_
In the present thesis a different point of view is adopted as re-gards the formulation of the open resonator problem. As point of depar-ture we choose the classical theory of cavity resonators, where the
angular frequenay oi is introduced as the separation constant occurring upon separating the time dependence of the field quantities from their dependence on the spatial coordinates. In the absence of sources (or, in our case, optically active media) non-trivial field distributions only exist for a specific sequence of values of u, the aharaateristia frequencies of the resonator. If losses due to either dissipation or radiation (or both) are present, the field amplitude decays exponen-tially in time and the characteristic frequencies are complex. \^en in the open resonator under consideration the optically active medium is present, the system operates in time in its steady state. The angular frequency is real and follows, amongst others, from the consideration that the power delivered by the active medium must compensate the losses. If the losses are not too high, the frequency of operation is expected to be close to one of the characteristic frequencies of the resonator. This approach of the open resonator problem has been adopt-ed, too, by BARONESS, BYLDYREV and FRADKIN^^ and KATSENELENBAUM and SIVOV^^.
In this picture, a mode of the open resonator is defined as a non-trivial solution of the source-free electromagnetic field equations in
Sect. 1 Introduction 3
complex form, satisfying the proper boundary conditions at the bounda-ries of the resonator system. As this formulates an eigenvalue problem, we expect the non-trivial field distributions only to exist at a
se-quence of complex frequencies, the characteristic frequencies defined above. The relevant field distribution associated with a particular characteristic frequency defines the corresponding mode. The determina-tion of the characteristic frequencies and the corresponding modes is the object of our investigation.
The rigorous treatment of the relevant electromagnetic boundary value problem is complicated in two respects, viz. being a vectorial problem and being a difficult diffraction problem. As the difficulties of the vectorial diffraction problem are of the same nature as those in the corresponding scalar problem, it is advantageous to investigate the corresponding scalar problem first. A second reason to do this lies in the fact that the practical applications of open resonator systems ly in the domain of optical frequencies, where the coupling between dif-ferent components of the field vectors is often negligible. Obviously, the scalar formulation discards polarization effects. As will be shown in Chapter II, the scalar problem becomes tractable, while the results are physically meaningful (Section 2 ) . The scalar treatment of the ex-act boundary value problem leads to either a system of inhomogeneous, simultaneous integral equations of the first kind or to a system of in-homogeneous differential-integral equations of the first kind (Section 3 ) . In the range of optical frequencies both systems of equations are very difficult to solve numerically. Therefore, approximate methods, in particular the modified Kirchhoff approximation, are considered. Appli-cation of the modified Kirchhoff approximation is shown to lead to sys-tems of simultaneous integral equations of the second kind with kernel functions that are certainly less singular than the ones occurring in the (differential-)integral equations of the first kind (Section 4 ) . Using the modified Kirchhoff approximation, several open resonators of the Fabry-Perot type are then studied in detail (Section 6 ) . The rele-vant systems of two simultaneous integral equations of the second kind are further simplified through the application of the Fresnel approxi-mation (Section 7 ) . Three types of open resonators of the Fabry-Perot
4 Introduction Sect.1
type are then considered: the strip resonator, the resonator with mir-rors of rectangular shape and the resonator with mirmir-rors of circular shape (Sections 8 and 9 respectively).
The resulting integral equations are of the Fredholm type and in some respects resemble the ones discussed by Fox and Li and others, only our interpretation is quite different. The equations are solved ntjmerical-ly. The numerical techniques employed are presented in Section 10 and a discussion of the results follows in Section 11. The obtained results for plane mirrors of circular shape are compared with measured data which have been obtained from experiments with a microwave model of this resonator (Section 12). Chapter II concludes with a detailed study of a ring-shaped open resonator consisting of a single ring.
In Chapter III, the electromagnetic formulation is taken up again. In the exact boundary value problem this leads, for perfectly conducting mirrors, to a system of simultaneous, inhomogeneous differential-inte-gral equations of the first kind in which the electric surface current densities on the mirrors are the unknown functions (Section 15). For the same reasons as pointed out in the scalar foirmulation the modified Kirchhoff approximation is next introduced (Section 16). Restricting our further considerations to resonators of the Fabry-Perot type, ap-plication of the modified Kirchhoff approximation yields a system of two homogeneous simultaneous vector integral equations of the second kind. For either plane parallel or paraboloidal mirrors this system can be reduced to a simpler system of equations. A separation into Carte-sian components then leads to two independent scalar problems of the type dealt with in Chapter II. The error made in this reduction is of order 0(d. ) + 0(d, ) as d. j - * " (.d. „ = radius of curvature of mirror M „ ) . Chapter III concludes with an investigation of the character of
the electromagnetic field of a particular mode in an open resonator with parallel plane mirrors. At points of observation in between and not too close to the surface of the mirrors the entire electromagnetic field is transverse up to terms of order 0(d ) as <i-+<» (d = distance between the mirrors).
C h a p t e r II
SCALAR DIFFRACTION THEORY OF OPEN RESONATORS
2.
Formulation of the problem, energy considerations and justification
of a scalar description
As has been outlined in Chapter I, the determination of the modes in an open resonator is an electromagnetic boundary value problem, in general requiring a vectorial formulation based on Maxwell's equations. In general the open resonator configuration to be studied can be de-scribed as follows: a single or a finite ntmiber of bounded, non-closed, reflecting surfaces ("mirrors") of vanishing thickness is embedded in a linear medium, the latter may be partially optically active. The domain occupied by the mirrors will be denoted by M; M may consist of a finite number of sub-domains M , M-, ..., M (see Fig.l).
Fig.l. Open resonator with mirrors M , M-, ..., M^^, V is the domain occupied by the optically active medium.
For the moment the active medium is considered to be a source of elec-tromagnetic radiation, with known source-current densities. Let V be
8 the domain in which the optically active medium is present and let J"=J(£,t) and K=K(r,t) represent the electric and magnetic volume
cur-6 Formulation of the problem Sect.2
rent densities respectively, differing from zero in \J only. Then the
electromagnetic field vectors satisfy Maxwell's equations (expressed in
Sl-units)
rot H - 3P/3t = J,
rot E + aB/at = - K,
(2.1)
where
^(r,t) = electric field vector,
H(.r^,t) = magnetic field vector, P(r,t) = electric flux density,
^ir_,t) = magnetic flux density.
In order to allow the medium to exhibit special effects, e.g.
electro-optical and/or magneto-electro-optical effects (TOUPIN and RIVLIN^^, CAROLL and
RIVLIN^^), the general formulation for a "Tellegen medium" is given. The
constitutive relations for such a medium are written as (KURSS^'')
V(.r,t)
B(r.t)4 4
a u
Ur,t)
H{r,t)
(2.2) where £ = permittivity tensor, ^ = permeability tensor,£ = inverse magneto-electric tensor,
Q_ = direct magneto-electric tensor.
For a linear medium the elements of these tensors are linear operators
of the convolution type.
On the assumption that in the steady state the optically active
Sect.2 Formulation of the problem 7
E, H, V, 8, J, K(r,t) = Re{ff, H, D, B, J, X(r;a)) exp(-!;ut)}, (2.3)
where u = angular frequency, and i = imaginary unit. In (2.3) the angu-lar frequency u is real. The corresponding Maxwell's equations in com-plex form are obtained as
rot H + iwD = J,
rot E - ioB
(2.4)
The complex form of the constitutive relations (2.2) is
D(.r;i^) S(^';i») E.(r;ü)) i(r';u) jl(r>;u) ^(.r;ui) £'(r;u)) fl(r;cü) (2.5)
The elements of the complex tensors gj ^, ^ and ^ are now functions of the position vector r and the angular frequency lo (KURSS ) .
On the mirrors boundary conditions are imposed. To start with we take the mirrors to be perfectly conducting; then ^xff = ^ on both faces of the mirrors (n = unit vector in the direction of the normal to M, see Fig.l). At infinity the electromagnetic field has to fulfil the ra-diation condition. The mathematical formulation of the rara-diation condi-tion can only be given if the medium outside a sphere of finite radius is essentially free space (cf. MÜLLER^^).
The determination of the electromagnetic field vectors E_ and H_
from the field equations (2.4), with given j/ and X and the constitutive relation (2.5), together with the boundary conditions on the mirrors and the radiation condition at infinity can be considered as the excitation problem of the open resonator. Now it is to be expected that in this excitation problem resonances occur at those angular fre-quencies which are close to the (complex) values oi of u for which the source-free problem (which is obtained upon setting « 7 = 0 and K = 0 i.n
(2.4)) has non-trivial solutions. The latter values to are called the characteristic frequencies or eigenfrequencies of the system. The cor-responding solutions, the natural modes, can be considered as the free
8 Formulation of the problem Sect.2
oscillations of the open resonator. In view of the uniqueness theorem in electromagnetic diffraction theory (cf. WILCOX^^) ^^ j-j^g absence of sources only the trivial solution exists when for complex values of ti) we have Im(ii)) i C Therefore, the characteristic frequencies are ex-pected to have a negative imaginary part. This is in accordance with the physical picture, where as a result of the leaky character of the open resonator energy is radiated away to infinity at the expense of the stored energy. This causes, sources being absent, the amplitude of the free oscillations at each point in space to decay exponentially in time. The latter decay in time is described in terms of the damping coefficient &, introduced through
01 = u - i6 with 6 > 0. (2.6)
a
At this point it is useful to introduce some characteristic quantities which are important in the theory of the classical damped oscillator, viz.; the quality factor Q, the bandwidth B and connected with it the frequency response curve of the amplitude. By analogy with free oscil-lations in low-frequency lumped element electrical networks we intro-duce the quality factor Q and the bandwidth B of a particular free oscillation as
Q de|^ u/26, (2.7)
B def^ u/Q. (2.8)
In case the quality factor Q is large compared with unity, the band-width B can be interpreted as the halfband-width of the frequency response curve (or resonance curve) of the amplitude of the electric field vec-tor E (cf. BORGNIS and PAPAS^^) . it will now be shown that these quanti-ties can be related to the time-averaged values of stored energy and radiation loss, when interpreted in a proper way. To this aim
Poynting's theorem in complex form is applied to a domain V, which is bounded by the mirror surfaces M , M_, ..., M^ and a supplementary boundary surface Z of the resonator (see Fig.l). By virtue of the boundary conditions on the mirrors we obtain, in the absence of
Sect.2 Formulation of the problem 9
sources,
i Re jj^ CE>^H*)-n d4 = 26 Re{///(^( Jff*-B + {E-D*) dV] +
(2.9)
- 0)
Im[fJ!^^QH*'B - iE-D*) dV}.
Contrary to the case of sinusoidal oscillations (where u is real, a
hence 6=0) neither ExH* nor the terms H*'B and E'D* appearing in (2.9) actually have the meaning of time-averaged values of power flow density or stored energy density. However is is easily seen that
i Ke{E-E*} = <exp(26t)E-E>y (2.10)
where
<exp(26t)E.E>„ def ^ ƒ * 0 * ^ exp(26t) E(t)-E(t) dt, (2.11) i 1 C Q
with T def 2Tr/ü).
Let us now introduce the quantities
<P>y def I Re{jj^(ExH*)'n dA} = <exp(26t) ƒƒ j. (ExH)-^ dA>^, (2.12)
<V>y def Re{ƒƒƒ^,(Jff*•B + {E'D*) dV] =
(2.13)
= <exp(26t)
jjf^,a^i^B
+
iE'V) dV>^
and
then (2.9) can be rewritten as
<P>^=26<W>^-<Q^^>^.
(2.15)
In case the damping factor 6 is very small compared with u), the factor exp(26t) is almost equal to exp(26t„) in the entire interval
10 Formulation of the problem Sect.2
t„ < t < t^+T and hence the quantity <P>m is approximately equal to the "time average at t=t'' over a single period of the power flow
<ƒƒ (ExH)-n dA> leaving the open resonator system in the form of radi-ation loss. In the same way the quantity <Q >_ is approximately equal to the "time-averaged" power dissipated in the medium in f, while the quantity <W> is approximately equal to the "time-averaged" stored en-ergy <///ii(|H*B + iE'V) dV>„ in the resonator system. Then the quality factor Q defined by (2.7) can be rewritten as
CJLJ 1
'^-üU<V>,.<Q^>,)l.'
^2.16)
all averages taken at t=t_, while
U) = ü)(l - t:/2Q). (2.17)
The whole picture of an open resonator sketched till now can be employed to formulate the problem in a slightly different way. In that formulation no sources are introduced, but from a certain instant t=0 on, the system in which a certain initial field is present, is left to itself. Through damped oscillations the stored energy is then radiated away to infinity. The angular frequencies of these damped oscillations are identical to the presently introduced characteristic frequencies of the system. This point of view has been adopted by JANSE^"*.
The starting point in this section has been the formulation of the open resonator problem as an electromagnetic boundary value problem. The rigorous treatment of this problem is complicated in two respects, viz. being a vectorial problem and being a difficult diffraction
prob-lem. Now the difficulties of the diffraction problem will be of the same nature as in the corresponding scalar problem. This furnishes one reason for considering the corresponding scalar problem first. A second reason is that the practical applications of open resonator systems ly in the domain of optical frequencies where coupling between different components of the field vectors is often negligible. In such a scalar formulation polarisation effects are evidently discarded. As will be shown in the following sections, the problem becomes tractable, whereas
Sect.2 Formulation of the problem
n
the results will be physically meaningful.
In stead of with an electromagnetic field satisfying Maxwell's equations (2.1) and the constitutive relations (2.2) we are now dealing with an optical wave motion described in terms of the scalar wave func-tion u = u(.r,t), being a single component of the electromagnetic field vectors. It is noted that this can only be true when we are dealing with a simple isotropic medium. The description of the behaviour of the open resonator is analogous to the one given at the beginning of this section. The relevant physical property of the medium is now the veloc-ity of propagation, denoted by a, which is taken to be constant
throughout space. Further, W = w(r,t) represents the volume source den-sity of the optically active mediimi (differing from zero in V only). The scalar wave function u satisfies the inhomogeneous scalar wave equation
V^w - c"^(3^u/3t^) = - w. (2.18)
As boundary conditions on the mirrors we choose either u = 0 or n-Vw = = 0 on both faces of the mirrors. At infinity, the scalar wave motion consists of waves travelling away from the resonator system.
As before, on the assumption that in the steady state the optical-ly active medium generates monochromatic coherent waves we write
u,w(r,t) = •Re{U,W(r;ui) e-ap(.-i(Mt)]. (2.19)
Then U satisfies the inhomogeneous scalar Helmholtz equation
V^ü + (t//a^)U = - W, (2.20)
the boundary conditions f/ = 0 or rl'VU = 0 on both faces of the mirrors and the radiation condition at infinity. Completely in accordance with the electromagnetic formulation the characteristic frequencies u and the natural modes are obtained upon setting W = 0 in (2.20). With re-gard to the radiation condition it is noticed that the scalar wave function of a natural mode at a great distance from the resonator sys-tem must behave as
12 Formulation of the problem Sect.2
u (r,t) = Re{^ (P) expfiu (.r/c-t))/U-nr}[\ + 0(r )) as r->•<». (2.21)
The right-hand side of (2.21) vanishes for fixed r as t-*<» (? = r/r de-notes the unit vector in the direction of r ) .
In the present section the general formulation of the open resona-tor problem has been introduced. In order to illustrate this concept,
an example of such a resonator system is given here. In this example the natural modes can be determined analytically. Basically the domain
exterior to a perfectly conducting sphere is an open resonator. Meant to be an illustration it is sufficient to consider the scalar analogue of the electromagnetic problem. In this analogue a scalar function U has to be found, such that the following conditions are satisfied: the homogeneous Helmholtz equation (obtained upon setting d' = 0 in (2.20))
exterior to a sphere of radius a, the boundary conditions either U = 0
or n'VU = 0 on the surface of the sphere while at infinity U must be-have according to (2.21). Introducing spherical polar coordinates r, 6 and (() with the centre of the sphere as origin, the general solution of the homogeneous Helmholtz equation fulfilling the condition at infinity is obtained as
U(.r,e,'\>)' ' '•n=0 ^m=0 = r n r ni^ cos(m<\>) + B sin(m<i,)]"^ mn mn ' n ^ •' p'^fcosO)) * (2.22) * ;; ^ ' ^ u r / e ) .
In (2.18), P fcos(9)) denotes the associated Legendre polynomial of " (1)
degree n and order m and h (uir/e) represents the spherical Bessel
function of the third kind and order n. At the surface of the sphere the boundary condition for the first boundary value problem (V = 0 at
r = a) leads to the relation
h ^'\u a/a) = 0 (2.23)
n
o
whereas for the second boundary value problem (dU/Sr = 0 at r = a) we
Sect.3 Integral equations J3
8/3r(;z ^'^(u r/a)] = 0 at r = a. (2.24)
The characteristic frequencies u in both cases are found as the roots of (2.23) and (2.24) respectively. In Table I some characteristic fre-quencies (viz. those that can be found directly by inspection) together with the corresponding quality factor Q and wavelength X (A def 2-na/ta)
are given (cf. STRATTON^S).
Table I. The angular frequency u and the damping factor 6 express-ed in a/a, the quality factor Q and the wavelength A of some low order natural modes exterior to a sphere of radius a.
y • 0 on the sphere 3i//3r - 0 on the sphere
n laa/a 'tala Q A/a '^Ic ia/o Q k/a
0 1 2 3 0 •0.87 0 •1.75 1 1.50 2.26 1.87 0 0.29 0 0.47
»
7.22.
3.59 0 •1 0 '1 29 not 1 1 1.28 1.36 0 0.5 0 0.47 calculated.
6.28 4.873. Formulation of the exaat problem in terms of integral equations
In this section the scalar open resonator problem, formulated in Section 2 will be reduced to the solution of a system of simultaneous integral equations. The main tool in this reduction is the Helmholtz integral representation for scalar wave functions (cf. BAKER and COPSON^^). If/If denotes the total number of mirrors the Helmholtz rep-resentation of U can be written as
U = U^ + f , U .
(3.1)
0
^n=\ n
14 and Integral equations y (^p) dfii ///i; W(r)G(.rJr) dV ^o^-P Sect.3 (3.2)
y„(£p) ^ f l/y {\(i:)G^(2:plz) - *„(£)(^-V)G(_rp|£)} dA n ("=1,2 N), (3.3) in which and \(r) = {(«.V) !/(£)}_
*„(£) = {y(r)r
with r on M with r on M (3.4)-(3.5)denote the jump of ^'Vü and U respectively, when crossing the mirror M from the "dark" to the "illvmiinated" side (see Fig. 2) and
Fig,2. Detail of an open resonator system, indicating the "dark" and "illiminated" side of the mirror M .
n G(.rr,\r) def^ (4Tr7?) expiiuiR/c), with if = \rp-r\ = {(Xp-x)^ + (.yp-y)^ + (Zp-s)^}^ i 0. (3.6) ( 3 . 7 )
Sect.3 Integral equations 15
In case the wave function itself vanishes on all mirrors we have $ = 0 . Reusing the boundary conditions we obtain
lU ^^M \(ü)«(i:pli:) ^ = - ^o<^p)
(3.8)
when rr,_ p on M ^ \ (m=\ ,2,... ,N) » »
Equation (3.8) constitutes a system of inhomogeneous, simultaneous in-tegral equations of the first kind, from which "f (n=l,2 N) can,
for given W, in principle be determined.
In case the normal derivative of the wave function vanishes on all mirrors we have T = 0 . Reusing the boundary condition in this case we have
i l (^^p-^'p) J!M *„(r)(«-V)G(rp|r) dA = (np'^p)UQirp)
" (3.9) when rp on M (m=l ,2,... ,A?) .
Equation (3.9) represents a system of inhomogeneous differential-inte-gral equations of the first kind, from which $ (n=l,2,...,N) can, for given W, in principle be determined. It is noted that in (3,8) and (3.9) the limiting value of the left-hand side when the point P of ob-servation approaches M has to be taken.
In accordance with the theory developed in Section 2, we introduce now a "mode" of the open resonator system as a non-trivial solution of the homogeneous systems of equations corresponding to either (3.8) or (3.9). The complex values of oj for which these homogeneous systems of equations have non-trivial solutions are consequently the characteris-tic or "mode resonance" frequencies. In view of the uniqueness theorem in scalar diffraction theory (cf. WILCOX^^) non-trivial solutions can only exist when Im(u ) < 0, and hence 6 > 0.
In the practical application of open resonators in lasers the wavelength of the forced oscillations is usually very small compared with the geometrical dimensions of the resonator system. In this range of frequencies the available analytical and/or numerical techniques are
16 Kirchhoff approximation Sect.4
expected to lead, only with a considerable amount of effort, to solu-tions of the homogeneous integral equasolu-tions corresponding to (3.8) and (3.9). Therefore, approximate methods and in particular the (modified) Kirchhoff approximation will be investigated.
4. The modified Kirahhoff approximation
In the Kirchhoff approximation it is assumed that both the wave function and its normal derivative vanish at the "dark" face of each mirror. Consequently, in this approximation
f K «-vy on the "illuminated" face of M (4.1)
n = — n
and
* K Ü on the "illuminated" face of M . (4.2)
n = n The modified Kirchhoff approximation takes, in addition, into account
that $ = 0, in association with the first boundary value problem and 4" = 0, in association with the second boundary value problem. Integral equations of the second kind from which the remaining unknown quanti-ties can be determined are then obtained upon letting the point P of observation in the Helmholtz representation of either IJil or U approach the illuminated face of the mirrors.
When $ = 0 this procedure leads to n ^
J>f^(rp) = i^^'V^) ^^ \{r)G{r^\r) dA +
'" (4 + L A « ( ^ P ' V ^ / M ^n^^^^^^pl^^ ^ when rp on M^,
• n
where -ff denotes the Cauchy principal value of the integral. When f = 0 the procedure leads to
Sect. 5 Open resonator systems 17
- I J. ! II * (r)(«'V)G(r„ r) dA when r^ on M . n
Equations (4.3) and (4.4) replace in the modified Kirchhoff approxima-tion the homogeneous integral equaapproxima-tions corresponding to (3.8) and (3.9), respectively. It is observed that (4.3) and (4.4) are integral equations of the second kind and that their kernel functions are cer-tainly less singular than those in (3.9). In case the radii of curva-ture of M are large compared with the wavelength, the first term at the right-hand sides of (4.3) and (4.4) can be neglected; for plane mirrors this term vanishes exactly.
6, A survey of passive open resonator systems in free space
The various open resonator systems which are of interest today can be grouped into three classes: (i) the conventional Fabry-Perot type resonator with two mirrors, (ii) the multireflector resonator and (iii) the ring-shaped resonator. Open resonators of this kind are employed in optically active devices such as lasers and in other optical systems. In view of these small-wavelength applications an analysis of their properties on the basis of a scalar description of the optical waves in the system is justified. The dimensions of the configuration are usual-ly large compared with the wavelength of the optical waves.
The most common laser resonator is of the conventional Fabry-Perot type. It consists of two either curved or plane mirrors facing each other and successively reflecting the waves. The cross-sections of the mirrors are usually either rectangular or circular in shape. Resonators of this type are shown in Table II. An open resonator consisting of two parallel plane mirrors is shown in (a,l); in (a,2) one of the mirrors is tilted. An open resonator with curved cylindrical mirrors is shown in (a,3); (a,4) shows a flat-roof resonator. In (a,5) one of the mir-rors of the flat-roof resonator has been turned around the optical axis. Various resonators consisting of mirrors of circular cross-sec-tion are shown in (b,l)-(b,5); plane mirrors are shown in (b,l), spher-ical mirrors in (b,2). If both mirrors are of equal curvature and the radius of curvature is equal to the distance between the mirrors, the
18 Open resonator systems Sect.5
Table II. Examples of open resonators of the Fabry-Perot type with mirrors of (a) rectangular cross-section, (b) circular cross-section.
M
(a)/
^
A
...^
/
(b)I. Plane mirrori I. Plane m i r r o n
A
/
(f
2. Plan* B i r r o r i , on« a i r r o r t i l c * d 2. Spherical n i r r o r i with d i f f e r e n t radii of curvature
k. Flat-roof reaonator 4. Conical mirrors
3. Flat-TOOf resonator with crossed edges 5. Plane mirrors with output-coupling apertures
Sect.5 Open resonator systems 19
resonator is called "confocal". An open resonator consisting of two conical mirrors is shown in (b,4), while in (b,5) an example is given of an open resonator having plane mirrors with output-coupling aper-tures .
If both mirrors of the conventional Fabry-Perot type of resonator are thought to be cylindrical and of infinite extent in one direction we arrive at a so-called strip resonator. Although the strip resonator cannot be realized physically, the analysis of it is included in our investigations because of its mathematical simplicity.
Multireflector optical resonators are employed in various optical systems. An example of a symmetrical four-mirror optical cavity is shown in Fig.3. A configuration of this type is employed in ring lasers (cf. MACEK and DAVIS^^).
Fig.3. A schematic diagram of a symmetrical four-mirror optical
cavity.
Fig.4. A ring-shaped resonator consisting of two coaxial mirrors
20 Fabry-Perot type Sect.6
The ring-shaped open resonator, also called cylindrical
Fabry-Perot resonator, is obtained by rotating a conventional Fabry-Fabry-Perot
resonator around a suitably chosen axis. An example of a multi-cylin-drical Fabry-Perot resonator consisting of two coaxial mirrors M and M„ is shown in Fig.4. For this type of open resonator applications have not been realized as yet, although a novel high-power single frequency laser construction using such a multi-cylindrical Fabry-Perot resonator has been proposed by Kumagai et al. (KUMAGAI et al.^^).
6. fhe Fabry-Perot type of open resonator
A cross-section of the conventional Fabry-Perot type of open res-onator with two mirrors is shown in Fig.5. Here M and M„ denote the
Fig.5. Cross-section of a conventional Fabry-Perot resonator.
two perfectly reflecting mirrors; for the moment their shape is arbi-trary. A Cartesian coordinate system is chosen as indicated in Fig.5. In the exact formulation of the problem (cf. Section 3) we restrict ourselves to the case where the wave function vanishes on the mirrors
(first boundary value problem). The differential-integral equations of the corresponding second boundary value problem are much more difficult to handle (cf. Section 3, eq. (3.9)). As we are interested in the modes and their characteristic frequencies the homogeneous system of equa-tions corresponding to (3.8), i.e.
S e c t . 6 Fabry-Perot type 21 (6.1)
J/M, ^ ( £ , ) « ( i : p . | l £ , ) ^ 1 - J/M^ ^2(^2>^(i:p,il^2)
^ 2
= °
when rp . on MJ/M, *,(i:,)c(£p.2li^i) ^ 1 * //M^ ^2(i!2>^(i:p.2l2!2>
<^^2 = °
when rp . on M»has to be studied. The system of equations (6.1) constitutes a system of two homogeneous integral equations of the first kind for the unknown functions t. and *„• The kernel functions G(£p il£i) ^nd (3(rp y\r^) in these equations exhibit an integrable singularity. For the calculation of the optical wave function in any point in space we employ the inte-gral representation
^(üp) = I/M *i(üi)<^(i:pli:i> ' ^ i * IIM ^2^i:2^'^^^pl^2^ ^^2' ^^-2)
which is obtained from (3.1)-(3.3). When the modified Kirchhoff approx-imation (cf. Section 4) is applied we assume at the same time that the curvature of both M and M is so small that the contribution of the
(Cauchy principal value) integral over the mirror on which the point of observation is chosen, can be neglected. Then the condition *. = <t„ = 0
(associated with the first boundary value problem) leads to (compare (4.3))
(£j) = 2(n,-Vj) //^ '^2^^2^^^-l 1-2^ '^^2 w h e n ^ i on Mj ,
(6.3)
^^(.r^) = 2(n^-^^) jj^ *|(£,)G(£2lri) ^ | when r^ on M^.
The optical wave function at any point in space can again be calculated with the aid of (6.2). The case "i. = '1^ = 0 (assc
boundary value problem) leads to (compare (4.4))
with the aid of (6.2). The case t . = *_ = 0 (associated with the second
*j(P|) = -2 jjj^ *2^-2^^-2''2^'^^-l 1-2^ ^ 2 "^en £ | on Mj ,
^ (6.4) *2^-2^ = -2 jj^ *,(i:|)(^,-V,)^(£2li;p '^i when £ 2 on M^.
22 Fabry-Perot type Sect.6
In this case the optical wave function at any point in space can be calculated with the aid of the integral representation
^(£p) = - JJM *i(i:i)(^i"'i><^<^£pl£i> " ^ i +
' (6.5) - J/M *2^£2^^^2*'2^'^(^pl£2^ ^V
which is obtained from (3.1)-(3.3).
As has been outlined in Section 3 and Section 4 a "mode" of the resona-tor is obtained as a non-trivial solution of the integral equations (6.1), (6.3) or (6.4). The complex values of u for which these equa-tions have non-trivial soluequa-tions are called the characteristic frequen-cies of the resonator system. For the relevant "mode" in a Fabry-Perot type of open resonator it is usual in the literature to introduce the power (or diffraction) loss per transit. In terms of the complex wave number k def ui /a = (ai—i6)/c, the power loss per transit of a mode is defined as
P = 1 - exp{2Im(fc)d}, (6.6)
while the mode resonance frequency is given by
V = u)/2iT = Re(fe)c/2Ti. (6.7)
The resonator systems under consideration either do or do not have a plane of geometrical symmetry. When this type of symmetry is present it is advantageous to introduce the wave functions that are either even or odd with respect to the plane of symmetry, which is chosen to be located at 3 = 0. Thus, we have
üg ^(£p) ief \{U{rp) *- Uir^p)}, (6.8)
fg ^(£,) def H'i'Cl!,) ' 'Kr,')}, (6.9)
Sect.6 Fabry-Perot type 23
*e o^-l^ dl£ H*(Z,) - *(£,')>, (6.10)
where r J = (.x,y,z),r^'= (x,y,-z), r-p = (.Xp,y-p,Sp), r^' = (a;p,j/p,-3p)
and where the upper and lower sign correspond to the even and odd wave functions respectively. The functions U , t and $ represent the
e,o e,o e,o even(e)/odd(o) part with respect to 2 = 0 of f/, T and $ respectively. Through these functions the systems of simultaneous integral equations for configurations with a plane of symmetry can be transformed into single integral equations for the even and odd wave functions separate-ly. From (6.1) we obtain
/ / M , ^,..(Z,)tG(rp^,l£,) *G(rp^,|r,')}d4, = 0
(6.11) when rp on M ,
being a homogeneous integral equation of the first kind. Equation (6.3) is transformed into
'e,o^^2^ = ^2(n2-V2) ƒƒ, ^^^^(r^)G(r^\r^) dA^
(6.12) when r, on M„,
and (6.4) into
%.o^^2'^ - ^2 J/M, *,,,(£i)(Bi-V,)G(r2|r,) dA^
(6.13) when r- on M„.
In these equations the upper and lower signs correspond to the even and odd wave functions respectively. The integral representations for the optical wave function (6.2) and (6.5) are in the same way transformed into
"e,o^^F^ = / / M , %.,(i:,){G(£pli:,) * G(rp'l-i)) dA^, (6.14)
24 Fresnel approximation Sect.7
^.,c(ip) = - / / M , \,o^L,)^^,-^,^^^^Lp\L,) 'G(rp'|r,)}d^,,(6.15)
when H» = 0. e,o
7. The Fresnel approximation
In many applications the transverse dimensions of the mirrors of the Fabry-Perot type of open resonator are small compared with their longitudinal distance of separation. When this is the case the kernel functions in the integral equations (6.3) and (6.4) pertaining to the modified Kirchhoff approximation (or (6.12) and (6.13) for a resonator with a plane of syimnetry) can be approximated. Let the 2-axis of the Cartesian coordinate system be chosen along the "center line" of the two mirrors and let i; = ^Ax,y) and C, = ^r.{x,y) represent the devia-tions of M and M from suitably chosen reference planes (see Fig.6).
reference plenes
Fig.6. Cross-section of a Fabry-Perot type of open resonator indicating center line and reference planes.
These reference planes are perpendicular to the center line; their point of intersection with the center line coincides with the point of intersection of the corresponding mirror with the center line (Fig.6). Let d be the distance of separation of the reference planes. For large values of d we then have
Sect.7 Fresnel approximation 25 (x -X ) + iy -y ) Ir^-r^l = d - U^+K^) + • — ^ — + 0(d ^) a s d ^ - ( 7 . 1 ) 2 d and -1
^ ( £ l Iü2^ " '^(l!2'-l^ ° (4Tr<i)~ exp(.iuid/o) x
( 7 . 2 )
exp[lï;{-r - r + _ J — 2 L _ 2 _ }]{i + o ( d - ' ) }
'^ ' '^ 2 d
as d^oo.
Introducing the same approximation in the kernel functions occurring in (6.3), (6.4), (6.12) and (6.13) we obtain (^,-V,)G(r, Ir^) dA^ = {(3c,/3j:,)^+0;,/3i/,)^+l}^ = {.ii^/a)G(,r\r) i-5 ^ j- dx. dy x (7.3) {(3^,/3x,) +(3C,/3!/,) +1}^ X {1 + 0(d"')} as d-"-" and (.52*'2^^^-l 1-2^ '^2 " <^"/o)G(£i l£2^ "^2 ^'^2 ^'+0^"')} (7.4) a s d -»•»».
In the right-hand sides of (7.3) and (7.4) the expression (7.2) for
Gix^. |r„) has to be substituted. In obtaining (7.3) and (7.4) it has been assumed that c. = ?,(^.2/) and C, = K^^x,y) are single-valued and continuously differentiable and that their first-order partial deriva-tives with respect to x and y are of order 0(1). Let, further M ' and M_' denote the projections of M and M- on the reference planes C, = 0 and ?„ = 0, respectively. Then the substitution of (7.3) in (6.3) and introduction of x., y and x-, y- as variables of integration leads to the system of approximate integral equations
26 Fresnel approximation Sect.7
(7.5) when (a;j,j/|) in M^ ',
=2(^2'^2^ ° ^^ ^"^Ld ^^M ' ^(^2'^2l^l'^l^-l^^l'^l^ '^1 '^^l
' (7.6)
when (Xy.yj^ i" M ' ,
where
K(x^,y^\x^,y2) = K(x ^,y ^\x ^ ,y ^) dej,
(.x^-x^) +(y,-yo'> •r^2^ " 23 2./,, „ ^2 (7.7) i e i exp(i/c{-(C,+C,) + ,J }) and =„ dSi '*'„t(3^„/8^„)^ + (3^„/32/„)^ + n ^ w i t h m = l , 2 . (7.8) m ' ' m m m m m
Substitution of (7.4) in (6.4) leads to the system of approximate integral equations *l(x,.J/|) = -^fe ^''l^d ' ^ ^^M ' ^(a;|,i/, |x2,J/2)*2^^2'^2^ '^2 ^^2
^
(
when (Xj,y^) in M|',*2(a;2.i/2) = - ^ ^ - ^ ^ ^ f S ^
J / M.
A : ( X 2 , I ^ 2 I ^ 1 ' ^ P * 1 ^ ' ^ I ' ^ 1 ^'^l "^^1
' (7.10) when (x ,1/2) in M^' .Substitution of (7.3) and (7.4) in (6.12) and (6.13) results in the single integral equations
M
^ = ^ i k ^ ^ ^ J L . K(x^.yJx^,y^) x e , o ' " 2 ' ^ 2 ' ""^ 2TTd J J M , ' " v - 2 " ' 2 ' " ^ l • " 1( 7 . 1 1 ) X H^ ^ ( X j , j / , ) dXj di/j when (.x^.y^) in M^',
when * = 0 and
S e c t . 8 S t r i p r e s o n a t o r s 27
(.X y ) = T ifc -^^^I^T J L , K(.x y \x ,y )
e,o^ 2 ' " 2 ' 2Trd 2>»2 I"*"] •» 1' ( 7 . 1 2 ) X *g o ^ ^ l ' ^ l ^ ' ^ l "^^l " ' ^ ^ " (^2*^2^ ^ " ^ 2 ' ' when f = 0 ; f u r t h e r m o r e , i n t h e e x p r e s s i o n ( 7 . 7 ) f o r X(x , j / | x , z / . ) we h a v e C,, = t . . I t i s n o t e d t h a t i n t h o s e a p p l i c a t i o n s w h e r e 3c / 3 x • ^ m mand 35 /dy w i t h m = 1 , 2 , a r e s m a l l we h a v e 5 (x.y) ~f ( x . w ) and
m °m m " m ' "
5 ( x , w ) = T ( x . w ) .
In the subsequent sections the systems of equations (7.5)-(7.6), (7.9)-(7.10) and the single integral equations (7.11) and (7.12) have served as the basis for the calculation of the characteristic frequen-cies and the field distribution on the mirrors of the modes in a number of open resonators of the Fabry-Perot type.
8. Strip resonators
The strip resonator, although physically not realizable, is the simplest resonator configuration from a mathematical point of view. Therefore, a short discussion on the relevant integral equations is in-cluded here. The mirrors of the resonator are of infinite extent in one direction which is chosen to be the !/-direction (see Fig.7). The wave functions to be considered are independent of y ("two-dimensional wave
;,-C,(a:) t , ; , ( x )
-Fig.7. Geometry of the (infinite) strip resonator with a plane of symmetry (3 = 0 ) .
28 Strip resonators Sect.8
functions"). We confine the discussion to configurations having a plane of symmetry (the plane 2 = 0 ) .
The definition of a mode and its characteristic frequency, as well as the derivation of the relevant integral equations for a two-dimen-sional open resonator is analogous to the corresponding theory of three-dimensional resonator systems given in the preceding sections. We are dealing now with a wave function U that has to satisfy the two-di-mensional Helmholtz equation, appropriate boundary conditions on the mirrors and the radiation condition at infinity. Consequently, the for-mulation of the problem in terms of integral equations starts now with the Helmholtz representation of a two-dimensional wave function U, that can be written as
U = Un * f , " >
(8.1)
0 ^n=l n where^O^^P^ ^ ^ J/s (/(r)r(rplr)
dA
(8.2)
s
and f/„(£p) def j ^ {*„(£) r(rp|r) - *^(r) (n.V^)r(rp |r)} ds. (8.3) nIn (8.2) and (8.3) the two-dimensional position vectors are defined as
ru = Xni + z„i and r = xi + zi , while V^ def (i 3/3x + i 3/32). In —P P—X P—3 — —X —3 T — —X —3
the same way as before the functions 4" (r) and $ (r) denote the amount
•' n — n —
by which n'Vf/ and U respectively jump across M (cf. Section 3). The function ^(rp\r) represents the two-dimensional Green's function in free space
T(rp\r) = (I;/4)//Q^'^W?), (8.4)
where H^ denotes the Hankel function of the first kind and order zero and
Sect.8 Strip resonators 29
Comparing (3.1)-(3.7) with (8.1)-(8.5) it is clear that the two-dimen-sional analogue of the subsequent equations derived in Section 3-7 is obtained by replacing: (a) the surface integrals over the mirrors by line integrals over their cross-sections, (b) the Green's function G(rp|r) given in (3.6) by r(rp|r) given in (8.4) and (c) the operator V by V^.
Proceeding in this way we obtain for the strip resonator with a plane of symmetry in case the wave function itself vanishes on the mir-rors as the exact integral equation (of the first kind)
/M,
\,o^L,)^nrp^^\r^)
^ r(rp^, |r,')} da, =0
(8.6) when rJ on M .
More in detail, (8.6) can be rewritten as
^M, ^..o(^i)^(^/^>^o^'^(^li:p,r^il>*(^/^)V^^^b,i-i:i'l>^^«i =
(8.7) = 0 when r, on M,, with |rp^,-r,| = {(x2-x,)^ + [K{X^)-K{X^)]'^]^ i O (8.8) and |rp ,-£,'! = {(x2-x,)^ + [-d+c(x2)+ax,)]^}^ i 0, (8.9) while ds^ = {(3c/3x,)^ + 1}^ dx,. (8.10)When the modified Kirchhoff approximation is applied, considerations analogous to those in the three-dimensional case lead to the integral equations of the second kind
30
Strip resonators
Sect.8
''e.0^^2^
=
'
^^^2-'T2> ^ M ,^..^(£i)''(i:2l^l) '^«l = °
(8.11)
when r- on M ,
in case $ = 0 and
e,o %,O^^2'>= ' 2
JM*g,,(£,)(^,-VT,)r(i2l£i)
ds^
= O
(8.12)
when r- on M ,
in case T = 0. In the integral equations (8.11) and (8.12) the
con-tribution from the principal value integrals following from (4.3) and
(4.4) respectively, is neglected. Further considerations as to this
point are given at the end of this section. Subsequent application of
the Fresnel approximation for large values of d (Fig.7) leads to
(n2-V^2>r(l!2l^P "^^1 = (^/^)(f2'^T2^^0^'^^^l-2"-l'^ '^^l °
,,.
{l + (3^/3a:,)2}^ ,
i,^ Uk\r^-r,
I) ^ - T {1 + 0(d'')}
dx.
(8.13)
^
{l+(3c/3x,)^}*
as d-*",
and
(^,.V^,)r(r2|2:,)
ds^
= (i/4)(^,-V.,,,)ffQ^'\fe|r2-2:, I)
ds^
= - {ik/tt)H^^^\k\r^-r^\){\ + 0(d~')} dx, as d-^».(8.14)
Introducing for the Hankel functions of order one occurring in (8.13)
and (8.14) the asymptotic representation for large values of their
ar-guments
g,^'^fe|r2-r,|) ^
{^^^l _^
|} exp(ifelr2-r:, I - i 3 w / 4 } , (8.15)
and substituting the result in (8.11) and (8.12) we obtain the integral
equations
Sect.8 Strip resonators 31 =..0<V = ^ é ^ ^ ^'^P^^^'^ - ^^/^) ^-a ^(^2l^l)=.,o(^l) ^ 1 (8.16) when -a < x.< a, and
%,o(^2) = M ^ ) ^ -P(^^^ - ^-/^) C ^(^2l^l)*e,.(^l) '^l
(8.17) when -a < X < a. In (8.16) and (8.17) (a:2-a;,)^^(X2k,) def^ exp[i:J:{-c(X2) - C(a;,) + 2d ^^ ' (8.18)
and
H ^(x) def -f^ ^(x){(3t/3x)' + 1 } ^ (8.19)
Denoting the unknown function by X (x), i.e.
(8.20)
the integral equations (8.16) and (8.17) are rewritten as
%,/e,o^''2^ = i-a ^^^2l''l^^e,o^''l^ '^\ ''''^" -a<x^<a. (8.21)
If (8.16) applies (i.e. $ = 0 ) the factor a , being an eigenvalue of the homogeneous integral equation (8.21), must satisfy the condition
(a^ ^ ) " ' = Hk/2^d)^ expiikd - iTi/4); (8.22)
condi-32 Strip resonators Sect.8
(a )
^ e,o' ±(fe/2Trd)^ exp(^fed - i;Tr/4). (8.23)
The problem of the strip resonator has now been reduced to the determi-nation of the eigenvalues a = a (fe) of the integral equation
(8.21) as a function of k and the subsequent determination of the com-plex wave number k from (8.22) or (8.23). The superscript n (n = 0,1, 2,...) indicates which eigenvalue of the integral equation is under consideration; the eigenvalues are ordered according to decreasing ab-solute value, n = 0 refers to the largest eigenvalue in magnitude. The corresponding eigensolution determines the field distribution (on the mirrors) of the "(M)-th mode" of the strip resonator.
Because of their practical interest, strip resonators with circu-larly curved mirrors (in general with different radii of curvature) will now be discussed in more detail. In this particular case the
devi-ation function of mirror M (see Fig.8) is given by
t,(x) = d,{l - (l-x^/d,^)h
when -a < x < a, (8.24)where d. denotes the radius of curvature.
-d/2
Fig.8. Geometry of the strip resonator with circularly curved mirrors.
Sect.8 Strip resonators 33
C,(x) = x^/2d, - x^/8d,^ + 0{(x/d,)^}
(8.25) as x/d, -»• 0 when -a < x < a.
Disregarding mirror aberration (all terms except the first at the right-hand side, the influence of which has been studied by Lazutkin (LAZUTKIN'*°)) the deviation function is given by
2
;,(x)=x/2d, w h e n - a < x < a . (8.26)
From (8.26) we see that in the order in which the Fresnel approximation (see (8.13)) holds, this parabolic mirror cannot be distinguished from a circularly curved mirror with the same "curvature".
This section is concluded with an estimation of the error intro-duced by neglecting of the principal value integral in (8.11) and
(8.12). To this aim we rewrite the integral equation (8.11), now in-cluding the principal value integral and obtain with (8.19)
=.(^2) = * I C =.(^,)(«2-V^0^'^(^l^2-i:il> ^ 1
^
(8.27)
+ ^j'' H (x)(«,.V-)«-^'^(A:|r--r|) dx when-a<x,<a.
In the case of parabolic mirrors l^^-^| in the second integral is given by
l2:2-i:l = ^ix^-x)"^ + (x^/2d,-X2^/2d,)^}^ i 0. (8.28)
In order to estimate the second integral in (8.27) we choose x_ = 0 and write for this integral
fex /2d
I - - \ j^^ ^.^^x)H^^'hk\r^-r\) ^
^ dx.
(8.29)
Replacing the Hankel function in (8.29) by the integral representation
34 Strip resonators Sect.8
where A = (k^-qh^ with Re(A) i O and Im(A)i0, leads to
I = -(.a/d.) /-°°L' 5^(x)exp(iA|r„-r|) ^^ r-r dx d^?. (8.31)
' " " ^ ^ {l+(xa/2d,)''}*
Separating the integration interval with respect to q into the inter-vals (0,fe) and (fe,"), it is easily shown that
\l\ < (a/d,){;4 + o[(a/d,)^]} when a/d, ->-0. (8.32)
Hence the principal value integral J = 0(.a/d ) when a/d -+0. The error made by neglecting the principal value integral in (8.11) is of the same order of magnitude as the error introduced by the application of the Fresnel approximation.
In the case of flat-roof mirrors, the presence of a ridge on the mirror surface (see Fig.9) may cause difficulties. In fact, integral
Fig.9. Geometry of the strip resonator with flat-roof mirrors (a/2 = arctan(A/a)).
equation (8.27) is no longer valid in the point x„ = 0. In the numeri-cal program no difficulties arise as long as the point P of observation is not chosen exactly on the ridge. Another problem is the contribution from the principal value integral in (8.27). This contribution has to be taken into account and will now be determined. The deviation func-tion ? for a flat-roof mirror is given by
C(x) = (A/a)|x|, w h e n - a < x < a . (8.33)
Sect.8 Strip resonators 35
|r2-ü, I = {(a;2-x,) + [d-(A/a)|x2|- (A/a)|x, | ] ^ } % (8.34)
and |2!5~Z I i"^ '-^^ second integral in (8.27) by
1^2-1:1 = {{x.^-x)'^ * (A/a)^(|x|-|x2|)^}^. (8.35)
With (8.33)-(8.35) the integral equation (8.27) is converted into
kid-U/a) (|x2|+|x, |-sign(x2) (X2-X, )) dx + '-2 -1 I (8.36) t fO - , .„ (I)/, I u fe{-2(A/a)x} , • r, y J - (x)H, (fc r„-r ) r—^—r— dx when 0 < x „ < a , 4 •'-a e l '—2 —' l2!2~—I •i (• a „ - ,,, (1)/, I K fe{2(A/a)x} j , n 4 io =e^^^^l (^1^:2"^ I ^ 1^ -p I ^ "hen -a < X2 < 0.
It is observed that the integrals in (8.36) resulting from the princi-pal value integral in (8.27), have exactly the same form.
In order to estimate these integrals we choose x , = 0 and select one of them, e.g.,
^ = - 1
!^-^Ax)H,^'\kxU*Wa)hh
-21AZ£) dr. (8.37)
^ ° ^ '
{\HA/a)h^
Replacing the Hankel function in (8.37) by the integral representation (8.30) we obtain
J ^ ^ ^ ^ \ , -!• ƒ " { Ji'- (x)exp(iAx{l + (A/a)^^)dx} dq. (8.38) {l + (A/a)'^}^ ^
Performing the integration with respect to x and seperating the inte-gration with respect to q into the intervals (0,fe) and (A:,"») we are led to the result
III = (A/a)fl + 0{(A/a)^)]W when (A/a)-> 0. (8.39)
where W is a real positive constant. Hence I = 0(A/a) when (A/a)-<• 0 or
36 Strip resonators Sect. 8
I = 0(a) when a -*• 0. (8.40)
In Table III the deviation functions c, = ^(x) of some strip resonators are given.
Table III. Deviation functions 5 = C(x) of some strip resonators (the first three with a plane of symmetry).
-d/2 ' l " l X 0 d z *^2 "2 in a a I. Plane mlrrora. 2. Parabolic mlrrora.
\ •
••1
7
c - ax) ' * ' < i / 2The resonator is called confocal if
3. Flat-roof airrore,
i/2 - erctan(A/a)
7
/ ^ ./2 - arctanfA/,
Sect. 9 Rectangular cross-sections 37
9. Mirrors of rectangular or circular shape
In this section we investigate in more detail the integral equa-tions (7.5)-(7.6) and (7.9)-(7.12) for the Fabry-Perot type of open resonator in the modified Kirchhoff and Fresnel approximation in case the mirrors are of either rectangular or circular shape. The shape of the mirrors is called rectangular if the projections M ' and M-' of M. and M„ on the reference planes C, = 0 and C, = 0 respectively are rec-tangles; the shape of the mirrors is called circular if the projections M ' and M.' are circles.
Let in the case of rectangular mirrors M ' and M ' occupy the re-gions -a, <x < a , -b <y <fc, and -a2<a;2<a2, "'^2*^2 "^^2 ^^^spective-ly. Further we restrict our considerations to mirrors, the deviation functions t = i; (x,v) of which can be written as
m m °
C^(x,j/) = C^(x) + n^(j/) with m = 1,2. (9.1)
Then the kernel function (7.7) is of the form
X(x2.i/2l^l'^l^ ° -^x^^l'^2^^V^^1'^2^ (9.2)
Due to this structure a separation of variables takes place which en-ables us to write ^„(.x^,y„) m m m
<
> = Xjx)ï(y) ^ m m m ^m ^„(x„,y„) m mm ^ with m = 1,2. (9.3)Substitution of (9.1)-(9.3) in (7.5)-(7.6) and (7.9)-(7.10) leads to the system of integral equations
a, 2 ^, (a:,) = /_^2 if^(x, \x.^X^{.x^ dx^ when -a, < x , <a,. (9.4)
38 Rectangular cross-sections Sect.9
and
^1,2 ^l^^P = /-i ^j/(i/ili/2)^2^^2^ ^^2^2 ''^^^ "^1 ^2^1 "^^1' ^^-^^
^2,1 ^2^2^2^ = ^-fc ^y(J'2l^l^^l^^l^ '^ï^l when -fc2<K2"^2- ^^'^^ In case $ = 0 the factors a, „, a. ,, g, „ and 6„ , must satisfy the
m l , z z , l l , 2 2,1
condition (compare (7.5)-(7.6))
^"l 2^1,2^~' ° ^"2 1^2 1^~' " (.ik/2T^d)expiikd); (9.8)
in case H* = 0, these factors must satisfy the condition (compare (7.9)-(7.I0))
•^"l 2^1 2^"' " ^"2 1^2 1^"' " -(ife/2TTd)exp(ifed). (9.9)
In (9.4)-(9.7) the kernel functions are given by
X^(x, 1x2) = .^^(x^lx,) iif exppfe{-[5,(x,)+C2(x2)) + (x,-X2)^/2d} (9.10)
and
•^j,(i/l \y2^ = '^y^^2^^0 iSi exp ^'^t-(n,(!/,)+n2(j/2)l + (y ^-y 2)^ / 2d]
(9.11)
The problem has now been reduced to determining the eigenvalues
M (") o ("j) J r, ("j) ü ..1. .. ^ • ^ 1
a, , , a„ , , 6j „ and g. from the systems of integral equations (9.4)-(9.7) as a function of the wave number k and the deter-mination of k from the relations (9.8) or (9.9). The superscripts n and
m in,m = 0,1,2,...) indicate which eigenvalue of the integral equations is under consideration; the eigenvalues are ordered according to de-creasing absolute value, n = m = 0 refers to the largest eigenvalue in magnitude. The corresponding eigensolutions determine the "(n,m)-th
mode" of the open resonator.
For resonators having a plane of symmetry, we have a. = a„ and
Sect.9 Rectangular cross-sections 39
= n, (!/) = r\2(.y) then substitution of H (x,y) = X(,x)y{y) in (7.11) or $ (.x,y) = X(,x)Y(y) in (7.12) leads to two single integral equations of the second kind
a Xix.) = ƒ K (x, |x„)Z(x„) dx„ when -a<x, < a, (9.12)
1 ' -a X \ ' 2 2. L 1
6 y(i/,) = \ \ K^iy^ li/2)^(2/2^ ^^2 ^^^" '^^^^l ^*- ^^''^^
In case * = 0 the factors a and g must satisfy the condition (com-pare (7.11))
(a B) = + {2ik/h-nd)expiikd) (even case),
(a g) = - (2ik/U-nd)expiikd) (odd case);
(9.14)
in case 4" = 0 , these factors must satisfy the condition (compare (7.12))
(a B) = - i2ik/hT[d)expiikd) (even case),
(9.15) (a g)~ = + i2ik/^-nd)expiikd) (odd case).
The kernel functions in (9.12) and (9.13) are given by
Ü:^(X, 1X2) def exp(i;fe{-£(x,) - ^ix^) + (x,-X2)^/2d}} (9.16)
and
^y'-yilyz^ def exp[iA:{-n(z/,) - 11(1/2) + iy ^-y 2^^ / 2d]). (9.17)
The problem has now been reduced to determining the eigenvalues a and g from the integral equations (9.12)-(9.13) as a function of the wave number k and the subsequent determination of k from the re-lations (9.14) or (9.15). The superscripts n and m in,m = 0,1,2,...) indicate which eigenvalue of the integral equations is under consider-ation; the eigenvalues are ordered according to decreasing absolute
40 Rectangular cross-sections Sect.9
Table IV. Deviation functions C, ~(x,z/) = £. „(x) + n, yiy) of some resonators with mirrors of rectangular shape.
I. plane mirrora.
2. Plane a i r r o r s . one a i r r o r t i l t e d .
tjCXj) - 0 a/2 - e r c t « a ( l / I > , )
njCvj) - ( » / i j ) » j
3 . Mirrors curved i n one d i r e c t i o n .
C , t e , ) . x , V M , n , ( , , ) - 0 n.C»,) - 0 4 . Plat-roof a i r r o r e . C , < x , ) . ( » , / a , ) | x , I n , ( y , ) - 0 C 2 ( X j ) - ( 4 j / O j ) | i j l nj(»2> - 0
/
1 1•~H
/ f ]
f
r
r
A°^"
y/ '
5 Ir ^
'if
f
t
3 . Flat-roof reaonetor with crossed e d g e s .
C | ( l , ) - ( i , / a | ) | x , I - I J C K I J - O
