coupling loss theory of single-mode waveguide resonators

Coupling loss theory of single-mode waveguide resonators

Christopher A. Hill and Denis R. Hall

The past dozen years have seen several important publications on mode coupling losses in circular and rec-tangular waveguide lasers. It is frequently assumed that the laser mode is pure EH1 1 (quasi-TEMOO). Wenote a flaw in the widely quoted Laguerre-Gaussian mode expansion method as it originally appeared andshow how to reconcile it with later results. Also we summarize and try to remove several discrepancies inthe published accounts of how the EH1 1 loss behaves for the popular near-Case I reflector (i.e., a plane mirrorplaced within a few guide widths of the guide aperture).

1. Introduction

Waveguide gas lasers have now been studied fornearly fifteen years' and have proved useful over a widerange of applications. 2-4 Recently there has been muchinterest in the physics and techniques of transverseradio frequency excitation of waveguide lasers, for whichthe square or rectangular guide geometry is especiallysuitable.5 In addition to parametric studies6 and freshapplications, a considerable literature of waveguideresonator theory has developed.

At the center of this theory lie the equations de-scribing the propagation of radiation inside hollow di-electric waveguides. These were derived for circulargeometry by Marcatili and Schmeltzer7 and later forrectangular geometry by Laakmann and Steier8 andKrammer. 9 Also essential is an understanding of howradiation behaves as it is launched from one end of theguide and recoupled by a laser reflector. Thus the lit-erature contains a large subset devoted to the theory ofmode coupling losses.

The simplest and most common way to recouple laserenergy is to place a plane mirror perpendicular to theguide axis and very near the guide aperture. Thus thelaunched radiation is recoupled with little chance todiffract, attenuate, or scatter.

Because of its evident interest as a low-loss compactreflector, this Case I configuration (see Sec. II) has beenstudied several times by different researchers. Inparticular, several expressions have been published for

The authors are with University of Hull, Applied Physics Depart-ment, Hull, HU6 7RX, U.K.

Received 18 December 1984.0003-6935/85/091283-00$02.00/0.© 1985 Optical Society of America.

the coupling loss of the fundamental EH,1 transverseguide mode as a function of the resonator parame-ters.

At present, there is some confusion on this subjectbecause the published accounts do not all agree with oneanother. In this paper we will try to clarify the positionand at least summarize the published accounts andpoint out where they differ. First, we briefly review thecoupling loss literature to date (Sec. II). In Sec. III weexamine the important initial theory of single-modecircular-guide losses due to Abrams.10 In Sec. IV thevarious published expressions for plane-mirror couplingloss are tabulated and discussed. Finally, we suggestcertain modifications to the accepted theory and discusstheir implications for practical laser design.

II. Treatments of Coupling Loss in WaveguideResonators

A general waveguide reflector is sketched in Fig. 1.The mode coupling efficiency of such a reflector, for aparticular waveguide transverse mode, is defined to bethe fraction of the power carried by the launched guidemode which is recoupled into the same mode after re-flection. It equals I C 12, where c is the amplitude overlapintegral across the guide aperture of the initial and re-turned field distributions. The mode coupling loss isdefined to be F = 1 - I C 1

2.

The problem of determining the field distributionreturned to the aperture plane has been approached intwo main ways. The first is to express the initial fieldas a linear combination of free-space transverse modes,which are then propagated to and from the mirror ac-cording to the familiar Gaussian beam equations ofKogelnik and Li.'1 This approach assumes that themirror aperture is effectively infinite. It has an intui-tive appeal because, for circular'0 and square12 guidegeometries, the beam waist radius co, which charac-terizes the complete set of free-space modes, can bechosen so that the mode expansion for the fundamental

1 May 1985 / Vol. 24, No. 9 / APPLIED OPTICS 1283

2Z v

Fig. 1. General waveguide reflector.

EH 11 guide mode is dominated by the fundamentalTEMOO free-space mode. In both geometries the poweroverlap is 98% if coo is suitably chosen. Hence thebehavior of a launched EH11 mode may be expected tobe fairly similar to that of its approximatingGaussian.'0

This mode-expansion method was introduced byAbrams10 and developed by Chester and Abrams.13 Itwas also used in various forms by Henderson,' 2 Laak-mann and Steier,8 and Melekhin and Melekhina.14

In the second method the fate of the launched ra-diation is described by a scalar Fresnel-Kirchhoff dif-fraction integral. One of its advantages is that reflec-tion from finitely wide mirrors can be treated naturallyby restricting the range of integration over the mirrorplane. Degnan and Hall15 published an account of fi-nite-aperture mirror coupling losses as part of a generalstudy of circular-waveguide resonator modes. In thepresent paper we restrict ourselves to the EH 11 mode.Similar techniques were developed independently byAbrams and Chester.16 These papers confirmed theexistence of three reflector configurations which involvelow loss for EH 11, named by Degnan and Hall as follows(see Fig. 1 and Table I):Case I. Large radius of curvature mirrors close to

the guide (RM ''' ,Z O 0, ''' 0).Case II. Large radius of curvature mirrors whose

centers of curvature lie approximately atthe guide entrance (I 1).

Case III. Mirrors of moderate curvature at a dis-tance RM/2 (RM As 2b, Z = b, an 2.4).

These three types of waveguide reflector became fa-miliar in the literature. But, as Abrams and Chester16remarked, so many different variables are present in reallasers that it is almost impossible to make useful gen-eralizations. Either long computations involving sev-eral guide modes must be undertaken to characterize

Table . Waveguide Resonator Notation

Quantity Symbol

Halfwidth of square guide I aRadius of circular guide ISeparation of guide and mirror z (or d)Mirror radius of curvature RM (or c)Gaussian beam parameters wo, b = rc0/o

Dimensionless quantities:a 2/(Xz) N (Fresnel number of reflector)ka 2/RMzIRM /

a given resonator or gross simplifications must be ac-cepted. Thus these early papers, although theoreticallycomprehensive, were far from exhausting a subjectwhich continues to produce interesting and unexpectedresults. 1 7"18

Diffraction integral calculations were performed forrectangular guides by Avrillier and Verdonck.'9 LaterBoulnois and Agrawal20 noticed that, if infinitely widemirrors were assumed, the twofold rectangular-guideintegral could be reduced to a much simpler onefoldintegral, which yielded closed-form solutions for theCase I, II, and III reflectors. This promised a valuablesaving of the computational effort needed to modelwaveguide resonators.

A study of this coupling loss literature reveals certaindifferences of opinion, particularly regarding the CaseI or near-Case I reflector. These problems are treatedin Secs. III and IV.

Ill. Circular-Guide Coupling Loss Theory of Abrams

Abrams' 0 calculated coupling losses by expanding theEH11 field (given by a Bessel function) in terms ofLaguerre-Gaussian free-space modes and performingan overlap integral across the reentrance guide aper-ture.

The normalized EH11 mode is (0 < r < a):

E(r) = [ira2 JI(un)V1Jo(uiir/a)

- L Ap Op(r),p

fE2(r)27rrdr = 1.

The normalized Laguerre-Gaussian modes are

4dp(r) = A/-Lp(2r2/@O) exp(-r2 / 0),

3 2 /S p(r)2 rrdr = 1.f

(1)

(2)

(3)

(4)

The Ap are expansion coefficients. The field returnedto the guide by an infinitely wide mirror (radius ofcurvature RM) a distance z away is

E'(r) = /id-A Z ApLp(2r2IW2) exp [- 2 (1 + i ] exp(iop),

(5)where the Gaussian beam parameters on return are

X = beam radius,A3 = (7rW2)/X,

p = radius of curvature of beam phase front,and the on-axis propagation phase shifts (sometimesknown as the Guoy shifts2') are given by Abrams as

(6)op = (2p + 1)[tan-'(z/b) + tan-'(z/)].

The EH, 1 coupling loss is

rll(z,RM) = 1- If E(r)E'(r)27rrdr 2- (7)

Abrams evaluated the losses for various configurationsusing the first six modes in the Laguerre-Gaussian ex-pansion with wo = 0.6435a to ensure that 99.86% of theEH11 energy was accounted for.

1284 APPLIED OPTICS / Vol. 24, No. 9 / 1 May 1985

2a -£rm==:~T ' d

ZPzC.)

z3-s

SI d

Fig. 2. Coupling efficiencies for the EH,1 guide mode (reproducedfrom Degnan and Hall'5 with permission).

100

q 90

I 80zw 70

U 60

u 50z- 40

3 30U

20W 10

01. 5

Z / R

Fig. 3. EH1, coupling efficiencies [after Abrams' 0 with op accordingto Eq. (6)].

Essentially similar approximations and assumptionsunderlie both the Gaussian beam approach of Abramsand the scalar diffraction method of, for example,Degnan and Hall. Therefore, apart from the effects oftruncation in the mode expansion, purely computa-tional errors such as roundoff, and aperturing at finitemirrors, the two should predict the same coupling lossfor a given configuration.

Figure 2 is reproduced from Degnan and Hall'spaper' 5 and shows the EHI coupling efficiency plottedagainst normalized distance = zIRM for a mirrorhalfwidth of 11a and several values of the dimensionlessparameter a = ka 2/RM. Figure 3 shows the corre-sponding predictions (for infinitely wide mirrors) ofAbrams's method, which we have recalculated and

Reflected beamR= e; = rcW

2/I,

RM=2f I New waistI R=°° b= r " ', d

z -

Aperture plane Mirrorz= O plane

Fictitious waist planeof reflected beam

Fig. 4. Beam parameters of approximating Gaussian reflected froma spherical mirror.

plotted to the same scale. These curves show kinks inthe region z RM, which are absent from Fig. 2. Thea = 0.296 curves are quite different because in this onecase the mirror aperturing loss is significant.

We think that the discrepancy between these two setsof results is due to the fact that the Gaussian beamphase shifts are not quite as given by Eq. (6). ConsiderFig. 4, which shows diagrammatically the progress of aTEMoo beam as it is launched from the guide and re-flected from an arbitrary spherical mirror. Initially itis characterized by the beam parameters b = 7rw/X andradius of phase front curvature R = . At the mirrorit undergoes a beam transformation described by theequations in Ref. 11 and may now be considered to havea fictitious beam waist with b' = rw 2/X and R = ao, lo-cated a distance z' from the mirror and (z + z') from theaperture plane.

On propagating the beam back to the aperture plane,we have a new q parameters given by

I1 iq

where2 2

q = z+z(f-z)-b 2 ibf2

(f-z)+b2' (f-Z)2+b2

= (z + z') + ib'.

(8)

(9)

The phase shift associated with the pth Laguerre-Gaussian mode at a distance (z + z') from its (fictitious)beam waist is (2p + 1) tan-'[(z + z')/b']. We thereforesee from Fig. 4 that the net phase shift enjoyed duringa return trip to the mirror is

p = (2p + 1) [tan-1 () + tan-, (Z + Z -tans1'\}]

Evidently

= 1+ + (Z +z')2]1/2

A- Z + Z'

P be

(10)

(12)

For a plane mirror the p have a simple geometricalform. If a Gaussian beam has -rW2 (z)/X equal to b at itswaist in the guide aperture plane, b at the mirror adistance z away and : on its return to the aperture, wehave (see Fig. 5)


bm = b 11 + ()2 1 1/2

/ = b [1 + (2Z)2]1/2,

tan1 (5) + tan-1 (1) = A + A,tn 1 2zl =e A + A

(13)

(14)

(15)

(16)

Another important special case is the phase-matchedmirror, where Eqs (6) and (10) yield the same result:

Fig. 5. Beam parameters and phase angles for reflection from a planemirror.

100

90

80

9 70

C 60

u 50

, 40

b 308 20

5 10

0o

Z / R

Fig. 6. EHII coupling efficiencies [after Abrams' 0 with p accordingto Eq. (10)].

RM = z + (b 2/z),

z = -Z,

b'= b,

Op = 2(2p + 1) tan-l(z/b).

(17)

(18)

(19)

(20)

Figure 6 shows results based on Eq. (10). The agree-ment with Fig. 2 (a = 0.296 again excepted) is excel-lent.

IV. Coupling Losses for a Plane Mirror in aWaveguide Resonator

Each of the main papers mentioned in Sec. II containssome account of how the EHI, coupling loss behaves fora plane mirror placed very close to (within a few guidewidths of) the guide aperture. In Table II we havelisted these published expressions, both in the originalnotation and in terms of the normalized distance (zib).Here z is the guide-mirror distance, and b = 7rwo/ isthe Rayleigh range or confocal parameter of the ap-propriate free-space approximating Gaussian (see Sec.

Table 1. Published Expressions for Plane-Mirror EH1, Coupling Loss

Authors Geometry Method EH,1 coupling loss (%) Range

Kogelnik 2 3 TEMOO Gaussian beam equations 100(•) zlb << 1Coupling b

Abrams'0 Circular Laguerre-Gaussian 70 (Z)3/2 zlb < 0.4 (from graph)expansion, wo = 0.6435a b

Iz 13/2Degnan and Hall15 Circular Diffraction integral, 605 I-I or

finite aperture kka2J

57 @)3/ , 38.4N-3/2 z/(ka2) < 0.1, zb 0.48

Henderson1 2 Square Hermite-Gaussian 80( -I or 1310 I- 2 33N- 2 zib < 0.2expansion, wo = 0.7032a b ha

Avrillier and Verdonck'9 Square Diffraction integral, 161.8 1-2- (see Ref. 24) z/(ka 2 ) < 0.0125infinite aperture ha2

Boulnois and Agrawal20 Square Diffraction integral, ^i33 N-31 2 or 64.5 () N >> 1infinite aperture z HZ3/2

5 2 4


z

z

Table Ill. Conversion Factors for Coupling Losses

Normalized z a 2distance ka 2 N =

z/b dzv 2) 2 N-1 Ir a \@

zib = 4.83 = 0.769N-1 (circular)

= 4.04 (. = 0.644N-1 (square)

II). For circular guides10 we choose w0 - 0.6435a, andfor square guides' 2 Wo0 0.7032a. Conversion factorsfor the different notations are shown in Table III.

In the previous section we proposed a slight changein the circular-guide approach of Ref. 10. This changedoes not materially affect the results for very small (zib),and therefore does not bear on a striking feature ofTable II, namely the disagreement about the power lawobeyed by r11 (z << b). The practical consequences ofthis disagreement are considered in Sec. V; in this sec-tion we will outline how the various expressions ariseand how we think some of them can be reconciled.

Abrams's Fig. 6 in Ref. 10 is a log-log plot of rilagainst (zib) for various mirror curvatures RM. For RM= and small (zib) he shows a straight line of slope3/2 yielding approximately r = 70(z/b) 3 /2 %.

Degnan and Hall15 explicitly quoted

ri, i--t 605 2 % for-<01

This was a numerical approximation to the results oftheir diffraction integral calculations.

To our knowledge, these are the only two indepen-dent calculations for circular guides in the literature.

Because both circular and square guides have EH11modes which are excellent approximations to TEMoomodes, it is at least plausible that their plane-mirrorcoupling losses should (a) behave similarly and (b) be-have as TEMoo plane-mirror coupling losses. Now itis well known2 3 and easily checked that the pure TEMOOcoupling loss is proportional to (z/b) 2 for small (zib).

Three main accounts of EH11 losses for square orrectangular guides have appeared:

Henderson12 was concerned with transmission lossbetween a guide and a misaligned intracavity modula-tor, but if we take the modulator to have the same di-mensions as the guide then his results are directly ap-plicable. He derived approximate square-law expres-sions for each of the losses caused by separation, lateraldisplacement, and angular tilt. We have considered tiltlosses elsewhere.18 Henderson gives r 1 1 80 (z/b) 2%for zib < 0.2 (to within 10% accuracy). In a privatecommunication Henderson confirmed that this ex-pression in terms of normalized distance is correct, al-though the axes of his Fig. 4 in Ref. 12 appear to bemis-scaled.

Avrillier and Verdonck'9 outlined a general diffrac-tion method for square-guide EHmn losses and pre-sented results for EH11. After quoting Degnan andHall's result for circular guides they offered thesquare-guide expression

IZ2rl = 161.8

Boulnois and Agrawal20 queried this result, havingthemselves derived

rll 1/3N-3/2 524 Z 32)%24

as an asymptotic approximation based on a simplifiedonefold integral. Direct numerical evaluation of thatintegral readily yields a three-halves law. However,Avrillier and Verdonck's expression appears to be amis-statement of Henderson's earlier one and not sep-arately derived.24

In an attempt to resolve this problem of the exponentin the r11 near-Case I expression, we examined theAbrams method again. Clearly some care is neededbecause ri, (z << b) is calculated as the difference be-tween two very nearly equal quantities, namely unityand Ic 2 .

We recall that

ri,(z,RM) = 1 - IS E(r)E'(r)27rrdrI . (17)

From Eqs (1)-(5) and (9)-(11), after some manipula-tion, we find that the integral is

where

c = K ac o 4 _ Ap exp(i p) ap (z,RM),

K = [7ra 2 J2(u1)]-1/2

a = [1 + (z + Z') 21 -1/2

ap (z,RM) = f J° -a2x bJo a \2 b+)

a 2xb IX exp - 11+fIl dx.

I- 2b' pI

(18)

(19)

(20)

(21)

In its present form the normalizing factor is

\/ 2 /2 ja0 Jl~ull) (22)

and its square is 0.7682. But the coupling integralabove was originally normalized according to the modeexpansion procedure where

IA, 2 =1.p=O

With six modes in the calculation we have5

I lAp 2 = 0.9986.p=o

To keep the normalization

S E2(r)27rrdr = 1,


2

1.

V)

0

0U

3IwUN

LD

I

.5

0

-. 5

-1

-1. 5

-2

-2.5 , i -2 -1. 5 -1 -. 5 0 .

LOG Z / B )

Fig. 7. Plane-mirror EH1 1 coupling loss.

U)U)C

0~

0U

I

UN

CJ

-2.-1

LOG ( Z / B )

Fig. 8. Plane-mirror near-field EH1 1 coupling losses: A, normalized

so that r11 (z = 0) = 0; B, normalized so that i IA, 12 = 1 (truncationp=O

error remains).

we divide c by (0.9986)2 - 0.997; then the square of themultiplying factor is 0.77043.

In practice we obtain the results shown in Figs. 7 and8. Figure 7 represents our best estimate of the couplingloss r 1 (zlb) based on a multiplier of 0.77043. Thedistortion in the loss curve for very small (zib) is prob-ably due to roundoff in the microcomputer calculationsand to uncertainty in the various coefficients. Allowingfor this small error (-0.01%), we can say that the slopeis very nearly 2 for z less than a few millimeters and fallssmoothly with increasing (zib). The middle portion

(0.1 < zb < 0.4) is indeed fairly well described by Pi70 (z/b)3 /2%We can empirically compensate for roundoff and

other processes by adjusting the normalizing factor sothat r11 (z = 0) is more accurately zero. The resultsthus obtained (curve A in Fig. 8) show r 1 (z = 0) <10-6% and F, (z << b) 158 (z/b) 2%.

Finally, if the truncation error mentioned above is notcompensated for, curve B in Fig. 8 is obtained. Herethe loss as zib approaches zero is

1- E IAp 2) 0.28%,

and again a three-halves law appears at moderatelysmall z.

We conclude that, in the absence of normalization orcomputation errors, the small distance limit of the cir-cular-guide EH11 coupling loss with a plane mirror isgiven by P11 158 (z/b) 2% as shown by curve A of Fig.8. However, the three-halves law quoted in the litera-ture15 and inferred from Abrams's results is approxi-mately valid over a small range of (zib).

V. Discussion

We have drawn attention to a few discrepancies whichexist in the literature of waveguide laser coupling losses.We think we have settled one or two problems, butothers are as yet unresolved. In this final section we willdiscuss the extent to which our suggestions may affectpractical laser designs.

It is perhaps worth emphasizing again that the sin-gle-mode (EH1 1 only) approximation yields predictionsfor the behavior of real waveguide resonators whichcompare rather poorly with those of multimode iterativeor matrix-diagonalization techniques, and that thelatter require much the greater amount of computation.Therefore, the laser designer has a choice and a trade-offto make, and in many cases the single-mode approxi-mation is considered adequate. This means that theaccuracy of the various (contradictory) expressions wehave discussed is still a live issue.

We have proposed that the account of circular-guidecoupling losses due to Abrams'0 should be modified intwo slight respects. First, the on-axis phase shift en-joyed by the Laguerre-Gaussian beams in the free-spacemode expansion should be as in our Eq. (10), not Eq. (6).It is clear from inspection that these two forms do notdiffer significantly when z << b; also, as mentionedearlier, they do not differ at all when the beams arephase-matched to the mirror, as in the important specialconfigurations characterized by Degnan and Hall asCases I, II, and III. The discrepancy is marked only forlarger values of zib, as seen in Fig. 3, and these are foundrelatively rarely in waveguide devices.

The second matter is both less fundamental and, inpractice, more important. We suggested that, when thecoupling loss integral is suitably normalized to accountfor the fact that not all the EH11 mode energy is con-tained in a finite free-space expansion, the loss F11(z)obeys a square law (not a three-halves law) in the limitof small z.


I1

35

30 +

U)U)C

-J

z-JLo

0

IUJUN

25

20 .

15 +

10 +

5.

0 . I . 2 .3 . 4 .5

Z / B

Fig. 9. Approximate EH11 coupling losses (square guide): A,Boulnois and Agrawal,20 riu = 64.5 (z/b) 312%; B, Henderson,1 2 F.

= 80 (z/b) 2%.

7

6U)Un0-J

LOz-J0

Iaw

5

4

3

Z / B

Fig. 10. Approximate EH11 coupling losses (circular guide): A,Degnan and Hall,15r P = 57 (z/b)3/2%; B, Hill and Hall, riu = 158

(z/b) 2%; C, actual loss.

This led us to a more comprehensive review of pre-vious studies, which revealed the discrepancies sum-marized in Table II. Some of these are trivial, and atleast one [the disagreement between Henderson (squarelaw) and Boulnois and Agrawal (three-halves law) forthe square waveguide] has defeated us. But, on theassumption that our square-law proposal is correct forboth circular and square guides, it is interesting toconsider the numbers which might arise in practice(Figs. 9 and 10).

For very small z the differences between the pub-lished loss expressions will be unimportant. But in

many waveguide devices the total round-trip dissipativeloss, apart from coupling losses, is <1-2%. For mod-erate z of (typically) several millimeters, the differentpower laws predict square-guide coupling losses whichdiffer by >1/2%. Even such apparently small differencescan seriously affect laser performance as described bya Rigrod-type analysis.2 5' 26 In some cases, for examplewhere an intracavity Brewster window is used, larger(zib) values 0.1 may occur, and extrapolation ac-cording to an incorrect power law would be unwise.Apart from their theoretical interest, therefore, ourresults have certain practical implications.

One of us (Chris Hill) was supported during 1981-84by a SERC CASE studentship with Culham Labora-tory.

References1. P. W. Smith, "A Waveguide Gas Laser," Appl. Phys. Lett. 19,132

(1971).2. J. J. Degnan, "The Waveguide Laser: A Review," Appl. Phys.

11, 1 (1976).3. R. L. Abrams, "Waveguide Gas Lasers," in Laser Handbook, Vol.

3 (North-Holland, Amsterdam, 1979).4. P. W. Smith, 0. R. Wood II, P. J. Maloney, and C. R. Adams,

"Transversely Excited Waveguide Gas Lasers," IEEE J. QuantumElectron. QE-17, 1166 (1981).

5. J. L. Lachambre, J. Macfarlane, G. Otis, and P. Lavigne, "ATransversely RF Excited CO2 Waveguide Laser," Appl. Phys.Lett. 32, 652 (1978).

6. C. J. Baker, D. He, P. J. Wilson, and D. R. Hall, "DischargeScaling in rf-Excited Waveguide CO2 lasers," in Technical Digest,Conference on Lasers and Electrooptics (Optical Society ofAmerica, Washington, D.C., 1984), paper THL3.

7. E. A. J. Marcatili and R. A. Schmeltzer, "Hollow Metallic andDielectric Waveguides for Long Distance Optical Transmissionand Lasers," Bell Syst. Tech. J. 43, 1783 (1964).

8. K. D. Laakmann and W. H. Steier, "Waveguides: CharacteristicModes of Hollow Rectangular Dielectric Waveguides," Appl. Opt.15, 1334 (1976).

9. H. Krammer, "Field Configurations and Propagation Constantsof Modes in Hollow Rectangular Dielectric Waveguides," IEEEJ. Quantum Electron. QE-12, 505 (1976).

10. R. L. Abrams, "Coupling Losses in Hollow Waveguide LaserResonators," IEEE J. Quantum Electron. QE-8, 838 (1972).

11. H. W. Kogelnik and T. Li, "Laser Beams and Resonators," Appl.Opt. 5, 1550 (1966).

12. D. M. Henderson, "Waveguide Lasers with Intracavity Elec-trooptic Modulators: Misalignment Loss," Appl. Opt. 15, 1066(1976).

13. A. N. Chester and R. L. Abrams, "Mode Losses in Hollow-Waveguide Lasers," Appl. Phys. Lett. 21, 576 (1972).

14. G. V. Melekhin and G. P. Melekhina, "Matching Losses inWaveguide Cavities. Two-Dimensional Problem," Opt. Spec-trosc. 52, 527 (1982).

15. J. J. Degnan and D. R. Hall, "Finite-Aperture Waveguide-LaserResonators," IEEE J. Quantum Electron. QE-9, 901 (1973).

16. R. L. Abrams and A. N. Chester, "Resonator Theory for HollowWaveguide Lasers," Appl. Opt. 13, 2117 (1974).

17. R. Gerlach, D. Wei, and N. M. Amer, "Coupling Efficiency ofWaveguide Laser Resonators Formed by Flat Mirrors: Analysisand Experiment," IEEE J. Quantum Electron. QE-20, 948(1984).

18. C. A. Hill and D. R. Hall, "Waveguide Resonators with a TiltedMirror," in preparation.


A

O.

19. S. Avrillier and J. Verdonck, "Coupling Losses in Laser Resona-tors Containing a Hollow Rectangular Dielectric Waveguide,"J. Appl. Phys. 48, 4937 (1977).

20. J.-L. Boulnois and G. P. Agrawal, "Mode Discrimination andCoupling Losses in Rectangular-Waveguide Resonators withConventional and Phase-Conjugate Mirrors," J. Opt. Soc. Am.72,853 (1982).

21. A. Siegman, Lasers (University Science Books, Mill Valley, Calif.,1983).

22. The expression for what we have called z' is misprinted in Ref.10 (the -b 2 term is missing).

23. H. W. Kogelnik, in Quasi-Optics (PIB Symposium Proceedings,New York, 1964; J. Fox, Ed.). For z << b the loss is r loo(z/b)2% when evaluated across the infinite aperture plane, andc98.4 (z/b) 2% when evaluated for wo = 0.6435a and 0 < r < a.

24. Henderson's result' 2 converts as follows:

20 S2%= 20 2 = 20 z- .4-IaI2'2() b ka2 wo)]

where wo = 0.7032a. This equals 20 X (8.089)2 (z/ka 2 )2= 1310

(z/ka 2 )2 or 33.1N- 2 %. The factor of 161.8 quoted in Ref. 19equals 20 X 8.089.

25. W. W. Rigrod, "Saturation Effects in High-Gain Lasers," J. Appl.Phys. 36, 2487 (1965).

26. D. He and D. R. Hall, "Influence of Xenon on Sealed-off Opera-tion of RF-Excited CO2 Waveguide Lasers," J. Appl. Phys. 56,856 (1984).

Meetings Calendar continued from page 1255

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10-14 Robot Manipulators, Computer Vision & IntelligentRobot Systems course, Edinburgh Off. of SummerSession, Rm. E19-356, MIT, Cambridge, Mass.02139

10-14 Using the ADA Programming Language course, AnnArbor Eng. Summer Confs., 200 Chrysler Ctr., N.Campus, U. of Mich., Ann Arbor, Mich. 48109

10-14 Applied Numerical Methods Using Personal Computerscourse, Ann Arbor Eng. Summer Confs., 200 ChryslerCtr., N. Campus, U. of Mich., Ann Arbor, Mich.48109

10-14 Introduction to Techniques For Information ExtractionFrom Remotely Sensed Data course, Wash., D.C. P.Vidal, Geo. Wash. U., Cont. Eng Ed., Wash., D.C.20052

10-14 Laser Safety: Hazard, Inspection & Control course,Wash., D.C. Laser Inst. of Amer., 5151 Monroe St.,Toledo, Ohio 43623

11-14 Image Science & Technology ICO Conf., Helsinki P.Oittinen, Helsinki U. Technology, Lab. of GraphicArts Tech., Tekniikantie 3, 02150 Espoo 15, Fin-land

12-14 Workshop on Optical Fabrication & Testing, OSATech. Mtg., Cherry Hill OSA Mtgs. Dept, 1816 Jef-ferson Pl., N. W., Wash.,D.C. 20036

17-18 Laser Nurse Seminar & Workshop, Chicago Laser Inst.of Amer., 5151 Monroe St., Toledo, Ohio 43623

17-19 43rd Ann. Device Research Conf., Boulder L. Toma-setta, Vitesse Electronics, 741 Calle Plano, Camarillo,Calif. 93010

17-19 Int. Conf. on Chemical Kinetics, Gaithersburg J. Her-ron, A147 Chem. Bldg., NBS, Wash., D.C. 20234

17-21 Database Management Using Personal Computerscourse, Ann Arbor Eng. Summer Confs., 200 ChryslerCtr., N. Campus, U. of Mich., Ann Arbor, Mich.48109

17-21 Infrared Technology Fundamentals & System Applica-tions course, Ann Arbor Eng. Summer Confs., 200Chrysler Ctr., N. Campus, U. of Mich., Ann Arbor,Mich. 48109

18-21 Instabilities & Dynamics of Lasers & Nonlinear OpticalSystems Mtg., Rochester OSA Mtgs. Dept., 1816Jefferson P., N. W., Wash., D. C. 20036

23-30 Soc. of Women Engineers Ann. Natl. Convention, Min-neapolis G. Hinschberger, P.O. Box 9542, Minneap-olis, Minn. 55440

24-28 7th Int. Conf. on Laser Spectroscopy, Maui T. Hansch,Physics Dept., Stanford U., Stanford, Calif. 94305

24-28 Int. Conf. on Fourier & Computerized Infrared Spec-troscopy, Ottawa L. Baignee, Conf. Services Office,Ottawa, Ontario KIA OR6, Canada

24-28 Robotics: Concepts, Theory & Applications course, AnnArbor Eng. Summer Confs., 200 Chrysler Ctr., N.Campus, U. of Mich., Ann Arbor, Mich. 48109

24-28 Advanced Infrared Technology course, Ann Arbor Eng.Summer Confs., 200 Chrysler Ctr., N. Campus, U. ofMich., Ann Arbor, Mich. 48109

24-29 Fourier & Computerized Infrared Spectroscopy nt.Conf., Ottawa Natl. Res. Council of Canada, L.Baignee, Conf. Services Off., Ottawa, Ontario, CanadaKlA OR6

24-5 July Applied Optics Summer course, London J. Dainty,Optics Sec., Blackett Lab., Imperial Coll., LondonSW72BZ, England

24-5 July Applied Materials Technology: Materials Processing forProcess-Sensitive Manufacturing course, EdinburghOff. of Summer Session, Rm. E19-356, MIT, Cam-bridge, Mass. 02139

10-14 Fundamentals & Applications of Lasers course, Wash.,D.C. Laser Inst. of Amer., 5151 Monroe St., Toledo,Ohio 43623


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coupling loss theory of single-mode waveguide resonators

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