Misalignment sensitivity of optical resonators

R. Hauck, H. P. Kortz, and H. Weber

The effect of mirror misalignment of spherical resonators is investigated experimentally and compared tofirst-order perturbation theory. An expression D is derived, which characterizes the misalignment sensitivi-ty of any spherical resonator. It is proved experimentally that this misalignment sensitivity depends on theeffective resonator length L* and the gi parameters only.

1. IntroductionBy adapting the diameter of the TEMoo mode to the

diameter of the active medium, the efficiency of a laseroscillator can be increased considerably.1 This requireseither a large mirror distance L or an optical resonatoroperating near the limit of stability. In any case, theresonator becomes very sensitive to a misalignment ofthe mirrors. From symmetry we may deduce that theincrease of diffraction loss due to misalignment is pro-portional to the square of the mirror tilting angle i.Therefore, a suitable expression for the loss factor Viper resonator bounce is

Vi = Vo[j - (au/aoi)2j, (1)

where i indicates mirror Si, which is tilted by an angleci with respect to the resonator axis. The misalignmentsensitivity of the resonator is characterized by aei Inthe following sections the relation between aoi and theresonator parameters is investigated experimentally andtheoretically.

I. TheoryThere are only a few papers dealing with the influence

of misalignment on diffraction losses and field distri-bution of optical resonators. Numerical calculationswere carried out for special systems such as symmetricor confocal resonators,Z: 3 unstable strip resonators,4 andplan-plan resonators.5-8 Berger et al. 9 used first-orderperturbation theory. But they assume that the fielddistribution of the infinite mirror is not disturbed by theaperture of the system. Therefore, their result is a verycrude approximation. Nevertheless, it is the only onederived for a generalized spherical resonator (see Fig.7).

The authors are with Universitaet Kaiserslautern, Physik, Postfach3049, 6750 Kaiserslautern, Federal Republic of Germany.

Received 15 October 1979.0003-6935/80/040598-04$00.50/0.Oc 1980 Optical Society of America.

The laser oscillator consists of two spherical mirrors,radii of curvature Pi and P2 in a distance L, and the ac-tive medium with length and refractive index n. It isassumed to be homogeneous. The mode properties ofthe resonator are characterized by the effective lengthL*, L* = L - /n, and the g parameters, gi 1 -L*/pi (Fig. 7).For infinite mirrors, the spot size of the TEMoo modeis given by'0

W2=?XL* 1gj )/2

i r -( -g1g2))(2

The resonator axis is defined by the two centers ofmirror curvature Ml and M2. If mirror Si is tilted byan angle ai, the resonator axis is rotated by an angle Oi,and the centers of the field intensity patterns areshifted. A simple geometric consideration delivers therelations

1-gEoi = i g1 - 1g2

Aii = aigjL*/(1 -912),

Aij = ajL*I(1 - gg2) i ;,d j.

(3)

(4)(5)

Aij means the displacement of the intensity pattern atmirror Si, if mirror Sj is tilted by aj.

Near the limit of stability (1 * 2 ' 1), the beamsteering angle Oi and the displacement Aij may becomeconsiderably large. Nevertheless, as long as infinitemirrors are considered, the resonator remains aligned,and there are no diffraction losses. But if a limitingaperture is inserted into the resonator, e.g., the activemedium or a mode selecting pinhole, diffraction lossesoccur and increase rapidly with increasing mirror tiltangle. For simplicity, we assume a pinhole of radius aito be placed directly in front of each mirror.

Tilting a mirror is equivalent to a displacement of thepinhole.3 For a system with only one pinhole, Bergeret al.9 calculated the dependence of diffraction lossfactor V on the pinhole displacement A. A first-orderperturbation theory for the TEMoo mode delivers

598 APPLIED OPTICS / Vol. 19, No. 4 / 15 February 1980

M1

Si C1 C2 52

I -I'L ---I

OA'

OA

Fig. 1. Misaligned spherical resonator. A tilt ofmirror S2 by a2 shifts the resonator axis by 02 andresults in displacement A12 and A22 of the intensity

patterns on both mirrors.

V = 1 - [1 + 2(A/w)2 (a/w)2 ] . exp[-2(a/w) 2 ], (6)

where a = pinhole radius,w = beam diameter of the TEMOO field

pattern at the pinhole, andV = loss factor per resonator bounce.

Generally a resonator has limiting apertures on bothmirrors. Then the loss factor by tilting mirror Si isgiven by

Vi = (Vii _ Vji) 1/2 i 54 j, (7)

with

Vji = 1 + 2 -/j) -a)2 exp -2 (a)]* (8)

For small losses (1 -Vji, 1 - Vii << 1), Eq. (7) combinedwith Eq. (8) can be approximated by

Vi -Vo _ 11(wj (w~1 )]ex ajt)]

[( -i) ( i)] * exp -2 ( i)] (9

V0 is the loss factor of the aligned system with

Vo 1 - 1 {exp [2 (aiJ2I + exp [-2 (~L)2I1 (10)21 I- [ (i) Waj2]

For minimizing diffraction losses on the one hand andpreventing multimode oscillation on the other hand, itis reasonable to use pinhole radii a bit larger than thebeam radii. Experimentally the following was found"for spiking laser systems:

ai = Swi, S = 1.2. (11)

Combining the above equations with Eqs. (2), (4), and(5), we finally get

vi= - S2 _ Di) (12)

withVo = 1 - exp(-2S2 )],

7rL* (gJ\1/2 1+glg2 (13)

r g (1 - 9192)3/2

Equation (12) represents the diffraction loss factor Viper resonator bounce, if mirror Si is tilted by ai. Wenotice that the losses (a) increase with the square of thetilt angle, and (b) are characterized by the misalignment

sensitivity Di, which according to Eq. (13) depends onthe resonator length L* and the gi parameters. Com-paring Eq. (12) and Eq. (1), we obtain for a0i

[exp(2S 2) - 1]1/2

atoi =~ SD,(14)

If a mirror is tilted by an angle a = 1/Di, additionallosses of -10% are caused if s = 1.2. This gives a clearidea of the meaning of the misalignment sensitivity Didefined by Eq. (13). It should be kept in mind, how-ever, that low-gain lasers are affected much more by anadditional loss of 10% than are high-gain lasers. Thus,misalignment sensitivities of different resonator con-figurations may be compared only if their gains areabout the same.

If both mirrors are misaligned, the losses proportionalto D? are summed up. Therefore, the misalignment ofthe complete system is defined as D = (D2 + D )1/2 andis given by

D = [(rL* 1 +g1g2 g +g2 11/2 (15)

I1IV XJ I (1-gg2 )3/2

(glg2 )1/2]

where D is a number characterizing any spherical res-onator with respect to its sensitivity against mirrortilting. High value of D means high misalignmentsensitivity. The most insensitive resonator is thesymmetric confocal one with g, = 2 = 0 and

= (2xrL*)1/2 (16)

But, from the stability diagram and from Fig. 2, we learnthat gl = = 0 represents a discontinuity. Small de-viations from symmetry may cause high losses and highmisalignment sensitivity. Figure 2 shows the depen-dence of the relative sensitivity DIDo on the gi param-eters.

Ill. Experimental InvestigationsThe influence of mirror misadjustment on laser

output and field distribution was investigated by vari-ous authors.9"12-' 5 But no systematic measurementswere carried out. Neither diffraction losses nor therelations of Eqs. (12)-(15) were measured.

For the present measurements a pulsed Nd:YAGTEMoo laser in the spiking mode was used. This laserhas the advantage of high gain, consequently, high lossesand high tilting angles are admissible. The relative lossfactor Vi was determined from the time delay betweenstarting the flashlamp and onset of laser oscillation.

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The absolute value of Vi could be obtained from thefrequency of the relaxation oscillations (spiking fre-quency). The details of this method were published byJunghans et al.16 The experimental results of onemirror configuration are shown in Fig. 3. By least-squares fit the experimental values are approximatedby a parabola, and a0i is determined. In Fig. 4 the ex-perimental values a0i of various resonator configura-tions are plotted against the misalignment sensitivityDi defined by Eq. (13). Results obtained by Freiberg

and Halsted' 4 from power measurements are includedand fit well. The experimental results confirm that themisalignment sensitivity of spherical resonators de-pends on one parameter Di only.

The deviation from theory is beyond the experi-mental error. It may be explained by the insufficiencyfirst-order perturbation theory. This is confirmed bycomparing the diffraction losses of the aligned confocalsymmetric resonator from Eq. (10), al = a2 = a:

Vo = 1 - exp(-27rF)

to the higher-order approximation of Slepian17

(17)

92

2

00 1 2 3 4 5

91 -

Fig. 2. Lines of constant misalignment sensitivity DIDo in the g1,g2

diagram. Do = (27rL*/X)l/2 .

Vo = (1 - 167r2 ) F exp(-4xrF),F = a2 /XL*. (18)

Furthermore, the experimental results indicate thatsensitivity parameter D [Eq. (13)] is a suitable param-eter to describe the alignment stability of a resonator.

The upper limit of D, which still permits stable laseroscillation, depends mainly on the mechanical stabilityof the system. Thermally induced beam deflection bythe laser rod may also be important. With our experi-mental setup,' 8 consisting of a granite optical bench,AOM mirror mounts (Aerotech, Incorporated type110-3), pinholes, and laser cavity, an angular stabilityof

amin = 2 X 10-" rad = 4 sec

is achievable. This means resonators up to Dmax = 5X 104 are admissible for stable fundamental mode os-cillation.

IV. ConclusionIt was shown that the misalignment tolerance angle

a0i of a spherical resonator mirror Si is a function of oneparameter Di, which depends on the effective resonatorlength L* and the well-known g parameters. The

V2

2 4 c2 0- rad

Fig. 3. Total Xs factor V2 vs tilt angle a2: (L* = 125.8 cm, g1 = 1.32, g2 = 0.75).

/ Vol. 19, No. 4 / 15 February 1980

-4 -2 0

600 APPLIED OPTICS

105

Li

104

l03

10-5 10- 4 10- 3

aoi/rad -

Fig. 4. Experimental values of angle aoi vs misalignment sensitivityDi: , X, and + represent the work of Freiberg and Halsted14 ; +

represents this work.

discrepancy between experimental results and theoryis beyond experimental error and probably due to thelimited validity of perturbation theory. Furthermore,it was deduced that the misalignment sensitivity of anyspherical resonator can be characterized by one pa-rameter D, which again is a function of effective reso-nator length L * and the gi parameters. The reciprocalvalue of D is the tilt angle, which increases the losses by-10%.

This work was supported by the Deutsche For-schungsgemeinschaft.

This paper is based on a presentation made at themeeting of the German Physical Society, Berlin.' 9

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