Vol. 7, No. 7/July 1990/J. Opt. Soc. Am. B 1251

Sensitive diffraction-loss measurements of transverse modesof optical cavities by the decay-time method

Y. Le Grand, J. P. Tach6, and A. Le Floch

Laboratoire d'Electronique Quantique-Physique des Lasers, Universit6 de Rennes I, UA Centre National de laRecherche Scientifique 1202, F 35042 Rennes Cedex, France

Received March 30, 1989; accepted January 15, 1990

A simple method is proposed to measure the diffraction losses for the first few modes of stable low-loss opticalresonators now available with the state-of-the-art coatings. The optical decay time of each cavity mode is iso-lated, permitting the determination of the cavity's diffraction losses. The sensitivity of the method, whichmight reach the parts-in-106 range, emphasizes the need for more detailed theoretical calculations for high-quality laser cavities, in which low diffraction losses can limit the performance.

Because of recent progress in fabricating high-qualityoptical coatings by ion-beam sputter deposition, opticalresonators characterized by low losses are now available.As transmission, absorption, and scattering losses becomelow, diffraction losses, even for vanishing values, cannotbe disregarded in high-finesse cavities' or in low-gainlasers, such as free-electron lasers,2" and ring laser gyro-scopes. Since the early years of laser physics, calcula-tions5 and measurements 9 of diffraction losses of opticalresonators have constituted a serious problem. On theone hand, calculation of low diffraction losses (<0.1o) wasnot performed by Fox and Li5"0 because such calculationwould have required a long computation time and was notof interest during the 1960's. On the other hand, experi-mental tests, which have been quite limited, are difficultto perform because diffraction losses are generally smalland are thus easily obscured by other mirror and scatter-ing losses, Brewster window losses, effects of cavity mis-alignment, and the like." Surprisingly, the only reliablevalues were given by experiments with laser cavities inwhich, owing to the active medium, the diffraction lossescan be frequency dependent.' 2 A satisfying comparisonbetween experiment and theory, for rather large values ofdiffraction losses, has been obtained for the TEMoo mode9

but not for the TEMoi mode, the oscillation of which wasconfirmed by stretching a hair over the resonator aper-ture.8 However, the methods used are clearly inadequatefor measurements below 0.1%, and new techniques mustbe developed for cavities with high-reflectivity mirrors.To avoid these problems we suggest performing measure-ments directly in the passive cavity, a procedure consid-ered theoretically by Fox and Li. Recently the decay-timemethod has proved to be straightforward and convenientfor determining losses due to transmission, absorption,and scattering reaching the parts-in-106 (ppm) range inhigh-reflectivity mirror coatings.' 3"4 Usually the mirrorsurfaces are relatively large compared with the cross-sectional area of the excited modes, so the diffractionlosses are low. Consequently, the decay time for all modesis practically the same, so the mode matching is not ex-tremely critical.' 3 Now, if the mirror surfaces are not toolarge, the excited modes will have different decay times

because they have different diffraction losses, their trans-verse extension being different. In this case, the fre-quency nondegeneracy of the transverse modes allows usto isolate each decay. Here we investigate this newmethod, based on optical cavity decay time, of measuringthe diffraction losses of stable low-loss laser cavities.

Let us recall the basis of light decay in an optical cavity.When a probe beam is injected into a resonant cavity andis abruptly switched off, the decay of the transmitted in-tensity (i.e., the output power) is described by' 5

I(t) = Io exp(-t/T'), (1)

where Io is the intensity transmitted at resonance and T'

is the decay time of the cavity that characterizes its totallosses. For cavities with large mirrors, the decay time isgiven by

T' -/2 ln[R(1- p)], (2)

where T/2 is the transit time of the stored photons, R isthe average reflectivity of the mirrors, andp is the single-pass propagation losses (atmospheric losses). In the caseof high-reflectivity mirrors (R = 1) and small propagationlosses (p << 1), the total cavity losses L' = 1 - R + p areobtained from T' by using the relation

L' = dct', (3)

where d is the cavity length.When diffraction losses are taken into account, the total

cavity losses L' are written as

L" = dcTn, (4)

where 7" is the new decay time. The diffraction losses re-sult either from the finite mirror aperture or from an extraaperture inserted into the cavity. For a given aperturethe single-pass diffraction losses LD of a given mode are

(5)

Determination of LD as a function of the aperture diame-ter is then straightforward. It consists first of measuringthe decay time T' of the fundamental mode (TEMoo) of the

0740-3224/90/071251-03$02.00 C 1990 Optical Society of America

Le Grand et al.

LD = d ( _ 1).C T' T"

1252 J. Opt. Soc. Am. B/Vol. 7, No. 7/July 1990

I I I I I I I

TEMoO--

TEM 01

TEM 02-- - --- - --- -

I I I I I I I I I

TIME (5 msec/div)

TIME (5 usec/div)Fig. 1. (a) Typical spectrum of the cavity (upper) and tlrm -nln 11nu tho 4. rnlt_nnt__ Acxfinr4in l-Xr 1

decay-time values of the different modes are transferredfrom the digitizing oscilloscope to the microcomputer,which selects and averages the decay time of each mode.

The probed cavity is a near-hemispherical resonatorconsisting of a plane input mirror and a concave output

a ) mirror (curvature R, = 300 cm). Both mirrors have high-reflectivity coatings, the plane mirror being coated for adiameter of 6 mm and the concave one for 8 mm. Themirror spacing d is adjusted to permit our experimentalresults to be compared with the diffraction-loss calcula-tions made by Li' for a symmetrical resonator with theassumption that the edge of the mirrors forms the limit-ing aperture. Indeed, it must be noted that the round-trip losses of a near-hemispherical resonator are the sameas the losses per transit for a symmetrical resonator oftwice the length. So the Fresnel number N and theparameterg to be considered are given here byN = a/2Adand g = 1 - 2d/R,, where a is the radius of the aper-ture placed adjacent to the curved mirror and A is the op-tical wavelength. Owing to the circular geometry of theresonator, the modes of our cavity are of the Laguerre-Gaussian type, as verified by a visual inspection of the

(b) mode patterns. These modes are called TEM,1, where pand are the radial and angular mode numbers. For d =270 cm (i.e., g = 0.8) and without an aperture, the cavitydecay time ', which is measured for the TEMoo mode,leads [see Eq. (3)] to total round-trip losses of 1800 ppm, inagreement with the reflectivity (>99.9%) of our mirrors.This value is unaffected by the diffraction losses sincethese losses estimated from the simple truncation formula(see, e.g., Ref. 16) are 4 x 10-2 ppm. Then the measure-

le wave- ment of T" with the aperture in the cavity allows us toUL lIIL aipplCJU I.U I1I6 aUULU-UpLIU Ul-lULVU I,.iUW i. .VIL L111

cavity output intensity reaches the preset threshold level (dashedline), the comparator circuit generates an electrical gate, whichswitches off the acousto-optic deflector for a time (1 msec)much longer than the decay time to be measured. (b) Sepa-rately recorded decay curves for the three lowest-order modes ofthe cavity.

mirrors spaced so that its diffraction losses are actuallynegligible (<1 ppm), and second of measuring the decaytime 'r" associated with each transverse mode when theaperture diameter is varied. From these two measure-ments the diffraction losses can be determined.

In our experimental setup, the light source is a 10-mWmonomode He-Ne laser that uses a Fox-Smith interfer-ometer and emits at 633 nm. The laser beam is focusedonto a fast acousto-optic deflector, and the deflected lightis collimated by a second lens to inject the energy into thelow-order transverse modes of the cavity being studied.The output cavity mirror is mounted upon a piezoelectricstack in order to sweep the cavity frequency across thedifferent transverse modes. The light transmitted by theresonator is detected by a photomultiplier tube preceded bya narrow-band filter. When the detector output reaches apreset threshold level, which corresponds here to the in-tensity of the weakest investigated mode [see Fig. 1(a)], acomparator circuit generates an electrical gate, which isapplied to the acousto-optic deflector to switch the injectedlight off abruptly (in -20 nsec). The different light de-cays appear successively [Fig. 1(b)] and are recorded bya 200-MHz digitizing oscilloscope coupled with a micro-computer, which controls the acquisition process. The

z

Z

PO

0

P4

C,)

0

-4

FRESNEL NUMBER1.5 2.0 2.5

N3.0 3.5

4.0 4.5 5.0 5.5 6.0 6.5 7.0

APERTURE DIAMETER 2a (mm)Fig. 2. Diffraction losses per round trip versus the aperture di-ameter 2a, or the Fresnel number N for the TEMOO, TEM01, andTEM02 modes and for gf = 0.8. The stars indicate the results ofLi's numerical calculation as published by Rilgrod.2 The solidcurves correspond to the values deduced from Li's graphs.'" Theexperimental data points, designed by filled triangles and filledand open circles, are linked by the dashed curves, except thosefor the TEM02 mode.'7

z:-I-

OFF

ON

.C

I-

z

I-z

Le Grand et al.

(

Vol. 7, No. 7/July 1990/J. Opt. Soc. Am. B 1253

determine the diffraction losses from Eq. (5). Our com-plete experimental results for the diffraction losses of theTEMoo, TEM 01, and TEM02 modes1 7 versus the aper-ture diameter 2a are shown in Fig. 2. Measurements forthe TEMoo mode are performed down to nearly 0.01% todemonstrate the sensitivity of the method. Experimentaldata are compared with the theoretical results for IJg =0.8, which are restricted to values above 0.1%, given by Lifor TEMoo and TEM01 modes. 0 The slight discrepanciesbetween our experimental data and Li's results may be at-tributed to the small ripples occurring in the LD-versus-Ncurves calculated by Gromov and Trashkeev.' 8 To ourknowledge, no experimental results for the TEM02 modediffraction losses have been published, and theoreticalpredictions seem to be available only for the confocal5 9

and plane-parallel geometries.2 0 2'To conclude, the decay-time method has been shown to

be a simple and efficient method of investigating a basicproperty of optical cavities, i.e., the diffraction lossesassociated with the different low-order modes. Thismethod is well suited for ultrahigh-finesse optical cavi-ties, for which diffraction losses can reach values com-parable with residual losses of mirrors. For example, inour case the estimated round-trip diffraction losses forTEMoo, TEM 01, and TEM02 modes are 4 x 10-2, 1, and10 ppm, respectively. Thus great care must be taken tomatch the injected laser beam to the passive cavity whenthe decay time is used to isolate the reflectivity of mirrorsin the ppm range. Note that the experimental results ob-tained here for the Laguerre-Gaussian modes can also beobtained for Hermite-Gaussian modes by inserting aBrewster window into the cavity. In addition, the possi-bility of measuring the diffraction losses for the differenttransverse modes with great sensitivity allows one to testprecisely the validity of theoretical models.

ACKNOWLEDGMENTS

We thank M. J. Jorrot and M. P. Robert, Laboratoire deRecherche de la Soci6t6 Frangaise d'Equipements pour la

Navigation Agrienne, for providing the high-finesse cavitymirrors. This research was supported by the Centre Na-tional de la Recherche Scientifique, the Direction desRecherches, Etudes et Techniques, and by the Etablisse-ment Public Regional.

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