Coupled optical -resonators for the enhancement oflaser intracavity power
Y. C. See, Shekhar Guha, and Joel Falk
A double-cavity optical configuration designed to maximize the power available from cw lasers is reported.A passive supplementary cavity is strongly coupled to the main active laser cavity. It is shown theoreticallythat higher powers are available inside the passive cavity than those achievable by operating the same laserin a two-mirror configuration designed for optimum coupling. Furthermore, the power inside the passivecavity is equal to or greater than the useful circulating power in the two-mirror configuration. Key aspectsof the theory are verified with a high power cw argon laser.
1. Introduction
High power cw lasers are useful for studying nonlinearoptical effects. The maximum cw power available froma laser is limited by the laser's gain and by laser satu-ration. In this paper we report an optical configurationwhich allows cw powers in excess of those available witha conventional two-mirror optical resonator. Basicallythe optical configuration consists of two optical cavitiescoupled together through a common, partially trans-mitting mirror (or window). This cavity, shown in Fig.1, will be referred to as a double cavity. Mirrors Ml andM2 form the main cavity which contains the activemedium. Mirrors M2 and M3 form an auxiliary (sup-plementary) cavity which is passive. The nonlinearelement is placed in the supplementary cavity. Pre-vious investigators have used a similar optical ar-rangement in efforts to provide either increased laserpower for nonlinear optical experiments- 6 or increasedsensitivity for multiple pass absorption spectros-copy.7 - 9
Yarborough et al. reported enhancement of opticalsecond harmonic generation using a multipass opticalsystem similar to that in Fig. 1.1 Ammann and co-workers2 4 reported stimulated Raman scattering using
When this work was done all authors were with University ofPittsburgh, Pittsburgh, Pennsylvania 15261: S. Guha in the De-partment of Physics & Astronomy and the others in the Departmentof Electrical Engineering; Y. C. See is now with Texas Instruments,Inc., Dallas, Texas 75265 and S. Guha is now with University ofSouthern California, Los Angeles, California 90007.
a double-cavity configuration. Gurski used opticalfeedback from a highly reflecting mirror to enhancelaser power in a LiIO3 upconverter. 5 See et al. operateda double cavity to enhance the laser power in an up-conversion process and to overcome power instabilitiesdue to thermal effects.6 Kyle and Schuster used asecondary laser cavity to provide multiple paths formeasurement of weak absorption.9 In none of thesereferenced works was the power resulting from thepresence of feedback discussed quantitatively.
The theory of operation of laser cavities similar to thedouble cavity of Fig. 1 has been reported. Earlier workcan be divided into two main categories. The firstcategory was concerned with obtaining single longitu-dinal-mode operation from a multimode laser.10-13Smith has summarized various double-cavity schemesused for mode selection and discussed their relativeadvantages.' 4 In all the single-mode work only thepower external to the laser cavities was discussed. Thesecond category of earlier work investigated the stabilityof the active cavity in the presence of coupling to thesupplementary cavity.15 16 This work considered weakcoupling between cavities whose lengths were almost thesame.
We shall discuss the operation of the double cavityand calculate the powers in both the active and passivecavities. Strong coupling between the cavities will beconsidered. The powers will be calculated for variousvalues of gain, loss, coupling coefficient, and ratios ofcavity lengths. Multilongitudinal-mode lasers withmixed inhomogeneous and homogeneous broadeningwill be considered. We will compare the power avail-able in the double cavity with that obtainable in anoptimally coupled single cavity and in an intracavityconfiguration. Optimization of the power in the passivecavity will be discussed. We shall present numericaland experimental results in support of the theory.
In our theoretical derivation of laser power, we makethe following assumptions:
(1) The two cavities shown in Fig. 1 are coupled to-gether through a common transmitting mirror (orwindow) M2. The amplitudes of the reflectance andthe transmittance (of the electric field) of mirror M2 aredesignated r and t, respectively. Both r and t are takento be positive,'7 and r2 + t2 = 1. Mirrors M1 and M3are assumed to be high reflecting (HR) mirrors so thatri = r3 = 1 and t = t3 = 0.
(2) The radius of curvature of mirror M3 is chosen tobe equal to the radius of curvature of the wave frontwhen it reaches mirror M3. This guarantees that theTEMoo modes of the two cavities are matched.
The active cavity is characterized by its cavity lengthLa, unsaturated gain coefficient go, power loss coeffi-cient Li, and saturation parameter PS.1 8 The passivecavity is characterized by its cavity length Lp and its lossparameter A. The parameters go, Li, and A are allsingle-pass quantities relating to intensity. A is aquantity which includes reflection, absorption, andscattering losses.
B. Effect of the FeedbackBecause of the presence of the HR mirror M3, the
wave propagating from the active cavity to the passivecavity will undergo multiple reflections. Each roundtrip through the passive cavity alters the electric fieldby (1 -a) 2 exp(2iop), where
RE represents the effective reflectivity seen by thewave incident upon M2, and TE is the effective trans-missivity through the mirror M2 to the passive cavity.The loss experienced by the laser, in addition to theinternal loss Lg. is thus-(1/2) lnRE. As can be seen fromFig. 3, RE and TE each have minima and maxima as afunction of phase. In a multimode laser there can bemany longitudinal modes oscillating simultaneouslywithin the gain profile. These modes are separatedfrom one another by c/2L0 Hz and thus have differentphase shifts p. Modes which oscillate in the absenceof the supplementary cavity feedback but which sufferfrom a high loss due to the feedback will be extin-guished. On the other hand, modes which experiencelower losses in the presence of the feedback will tend tooscillate and suppress nearby modes having higherlosses. Hence, the supplementary cavity behaves likea Fabry-Perot etalon.
E
r,t
Fig. 2. Electric fields
E (-a) e24,pp
aL
r3 l
t3=O
at the interface of the active and passivecavities.
2irOP -- rLP1.0
(1)
is the phase shift introduced by a single pass throughthe passive cavity, and a is the loss coefficient for theelectric field. The power loss A is related to the fieldloss a via (1-A) = (1-a) 2 .
Figure 2 shows the fields that exist at mirror M2 Inthe steady state the electric fields and powers at theinterface of the two cavities can be determined fromsums of infinite series. The effective power reflectioncoefficient seen by the active cavity and the effectivetransmission coefficient into the passive cavity can beshown to be
R = 4-T(1 - A)2
1 + R( - A)2 + 2rR(1 - A) cos2O,
=I - TA(2-A)1 + R( - A)2 + 2R(1 - A) cos2op
TE =T1 + R(1 - A)2 + 27R(1 - A) cos2(p
where R = r2 and T = t2 .
z 0.5
O
wF-
(2)
(3)
(4)
7r 27r 37r p
(a)
(b)
Fig. 3. Effective reflectivity (RE) transmissivity (TE) as a functionof phase shift (p).
Smith,' 9 in a discussion of the output power from a6328-A He-Ne laser, considers the saturation behaviorof a laser that is both homogeneously and inhomo-geneously broadened. Each broadening mechanism ischaracterized by a linewidth and a line shape. Modecompetition and hole burning effects take place withina broadened Lorentzian (homogeneous) linewidth suchthat the total multimode power present is the same asif only one mode were oscillating and the others weresuppressed. Smith terms this single mode an equiva-lent mode. The total laser power output depends onboth the number of equivalent modes and the excitationparameter X which is defined as the ratio of the unsat-urated gain at the line center to the total loss. Smithrelates the excitation parameter to the power of a par-ticular equivalent mode as
7 = 1+ 2PI /2 I RW(o + i'1AvD) I.
t [ t1 (AVD) ( s ]
(5)
P1 is the power of that particular equivalent mode lo-cated halfway from the line center to the gain boundary.The gain boundary is the frequency where the unsatu-rated gain equals the loss. The parameters y' and AVDare the low power Lorentzian linewidth and the Dopplerlinewidth, respectively. P is the laser's saturationparameter. The factor of 2 in front of P, takes intoaccount the fact that the equivalent modes are placedsymmetrically with respect to the center of the Dop-pler-broadened gain curve. The gain boundary definesthe oscillating bandwidth AVD V 20 RW(z) is thereal part of the complex error function which is definedby
W(z) = [1 + erf(iz)] exp(-Z 2). (6)
The total power PT is found by multiplying the powerin this equivalent mode by the total number of equiva-lent modes within the oscillating bandwidth, or
PT = PAVID , (7)
where
h= 'i +i)'/P2 (8)
is the power-broadened Lorentzian linewidth.Smith has found that the total power PT determined
from Eq. (7) can, for a wide range of Y/'IAVD, be ap-proximated by a linear relationship between PT and 7as
PT BP.(n -1), (9)
where B is a proportionality constant. We checked thevalidity of this approximation for the parameters ap-propriate to the argon laser used in the experimentsreported later in this paper. Figure 4 shows a goodagreement between the linear approximation and theexact solution [Eq. (5)].21 In the subsequent treatmentof the power in the double-cavity argon laser, we shalluse the linearized equation.
100
80
60or
0.40
0 1 1 I , I0 10 20 30
77
Fig. 4. Total power [from Eq. (5)] in the two-mirror argon laser ('= 500 MHz, AVD = 4200 MHz, 0i = 3).
_I 1_
(a )
(b)
Fig. 5. Power spectra of the argon laser (550 MHz/division): (a)single cavity; (b) double cavity.
D. Double Cavity
As noted earlier, the laser power in the double cavitydepends greatly on the detailed laser mode structure.The argon laser used in our experiments has a modespacing c/2La of 71.7 MHz so that many modes shouldoscillate simultaneously within the 5-GHz argon fullinhomogeneous bandwidth. Figure 5(a) shows a singletrace of the spectrum of the argon laser operating in asingle (two-mirror) cavity. Greater than thirty axialmodes are visible in Fig. 5(a).
The mode structure with the double cavity is sub-stantially different from that described above. The
observed mode structure is shown in Fig. 5(b). Notethat many modes observed in the single cavity aresuppressed in the double cavity. The instantaneouscirculating power for each equivalent mode of the activecavity can be written from Eqs. (7) and (9) as
go1Pi(Op) = BP. ( - -] (10)AVD
Oh
where P1 now is a function of the reflection coefficientRE which is itself a function of the phase shift op. Thephase shift op is not a constant but varies from oneequivalent mode to another.
If op were fixed, the total power could be calculatedfrom Eq. (10). We find an approximate value of PT byaveraging P over possible values of p and thenmultiplying by the number of equivalent modes. Weassume that within each free spectral range (FSR) of thepassive cavity, only the one mode which has the leastloss will oscillate with significant power while othermodes will be suppressed because of mode competition.Any mode whose frequency is within one-half of a modespacing of a loss minimum, i.e., near the peak of RE (Fig.3), will oscillate. This frequency range corresponds toa phase shift range p of I(dr/2)(Lp/La). Thus thecirculating power can be written as an average over therange of phase Op, Op = (/2)(Lp/La), or
P., = OBP f go -1 dfi20,, _0 S[L - /2nRE (p) I (11)
The overbar indicates the average over phase. Thepower in the passive cavity is given by PUP(oP) =P (p) TE. The power in the passive cavity can be ex-pressed by a similar phase average:
1 (' Bg oPsu = -J BP, L - 11 TE(op)dop. (12)20p EP i - /2 nRE (p )
In the experimental case to be considered Lp < (/3)Laand, therefore, Op S r/6. Within this phase range RE
I so that-lnRE _ 1-RE-Equations (11) and (12) can be integrated analytically
but yield equations which are still too complicated tobe useful. We have evaluated the integrals on a digitalcomputer and have found that, when Op < 7r/6, thevalues of the integrals differ by <10% from their limitingvalues as Op goes to zero. This is true for any go, Li, andA to be considered in this paper. Thus with the inte-grals approximated by their limiting values, Eqs. (11)and (12) can be simplified to
P. = BP, I -1) (13)
where
TA (1-)Leff[1 + (-A)] 2 (14)
is the effective loss seen by the active cavity, and
Psup = Pc Teff, (15)
where
Teff .- TPc= [1 + V (1 - A)]2 (16)
is the effective transmission from the main cavity to thesupplementary cavity.
Thus the circulating power in the active cavity can beobtained by simply substituting for the usual mirrortransmission loss an effective loss Leff. The supple-mentary power is found from the effective transmissionTeff and the circulating power. Equations (13) and (15)are similar to those which describe a single-cavity laser.Note that the effective transmission Teff does not de-pend on laser parameters go, Li, and P. This inde-pendence will be verified experimentally. Furthermorenote that it is possible to maximize the supplementarypower by choosing an optimal mirror transmission T forany given loss in the passive cavity.
E. Optimization of Supplementary Power in theDouble Cavity
In practice, the loss A is always nonzero. The sup-plementary power P5Up [Eq. (15)] can therefore be re-written as
BP., (L, go L
A= 1 IA V +Leff jA 2i~
(17)
This power can be maximized by the correct choice ofLeff:
(Psup)max = BPSL ( 1)2
which occurs with
Leff = (Leff)opt = i- i.
(18)
(19)
The maximum value of Leff is A(1 - A/2) [Eq. (14)].Consequently, the optimized Psup [Eq. (18)] cannot beachieved under some circumstances. This problemarises when the loss A is too small for Leff to approachA/gL - Li for any T 1. In this condition, Pup ismaximized with T = 1.
F. Comparison of the Optimized Double Cavity withSingle Cavities
We will show that the optimized Psup is the highestpower obtainable in any of the three cavity configura-tions shown in Fig. 6. Figure 6(a) shows the doublecavity and Fig. 6(b) the usual single-cavity configura-tion. For a single-cavity laser the available pump poweris the output power from the laser. It is independentof A and has a maximum value given by
(Pout)max = 2BPLi (d - 1)2, (20)
which occurs when T = Topt = 2(`goLd - Li).The ratio of the maximum power available in the
double cavity to that available as power output from thesingle-cavity laser is
(Psup)max 1
(Pout)max [1 - (1 - A)2](21)
This ratio is always greater than unity and increases as
A decreases. In practice the loss A is usually small.Thus the enhancement of power over the external singlecavity could be substantial. For example, with A = 0.3,the power would be doubled. For large A the feedbackis small, thus the enhancement is insignificant.
Figure 6(c) shows an internal two-mirror single cavity.In this cavity the useful power is the power circulatingbetween the end mirrors M1 and M2. We can write thedouble-cavity supplementary power in a manner thatpermits comparison with the internal configuration:
(Pu)max = BPS [ go gL 11 (Leff)t. (22)
( ~ A~1Li + (Leff)optI
2/
Since Leff has an explicit dependence on T, optimizingLeff is equivalent to optimizing T. Thus
(Psup)max -- L + ( L - 1) (Leff)- (23)
A (1--) _
Equation (23) must be true for all T, thus with T = 1 (R= 0), Eq. (23) becomes
single cavity. Thus for all cases the double cavity canproduce power equal to or greater than the single-cavityinternal configuration.
G. Numerical and Experimental Results
The laser used to test the predictions of the theorydeveloped in this paper was a Spectra-Physics model170 argon laser. The single cavity was formed by ahighly reflecting flat mirror and Littrow prism (used toselect the 514.5-nm argon laser line) and a 13.8% highlyreflecting 6-m radius mirror. The supplementarycavity was formed by the same 6-m radius mirror, a25-cm focal length AR-coated lens, and a 30-cm radiusof curvature mirror. The lens was positioned inside thesupplementary cavity so that the TEMOO transverseprofile resulting from the single cavity was not changedby the addition of the supplementary cavity.6
Our double-cavity argon laser has cavity lengths La= 2.09 m and LP = 0.8 m or a ratio of Lp/La = 1/2.6.Using the Tropel model 2401 scanning Fabry-Perotspectrum analyzer [free spectral range (FSR) = 7.5GHz], we observed the optical spectrum of both thesingle-cavity and the double-cavity laser. Figures 5(a)and (b) show typical spectra for the single and doublecavities, respectively. Even though the mode selectionis not perfect, the selection and suppression of modesare clearly seen.
The laser parameters go, Li, and P8 fro our Ar-ion laserwere determined. The excess unsaturated gain (go -Li) at various argon plasma currents was obtained byfinding the minimum loss necessary to extinguish thelaser at each current. A pair of quartz windows wereplaced inside the single cavity. They were mounted onrotary mounts so as to allow variation of their reflectionlosses. The quartz windows were mounted so thatdisplacement of the laser beam was avoided. The la-ser's loss and saturation parameters were obtained froma measurement of circulating power in the single cavityas a function of intracavity loss and current. The in-sertion loss was varied either by using mirrors of dif-ferent transmission or by rotating the quartz windows.Using Smith's linear approximation [Eq. (9)] we find
16(Psu0max I BPs go 1>BP, ( g (24)
A ~ - j~ Lj + A/But the right-hand side of Eq. (24) is simply the cir-
culating power produced in a single-cavity internalconfiguration. Thus for any A 0, the optimizeddouble cavity actually provides more power than theinternal single cavity. Figure 7 shows the theoreticallypredicted [Eqs. (14) and (17)] supplementary power(Psup) for our argon laser vs T for different losses A. Wesee that (Psup)max can be significantly higher than thepower in either of the single cavities.
When A is small so that Leff can no longer approachits optimal value of gLi - Li, the maximum supple-mentary power PSUP obtainable is achieved when T isunity. However, T = is equivalent to an internal
12
0-30'
0Z
8
4
o 1~~~~~~~100 1 I II I0 0.2 0.4 0.6 0.8 1.0
Transmission (T )Fig. 7. Supplementary power (PSup) vs coupling mirror transmission
Fig. 8. Circulating power P, in the single cavity as a function of lasergain go. The theoretical curves are plotted for BPS = 9 W and Li =
0.06.
variable loss are 6 mm thick. We measured the ab-sorption of these slides at the laser wavelength (5145 A)with a Cary-118 spectrometer. The two windows haveabsorption coefficients of -0.008 cm-'. This absorp-tion could cause significant thermal lensing and wedgingeffects at high cw powers leading to the discrepancybetween theory and experiment at high power.6
Figures 10 and 11 show the circulating and supple-mentary powers, predicted and measured, in the dou-ble-cavity argon laser. Figure 10 shows a theory-ex-periment comparison of the circulating power PC as afunction of gain go and loss A. The agreement betweentheory and experiment is good. Figure 11 shows Psupvs go for various losses. This comparison between thetheory and the experiment is difficult because Psup doesnot change significantly with different losses at theparticular mirror transmission (13.8%) that was avail-able. Again the agreement at higher gain and lower Ais poorer because of the high circulating power and theassociated thermal effects.
10(
0.16
0
ca.0.08U,
Of
- 0 0.2 0.4 0.6 0.8 1.0
Loss (A)Fig. 9. Ratio Psup/Pc as a function of supplementary cavity loss at
various laser gains.
80
60
3
40a!,
20
0
90
Fig. 10. Circulating power P, in the double cavity as a function oflaser gain go and supplementary cavity loss A.
P= BP Lg - i)- (25)
Thus when the circulating power P is plotted vs go, weobtain the loss parameter Li and the saturation pa-rameter BPS from the P = 0 intercept and the slope,respectively. Figure 8 shows a plot of the experimen-tally measured power as a function of laser gain and thevalues of power predicted from Eq. (25).
As a check of the double-cavity theory we measuredP, and P as a function of laser gain and supplementarycavity loss. We have shown that the ratio of Psup/Pcshould not depend on laser parameters; Figure 9 showsthe experimentally determined variation of the ratioPsUP/P, as a function of the insertion loss A at variousplasma currents. The agreement between theory andexperiment is very good at lower plasma currents andis reasonable at high currents. For small losses and athigh currents the cw circulating power approached orexceeded 100 W. The quartz slides used to provide
8
0
3
In0.
6
4
2
o- 0.2 0.4 0.6 0.8 1.0
Fig. 11. Supplementary power Pup in the double cavity as a functionof laser gain go and loss A.
L effFig. 12. Circulating power P, in the double and single cavities as Efunction of the effective loss Leff. In the single cavity Leff = T/2.
What is the significance of the parameters Leff andTeff defined in Eqs. (14) and (16)? In the double cavitythe supplementary power Psup can be related to thecirculating power P, through a single parameter Teff,and the effective loss Leff [Eq. (14)] in the double cavityplays the role of T/2 in the single cavity. In Figure 12we plot the circulating power in the active cavity vs thiseffective loss Leff for both the single and the doublecavities. It can be seen that for equal Leff and go ap-proximately the same circulating power occurs in bothconfigurations. Thus the powers in the double cavitycan be characterized by only two parameters Leff andTeff.
I1. Discussion and Conclusions
To maximize the power available from cw lasers andthus increase their usefulness for nonlinear optics, wehave developed a double-cavity laser configuration.This configuration is characterized by the coupling ofa passive supplementary cavity to the main laser cavity.Placement of optical elements in the supplementarycavity rather than in the main cavity reduces their effecton available power.
The circulating power P, in the active cavity and thesupplementary power PUP in the passive cavity can becalculated by considering the effect of feedback re-sulting from the coupled cavities. It was shown that theequations for P, and Pup are similar to those thatgovern the single cavity if an effective loss Leff and aneffective mirror transmission Teff are defined. Both ofthese two quantities are functions only of the loss in thesupplementary cavity and the transmission of thecoupling mirror. It was also shown that the ratio ofPsup/Pc does not depend on laser parameters but is onlya function of the loss A.
The supplementary power is the actual pump poweravailable for nonlinear optical processes. This powercan be optimized by choosing an appropriate Leff (or T)
for any given loss A. It was shown that the supple-mentary power Psup can always be made greater thanor equal to the circulating power in the internal cavity.Thus the double cavity should find many applicationsin nonlinear optics.
The authors thank NASA Goddard Space FlightCenter for support of the work reported in this paperand R. Moshrefzadeh for a careful reading of the man-uscript.
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