Modifying the one-dimensional power-couplingformalism to model a cw Nd:YAG laser
T. B. Simpson and P. G. Coakley
A lamp-pumped, cw Nd:YAG laser cannot be modeled adequately by the one-dimensional power-couplingformalism without assuming intensity-dependent cavity losses. The source of the intensity dependencehas not been determined but may be due to absorption saturation in the gain medium or to aperture andbeam profile effects.
Key words: lasers (neodymium), resonators.
One of the most appealing and pedagogically success-ful models of laser operation is the basic one-dimensional linearized laser-power-coupling model,'which is a limiting form of the more broadly applica-ble Rigrod Formalism.2 There has been excellentagreement between the model and experimental re-sults, even in a cavity with only a single transversecavity mode oscillating.3 In a homogeneously broad-ened medium, four parameters are used to describethe laser medium: the length L, the small signalpower gain go, distributed losses ao, and saturationintensity Isat. Laser intensity can be specified withthe end mirror reflectivities R, and R2. In the smalloutput-coupling limit, the model gives an equationfor the output laser intensity Iout of
21sat ( 2goL _'out = 2 \2 L + i + -2 (1)
with 8l = 1 - R 1, 2 = 1 - R2, and R2 assumed to beoutput-coupler reflectivity. Here we report that theuse of Eq. (1) can provide misleading informationabout the parameters of the laser cavity. It wasnecessary to make the round-trip cavity losses depen-dent on the circulating laser intensity to adequatelymodel an operating laser.
We attempted to use Eq. (1) to model an operating,lamp-pumped cw Nd:YAG laser. Using a proceduresimilar to that of Rice et al.
3 we inserted a rotatingoptical flat to provide a variable, controllable loss in
The authors are with JAYCOR, P.O. Box 85154, San Diego,California 92186.
the laser cavity. The laser, shown schematically inFig. 1, consists of a high reflecting mirror (T =0.0002), which has a 1.5-m radius of curvature, a6.5-cm-long by 3-mm-diameter Nd:YAG rod pumpedby a 750 W tungsten lamp, a plane-parallel fusedsilica optical flat on a motorized rotation stage, and aflat output coupling mirror, either T2 = 0.0 17 or T2 =0.002. In addition, a 1.5-mm-diameter pinhole couldbe placed between mirror #1 and the laser rod to limitlaser oscillation to the TEMOO mode. The cavitylength is approximately 45 cm. Resonatorg parame-ters associated with the high-reflector and outputcouplers are 0.7 and 1, respectively, making the cavitya half-symmetric, near-planar resonator. 4 A photo-diode measures the leakage laser output throughmirror #1. Its signal is proportional to the circulat-ing laser power. Because of the insertion of theoptical flat, Eq. (1) must be modified to describe thelaser output:
'out = 5 2a (2ckL+ +l 2goL 1), (2)
where lo has been added to account for any additionalangle-independent cavity losses;
tan 2 0i - sin-'(! sin Oi)j
tan2 0O + sin-'(! sin )(3)
and represents the extra round-trip cavity losses dueto the four-surface reflections for the polarization inthe plane of incidence. Losses for the orthogonalpolarization are too high to permit laser oscillation.In Eq. (3), O is the angle of incidence of the optical
Fig. 1. Schematic of the laser configuration with the intracavityrotating optical flat. A photodiode monitors the circulating laserpower by using the leakage current through the highly reflectingmirror #1. A second photodiode monitors the pump lamp inten-sity.
axis with the plane-parallel flat and n is the refractiveindex, 1.4497 for fused silica at 1064 nm. To deter-mine Isat, go, and c-0, the output power was monitored,as a function of the optical flat angle, as the flat wasrotated from the threshold angle less than Brewster'sangle to the other threshold angle. Using a least-squares fitting procedure and the known values of theother parameters in Eq. (2), we calculated the threeunknown values.
One of the preliminary tests of the apparatusinvolved replacing the high-reflecting output couplerwith the 0.017 output coupler while all other condi-tions remained unchanged. This change should af-fect only round-trip cavity losses. However, the bestparameter fit to the data produced radically differentresults, with the small signal gain and saturationintensity parameters varying as significantly as thecavity losses. With either output coupler the datacould be fit to a curve well within experimentalaccuracy and close to the fit obtained by Rice et al.3 intheir similar configuration. However, the resultswith the two different values of T2 were not consis-tent and, further, the calculated round-trip lossesassociated with the higher transmission output cou-pler did not even account for the transmission lossfrom the laser cavity. A variety of experimentalchecks were performed to verify that there were noprocedural problems. We were concerned that beamdisplacement on the output coupler, because of therotation of the optical flat, could be artificially induc-ing extra losses. Perhaps the strongest validation ofthe experimental procedure is summarized in Fig. 2.The circulating laser power Pcirc is plotted as afunction of angle-dependent loss that is due to therotating optical flat where the reflection losses madePcirc identical for the two output couplers. Exceptfor the additional 0.012 loss that is required with thehigh-reflecting coupler to equal the decreased reflec-tivity of the other output coupler, the two curves areessentially identical.5 This shows that the addedloss of the second output coupler can be equally addedby rotating the optical flat, while all other cavityparameters remain unchanged. We therefore con-cluded that the one-dimensional formalism did notadequately model the laser system, even though itcould produce good fits to the data.
To better understand why the model was failing,
a
I-
2
a
t - LOSS
0.017 0.022 0.027
2 a2Dm
1 a
e
0.005 0.010 0.015t - LOSS
Fig. 2. Circulating laser power as a function of loss due to therotating optical flat for two different output couplers. The uppertrace corresponds to the 0.002 transmission output coupler, andthe lower trace corresponds to the 0.017 transmission outputcoupler. Two sets of data for each trace reflect angles less thanand greater than Brewster's angle. Except for the difference inloss required to match the different coupler reflectivities, 0.012, thecurves are essentially equivalent.
we made some modifications in the definition of theparameters, namely, that the saturation intensityand the distributed losses be allowed to be a functionof the circulating laser intensity. Our reasoningbehind these modifications was that, first, if theNd:YAG was not following true homogeneous broad-ening, the saturation behavior of the laser would bemodified. Second, some of the distributed lossescould undergo saturation. Both of these changes areconsistent with the general assumptions of the one-dimensional linearized laser-power-coupling model aslong as the round-trip losses are small and transversevariations are ignored. We also made the assump-tion that the small signal gain in the laser rod wasproportional to the pump lamp intensity in the visibleand near-IR part of the spectrum. With thesechanges, Eq. (2) becomes
82'sat(Icirc)'out 2
E BIpump
lC(Icrc) + I(4)
where circ is the circulating intensity in the lasercavity, Ipump is the pump lamp intensity, B is aproportionality constant, and C(Icirc) is the intensity-dependent cavity loss without the optical flat:
d(Icirc) = 2O(Icirc)L + 1 + 2 + 10- (5)
For a constant ire, Eq. (4) can be rewritten as a linearrelation between pump and 1:
The intercept in Eq. (5) yields the cavity losses whilethe slope contains the information about gain andsaturation. If the inverse slope varies linearly withIcire then the gain medium is following the prescrip-tion for a homogeneously broadened medium and Iatis independent of Icirc
Instead of generating a single data curve of laserpower versus optical flat angle, we generated a familyof curves at different pump intensities. For each ofthe curves the loss was calculated at specified powerlevels. The loss versus Ipump data at each power levelwere collected. Figure 3 is a representative set ofdata points, and the solid line is the linear least-squares fit. The linear relation bewteen Ipump and Iat a given Icirc, which was predicted in Eq. (6),matched the data quite well under all conditions.The linear least-squares fit coefficients were thenused to determine the laser parameters.
The inverse slope and negative intercept are plottedas a function of laser power in Figs. 4(a) and 4(b),respectively. Two sets of data, one for the laser inmultimode operation and one corresponding to theTEMOO operation, are shown. Both sets of inverseslope data approximately follow a linear increase withincreasing power. The small differences betweenthe two sets of inverse slope data are within experi-mental reproducibility. However, both sets of inter-cept data show a strong dependence on circulatinglaser power. The round-trip cavity losses decreasewith increasing circulating laser power. A key fea-ture of this data is the consistently larger round-triplosses for the single-mode laser cavity relative to themultimode laser cavity. Even at threshold, wherethe circulating power is negligibly small, the single-mode cavity has approximately 60% greater losses.
If we attempt to interpret the data literally, theinverse slope data, Fig. 4(a), indicates that the Nd:YAG gain medium is acting in accordance with homo-geneous broadening. This is consistent with theoriginal assumptions of Eq. (1). However, both theTEMOO and multimode intercept data, Fig. 4(b), indi-
0.06
0.04
0.02.
0.004.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
IPUMp (ARB. UNITS)
Fig.3. Rotating flat losses required for a circulating power of 0.76W as a function of pump lamp intensity. The linear fit, predictedby Eq. (5), is very good.
400
PzmD
-
cnusInw
z
300
200
100
C
0.061
InIn
0-IC.
0.04*
0.02
0.000
0 5 10 15 20 25
PCIRC (WATS)
(a)
5 10 15
PCIRC (WATTS)
(b)
20 25
Fig. 4. (a) Eq. (5) inverse slope and (b) negative intercept valuesfrom the data as functions of circulating laser power for single-mode (open circles) and multimode (filled circles) operations. Seetext for details.
cate that the distributed losses are undergoing satura-tion, if our reasoning for making the cavity lossespower dependent is correct. This contradicts theassumptions of Eq. (1). Decreasing absorption losseswith laser power has been inferred in a diode-pumpedNd:YAG oscillators but, to the best of our knowledge,has not been explicitly measured. There is a furthercomplication with the consistently higher losses calcu-lated from the TEMOO data. The difference in cavitylosses between the TEMoo and multimode data sets,extending down to threshold cannot be explained asarising from only a power-dependent distributed losscoefficient.
The one-dimensional analysis does not, of course,include transverse effects. The presence of aper-tures, the Nd:YAG rod in the multimode case, and the1.5-mm-diameter aperture in single-mode operation,cause cavity loss through truncation of the wings ofthe circulating laser modes,4 and diffraction of thecirculating field patterns. 7 The larger resonatorFresnel number associated with the multimode cavityyields lower round-trip losses for each of the cavitymodes.4 In our notation in Eq. (5), depends on the
laser configuration. We observed that the multi-mode configuration initially oscillated in the TEMOOmode at threshold. The difference in cavity losses atthreshold is reasonably consistent with the reducedlosses expected for the TEMOO mode because of thelarger resonator Fresnel number.4 In addition, wemeasured changes in the beam size outside theoutput coupler. Holding other parameters constant,we varied the circulating power level by changing thepump lamp intensity. Near threshold, the single-mode beam diameter measured 1.1 mm [exp(-2)points]. This diameter decreased to 0.9 mm at acirculating power level of 9 W. For the multimodeconfiguration changes were smaller, from an exp(-2)diameter of 1.4 mm just above threshold to 1.25 mmat a circulating power of 26 W. A changing beamsize with power means that aperture diffraction andlosses will vary. Therefore it may not even be cor-rect to associate the power dependence of the cavitylosses, Fig. 4(b), with saturation of the distributedlosses in the gain medium. Some, or all, of the powerdependence in the cavity losses may be in the parame-ter l0 in Eq. (5).
It is beyond the scope of this paper to differentiatebetween these and other possible mechanisms. Ourmain point is to emphasize that it was inappropriateto associate the cavity losses with only an intensity-independent distributed rod loss and mirror couplinglosses, as modeled in Eq. (1). Care must be taken ifthe power-coupling formalism is to be used to param-eterize an operating laser. The observations in thiswork may be relevant to recent modeling of diode-pumped Nd:YAG lasers.6 8 Our data agree with theassertion that cavity losses decrease under lasingconditions made in Ref. 6, but we cannot rule out thepossibility that the change in round-trip losses may
be due to aperture effects rather than absorptionchanges. A model of laser operation that includesthe decreasing cavity losses accounts for the increas-ing slope efficiency with input power observed in thediode-pumped systems. Further work is necessaryto fully clarify the source of the changing cavity lossesand the utility of a modified power coupling formal-ism.
The authors thank S. Eric Wheatley and RolandLeadon for useful discussions and Steven Niederhausfor technical assistance. We also thank the refereefor a careful reading of the original manuscript anduseful comments. This work was performed undercontract DNA001-88-C-0092 from the Defense Nu-clear Agency.
References and Notes
1. See, for example, A. E. Siegman, Lasers (University Science,Mill Valley, Calif., 1986), Secs. 12.3 and 12.4.
2. W. W. Rigrod, "Saturation effects in high-gain lasers," J. Appl.Phys. 36,2487-2490 (1965).
3. R. R. Rice, J. R. Teague, and J. E. Jackson, "Dynamic couplingcharacterization of TEMOO Nd:YAG lasers," J. Appl. Phys. 46,2716-2720 (1975).
4. Ref. 1, Chap. 19.5. Differences in scattering and absorption by the two output
mirrors can account for the measured transmission differencein 0.015 and loss difference of 0.012.
6. S. C. Tidwell, J. F. Seamans, C. E. Hamilton, C. H. Muller, andD. D. Lowenthal, "Efficient, 15-W output power, diode-end-pumped Nd:YAG laser," Opt. Lett. 16, 584-586 (1991).
7. P. Belland and J. P. Crenn, "Changes in the characteristics of aGaussian beam weakly diffracted by a circular aperture," Appl.Opt. 21, 522-527 (1982).
8. R. Burnham and A. D. Hays, "high-power diode-array-pumpedfrequency-doubled cw Nd:YAG laser," Opt. Lett. 14, 27-29(1989).
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