Relaxation Oscillations in a Laser with a Gaussian Mirror

Relaxation oscillations in a laser with a Gaussian mirror

Agnieszka Mossakowska-Wyszynska, Piotr Witonski, and Paweł Szczepanski

We present an analysis of the relaxation oscillations in a laser with a Gaussian mirror by taking intoaccount the three-dimensional spatial field distribution of the laser modes and the spatial hole burningeffect. In particular, we discuss the influence of the Gaussian mirror peak reflectivity and a Gaussianparameter on the damping rate and frequency of the relaxation oscillation for two different laserstructures, i.e., with a classically unstable resonator and a classically stable resonator. © 2002 OpticalSociety of America

OCIS codes: 140.0140, 140.3410, 140.3430, 140.3460.

1. Introduction

Unstable laser resonators that use variable reflec-tance have been proposed1–13 and successfully imple-mented in many gas and solid-state lasers14–22 toobtain diffraction-limited light beams with high en-ergy �or power�. In the most theoretical approachesthe properties of the cavity with the Gaussian andsuper-Gaussian mirror have been studied.1–22 Morerecently, the nonlinear operation theory of a laserwith a Gaussian mirror was developed23,24 that pro-vides a simple way to obtain output power energycalculations. A closed-form relationship for the out-put energy as a function of the transverse-mode in-tensity profile and pump energy has been derived forfree-running pulse lasers.

Finally, a semiclassical model of the nonlinear oper-ation of a laser with a Gaussian mirror was devel-oped.25 Modified nonlinear self-consistent equationsincluding transverse and longitudinal field depen-dence, gain saturation, spatial hole burning, and non-linear dispersion effects have been derived. Withthe help of an energy theorem, we present an ap-proximate solution for steady-state single-mode op-eration. This solution reveals the influence of the

A. Mossakowska-Wyszynska �[email protected]�,P. Witonski, and P. Szczepanski are with the Institute of Micro-electronics and Optoelectronics, Warsaw University of Technology,ul. Koszykowa 75, 00-662 Warsaw, Poland. P. Szczepanski is alsowith the National Institute of Telecommunications, ul. Szachowa1, 04-894 Warsaw, Poland.

Received 25 May 2001; revised manuscript received 29 October2001.

0003-6935�02�091668-09$15.00�0© 2002 Optical Society of America

1668 APPLIED OPTICS � Vol. 41, No. 9 � 20 March 2002

Gaussian mirror parameter and the parameters ofthe system characteristics on the power efficiency ofthe laser.

We present a systematic study of unsteady oper-ation of a laser with a Gaussian mirror. We modifythe time-dependent laser rate equations to includespatial field dependence of laser modes �character-istic for a Gaussian mirror laser� and the spatialhole burning effect. It is worth noting that, withthis kind of laser, in contrast with the laser struc-tures having cavities with a classical mirror, thelaser modes comprise two counter-propagatingwaves with different transverse and longitudinalfield distributions. We derive a small-signal linearanalysis for relaxation oscillations. Note that, ingeneral, relaxation oscillations can be a valuabletool for analysis of various laser parameters asspontaneous lifetimes of the active medium andreal cavity losses. For our calculations we usedthreshold field approximation to obtain approxi-mate expressions for frequency and damping of therelaxation oscillations as a function of the systemcharacteristic parameters, i.e., distributed losses,arbitrary mirror reflectivity, Gaussian mirror pa-rameter, and geometry of the cavity. We espe-cially discuss the influence of the Gaussian mirrorparameter and spatial hole burning effect on thebehavior of relaxation oscillations. We obtainedlaser characteristics for a Nd:YAG laser systemthat operates at 1.06 �m having classically unsta-ble and classically stable cavities.

In Section 2 we derive the expression for the relax-ation oscillation parameters of a laser with a Gauss-ian mirror. In Section 3 we discuss the relaxationoscillation laser characteristics, and we present ourconclusions in Section 4.

2. Theory

The investigated structure is presented in Fig. 1,where the whole volume between mirrors is filled outby an active medium.

Similarly, as in Refs. 26–28, we start our analysisby writing the coupled laser rate equations for single-operation mode in the well-known form

dNdt

� �Imnq�rT, z, t�

Is

N�rT, z, t��

�N�rT, z, t�

�� pr,

(1)

dQdt

� � d�Imnq�rT, z, t�

Is

N�rT, z, t��

�Q�Q

, (2)

where N�rT, z, t� denotes the inversion density,Imnq�rT, z, t� describes the total intensity of the mnqthlaser mode in the cavity, Is is the saturation intensity,� is the inversion population lifetime �for nonradia-tive and radiative lifetimes29�, pr is the excitationrate, Q denotes the number of photons in the mnqthlaser mode, �Q is the cavity lifetime of the photons,and the transverse coordinates are described by rT.The integral in Eq. �2� is carried out over a volume ofthe active medium.

For a laser with a Gaussian mirror, the electricfield of the laser mode is represented by two counter-propagating waves having different transverse aswell as longitudinal spatial distribution.30 Thus,the general multimode field can be written in thefollowing form:

E�rT, z, t� �12 �

m,n,qEmnq�rT, z, t� � c.c.

�12 �

m,n,qRq

mn� z, t� AmnR�rT, z�exp�ikmnqz�

� Sqmn� z, t� Amn

S�rT, z�exp��ikmnqz�

� exp��i�mnqt� � c.c., (3)

where Rqmn and Sq

mn slowly vary in an opticalwavelength and are the complex amplitudes of thecounter-propagating waves of the mnqth lasermode, Amn

R and AmnS describe the transverse field

distributions of the counter-propagating waves ofthe mnqth laser mode. The transverse field distri-butions vary slowly at optical wavelengths alongthe laser axis z �this is equivalent to the paraxialapproximation� and for a resonator with a Gaussian

mirror there are two different waves that travel inthe positive and the negative directions along thelaser axis. The wave number of the mode in thepassive cavity is denoted by kmnq and �mnq is theoscillation �angular� frequency. Note that, in gen-eral, the laser modes in Eq. �3� are power non-orthogonal, i.e.,

� d�Emnq�rT, z, t� Emnq*�rT, z, t� � �mnq.

Furthermore, we assume that central tuning forthe laser mode is equal to the oscillation �angular�frequency � �mnq �which is the resonance fre-quency of the laser transition�, and that the complexamplitudes R and S of the counter-propagatingwaves are proportional to the threshold field distri-bution. Thus, according to Ref. 30, we have

�Rq� z�� Mqmn�exp��q z�,

�Sq� z�� 1�c

�Mqmn�exp��q z�, (4)

where �Mqmn� is the real amplitude and �q is the

propagation constant that equals

�q �1

2Lln� 1

�c�geff� . (5)

The laser length is denoted by L, �c is the amplitudereflectivity coefficient of a classical mirror, and �g

eff

describes the effective amplitude reflectivity coeffi-cient of the Gaussian mirror and can be defined by

�geff � �0��

�� d��exp��

rT2

wg2�exp��

rT2

wR2�L��

2

��

� d��exp��rT

2

wR2�L��

2 �1�2

� �0��


2

wS2�L��

2

��


2

wR2�L��

2�1�2

� �0

wS�L�

wR�L�,

where d� is the cross section perpendicular to thelaser axis, wg is the Gaussian mirror parameter,2 �0 isthe amplitude reflectivity at the center of the Gauss-ian mirror, and wR and wS are the Gaussian beamparameters for both counter-propagating waves Rand S, respectively.

It is worth noting here that the threshold fieldapproximation provides the results that remain ingood agreement with the exact solution over a rangeof laser parameters that are of practical interest.31,32

Fig. 1. Configuration of a laser with a Gaussian mirror.

20 March 2002 � Vol. 41, No. 9 � APPLIED OPTICS 1669

The boundary conditions for our laser structure canbe written in the following form:

�Rq�0� AmnR�rT, 0�� c�Sq�0� Amn

S�rT, 0��,

�1 � �c2��Sq�0��2 �

rT� drT� Amn

S�rT, 0��2 � PoutS, (6)

�Sq�L� AmnS�rT, L�� g

eff�Rq�L� AmnR�rT, L��,

1 � ��geff�2 �Rq�L��2 �

rT� drT� Amn

R�rT, L��2 � PoutR,

(7)

where the PoutR and Pout

S describe the power of thelaser emitted at the end of the laser. Taking intoaccount Eqs. �4� and the boundary conditions in Eqs.�6� and �7�, we can relate the amplitude Mq

mn of themnqth mode to the output power in the following way:

�Mqmn�2 �

Pout

�c

NmnS�0�

�c1 � �c

2 �Nmn

R�L�

�geff 1 � ��g

eff�2

, (8)

where the total output powers

Pout � PoutR � Pout

S,

NmnR� z� � �

rT� � Amn

R�2drT,

NmnS� z� � �

rT� � Amn

S�2drT

are normalization integrals carried out over the crosssection perpendicular to the laser axis, treating the zcoordinate as a parameter.

Here we analyze the transient behavior of thelaser, i.e., relaxation oscillations, which result froma small perturbation of the steady-state operationobserved far above threshold. Thus, in this regionof laser operation we can expect that the selectivityof the Gaussian mirror is sufficient to provide lasergeneration on a fundamental Gaussian mode.Moreover, far above threshold, i.e., for a relativelyhigh pumping level, coherent photons �resultingfrom stimulating emission� dominate. Addition-ally, the influence of the excess noise on the modeintensity is relatively low �the influence of the ex-cess noise that results from diminishing mode non-orthogonality when the pumping level increases;see Ref. 33�.

Thus for our approach we confine our considerationto the laser operation on a fundamental Gaussian

mode and omit cross-mode products in the mode in-tensity expression. So according to Eqs. �3�–�8�,Imnq�rT, z, t��Is can be written in terms of the outputpower as

Imnq�rT, z, t�Is

�Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�2

�Pout

PsB��Rq

mnAmnR�2 � �Sq

mnAmnS�2

� ��RqmnAmn

R�*SqmnAmn

S

� �SqmnAmn

S�*RqmnAmn

R�, (9)

where C � d��AmnR�2 � �Amn

S�2� and Ps is thesaturation power. Note that in Eq. �9� we take intoaccount the spatial hole burning effect. We intro-duce the phenomenological parameter � for two cas-es: the first, � 0, is addressed to the envelopefield approximation �coherent terms are neglected�and the second, � 1, includes the spatial holeburning effect. The normalization constant B isdefined as

B ��c

NmnS�0�

�c1 � �c

2 �Nmn

R�L�

�geff 1 � ��g

eff�2

.

Moreover, in Eq. �9� we also assume single-longitudinal-mode operation. In general, the spa-tial hole burning effect in a homogeneouslybroadened laser medium �as discussed in this pa-per� is likely to lead to the oscillation of multiplelongitudinal modes. However, if the mode compe-tition effects are weak, �i.e., the cross saturationterms are much smaller than the self-saturationterms in equations of motion that describe multi-mode operation �see, for example, Chap. 9 in Ref.34�, the laser operates in the so-called weak cou-pling region. In this case, our assumption ofsingle-mode operation can be justified since eachlongitudinal mode oscillates rather independentlyin a similar way.

On the other hand, when the mode competitionbecomes important, the description of the laser oper-ation becomes more complex, since in multimodeequations the cross-saturation terms should be in-cluded. Such an analysis is much more complicatedand is beyond the scope of this paper.

To obtain the small-signal linear oscillatory solu-tions of Eqs. �1� and �2� we used the perturbationmethod.26 We let N N0 � N and Q Q0 � Q,where the parameters with circumflexes representtime-dependent quantities much smaller than therespective steady-state parameters for N0 and Q0.Next, when these variables are inserted into Eqs.


�1� and �2� and products NQ are neglected, we ob-tain

dNdt

� �N� �1 �

Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�2C

�N0

�

QQ0

Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�2C,

(10)

dQdt

� C � d�Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�2

N�

.

(11)

The first equation for N is of the form

dNdt

� �b�rT, z, t�N � a�rT, z, t�Q, (12)

with

b�rT, z, t� �1� �1 �

Pout

PsB�Rq

mnAmnR

� SqmnAmn

S�2C , (13)

a�rT, z, t� �1�

N0

Q0

Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�2C.

(14)

The solution is

N � �a�rT, z, t�exp�b�rT, z, t�t �0

t

exp�b��Q��d�,

(15)

which is then substituted into Eq. �11�. For smallperturbations, the assumed form of solution is

Q � Q0 exp��t�cos��t�, (16)

with N�0� 0 and Q�0� Q0, where � is the damp-ing rate coefficient and � is the frequency of therelaxation oscillations. The integration providesconsiderable simplification for � �� . By equat-

ing like coefficients of cos��t� and sin��t�, we ob-tained the solution for the damping rate coefficientand frequency of the relaxation oscillations in thefollowing form:

� �1�

� d��RqmnAmn

R � SqmnAmn

S�4

� d��Rq

mnAmnR � Sq

mnAmnS�4

1 �Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�2C

,

(17)

�2 �cn

2�0

� � d�

Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�4C

1 �Pout

PsB�Rq

mnAmnR � Sq

mnAmnS�2C

,

(18)

where c is the free-space velocity of light, n is theeffective refractive index, and �0 is the small-signalgain. Note that in Eq. �18� we have two parameters,i.e., the small-signal gain �0 and normalized outputpower Pout�Ps, which are, for a given laser structure,related to each other. Because of the nonlinear gainsaturation effect the problem of finding this relationcan be analytically difficult and the usual approach isto choose a suitable approximation. To relate Pout�Psto �0, we applied the results obtained in Ref. 30.

For the homogeneous laser medium, when modecompetition is neglected, the small-signal gain �0 iscombined with the gain coefficient as follows:

� ��0

1 ��Rq

mnAmnR � Sq

mnAmnS�2

Ps

. (19)

In Ref. 30 it was shown that, with the help of theenergy theorem and the threshold field approxima-tion, the small-signal gain can be related to the out-put power and the laser system characteristicparameters in the following way:

2�0 �

� 1�c

� 1�g

eff �1�c� � ��g

eff � �c

�c��1 �

�L

�q

� dz� 1Nmn

R �rT

� drT

f �rT, z�exp�2�q z�� A00R�2

1 �Pout

PsB��rT, z�

�1

NmnS �

rT� drT

f �rT, z�exp��2�q z�� A00S�2

1 �Pout

PsB��rT, z� �

,

��rT, z� � exp�2�q z�� A00R�rT, z��2

�1

�c2 exp��2�q z�� A00

S�rT, z��2 � �1�c

A00R�rT, z� A00

S*�rT, z� � A00R*�rT, z� A00

S�rT, z�, (20)


where �L describes the distributed losses, f �rT, z� isthe normalized function, which describes the spatialdistribution of the small-signal gain and, in general,it depends on the pumping conditions.

In Section 3 we use Eqs. �17�, �18�, and �20� toobtain laser characteristics, revealing the influence ofthe real parameters of the laser structure on relax-ation oscillations in a laser with a Gaussian mirror.We especially discuss the effect of the Gaussian mir-ror peak reflectivity and a Gaussian mirror parame-ter.

3. Numerical Results

We present characteristics of the relaxation oscilla-tions in a Nd3�:YAG laser with a Gaussian mirror.The numerical results are obtained for the funda-mental laser mode TEM00 for various geometries ofthe laser cavity �classically unstable as well as clas-sically stable�. The spot size distribution of the leftand right propagating waves that create the lasermode are obtained with the help of the ABCD matrixformalism by an appropriate change of the referenceplane. In our calculations we assume uniformpumping of the active medium, i.e., f �rT, z� 1, andthe laser wavelength is � 1.06 �m. The inversionpopulation lifetime is � 260 �s. Moreover, we as-sume that the power is extracted through the Gauss-ian mirror and the reflectivity of the conventionalmirror is �c 1. We obtained the characteristics fortwo levels of distributed losses: �L 10�5 and �L 10�3 cm�1.

In our numerical analysis we consider two differentcavity configurations, classically unstable and a sym-metric one that is classically stable. In general, it isa well-established fact that a Gaussian mirror that isused as an output coupler in unstable laser resona-tors significantly improves the optical quality of theoutput laser beam. However, more recently it hasalso been shown that the implementation of theGaussian mirror in a laser having a classically stablecavity �for example, confocal geometry; see Ref. 30�can cause an increase in the output power level �for agiven pumping level� in comparison with an identicallaser having classical mirrors.

Figure 2 shows the dependence of the relaxationoscillation frequency � on the Gaussian mirror pa-rameter wg for four different Gaussian mirror peakreflectivities �0 and for two levels of distributed losses�L. It is worth noting that, with the increase inGaussian parameter wg, the Gaussian mirror tendsto be classical. We present the characteristics ob-tained for the classically unstable resonator with thespatial hole burning effect neglected, � 0. In thiscase the radii of curvature for both convex mirrors,conventional Rc and Gaussian Rg, are the same andare equal to Rc Rg �10 cm. The length of theresonator is equal to L 5 cm. As can be observed,with the increase of the Gaussian mirror parameterwg, the oscillation frequency � monotonically de-creases. Moreover, when the Gaussian mirror peakreflectivity �0 tends to unity, frequency � has asmaller value over the range of wg. Such laser be-

havior is directly related to the results obtained inRef. 30, in which steady-state nonlinear operation ofthis kind of laser was analyzed and can be explainedas follows. It is a well-established fact26,28 that thesquare of the relaxation oscillation frequency � isproportional to the small-signal gain see also Eq.�18� in the active medium required to maintain agiven output power level. In classically unstableresonators, while the Gaussian mirror tends towardthe classical one, the mode volume increases and themode extracts energy from the active medium in amore efficient way. Moreover, for the Gaussian mir-ror peak reflectivity �0 tending to unity the effectivemirror reflectivity becomes greater �it also increaseswith wg� and it simultaneously makes the Q factor ofthe laser cavity greater. Thus, these two effects re-sult in a decrease in the small-signal gain required tomaintain the given output power level. As a conse-quence the oscillation frequency � also decreases.Note that in this case the losses do not affect the lasercharacteristics since they remain much smaller thanthe available gain for a laser mode.

In Fig. 3, similar oscillation frequency characteris-tics are plotted for different output power levels andtwo loss levels. The laser characteristics are similarto those in the previous case. Again, the effect oflosses is negligible, and, as is obvious, the relaxationoscillation frequency � increases with the increasedoutput power level, since the small-signal gain re-quired to maintain a given output power also in-creases.

Figure 4 illustrates the behavior of the dampingrate coefficient � as a function of the Gaussian mirrorparameter for the same classically unstable resona-tor and four different Gaussian mirror peak reflec-tivity �0. The spatial hole burning effect is omittedand two levels of distributed losses are considered.As is obvious, the damping rate decreases when theGaussian mirror parameter wg increases, i.e., whenthe Gaussian mirror tends to be a classical one.Such behavior of the laser characteristics is again

Fig. 2. Frequency � of the relaxation oscillations plotted versusthe Gaussian parameter wg for two levels of distributed losses andGaussian mirror peak reflectivity �0 as parameters. Both mirrorsare convex.


related to the changes of the gain �required to main-tain a given output power level� in the active medium,which according to Refs. 26 and 28 determines thedamping rate of the relaxation oscillation.

Thus, the damping rate increases with the outputpower level, see Fig. 5, and is not noticeably affectedby the distributed losses.

In Fig. 6 the frequency of the relaxation oscillationsis plotted versus the Gaussian mirror parameter fortwo levels of the normalized output power level �Pout�Ps 0.01 and Pout�Ps 10� and for two values of thephenomenological parameters � 0 �with the spatialhole burning effect neglected� and � 1 �with thespatial hole burning effect included�. As is obvious,the spatial hole burning effect causes an increase ofthe relaxation oscillation frequency. In this case�i.e., for � 1� the active medium is not saturateduniformly by the laser mode and higher gain is re-quired to maintain a given output power level.These result in higher frequency of the relaxationoscillations as well as a greater damping rate; see Fig.7 �since � is also determined by �0�.

Next we obtained laser characteristics �Figs. 8–12�for a classically stable resonator having symmetricgeometry. In general, it is a well-established factthat a Gaussian mirror is used mostly as an outputcoupler in unstable laser resonators. It significantlyimproves the optical quality of the output laser beam�the cavity with a Gaussian mirror becomes stablewith Gaussian eigenmodes�. However, more re-cently it has also been shown30 that the implemen-tation of the Gaussian mirror in the laser having aclassically stable cavity �for certain resonator config-uration� can cause an increase in the output powerlevel �for a given pumping level� in comparison withthe identical laser having classical mirrors �see Ref.30 for a detailed discussion�. Moreover in this casethe influence of the gain saturation on relaxationoscillation parameters, � and �, is also evident, butthe gain saturation effect affects laser characteristicsin a slightly different way.

Figure 8 shows the dependence of the relaxationoscillation frequency � on Gaussian mirror parame-

Fig. 3. Frequency � of the relaxation oscillations plotted versusthe Gaussian parameter wg for two levels of distributed lossesand output power levels Pout�Ps as parameters. Both mirrorsare convex.

Fig. 4. Damping rate coefficient � of the relaxation oscillationsplotted versus the Gaussian parameter wg for two levels of distrib-uted losses and Gaussian mirror peak reflectivity �0 as parame-ters. Both mirrors are convex.

Fig. 5. Damping rate coefficient � of the relaxation oscillationsplotted versus the Gaussian parameter wg for two levels of thedistributed losses and output power levels Pout�Ps as parameters.Both mirrors are convex.

Fig. 6. Frequency � of the relaxation oscillations plotted versusthe Gaussian parameter wg for the spatial hole burning effect andfor output power levels Pout�Ps as parameters. Both mirrors areconvex.


ter wg for four Gaussian mirror peak reflectivities �0,for two levels of distributed losses �L, and with thespatial hole burning effect neglected. The symmet-ric resonator has two concave mirrors of the sameradius of curvature, i.e., Rc Rg 10 cm and thelength of the laser is equal to L 5 cm. Note that incontrast with an unstable resonator, compare Fig. 2,all the curvatures exhibit minima in � as a functionof wg.

This interesting behavior of the laser characteris-tics is again directly related to the results obtained inRef. 30, in which the steady-state nonlinear operationof this kind of laser was analyzed, which can be ex-plained as follows. It is a well-established fact that,for the laser structure, for a given output power�pumping level� and a given loss level, an optimaloutput mirror reflectivity exists providing maximalpower efficiency. It results in minimal small-signalgain required to maintain a given output power level.For a cavity with a Gaussian mirror, the effectivereflectivity of the Gaussian mirror is determined by

peak reflectivity �0 �Ref. 30� and Gaussian mirrorparameter wg �increases with both of them�, as wellas by the incident spot size on the mirror beam �whichagain depends on wg�. Therefore, the effective re-flectivity also depends on cavity configuration.

In particular it has been shown30 that for a sym-metric configuration an optimal value of the increas-ing Gaussian parameter wg exists, which yields theminimal gain required to provide a given outputpower level. The optimal value of wg results fromthe two opposite effects. On the one hand, the modevolume increases when wg decreases �the energy isextracted from the active medium more efficiently�.On the other hand, the effective reflectivity of theGaussian decreases simultaneously. Since relax-ation oscillation frequency � is directly related to thesmall-signal gain, the behavior of the frequency char-acteristics should be the same as those of the small-signal gain, which is what can be observed in Fig. 8�compare Fig. 6 in Ref. 30�. Furthermore, the min-ima of oscillation frequency � correspond to the op-timal values of the Gaussian mirror parameter wg�for fixed �0�, for which the laser has a maximumoutput power level for a given pumping rate of theactive medium and a fixed loss level.

The effect of the gain saturation is also manifestedby the oscillation changes of the frequency character-istics caused by the variation of the peak mirror re-flectivity �0. As can be seen in Fig. 8, for smallvalues of wg �the Gaussian mirror has a sharp reflec-tivity profile�, when the laser structure is under-coupled �i.e., the effective reflectivity of the Gaussianmirror is too small to provide optimal feedback� withan increase of �0 �effective reflectivity tends to opti-mal value�, the relaxation oscillation frequency � de-creases. The opposite situation is observed forgreater values of wg �for which the effective reflectiv-ity of the Gaussian mirror also increases�, when os-cillation frequency � increases when �0 tends tounity. In this region of operation the laser structureis beyond the optimal mirror reflectivity �it is over-coupled�. Thus, with increasing �0 one needs ahigher pumping level �i.e., greater small-signal gainavailable in the active medium� to maintain a givenoutput power level.

It is worth noting that the difference in behavior ofthe frequency characteristics for classically unstable,Fig. 2, and classically stable, Fig. 8, resonators resultfrom the fact that, in the former case for a givenstructure parameter, in the presented range of wgclassically unstable resonator is undercoupled.Thus, when wg increases the effective reflectivity ofthe Gaussian mirror monotonically tends to an opti-mal value.

In Fig. 9, similar oscillation frequency characteris-tics of a classically stable �symmetric� cavity for twopower levels and two values of the loss coefficient arepresented. The laser characteristics are similar tothose in the previous case, since they reflect the gainsaturation effects. As with a classically unstablecavity, the relaxation oscillation frequency � in-creases when the output power level increases as well

Fig. 7. Damping rate coefficient � of the relaxation oscillationsplotted versus the Gaussian parameter wg for the spatial holeburning effect and for output power levels Pout�Ps as parameters.Both mirrors are convex.

Fig. 8. Frequency � of the relaxation oscillations plotted versusthe Gaussian parameter wg for two levels of the distributed lossesand Gaussian mirror peak reflectivity �0 as parameters. Bothmirrors are concave.


as the loss level �since the small-signal gain requiredto maintain a given output power also increases�.Note that the losses affect the frequency characteris-tics more significantly for higher output power levels.

The effect of spatial hole burning on the relaxationoscillation frequency in a classically stable cavity isshown in Fig. 10, in which � is plotted as a functionof wg for two levels of the normalized output powerand two values of the � parameter, � 0 and � 1,respectively. Comparing the results in Fig. 10 withthose of Fig. 8, we observed that spatial hole burningaffects the frequency characteristics in a similar way,causing an increase in � for both cavities. The re-sult is obvious, since the physical mechanism of thisphenomenon is exactly the same for both cavity con-figurations �i.e., the small-signal gain required tomaintain a given output power level increases be-cause of nonuniform saturation of the active medi-um�.

Figure 11 shows the dependence of the dampingrate of the relaxation oscillation in a symmetric �clas-

sically stable� cavity as a function of Gaussian mirrorparameter wg with the Gaussian mirror peak reflec-tivity as a parameter. As is obvious, the dampingrate characteristics are similar to the relaxation os-cillation frequency characteristics, because �as wementioned before� they both follow the changes of thesmall-signal gain �required to maintain a given out-put power level in the laser structure�. Thus, withthe increase in output power level, as well as with thepronounced spatial hole burning effect, the dampingrate should be greater. This is what we observe inFig. 12, where similar characteristics are shown fortwo levels of the normalized output power and twovalues of �, � 0 �with spatial hole burning neglect-ed� and � 1 �with spatial hole burning effect in-cluded�.

It is worth noting here that, for a symmetric cavityconfiguration having a Gaussian mirror, for in-creased values of Gaussian mirror parameter wg,damping rate � becomes comparable with the � fre-quency of the relaxation oscillation �see Figs. 8–12�.

Fig. 9. Frequency � of the relaxation oscillations plotted versusthe Gaussian parameter wg for two levels of the distributed lossesand output power levels Pout�Ps as parameters. Both mirrors areconcave.

Fig. 10. Frequency � of the relaxation oscillations plotted ver-sus the Gaussian parameter wg for the spatial hole burningeffect and output power levels Pout�Ps as parameters. Bothmirrors are concave.

Fig. 11. Damping rate coefficient � of the relaxation oscillationsplotted versus the Gaussian parameter wg for two levels of thedistributed losses and Gaussian mirror peak reflectivity �0 as pa-rameters. Both mirrors are concave.

Fig. 12. Damping rate coefficient � of the relaxation oscillationsplotted versus the Gaussian parameter wg for the spatial holeburning effect and output power levels Pout�Ps as parameters.Both mirrors are concave.


Thus, for the assumption made in the derivation ofthe expressions in Eqs. �17� and �18� for � and �,respectively, i.e., � �� , is satisfied only for a Gauss-ian mirror having a sharp reflectivity profile �i.e., forsmall values of wg�. However, the quantitativeanalysis of the relaxation oscillation in a symmetriccavity having a Gaussian mirror with a soft profile�having greater values of wg� requires more rigoroussolutions of these equations.

4. Conclusions

We have presented a systematic study of the relax-ation oscillations in a laser with a Gaussian mirrortaking into account the spatial hole burning effect.Beginning with the time-dependent laser rate equa-tions, we derived approximate expressions for damp-ing rate � and oscillation frequency �. Numericalcalculations were carried out for a Nd3�:YAG laserstructure.

The laser characteristics reveal the influence of theGaussian mirror parameter wg, Gaussian mirrorpeak reflectivity �0, and the geometry of the cavity onthe relaxation oscillations. In particular it has beenshown that the Gaussian mirror parameter wg affectsoscillation frequency � and damping rate coefficient� in different ways in lasers with classically unstableand classically stable cavities. Moreover, the fre-quency characteristics reveal the gain saturation ef-fects and can be used to optimize the power efficiencyof the laser structure.

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Relaxation Oscillations in a Laser with a Gaussian Mirror

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