Numerical solution of the exact cavity equations of motion for an unstable optical resonator

Numerical solution of the exact cavity equations ofmotion for an unstable optical resonator

Mark S. Bowers and Stephen E. Moody

We solve numerically, we believe for the first time, the exact cavity equations of motion for a realistic unstableresonator with a simple gain saturation model. The cavity equations of motion, first formulated by Siegman["Exact Cavity Equations for Lasers with Large Output Coupling," Appl. Phys. Lett. 36,412-414 (1980)], andwhich we term the dynamic coupled modes (DCM) method of solution, solve for the full 3-D time dependentelectric field inside the optical cavity by expanding the field in terms of the actual diffractive transverseeigenmodes of the bare (gain free) cavity with time varying coefficients. The spatially varying gain serves tocouple the bare cavity transverse modes and to scatter power from mode to mode. We show that the DCMmethod numerically converges with respect to the number of eigenmodes in the basis set. The intracavityintensity in the numerical example shown reaches a steady state, and this steady state distribution iscompared with that computed from the traditional Fox and Li approach using a fast Fourier transformpropagation algorithm. The output wavefronts from both methods are quite similar, and the computedoutput powers agree to within 10%. The usefulness and advantages of using this method for predicting theoutput of a laser, especially pulsed lasers used for coherent detection, are discussed. Key words: Unstableresonator, equations of motion, mode analysis.

1. IntroductionSingle frequency laser sources are coming into wide-

spread application as transmitters for coherent opticalradars. Such radars are in various stages of develop-ment for both remote sensing and military applica-tions. The requirement for coherence and beam qual-ity of such a transmitter implies that the laser sourcemust operate, to a good approximation, on both asingle longitudinal mode and a single transverse modeof the resonator.

It is common for lasers designed for this applicationto be equipped with unstable resonators. Unstableresonators offer a large extracted mode volume andintrinsic discrimination against higher-order trans-verse modes, which is needed for good beam quality.For the coherent application, single transverse-modeoperation is also required to ensure single frequencyoutput. Our work has been focused primarily on largeaperture pulsed CO 2 lasers, for which the use of unsta-ble resonators is essentially mandatory, if reasonableextraction efficiency from the large volume discharge

The authors are with Spectra Technology, Inc., 2755 NorthupWay, Bellevue, Washington 98004-1495.

Received 24 October 1989.0003-6935/90/273905-11$02.00/0.© 1990 Optical Society of America.

is to be achieved. Some form of injection seeding isused to achieve single longitudinal mode oscillation.This paper develops a calculational methodology forevaluating the time dependent distribution of trans-verse modes in a laser equipped with an unstable reso-nator and containing a saturable and time varying gainmedium. While the near term motivation has beendesign of CO 2 lasers, the methods presented are gener-al and can easily be extended to other types of lasers,e.g., solid state.

Most of the analysis of unstable resonators has cen-tered on the purely optical properties of the bare (gainfree) cavity. The effects of a saturable gain mediumon the cavity field have been considered only to theextent of deriving a modified fundamental field distri-bution which includes the spatially varying, but timeinvariant, gain saturation.' The effects of gain havetraditionally been studied using relatively simple gainsaturation models. A few gain sheets are placed insidethe optical cavity, and a single wavefront is propagatedthrough the cavity in a manner pioneered by Fox andLi2 until the wavefront is unchanged on a round trippropagation. In a standing wave cavity, gain satura-tion is normally computed by storing the forward andbackward intensity distributions at the axial locationsof the gain sheets during a single propagation throughthe resonator, and after one round trip is completed,the gain is allowed to saturate using the stored intensi-ty distributions according to the gain saturation modelused. The procedure is the same for a ring resonator,

20 September 1990 / Vol. 29, No. 27 / APPLIED OPTICS 3905

except that summing of the two overlapping intensityfields is not required. This method was initially devel-oped for predicting the steady state cavity distributionand output beam characteristics of high power cw gaslasers.3 4

The Fox and Li approach of following a single wave-front as it propagates back and forth through the cavi-ty implicitly assumes that the gain changes slowlyenough to be treated as quasistatic over the time inter-val covered by a round trip propagation. Therefore,the method is inappropriate for pulsed lasers wherethe gain can change on time scales comparable with anoptical round trip. Such lasers do not necessarilyreach a combined steady state of the optical field andgain medium. Any detailed description of the outputbeam characteristics of such devices must include boththe spatial and temporal buildup and decay of thecavity field and laser gain to be realistic.

We now believe that such a detailed coupled gainkinetics and cavity field analysis is required for pulsedlasers used for coherent detection. For this applica-tion, it is important to predict accurately laser wave-form and pulse energy. However, of equal importanceis the characterization of intrapulse and interpulsefrequency shifts due to transverse mode hopping ormultitransverse mode oscillation. Multiple trans-verse mode oscillation of pulsed coherent CO2 sourcesusing unstable resonators has been widely observed,but to date its causes, sensitivities, and methods forcontrol have not been fully characterized or quanti-fied.5 The effects of mode-medium interactions (in-tensity driven time dependent phase changes) on fre-quency and output beam quality are similarly knownto be important in realistic coherent pulsed CO2 lasersand should be incorporated into any general method-ology.6

Almost a decade ago, Siegman7 formulated a methodfor solving the wave equation for the electric field in anunstable optical cavity. In this paper, we show a fullnumerical implementation of Siegman's formulationwith gain, which we term the dynamic coupled modes(DCM) method. We show that the method is bothpractical and effective in addressing the issues whichwe have encountered in the modeling of coherent lasersystems with unstable resonators.

Siegman's approach uses the bare cavity eigenmodesas a basis set for expanding the time dependent gainloaded cavity field. The time varying field is repre-sented as an expansion of the stationary bare cavityeigenmodes with time varying coefficients. In thisformulation, spatially varying gain and/or the index ofrefraction serves to couple the stationary bare cavitymodes and to scatter power from mode to mode. Sincethe saturation of the medium will be time dependent,the saturation-induced spatial gain variation will itselfvary with time and hence so will the mode couplingmatrix elements. The DCM method is capable ofcorrectly incorporating this complicated multifacetedtime dependence in a straightforward and natural way.The gain model can be made as complete and detailedas necessary, since the method intrinsically accommo-

dates a fully space and time dependent approach togain calculation. For test purposes, we have so faronly exercised the method with a simple saturable gainmodel. A fully time dependent molecular kineticsmodel of CO2 gain is currently being added withoutany fundamental restructuring of the methodology.Time varying intracavity phase changes as well as gainchanges are easily considered by allowing the gain to becomplex.

In this paper, we show the feasibility of using theDCM method for numerically predicting the outputcharacteristics of a pulsed laser using an unstable reso-nator. The field equation is formulated in the linearsusceptibility or rate equation approximation andsolved numerically for a realistic unstable cavity. Aspecific test case is shown using a graded reflectivityoutput mirror, but the method can also accommodate aconventional hard-edged output mirror. The onlylimitation in the present formalism is the assumptionthat all the diffractive losses in the cavity occur at theoutput coupler. A simple gain saturation model isused in which the gain is assumed to be radially sym-metric and saturate homogeneously, and the small sig-nal gain is assumed to rise linearly in time to a peakvalue, after which it remains constant. The resultantsteady state distribution is then compared with thatcalculated from the conventional Fox and Li iterativemethod using a fast Fourier transform propagationalorithm.8

The next section reviews the formulation given bySiegman7 of the DCM method for solving the waveequations for the cavity field in an unstable opticalcavity. Although most of the formulation has beengiven in Ref. 7, it it is included here for completenessand generalized further into a form suitable for compu-tation. Section III describes the resonator and gainmodel used in the calculation along with some simpli-fying assumptions and numerical methods used. Thisis followed in Sec. IV where we show the first numericalresults obtained from solving the exact cavity equa-tions of motion for a realistic unstable cavity. SectionV gives a comparison of the steady state solution fromthe DCM method with the more traditional Fox and Liapproach using an FFT propagation algorithm. Fi-nally, Sec. VI is a discussion of the usefulness andadvantages of using the DCM method of solution forpredicting the output characteristics of a laser thatuses an unstable optical cavity.

11. FormulationThe derivation of the equation of motion for an

unstable optical laser cavity in the DCM method fol-lows the analysis of Siegman7 and is reproduced herefor completeness. We confine ourselves in this studyto a standing wave cavity as in Fig. 1(a), although themodel can also describe a ring laser cavity. A singleround trip propagation through the cavity is repre-sented by the unfolded cavity cell depicted in Fig.1 (b).9 A wave originating at the reference plane z = 0,taken here to be at the output coupler, propagates tothe end mirror at z = L and then back to the output

3906 APPLIED OPTICS / Vol. 29, No. 27 / 20 September 1990

w0 propagating in the positive z-direction. They arewritten as

E(r,z,t) = RetE(r,z,t,) exp[i(cot-koz)] ,

p(r,z,t) = Ret(r,z,t) exp[i(w 0t -koz)], (2)

where E and p are complex vectors. The carrier fre-quency wo is arbitrary, but it is convenient to define itas 2woL/c = qo2ir, with q0 any axial-mode integer nearline center and ko = wo(ME)1/ 2. Under the paraxial andslowly varying envelope approximation, 0 Eq. (1) be-comes

Vt -2iko,\Z + + -)]l(r,z,t) =-Oz(r^ztt)V2I 2k(a? c C at J (3)

OUTPUTCOUPLER

PARAXIAL OPTICALSYSTEM

r- - I(AB)

-(CD

z=0

OUTPUTCOUPLER

Z= 2L

Fig. 1. (a) Standing wave unstable resonator cavity. (b) Model ofthe standing wave cavity characterized by a single diffractive aper-

ture (the output coupler) and a single paraxial optical system.

where ao = a/2E, c = /(Moe)1/2, and V2 is the Laplacianoperator for the transverse coordinates.

It is assumed that the response of the lasing mediumto the electric field is collinear. In this case the reso-nant susceptibility assciated with the laser transitionis a scalar (rather than a tensor), and E and p are in thesame direction. In this approximation, sometimescalled the linear susceptibility or rate equation ap-proximation,10 the polarization is approximated by

p(r,z,t) = i g(r,z,t)E(r,z,t)k,

(4)

for frequencies near line center, where g(r,z,t) is theusual power gain coefficient. We can drop the vectornotation and concern outselves with any one of thevecor components of E, assuming that there are noanisotropic or polarizing optical elements inside thecavity. Substituting Eq. (4) into Eq. (3) results in thescalar equation for the cavity field in the rate equationapproximation:

coupler reference plane at z = 2L. It is assumed thatthe only diffractive losses in the cavity occur at theoutput coupler located at z = 0 = 2L. The remainderof the optical elements in the cavity (which in thenumerical example below is just the end mirror) com-promises a paraxial optical system designated by itsABCD ray transfer matrix9 for propagation in the posi-tive z-direction. The output mirror is characterizedby a complex amplitude reflectivity function p(r),which includes the curvature of the mirror. The vec-tor r denotes the transverse coordinates (x,y) or (r,O)perpendicular to the z-axis.

A. Scalar Wave Equation for the Cavity Field

The full vector wave equation for the real valuedelectric field E(r,z,t) inside such a cavity is' 0

v2 -g~ a - a2 4)a~~t 2p~r~z,t),(1( J2-ya ato -twsd2E(r,z,t) = Ao t2 ^ 1

where the polarization vector p(r,z,t) represents thelaser atomic transitions. The conductivity a is includ-ed to represent a linear distribution of the losses insidethe cavity, excluding the outcoupling loss at the outputmirror. It is assumed that E(r,z,t) and p(r,z,t) arequasisinusoidal quantities with slowly varying ampli-tudes and phases referenced to some carrier frequency

It- 2ik 0 - + c 2 g(r,zt) +-- t#L(r,z,t) = 0. (5)

The boundary condition on the complex phasor ampli-tude E at the output coupler is

2(rz = 0,t) = p(r)E(r,z = 2L,t), (6)

where z = 2L indicates that the field is just incident onthe output coupler feedback plane and z = 0 indicatesthat the field is just leaving the plane.

Equation (5) for the cavity field, along with all theboundary and initial conditions satisfied by the field inthe cavity, is the fundamental starting point in thispaper for determining laser action. The analysis givenby Siegman7 proceeds by separating the time depen-dence from the spatial dependence by expanding theelectric field in terms of the diffractive stationary barecavity modes of the resonator. Before proceeding withthis, a discussion on the relevant mathematical proper-ties of the bare cavity transverse modes is in order.

B. Mathematical Properties of the Transverse Bare CavityModes

The diffractive transverse eigenmodes unm(r,z) ofthe bare cavity are solutions of the paraxial Helmholtzequation


OUTPUTCOUPLER

z=0

I-Z = 2L

I _ Iz I

S

-

l

l

I

I I

_ - - - - - -J

(v - 2ik, a)unm(rz) = 0, (7)

which satisfy all the boundary conditions in the cavity,and at the output coupler reference plane they satisfythe reduced boundary condition

p(r)unmn(r,z = 2L) = YnmUnm(rz = 0), (8)

where Ynm is the complex eigenvalue of the eigenmodeunm(r,z). The eigenmodes are normally calculated atthe output coupler feedback plane. At the feedbackplane, Eqs. (7) and (8) can be transformed into thefamiliar linear homogeneous Fredholm integral equa-tion for defining the eigenmodes'

'Ynmunm(r,z = 2L) = f p(r')K(r,r')unm(r',z = 2L)dr', (9)

where the integration is over the output coupler refer-ence plane. The right-hand side of this equation is thepropagation integral in the Huygens-Fresnel approxi-mation for one complete round trip through the reso-nator starting at the reference plane z = 2L. TheHuygens kernel K(r,r') in the Fresnel approximationis given in terms of the ABCD matrix elements inCartesian coordinates by'

K(r,r') = 2iB expj- iko [A(x' 2 + y' 2)

- 2(xx' + yy') + D(x2 + y2)]}, (10)

when the paraxial system is rotationally symmetric.The propagation kernel in Eq. (9) is in general not a

Hermitian operator. This is due to the nature of theboundary conditions imposed on the paraxial Hel-moltz equation for open-sided resonators.'," Hencethe eigenvalues Ynm will generally be complex, and thetransverse eigenmodes will generally not be power or-thogonal or self-adjoint to each other. Rather, themodes unm(r,z) are biorthogonal to an adjoint set ofeigenmodes vnm(r,z) at every plane z.11 The adjointset of eigenmodes have the same eigenvalues Ynm.Physically, nm(r,z) represents the eigenmodes goingin the reverse direction around the same resonator. Atthe output coupler feedback reference plane, thesemodes are the solutions to the integral equation

Ynmvnm(r,z = 2L) = p(r) KR(r,r')Vnm(r',z = 2L)dr', (11)

where KR(r,r') is the Huygens kernel in the reverse (ornegative) direction:

KR(r,r') = 2iB expj- i [D(x'2 + y 2)

- 2(xx' + yy') + A(x 2 + y2)]} . (12)

The biorthogonality of the transverse eigenmodes isexpressed mathematically by

S Pnm'(rz)Unm(rz)dr = n'nam'm, (13)

and this holds at all transverse planes within the reso-nator.12

For a standing wave cavity as shown in Fig. 1(a), A =D; therefore, propagating in the forward and reverse

directions around the resonator starting from the mid-plane of the aperture, p(r) is identical. A little consid-eration then shows that at any plane within the resona-tor the modes can be normalized so that vnm(r,z) =unm(r, 2L - z). Thus, in a standing wave cavity, thetwo sets of modes unm(r,z) and vnm(r,z) represent thesame set of transverse eigenmodes of the cavity. Thisis not true in general for ring resonators. The biortho-gonality of the eigenmodes is used below to obtain thecoefficients of a modal expansion of the cavity fields.

C. Cavity Equation of MotionReturning to Eq. (5) for the cavity electric field,

Siegman7 has defined a set of oscillation eigenmodesfor the cavity given by

Enm(rz) = nmunm(rz) expj[ln(1/nm)]z/2L. (14)

From Eq. (8), each of these eigenmodes is seen tosatisfy the boundary condition given in Eq. (6). Phys-ically, each of these modes represents the field thatwould be sustained by the cavity if it were filled with auniform gain medium and oscillating in a steady state.Each mode has lYnm12 less power after reflection off theoutput mirror, thus accounting for diffractive and out-put coupling losses. Defining

°0nm - iflnm 2 L ln(lnm), (15)

where anm and fOnm are real quantities which give thetime decay rate and frequency offset, respectively, ofeach transverse mode, the oscillation eigenmodes inEq. (14) can be rewritten as

Enm(rz) = unm(rz) exp[(anm - iflnm)(Z - 2L)/c]. (16)

The complex phasor amplitude which describes thecavity field is expanded as

E(r,z,t) = I Anm(z,t)Enm(r,z).nm

(17)

The functional dependence of the expansion coeffi-cients, Anm(Zt), on time and z is retained to allow fortemporal and spatial longitudinal buildup of the cavityfield. Since the functions unm(r,z) are the eigenmodesof a non-Hermitian operator, it cannot be rigorouslyguaranteed that they form a complete set. We simplymake the assumption that these modes can be used as abasis set and check this assumption numerically byexamining the convergence of the series expansion.Substituting this expansion into Eq. (5) and using Eq.(7) and the biorthogonality relation Eq. (13) result inthe coupled differential equations for the expansioncoefficients Anm(z,t):

(dt + nm iOn. + C 9 A(Znm(t)

2 > Gnmn'm(Zt)An'm.(Z,t), (18)nm

where we have defined the matrix

Gnm,n'm'(Zt) = exp[(am,- -nm) - (ln'm' - f3nm)( - 2L)/cI

X r drvnm(r,z)g(r,z,t)un',n(r,z). (19)


It can be seen that transverse variations in the gaincouples the nmth oscillation eigenmode to many othernm' modes via the coupling matrix given in Eq. (19).Since each oscillation eigenmode individually satisfiesthe boundary condition Eq. (6), the boundary condi-tion on the expansion coefficients is

Anm(0,t) = Anm(2L,t). (20)

Because of the boundary condition Eq. (20), each ofthe transverse amplitude coefficients Anm(Z,t) can beexpanded as a Fourier series of longitudinal modes:

Anm(Zt) = E Anmq(t) exp(-iirqz/L), (21)q

where each of the longitudinal modes is numbered withrespect to the central value qO so that (q + qo) is theactual axial mode number. Substituting Eq. (21) intoEq. (18) results in the reduced equation of motion forthe coefficients

d + [(aO + anm)-i(nm + L ]Anmq(t)

= 4L Gnmqsn,'q(t)An','q(t)4Ln',4q

with2L

Gnmqn'm'q'(t) = dz exp[iirz(q - q')/L]

X expi[(an - anm) - i(ln'm' - fnm)](Z - 2L)/cl

X f drv,,(rwz)g(rszt)untm(r~z).

Equations (22) and (23) are the individual tr;verse and longitudinal equations of motion in the:equation approximation. They explicitly show]nonuniform gain dynamically couples each transvlongitudinal mode coefficient Anmq(t) with olAn'm'q'(t). It can be seen from Eq. (22) that thequency of each mode is given by

&Jnmq = °° + L + fnm =( )L + nm-

The above formulation is quite general and apyto any optical cavity and gain medium for whichrate equation approximation is valid. The frequEdependence of the resonant susceptibility of the la:medium has been ignored in Eq. (4), which is valiline center. Thus effects due to frequency depencgain are not included, and it is assumed that the 1gain curve is sufficiently broad so that the transvand longitudinal modes of interest see essentiallysame gain as is usually the case for CO2 lasers.

D. Intensity and Outcoupled PowerThe intensity anywhere inside the cavity is give

the time average Poynting vector (in MKS units)

I(r,z,t) = 1 ( L) /2 I(r z t)12

The output intensity distribution Iout(rt) at the out-

put mirror can be obtained from the intracavity inten-sity distribution just incident on the output mirror:

Iout(r,t) = [1 - p(r)J2]I(r,z = 2L,t),

and the output power P(t) is then simply

P(t) = f I,,,(r,t)dr,

where the integration limits extend to infinity.

E. Circular Mirror Cavities

(26)

(27)

The formulation given above holds for an outputcoupler of arbitrary shape. For resonators with circu-lar mirrors, however, the above equations can be sim-plified somewhat by utilizing circular symmetry.

Let the subscript I denote the azimuthal mode indexand the subscriptp the radial mode index. The trans-verse eigenmodes in polar cylindrical coordinates canbe decomposed into azimuthal components

ulp(r,z) = u1p(r,z) exp(il0), (28)

and each azimuthal component radial mode is found tosatisfy the integral equation'

,lpulp(rz = 2L) = °B rp(r)u1p(r',z = 2L)J1 (°)Bf 43_

X exp o- (Ar'2 + Dr2) dr'p 2B I (29)

at the output feedback plane z = 2L, where J is the thorder Bessel function. The biorthogonality relationEq. (13) now reads

(23) rdr 2J d0vLP,,(r,z)ulP(r,z) exp[i(l - l')0] = 6ll6ppt

rate The complex phasor amplitude is expanded as

hjow-erse E(r,z,t) = 7 Alp(z,t)Elp(r,z),-L _r 1=-. p=o

(30)

(31)

fre- where the oscillation eigenmodes Elp(r,z) are given byan expression similar to Eq. (16) with the subscriptsnm replaced by Ip and Unm by Eq. (28). The amplitude

(24) coefficients Alp(z,t) are expanded in a Fourier series asin Eq. (21), and using the biorthogonality relation in

lies Eq. (30), the resultant equation of motion for the coef-the ficients with circular mirrors becomesncy dAJP + [(aO + alp) - + L]A.q(t

(25)

= G pq p'q(t)A1'p'q,(0)I,=-. p =o q =--

(32)

with2L

Glpq,'p'q,(t) = J dz exp[irz(q - q')/L]

X expf[(ap, - alp) - i(,,P, - fp)](z - 2L)/cI

X J dO exp[i(l' - 001 J rdrv1P(r,z)g(r,z,t)u1,p(rz). (33)

It can be seen that, as expected, any radial variation inthe gain will couple different radial modes, and any


azimuthal variation will couple modes of different azi-muthal symmetry.

Ill. Numerical ExampleThe cavity equations of motion in the DCM method

formulated in the previous section for a circular mirrorresonator [Eqs. (32) and (33)] have been solved nu-merically for a realistic optical cavity. This sectiongives a description of the resonator, the nonlinear gainmodel used, the numerical methods, and some simpli-fications that were made to reduce the computationaleffort.

The optical resonator used in this example is anunstable resonator currently being used in an existingTE-CO2 wind sensing laser. A list of parameters de-scribing the resonator is in Table I. The output cou-pler is a graded reflectivity mirror with a radially vary-ing power reflectivity profile given by

IP(r)12 = Ro[l - (ra)2]. (34)

Graded reflectivity output couplers in unstable reso-nators have recently been receiving some attentionand appear to be well suited for coherent laser radarapplications.' 3 They are also attractive from a com-putational standpoint because the eigenmodes arerather smooth functions, as shown below. Thus theresolution requirement for the accurate calculationand description of the modes and for the accuratecomputation of the integrals over the eigenmodes inEq. (33) is not excessive.

In the calculations presented here, a simple nonlin-ear gain saturation model is used in which the gain isassumed to saturate homogeneously and is given by

g~~r~z~t) = go(r,t) (5g(r,z,t) = 1 + [I(rzt) + I(r,2L- ZtI/Isat

where Isat is the saturation intensity, and go(r,t) is thesmall signal (unsaturated) power gain coefficient.The small signal gain is assumed to be radially sym-metric and rise linearly with time to a maximum steadystate value. Although any transverse variation of thesmall signal gain can be readily modeled, for simplicitygo is assumed to be uniform over the radial extent ofthe gain region, which was taken here to be infinite.The small signal gain is given explicitly by

go(r,t) = g(rt) gt) t < T. (36)

where is the time it takes the small signal gain toreach its peak value. The values of the parametersused in the present example are listed in Table I.

To simplify the calculation, only the = 0 azimuthalsymmetry modes were coupled in Eq. (32). This isequivalent to the assumption that initially the field ispurely radially symmetric, i.e., at t = 0, Alpq(t = 0) for = 0. Since the small signal gain is taken to be radiallysymmetric, Eqs. (32), (33), and (35) show that Alpq(t) =0 for I d 0 for all times t > 0 if initially Alpq(t = 0) = 0for 1 # 0. For further simplification, the couplingbetween different longitudinal modes was ignored.This was done by setting q = q' = 0 in Eqs. (32) and

(33). Therefore, only the coupling between 1 = 0 sym-metry transverse modes was considered.

The first step in solving the coupled equations Eq.(32) is to calculate the bare cavity eigenmodes. Themodes must be obtained at the output mirror z = 2L forcomputing the output power, and they must also beknown as a function of z where the gain is nonzero forcalculating the matrix elements given in Eq. (33). Forthe present problem, the modes ulp(r,z = 2L) werecalculated from Eq. (29) by discretizing the integralequation using a Gauss-Legendre quadrature and di-agonalizing the resultant matrix equation using stan-dard numerical techniques.' 4 As mentioned earlier,the modes ulp(r,z) are all that is required for a standingwave cavity since the adjoint eigenmodes v(r,z) arerelated to ulp(r,z) by vp(r,z) = ulp(r,2L - z) anywhereinside the cavity. The modes were then normalizedusing Eq. (30) and then computed at the required axialpositions z by propagating the mode from the outputmirror by directly solving the Huygens integral for thewave propagation using the same Gauss-Legendrequadrature algorithm.

There are other methods described in the literaturethat can be used to compute the bare cavity modes.'These include matrix techniques such as the Prony'5or Krylov' 6 methods coupled with the Murphy andBernabe method 7 for extracting the higher modes andthe virtual source theory for hard-edged unstable reso-nators.18 Any of these methods can be used to com-pute the transverse eigenmodes provided enoughmodes can be obtained for convergence of the electricfield. In this respect, the matrix techniques may notbe well suited for this application since they only givethe first few lowest loss modes of the cavity. Thediscretization and diagonalization method used herewere found to be quite adequate for computing themodes in the unstable optical cavity with a gradedreflectivity output mirror studied in this paper. Themethod of choice for a particular problem will, ofcourse, depend on the optical properties of the unsta-ble resonator under investigation.

The integration in time of the first-order coupleddifferential equations given in Eq. (32) was performedusing a standard fourth-order Runge-Kutta algorithm.The integral over the radial coordinate in the matrixelements of Eq. (33) was computed using a Gauss-Legendre quadrature. The integrand over the axialcoordinate z was found numerically to be almost con-stant when the coupling between longitudinal modeswas neglected and q = q' =0 in Eq. (33). Thus a simplerectangular rule was found to be sufficient for theintegration over the axial coordinate. Of course, theinclusion of different longitudinal modes may requirea more accurate integration scheme such as a Gaussianquadrature, which may also require the computationand storage of the bare cavity eigenmodes at manyaxial locations.

IV. ResultsThe intensity of the five lowest loss I = 0 symmetry

bare cavity modes incident on the output mirror com-


Table 1. Resonator and Gain Parameters Used In the Present NumericalExample

Resonator parametersType Positive branch circular confocalLength 3.1 mWavelength 9.11 amMagnification 1.26Ro 0.7a 1.5cm

Active volumeGain length 90 cmgo 0.0097 cm-1

Isat 1.75 X 105 W/cm2

ao 0X 0.3As

Gain region located 10 cm from back mirror

Table II. I = 0 Azimuthal Symmetry Elgenvalues for the First SevenDominant Bare Cavity Elgenmodes

p I-yI Phase (deg)

0 0.656 2.951 0.564 16.22 0.481 57.53 0.379 115.34 0.283 192.65 0.173 287.26 0.092 395.0

puted from Eq. (29) is shown in Fig. 2. These modesare normalized using the biorthogonality relation Eq.(30). The direct solution of Eq. (29) for the eigen-modes gives the modes only for r < a, where a = 1.5 cmis the radius of the reflectivity profile. The intensitybeyond the edge of the reflectivity profile is found bypropagating each mode once through the resonator.The computed eigenvalues YIp are listed in Table II.

It is critical that the computed eigenmodes satisfythe biorthogonality relation Eq. (30). How well thisrelation is satisfied numerically is a good check on theaccuracy of the calculated modes. It was found thatforty-five points were sufficient to resolve the quadra-ture in Eq. (29) to obtain the eigenvalues and eigen-modes on the surface of the output mirror. However,when this same quadrature was used in the Huygens-Fresnel integral to propagate the modes inside thecavity, the biorthogonality relation inside the cavitywas not well satisfied, especially for the higher-ordermodes. It was found that the modes at large values of rwere inaccurate. When the quadrature was doubledto ninety points, the biorthogonality relation wasfound to be well satisfied inside the cavity for the firstfive lowest loss modes. The fifth mode was the leastaccurate, satisfying Eq. (30) to better than 5 parts in athousand. The accuracy of the modes inside the cavi-ty can be increased further if more points in the quad-rature are used; however, ninety points were found tobe sufficient when the first five lowest loss modes areused in the basis set.

The coupled equations of motion for the modal am-plitudes Aopo(t) (1 and q are set to zero as explained in

1.50

N

.

0:j

C.

0I-

1o~a

1.25

1.00

0.75

0.50

0.25

0.00

1.50

1.25

1.00

0.75

0.50

0.25

0.00

0 1 2 3

r (cm)

0 1 2

r (cm)

Fig. 2. Intracavity intensity distribution incident on the outputmirror of the first five lowest loss 1 = 0 symmetry bare cavitytransverse modes. The modes are normalized using the biorthogon-

ality relationship.

the last section) were integrated in time using thecomputed bare cavity eigenmodes until the intracavityintensity distribution reached a steady state. Sincethe temporal evolution of the system to reach a steadystate was of secondary interest, the initial conditionsfor the modal amplitudes were chosen rather arbitrari-ly. The initial values were taken to be small so that theinitial intensity is many orders of magnitude less thanthe saturation intensity. It was found that the steadystate solution is insensitive to the initial conditions forsmall initial values of the modal amplitudes.

The time dependene of the absolute value of themodal amplitudes with the five lowest loss 1 = 0 trans-verse modes coupled is shown in Fig. 3. Also shown inthe time dependence of the small signal gain. In thisfigure, each modal amplitude was initially set to thesame small initial value. The modal amplitudes are inunits of volts. The absolute values of the amplitudeswere found to reach a steady state value at t = 1 Its for


1.5 15 1.5

10

0'-4

Pa

5

0

0

01.0 I-

U

_U

_4

covD

0.00 0.2 0.4 0.6 0.8 1

Time (usec)Fig. 3. Time dependence of the absolute value of the modal ampli-tudes using the five lowest loss I = 0 transverse modes in the basis set.The initial conditions are that each modal amplitude has the samesmall value. The numbers on the curves designate the modal indexpair p. Also shown for reference is the time dependence of the small

signal gain.

these initial conditions. The small signal gain reachedits maximum value at t = 0.3 As. Thus during thebuildup of the cavity field after 0.3 As, the gain is large,there is little saturation, and, therefore, the couplingbetween different transverse modes is small. Sincethe fundamental mode has the lowest loss, it builds upfaster from its initial value than the higher loss trans-verse modes.

An interesting result is obtained when the initialconditions are changed so that Aooo(t = 0) = 0, and therest of the modal amplitudes have the same initialvalue as used to obtain the results in Fig. 3. Initiallythen, it is assumed that the cavity field has equalamounts of all modes except no fundamental modecomponent. The temporal evolution of the modalamplitudes assuming this initial condition is shown inFig. 4. It is seen that initially the next higher loss p = 1mode dominates reaching a peak about twice its steadystate value around 0.7 /is. As the p = 1 mode saturatesthe gain, the transverse gain distribution becomes non-uniform, which allows for coupling between differenttransverse modes. Since the fundamental mode hasthe lowest loss, it begins to compete with the p = 1mode for the available gain, and it eventually becomesdominant. The p = 1 modal amplitude then decreasesfrom its maximum value, a mode crossing occurs, andthe oscillator settles into the same steady state opera-tion as that shown in Fig. 3 with different initial condi-tions.

Because the bare cavity eigenmodes do not obey theusual power orthogonality relationship, but rather abiorthogonality relationship, the absolute square of

10

I,

co0

'-.4

5

10

0

S

1.0 I-

tto0.5 -U)

U

0 [ /7 2 X _. . L ... 03 -7i0~~~~~~~~~.0 0.5 1 1.5 2

Time (sec)Fig.4. Same as Fig.3 with different initial conditions. Initially A0= 0 and Aoj,A 02,A03,A04 equal the same small value as used to obtain

the results in Fig. 3.

the modal amplitudes does not give the power contentof each mode. This can easily be seen from Eq. (31),where we have

J i(r,z,t)12dr = E E AlPArP, ElP(r,z)E,p,(r,z)dr E IAIPl2'P lp' 'P

(37)

since the modes Elp(r,z) are not power orthogonal.The modal amplitudes are defined mathematically inEq. (31), which shows that the electric field inside thecavity is a coherent mixture of the bare cavity station-ary modes with complex modal amplitudes as the ex-pansion coefficients. The modal amplitudes shown inFigs. 3 and 4 are given in their absolute units of volts.The absolute values shown in these figures give onlysome indication of how much of a particular modemakes up the total electric field, not the power contentin that particular mode.

The convergence of the DCM method with respect tothe number of I = 0 transverse eigenmodes used in theexpansion of the electric field was examined numeri-cally. Again, since the transverse eigenmodes are themodes of a non-Hermitian operator, it is not guaran-teed that they form a complete set. Figure 5 showshow the steady state intracavity intensity distributionincident on the output mirror changes for each addi-tion of the next lowest loss = 0 transverse mode in thebasis set. The distribution with five lowest loss modesis quite different from the bare cavity fundamentalmode. The intensity distribution is nearly the samewhen three, four, and five I = 0 lowest loss modes areused. Table III lists the calculated steady state outputpower and output coupling as a function of the numberof modes in the basis set. The results are well con-verged when the first five 1 = 0 lowest loss modes areincluded in the basis.


15

Table Ill. Convergence of the Steady-State Outcoupled Power and Output Coupling as a Function of theNumber of I = 0 Symmetry Transverse Modes in the Basis Set in the Present DCM Methoda

One Two Three Four Fivemode modes modes modes modes Fox and Li

Output power (kW) 328 471 526 532 534 489Output coupling 0.570 0.641 0.643 0.647 0.649 0.641

a Also listed for comparison are the output power and output coupling computed from the Fox and Limethod using an FFT propagator.

V. Comparison of the DCM Results to the Fox and LiMethod

The optical resonator and gain model used in thenumerical example of the last section were found toeach a steady state in the DCM method of solution.The steady state solution from the DCM method canthen be compared to the converged wavefront ob-tained from the Fox and Li iterative method.

The calculational procedure used here to find thesteady state gain loaded field distribution in the Foxand Li approach is described in Ref. 8. In this method,the laser cavity is divided into a number of axial seg-ments. A wavefront is propagated in free space fash-ion to each segment and multiplied by the gain orphase distribution. At a gain segment, the gain withinone half-segment is lumped into a single gain sheet.Thus the wavefront is acted on discontinuously as itpropagates through the laser cavity.

A fast Fourier transform algorithm is used to per-form the free space propagation between segments.As a single wavefront is propagated once through thecavity, it is acted on by a fixed gain distribution. Aftereach complete round trip propagation, the gain is al-lowed to saturate via Eq. (35) (without the explicittime dependence on go) using the stored intensity dis-tributions at each gain segment. The optical wave-front is then again propagated through the cavity actedon by the updated gain distribution. This procedureis repeated until both the gain distribution and wave-front are unchanged after a round trip propagation.This then gives the steady state gain and field distribu-tions inside the optical cavity.

The guard band used in the FFT calculation was setto G = 2.5, which is larger than the value obtained fromthe requirement given in Ref. 8 for the cavity Fresnelnumber considered here. The number of Cartesiansample points in each transverse dimensions was takenat 28 = 256, and the gain medium was divided into sixaxial segments. A run using 128 sample points andfive gain segments gave virtually identical results.The calculation was initially started with a uniformwavefront with initial intensity equal to the saturationintensity. The initial wavefront was propagated 30times through the resonator to obtain the steady stategain loaded distribution.

Figure 6 shows the comparison between the steadystate intracavity and laser output intensity distribu-tions calculated from the Fox and Li iterative methodto that computed from the DCM method using onlythe fundamental bare cavity mode and using five low-

1.0

0.8

0

._

)

a)

0.6

0.4

0.2

0.0

1.0

0.80

4._

CDr)

0.6

0.4

0.2

0.0

0 0.5 1 1.5

r (cm)2 2.5

0 0.5 1 1.5 2 2.5

r (cm)Fig. 5. Steady state intracavity intensity distribution incident onthe output mirror as the number of 1 = 0 symmetry transverse modesin the basis set is increased: (a) (..... ) fundamental mode, (---- )two modes, ( ) five modes; (b) (..... ) three modes, (- ) four

modes, ( ) five modes.

est loss 1 = 0 modes. The DCM method using fivetransverse modes agrees well with the Fox and Li re-sults. There are some small quantitative differences;however, the intensity distributions exhibit the samequalitative structure. The DCM results actually ex-hibit more pronounced structure than the Fox and Li


1.0

0.8

0.6

0.4

4.)

a)z-

0.2

0.0

0.5

4.

c

Cl-

a)

4.)

4.)

-04.

-0

0.4

0.3

0.2

0.1

0.0

0 0.5 1 1.5

r (cm)2 2.5

0 0.5 1 1.5 2 2.5

r (cm)Fig. 6. Comparison of the steady state gain loaded (a) intracavityand (b) output intensity distributions from the DCM method usingfive transverse modes in the basis set ( ) and using only thefundamental bare cavity mode ( ..... ) to that obtained from the Foxand Li iterative approach using a 3-D FFT propagation algorithm

(…)

results. A comparison of the output phase distribu-tions is shown in Fig. 7.

In Table III, the steady state outcoupled power andoutput coupling computed from the Fox and Li meth-od is listed along with the DCM results. The outputpower computed from the DCM method is -10% high-er than that from the Fox and Li method. The outputcoupling calculated from both methods agrees to bet-ter than 2%.

VI. Discussion and Conclusions

We have shown that the dynamic coupled modesmethod can be successfully used to compute the timedependent 3-D electric field inside a loaded unstablecavity. Calculations were performed with a realistic

0.4

0.2

a

a)co

Q4

0.0

-0.2

-0.4

0 0.5 1 1.5 2 2.5

r (cm)

Fig. 7. Comparison of the steady state intracavity phase distribu-tion at the output mirror from the present DCM method using fivetransverse modes in the basis set (-) to the Fox and Li approach

(…)

unstable resonator using a simple gain saturation mod-el. The DCM method explicitly shows that spatiallyinduced gain variations due to the nonuniform timedependent saturation of the medium serves to coupledifferent stationary bare cavity transverse modes.Thus the cavity field that develops in a laser equippedwith an unsatable resonator, in which all the diffrac-tive losses are assumed to be at the output coupler, willalways consist of a combination of transverse station-ary modes due to gain saturation.

The usefulness of the DCM method for doing anypractical calculations depends, first, on the number ofbare cavity eigenmodes needed in the basis set forconvergence and, second, whether one can accuratelycompute the necessary number of higher-order trans-verse eigenmodes of the bare cavity under consider-ation. The results in this paper are for a particularunstable resonator which uses a graded reflectivityoutput mirror. Although not explicitly shown in thiswork, there is no reason to believe that the DCM meth-od cannot be used for cavities employing a convention-al hard-edged mirror provided enough eigenmodes canbe calculated to converge the electric field, since themodes of a hard-edged and graded reflectivity unsta-ble resonator have the same mathematical properties.However, to substantiate fully the generality and use-fulness of the DCM method requires examining thedifferences between hard-edged and graded mirrors,the effects of magnification, Fresnel number, and dif-ferent gain saturation models.

The advantage of using the DCM method for com-puting the cavity field is that it offers a rigorous,soundly based, and convenient starting point for pre-dicting laser performance. It gives a much more com-plete temporal and spatial description of the cavityfield than the Fox and Li method, which is really onlyuseful for computing steady state conditions. It is


capable of giving deeper physical insight into the be-havior of a laser that uses an unstable resonator. Real-istic time dependent laser kinetics for computing gainand fluid dynamics for computing refractive indexchanges can be incorporated in a straightforward, logi-cal, and consistent manner, as will be shown in futurepublications. Thus time dependent phenomena suchas the waveform of a pulsed laser, intrapulse modebeating, and mode-medium instabilities can be calcu-lated rigorously using full knowledge of the spatialdependence of the cavity field.

The authors appreciate the helpful comments fromProfessor A. E. Siegman of Stanford University.

We authors would also like to thank T. R. Lawrence,A. K. Cousins, and A. L. Pindroh for helpful and stimu-lating discussions.

This work was supported by Spectra Technology,Inc., Internal Research and Development Funds.

References1. See K. E. Oughstun, "Unstable Resonator Modes," in Progress

in Optics, Vol. 24, E. Wolf, Ed. (North-Holland, Amsterdam,1987), pp. 165-387 and references therein.

2. A. G. Fox and T. Li, "Modes in a Maser Interferometer withCurved and Tilted Mirrors," Proc. IEEE 51, 80-89 (1963).

3. D. B. Rensch, "Three-Dimensional Unstable Resonator Calcu-lations with Laser Medium," Appl. Opt. 13, 2546-2561 (1974).

4. A. E. Siegman and E. A. Sziklas, "Mode Calculations in UnstableResonators with Flowing Saturable Gain. 1: Hermite-Gauss-ian Expansion," Appl. Opt. 13, 2775-2792 (1974).

5. G. M. Ancellet, R. T. Menzies, and A. M. Brothers, "FrequencyStabilization and Transverse Mode Discrimination in Injection-

Seeded Unstable Resonator TEA CO2 Lasers," Appl. Phys. B 44,29-35 (1987).

6. D. V. Willetts and M. R. Harris, "An Investigation into theOrigin of Frequency Sweeping in a Hybrid TEA CO2 Laser," J.Phys. D 15, 51-67 (1981).

7. A. E. Siegman, "Exact Cavity Equations for Lasers with LargeOutput Coupling," Appl. Phys. Lett. 36, 412-414 (1980).

8. E. A. Sziklas and A. E. Siegman, "Mode Calculations in UnstableResonators with Flowing Saturable Gain. 2: Fast FourierTransform Method," Appl. Opt. 14, 1874-1889 (1975).

9. A. E. Siegman, "A Canonical Formulation for Analyzing Multie-lement Unstable Resonators," IEEE J. Quantum Electron. QE-12, 35-40 (1976).

10. A. E. Siegman, Lasers (University Science Books, Mill Valley,CA, 1986), Chap. 24.

11. A. E. Siegman, "Orthogonality Properties of Optical ResonatorEigenmodes," Opt. Commun. 31, 369-373 (1979).

12. E. M. Wright and W. J. Firth, "Orthogonality Properties ofGeneral Optical Resonator Eigenmodes," Opt. Commun. 40,410-412 (1982).

13. A. Parent and P. Lavigne, "Variable Reflectivity Unstable Reso-nators for Coherent Laser Radar Emitters," Appl. Opt. 28, 901-903 (1989).

14. B. T. Smith et al., Matrix Eigensystem Routines-EISPACKGuide (Springer-Verlag, New York, 1976).

15. A. E. Siegman and H. Y. Miller, "Unstable Optical ResonatorLoss Calculations Using the Prony Method," Appl. Opt. 9,2729-2736 (1970).

16. W. P. Latham, Jr., and G. C. Dente, "Matrix Methods for BareResonator Eigenvalue Analysis," Appl. Opt. 19, 1618-1621(1980).

17. W. D. Murphy and M. L. Bernabe, "Numerical Procedures forSolving Nonsymmetric Eigenvalue Problems Associated withOptical Resonators," Appl. Opt. 17, 2358-2365 (1978).

18. W. H. Southwell, "Unstable-Resonator-Mode Derivation UsingVirtual-Source Theory," J. Opt. Soc. Am. A 3,1885-1891 (1986).


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Numerical solution of the exact cavity equations of motion for an unstable optical resonator