Closed-cavity solutions with partially coherent fieldsin the space-frequency domain

Anup Bhowmik

Closed or stable optical cavities, used frequently to determine the efficiency of high performance chemicallaser nozzles, are designed primarily for maximum multimode power extraction from the medium. The verylarge (>500) Fresnel numbers associated with such cavities have in the past necessitated their analyticalmodeling by representing them as plane-parallel Fabry-Perot or rooftop cavities. In this paper, a rigorous2-D scalar diffraction formalism of the closed cavity is presented in which quasi-monochromatic partiallycoherent fields in the space-frequency domain are used to obtain quasi-steady state but stable solutionsusing a simplified gain model. Small power fluctuations in the numerical iterative solution history that dis-plays no monotonic increasing or decreasing trends are interpreted as the redistribution of energy from onedegenerate set of high-order transverse modes into another. The degree of coherence in the second-orderspatial correlation function (or the mutual coherence function) required of the input fields which permitsuch solutions is presented. Further, it is shown that the upstream/downstream coupling in this closed cavi-ty occurs as a natural consequence of the physical model itself rather than through some artificial geometri-cal means, such as that introduced in the rooftop model. The axial variation in the resulting mode widthis in excellent agreement with the Hermite-Gaussian distribution predicted for the particular geometry ofinterest. The computed closed-cavity power variation with mode width using a simplified gain model showsqualitative agreement with experimentally observed trends; quantitative agreement is poor and is ascribedto the rudimentary nature of the gain model. In the limiting case of small Fresnel numbers (NF 1) thisprocedure yields, in the bare cavity, the well-known fundamental mode of the cavity when appropriate sym-metry constraints are applied.

I. IntroductionClosed or stable optical cavities are used routinely1l

as calorimeters for the measurement of the power ex-traction efficiency in saturation conditions of chemicallaser generators that produce excited species in a su-personic transversely flowing gaseous medium. Theexperimental closed-cavity power, along with its spec-tral and spatial distributions within the cavity are,furthermore, utilized in the refinement of adjustableparameters in phenomenological gain models used tocharacterize such media.7-10 Since closed cavities areused primarily for maximum multimode power ex-traction, the Fresnel number NF of a typical experi-mental cavity must necessarily be very large (NF >> 1)because it is required to stimulate the entire gain vol-ume despite length constraints imposed on it for prac-tical reasons.1 The difficulties associated with a

The author is with Rockwell International, Rocketdyne Division,6633 Canoga Avenue, Canoga Park, California 91304.

Received 31 March 1983.0003-6935/83/213338-09$01.00/0.© 1983 Optical Society of America.

physical optics representation of such a cavity, from thepoint of view of intuitive understanding as well ascomputational resource requirements, led traditionallyto the use of its plane-parallel Fabry-Perotl2 -14 andsometimes its rooftop15 approximations,16 so that someof the chemical kinetic and fluid dynamic properties ofthe chemical laser medium might be understood. In theFabry-Perot approximation, however, the neglect ofoptical coupling between the excited upstream specieswith those in the reabsorbing downstream region re-sults, as might be expected, in a constant value of thecomputed closed-cavity power with increasing modewidth once its peak value is reached.1718 In an attemptto reproduce the trends observed experimentally,Yang' 5 proposed the use of a rooftop/flat cavity usinggeometrical optics, wherein the upstream-downstreamoptical coupling is introduced artificially by changingone of the Fabry-Perot elements into a 90° rooftop re-flector. Even though the experimental trends arequalitatively reproduced by the use of this model, theinfluence of its arbitrary and strong coupling on thepredicted closed-cavity power and spectral distribution,and more importantly the uncertainties introduced inthe adjusted gain model parameters, remain undeter-mined. In an accompanying paper,19 the magnitude of

3338 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

these effects is examined with a chemical laser gainmedium model and a rigorous scalar wave formulationof the closed cavity developed in this paper. Thisclosed-cavity formalism alleviates the shortcomings ofthe above models by (1) preserving faithfully the cavitygeometry, (2) introducing the upstream-downstreamcoupling naturally through its Fresnel diffraction for-mulation, and (3) using randomly distributed noisesources to initiate a numerical, iterative procedurewhich yields stable closed-cavity solutions in the pres-ence of a simple gain model. The random input func-tions represent a space-domain noise, quite similar toa superposition of spatially dependent spontaneousemissions that very well might be the sources fromwhich the closed-cavity mode starts and builds up toeventually fill the available gain volume in an actualcavity.

In this paper we describe this rigorous 2-D scalardiffraction formalism and show, when cavity Fresnelnumbers are very large (NF >> 1), that by launchingnoisy fields which are partially coherent in the space-frequency domain quasi-steady state but stable solu-tions of the closed cavity may be obtained. The degreeof coherence or rather the incoherence required of theinput fields which yield such solutions is discussedquantitatively by computing the complex degree ofcoherence 20 of several generating functions. It is shownthat only one of these functions results in a resonatorsolution, while the others yield a beam-train solutionor one like it. As might be expected of very largeFresnel number cavities, no unique -solution in the senseof a self-consistent field distribution that is invariantfrom one iteration to the next is achieved in the presenceof a simple analytical representation of the gain. In-stead, a quasi-steady state is reached in which thecomputed closed-cavity power displays small fluctua-tions from iteration to iteration, with no increasing ordecreasing trends in the iteration history, regardless ofthe number of iterations. This small fluctuation is in-terpreted as the redistribution of energy from one de-generate set of (bare) cavity modes into another, whichmay be used formally to synthesize the loaded cavitymode. The circulation of transverse modes of highorder within the cavity is permitted by the high cutoff,frequency 2l of the cavity by virtue of its large Fresnelnumber; this is also the reason for its inability to dis-criminate between one degenerate set of modes fromanother. That this procedure indeed yields a resonatorsolution is evidenced by the fact that the axial variationof the transverse mode width is in agreement with theHermite-Gaussian distribution predicted for this cavity.In the limiting case of small Fresnel numbers (NF 1),the same procedure in the bare cavity yields the well-known22 symmetric (antisymmetric) lowest loss reso-nator mode as its self-consistent and unique solutionthroughout the cavity when the appropriate symmetry(asymmetry) constraint is applied.

The closed-cavity description and resolution re-quirements for its numerical representation are pre-sented in Sec. II. The 2-D mutual coherence functionis discussed briefly in Sec. III. The results of propa-

i

k, 1,2 -PLANE-

FLOWDIRECTION

x:=C I =~=

J SPHERICAL: GAIN PLANE.1MIRROR ISHEET IMIRROR

\ _-4-NOZZ-65 CM-65 CM-.-j

. 1= 150 CM

X= 2.8pmXc < 2.0 CM

aja2L L ~~~NF = . ld* Xd

1 ~~5x2 ZLEI 2.8x10

4x65

= 550

(a)

As _ .4 J (b)

Fig. 1. (a) Experimental closed cavity of interest in which very largepropagation Fresnel numbers NF 550 are encountered. The mu-tual coherence of an input field is evaluated at the curved mirror planefollowing its reflection and propagation to the focal plane, where thefield now acts as an extended quasi-monochromatic partially coherentsource. (b) Lens equivalent shows the predominantly geometricnature of the upstream-downstream coupling in this cavity, whereinFresnel diffraction effects are now incorporated by the present

formalism.

gating various noisy input fields are discussed or shownin Sec. IV, along with their mutual coherence functionsand these are compared with one another. From thiscomparison the requirement of the input fields whichyield closed-cavity solutions becomes clear. Closed-cavity solutions obtained with a simplified gain modelare discussed in Sec. V, where this simple gain model isalso described. Conclusions are presented in Sec. VI.

II. Numerical Representation of the Closed CavityThe experimental closed cavity of interest is a half-

symmetric spherical mirror cavity reduced to two di-mensions [Fig. 1(a)] by assuming that the principal gainvariation occurs in the plane of the figure. A furthersimplification results if we restrict ourselves only tothose systems whose axial gain variations are so smallthat the entire gain may be applied in one lump at astation located midway between the two mirrors. Thisloaded cavity may now be represented, for numericalcomputational purposes, by four equal propagationsteps [see Fig. 1(a)] with intermediate applications ofapertures, mirror diffraction and reflection losses,mirror curvatures and gain, as appropriate, at eachstation.

The propagation of the various input fields describedin Sec. IV, from one station to the next, is accomplishedformally in frequency space by evaluating the Hu-ygens-Fresnel-Kirchhoff diffraction integral numeri-cally.23 The computationally efficient DFT algorithm24

is used to convert the complex field back and forth be-'tween the real and frequency space. To assure ade-quate numerical resolution in this process, one needconsider the resolution of only those spatial frequenciesthat are equal to or less than the cutoff frequency 2 ' ofthe propagation step, approximately the propagation

1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS 3339

I1 l; 5 i

Fresnel number of that step. If mode widths are re-stricted to 2X, < 4 cm and 10-cm wide mirrors are used,the largest Fresnel number25 encountered in any stepin this cavity is NF S 550 (for X = 2.8 Mm). In general,the field curvature due to mirror curvatures must alsobe resolved and appropriate guard bands applied toprevent interband energy spillover. Following Sziklasand Siegman 26 one can show that, if a plane wave ofwidth 2Ga illuminates an element whose radius ofcurvature is R, the minimum number of points N, re-quired to resolve the field curvature introduced by it is

,given by

N1 > 4Ga 2/XR, (1)

where X is the wavelength and G is a guard band sur-rounding the field whose width is 2a. In our application2a 10 cm, and if a guard band 10% larger than thefield is used (G = 1.10) the minimum number of pointsN 1 required to resolve the curvature R = 300 cm is

N > 1310. (2)

To assure the resolution of the curvature in a propaga-tion step the minimum number of points needed is thenNG + N1 , and if 4 points/cyle are considered sufficientto represent the highest spatial frequency of interest,the required number of points is given by

N = 4(NF + N) 7440. (3)

The number of points actually used in the computationsbased on the radix-2 DFT algorithm was 8192, and a10% guard band was found empirically to suffice withno perceptible energy spillover from adjacent bands.

Now, if a plane wave is used as an input function tostart an iterative resonator computation, a beam-trainsolution results, with periodic geometric focusing27 ofthe field as depicted by the ray traces in the lensequivalent of the cavity [Fig. 1(b)] regardless of thenumber of iterations. This result is not surprising sincethe perfect spatial coherence of the plane wave inputfield is preserved by the repeated application of thelinear transfer functions representing the optical cavity,because no spatially uncorrelated transformations arecontained in them. Furthermore, a unit Fresnel regionnear the optic axis does not expand geometrically toultimately fill the stable cavity as it does in unstableresonators,28 since the effective closed-cavity magnifi-cation is unity. Hence, the field is relayed undispersedby the cavity from one iteration to the next with periodicdiffraction-limited imaging, and the cavity functionsmerely as an afocal relay system. Since it is well knownfrom experiments that radiation ultimately fills theclosed cavity, our problem reduces to one of findinginput fields that are dispersive enough to fill the cavity.As most lasers are believed to start from noisy, spon-taneous emissions that are spatially uncorrelated, wesearch for input fields that are also noisy but in thespace-frequency domain in which the cavity is beingrepresented.

Ill. Mutual Coherence FunctionThe noise content or the degree of coherence of spa-

tially noisy fields is quantitatively expressed by themutual coherence function 91,2 defined by29

1.,2 = Y(P 1,P2) = 1 Xmi

[I(P1)I(P2)I1 2

X dx I(X) exp[2-ri(p - P2)/P] (4)

P1P2

where X is the wavelength of the quasi-monochromaticradiation illuminating an extended source, x is thecoordinate of a point P located on this source, and Xminand xmax are its lower and upper bounds, respectively;the intensity due to an elemental length dx at P is I(x),and I(P1),I(P2 ) are the intensities due to this source atpoints Pi and P 2 located at distances P1 and P2, re-spectively, on a distant screen. This function was nu-merically evaluated at the 1,2 plane [Fig. 1(a)] forvarious input functions (see Sec. IV) by first propa-gating the field to the focal plane of the curved mirrorfollowing a single reflection from it.

IV. Noisy Input Fields in the Space-FrequencyDomain

A. Periodic Random NoiseThe random sum of sinusoidal spatial frequencies

given byNF

OX.) = E Am sin[mX. + kR(m)]m=1

(5)

appears to be a natural choice for the input field, be-cause it contains all the spatial frequencies of interestand they are added in some random fashion. Here xnis the spatial coordinate, OR (m) is a computer-generatedquasi-random number associated with the randomphase of mth spatial harmonic, and Am is its amplitude.The result of propagating this input field to the focalplane of the curved mirror [Fig. 2(a)] shows that thefocusing characteristic of the system is strongly evident.Its coherence function [Fig. 2(b)] shows that a highdegree of correlation (compared with the one discussedbelow) is present in this function. An attempt to obtaina resonator solution with it resulted in partial periodicfocusing of the field, so that this choice of the input fieldwas discarded since it was inappropriate for obtainingclosed-cavity solutions.

B. Aperiodic Random NoiseConsider the input field /(x) given by

(x) = Ki4o(x) exp[i0(x)K 21, (6)

where lo(x) and i(x) are, respectively, its randomlydistributed amplitude and phase, and the constantsK1,K2 represent the level of its amplitude or phasemodulation, respectively.

1. Amplitude ModulationIf K 2 = 0, a pure amplitude modulated field results,

and its use in resonator computations also showedstrong periodic focusing with little of the spatial dis-

3340 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

'-zwUI-

C-

-1.o -0.6 -0.2 0.2X.CM

0.6 1.0

0 I ' I I" ' . I * I * I. I .

-10 -6 -2 2 6 10 X 1-2x'CM

Fig. 2. Result of propagating a sum of spatially periodic functions,coherently summed with random relative phases, shows (a) that it actsas a high spatial frequency phase grating, and (b) its high degree of

correlation.

Now, as the level of phase modulation of the inputfield is increased by increasing the value of K, an im-proved degree of dispersion is obtained in the propa-gated field.30 The intensity distributions and themutual coherence functions are shown in Figs. 4, 5, and6 for K = 0.5, 1.0, and 2.0, respectively. Increased dis-persion with increasing values of K is evident from boththe appearance of off-axis energy and the reduction inpeak intensity at the Gaussian image point. A corre-sponding reduction in the degree of coherence is alsoseen. In all but the case of K = 2.0, the imaging char-acteristic of the field is clearly evident. However, whenK = 2.0, that is when the phase is allowed to vary ran-domly over the entire period 27r of the circular gener-ating function of the input field, the imaging charac-teristic disappears, and a totally dispersed field is ob-tained at the image plane following a single propagationstep. Its degree of correlation at this plane is such thatI 1,21 0.2, for all pairs of points whose spacing Ix'I 2R X/2d (R = mirror radius of curvature and d = inputaperture diameter).

Using this as the input field an iterative resonatorcalculation no longer displays a periodic focusing of the

-z

I-zUpersion that might eventually lead to a resonator solu-tion. This too was discarded.

2. Phase ModulationBy setting constants K1 = 1 and K2 = K and 4'o(x) =

Q(O), a complex constant in Eq. (6), a pure phasemodulated field with a constant amplitude results, givenby

+(x) = 4(O) exp[itk(x)K]. (7)

In discrete notation useful for computational purposesEq. (7) may be rewritten as

(xn)= (O) expliir[R(x,))-1/2]K}, ( < R(x) < 1), (8)

where R(xn) is a computer-generated quasi-randomnumber at the xnth spatial coordinate. A plane waveis obtained by setting K = 0 in Eq. (8), and the result ofpropagating it through the converging element (with a1-cm diam aperture) to its focal plane yields its dif-fraction-limited image [Fig. 3(a)]. The mutual coher-ence function 1, 121 of this image acting as an extendedsource is shown as the correlation of off-axis points witha point on-axis in Fig. 3(b). The function 1M1,21 iscomputed over a strip 2 mm wide, approximately equalto the size of a Fresnel zone. Figure 3(b) shows that thelevel of spatial correlation is very high, and as describedin Sec. II, an iterative calculation results in a beam-trainsolution of the cavity.

1.0

0.81

0.6

0.4

0.2

-10

-0.3 -0.2 -0.1 0X, cM

-6 - -2 2X-CM

0.1 0.2 0.3

6 1o X1 o-,

Fig. 3. (a) Diffraction-limited image is obtained when a plane wave[K = 0 in Eq. (8)] is propagated to the focal plane of the curved mirror,and (b) its high degree of correlation, which leads to a beam-trainsolution with periodic imaging, when a resonator computation is at-

tempted with it.

1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS 3341

(a),

10 L

5

0 AC

60 (a)

45

30-

15-

0

(b)

R A/2dI I I A

u _ q

4

3

I-

z 2

z

1'

(a).0.

0.

0.

0.

0 : I-1.0 -0.6 -02 02 0.6 1.a

X, cM

1.0

0.8

,I'

4-

0.61-

OA

0.2 -

-10 -6 -2 2 6 10 102X',CM

Fig. 4. (a) K = 0.5 in Eq. (8) shows the appearance of small amountsof off-axis energy following a single propagation step; (b) its degree

of correlation is very high, 11,21 2 0.75.

-

z

-

4-

-1.0 -0.6 -0.2 0.2X,CM

0.6 1.0

10 X 10-2

X'.CM

Fig. 5. (a) With K = 1.0 in Eq. (8) the propagated field shows in-creased amounts of off-axis energy accompanied by a reduction inpeak intensity. Imaging characteristic is still strongly evident. (b)Correlation of off-axis points with a point on-axis is reduced but still

high, ul,21 0.30.

I-

zLU2

Ot

4-

-1.0 -0.6 -0.2 0.2 0.6 1.0XcM

Fig. 6. (a) When K = 2.0 in Eq. (8) the imaging characteristic com-pletely disappears; (b) its mutual coherence is now ["1,21 < 0.2 forpoints lying outside a highly coherent region of width R/2d.

3342 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

(a);

16

12

8

4

a IL A - AA

-(b)

4 (a)

3

2

0

X '.CM

no_ X x | . I_ | t w t |. . . . . .

;

1

0.025

0.020

0.015

_ 0.010

L. 0.005z4 Oro

-0.005

-0.010

-0.015

-0.020 I

Xc =1.25cm

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5X,CM

Fig. 7. Parabolic approximation of the small signal gain profile usedin the resonator computations presented in this paper.

generator relative to the optical axis, and x0 is the zerocrossing distance measured from the gain generator.The loaded gain in the presence of an average two-wayirradiance I(x) is defined by

g(xn) = go(x.)I[1 + I(x)/IJ],

where

I(x.) = [Ifnc(x.) + Ifnc(x.] exp[g(x)Lg] - 1/g(x.).

(10)

(11)

In Eq. (11), IP+)(xj) and IP-)(x) are the irradiances inthe forward and return passes, respectively, g(xn) is, inthis iterative procedure, the value of the loaded gainobtained by a similar procedure in the previous iterationstep, and Lg is the length of the gain region. Thecomplex field amplitude Ct ran following each passagethrough the gain sheet is given by

0(440(x.) = i,(')(xj) exp[g(xn)Lg/2], (12)

4 where (+) or (-) denotes, respectively, the forward orreturn direction of propagation. If Pi and P2 are the

0 3 reflection coefficients of the mirrors, the closed-cavity

, 2 power Pc is given in the quasi-steady-state condition< ~~~~~~~~~~by

1 = (1 - l I l) (Xn) 12 + ( -. P2) E | +) (xn)I2 (13)M ~~~~~~~~~~~~~~~n n

LU0 (Mirror scattering losses at the wavelength of interest

< -1 are negligibly small compared to p1 and P2)oLU ~~~~~~~~~~~~~Figure 7 shows the small signal gain profile used in

-2 this paper with x0 = 3c m, gmax = 0.0221 cm-' [Eq. (9)],and I, = 983 W/CM2 [Eq. (10)]. The phase modulated

-3 field in Eq. (7), with K = 2.0, was used to start numerical-1.0 -0.6 -0.2 0.2 0.6 1.0 iterative solutions for the closed cavity in the presence

X, CM of this gain. A typical computer-generated noisy inputFig. 8. Typical input field random phase distribution. field (in the space domain) used to start the computa-

tions is shown in Fig. 8. Note that the random varia-field, and the entire cavity is filled in a single propaga- tions in phase 0 are such that -7r < 0 < r.tion. The resulting quasi-monochromatic field is par- An iteration history in a typical computation withtially coherent in the space-frequency domain since by mirror reflection coefficients Pi = 0.992 and P2 = 0.998definition 1 complete spatial incoherence and complete is shown in Fig. 9. In this particular case it is seen thatspatial coherence (to second order) occur only when a relatively large number of iterations is required to1u1,2 1 = 0 and 1, respectively, for all points Pi and P2 in achieve steady state. This is because, except for thethe field. first few iterations, the cavity diffraction losses at the

V. Simple Gain Model and Closed-Cavity Solutions

A. Gain Model and Closed-Cavity PowerIn a subsequent paper19 the effects of incorporating

into this closed-cavity formalism a multilevel, chemicallaser medium in rotational nonequilibrium are inves-tigated. In this paper, a simple, analytic form of thegain is used to simulate qualitatively the aggregatefeatures of a multitude of chemical laser transitions bya single, weighted average gain profile so that theunderlying principles of the cavity formalism may beverified. This simple, parabolic profile of the smallsignal gain is given in discrete notation at the xnthcoordinate by

go(xn) = 4gmax(xn + Xc-Xo/2)0/X + gmax, X > -Xc, (9)to, X i -Xe,

where gmax is the peak gain, X, is the location of the galn

L I 1.251I - 70 80 90 100

1.26 QUASISTEADY-

1.0 -STATE

0.01 1 1 , , , , I I0 10 20 30 40 50 60 70 80 90 100

ITERATION NO.

Fig. 9. With mirror reflection coefficients near unity (Pi = 0.992, P2

- 0.998) the iteration history shows slow convergence to quasi-steady

state but stable solution.

1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS 3343

, , , I I , I~~~~~~~~~~~~~~~~~~~~~~~~~~

8000

z

z-U 4000

Lu

cc 2000

0-6.0 -4.0 -2.0 0.0 .0 4.0 6.0

X, CM

Fig. 10. Irradiance distribution at curved mirror (Xc = 1.5 cm) inquasi-steady-state conditions.

(PLANE MIRROR)1600

Z 1200

z

Ui

800

400-

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0X, CM

Fig. 11. Quasi-steady-state irradiance distribution at plane mirror(X = 1.5 cm) shows waist formation (compare with Fig. 10).

mirrors by virtue of the large cavity Fresnel number aresmall and the round trip cavity losses are due primarilyto the reflection losses at the mirrors and the diffractionloss at the intracavity aperture representing gain gen-erator hardware. The fractional loss per round trip isgiven approximately by (1 - P1) (1 - P2), which in thiscase equals 1.6 X 10-5 ( = 0.992, P2 = 0.998). Thesharp decline in power in the first few iterations is dueto the rapid filtration of spatial frequencies exceedingthe cavity cutoff frequency. At higher reflection lossessteady state was achieved in a fewer number of itera-tions. For example, with P = P2 = 0.832, fifty iterationswere sufficient to reach steady state.

Closer examination of the last thirty iterations underincreased resolution of the ordinate scale (Fig. 9, insert)reveals that small power fluctuations, whose amplitudeis < 0.5%, persist even when the number of iterationsis as large as 100. These fluctuations are interpretedas due to the redistribution of energy from one degen-erate set of spatial frequency components into another.Circulation of various combinations of spatial frequencycomponents, which are amplified to different degreesby the spatially varying gain, is permitted by the cavitybecause of its very large Fresnel number and conse-quently its inability to discriminate between one setfrom another.

The irradiance profiles (at any station in the cavity)also show fluctuations in the details of their spatialfrequency content but their transverse dimensionsthroughout the cavity remain unchanged from one it-eration to the next. Consequently, as seen from theiteration history in Fig. 9, the mode volume remainsrelatively constant (within ±0.5%).

Hence, these noisy input fields yield unambiguoussolutions (except for small fluctuations) for the gross ormacroscopic properties of the system, such as theclosed-cavity power or the intracavity diffraction lossesin a statistical sense, and the iterative solutions arelargely independent of the initial conditions such as themicroscopic details of the spatial distribution or initialamplitudes of the input field. The solutions arenonunique, however, to the extent that no self-repro-ducing field distribution throughout the cavity is ob-tained that is invariant from iteration to iteration.Nevertheless, for a given set of cavity parameters, aquasi-steady state is achieved in which no monotonicincreasing or decreasing trends are observed, and thisstable state is regarded as a solution of the closed cavity.That indeed a cavity solution is obtained by this pro-cedure is testified, as shown below, by the fact that inany iteration the axial variation in the transverse modedimension is in agreement with that predicted for thehalf-symmetric cavity by Gaussian beam analysis.

The relative irradiance distributions at the curvedand the flat mirror are displayed in Figs. 10 and 11, re-spectively, in quasi-steady-state conditions for X = 1.5cm, while Fig. 12 shows the radiance distribution at thegain station. The very high spatial frequency contentin the solution is evident in all the figures.

Now, using Gaussian beam analysis it is easy to showthat in the given cavity geometry [Fig. 1(a)] the funda-mental mode waist, which occurs outside the cavity, is0.1156 cm in radius ( = 2.8 lam) so that the ratio of the

4000

>.3000-

z

w2000i

100,01

0.0: -2.& -1.0 0.0 1.0 -2.0

X, CM

Fig. 12. Quasi-steady-state two-way average intensity distribution1 used in Eq. (10) for computing the loaded gain (X = 1.5 cm).

3344 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

spot radius at the curved mirror to that at the planemirror is 1.402. This ratio applies to modes of all ordersin the bare cavity. If the quasi-steady states found hereare to be ligitimate cavity solutions, they too must sat-isfy this condition for they may be constructed from anappropriate linear combination of the bare cavity modesin any given iteration. From Figs. 10 and 11, this ratiois found, for the quasi-steady-state solution, to be2.00/1.43 = 1.40, which is in excellent agreement withthe ratio for the fundamental mode for this cavity. Thisformation of a waist with correct spot sizes at the twomirrors is the strongest evidence in favor of thesequasi-steady states as being the legitimate solutions ofthe cavity, and this characteristic of the closed-cavitymode has been observed experimentally. 32

B. Closed-Cavity Power vs X,The computed closed-cavity power variation with Xc

is shown for two sets of mirror reflection coefficients (Pi= 0.992, P2 = 0.998, and P1 = P2 = 0.832) in Fig. 13(a).The decline in power for Xc > 1.4 cm shows the pres-ence of the upstream-downstream coupling in thiscavity, which now occurs as a natural consequence of thecavity formalism itself, whose basis is the fundamentalprinciple of wave propagation in the cavity, that is, theHuygens-Fresnel-Kirchoff diffraction principle.Comparison of the predicted closed-cavity power vari-ation with Xc and experimental results in Fig. 13(b)shows poor quantitative agreement. This is to be ex-pected since an oversimplified single transition gainmodel is used to describe the multiple transitions of aflowing gain medium, and the cursory selection of itsparameters does not optimally reproduce the aggregatesmall signal gain profile or the saturation characteristicsof the medium. The simple model is used primarily tovalidate the cavity optics formalism and for purposesof illustration. Results of a more detailed gain modelare compared to experiment in a subsequent paper.

VI. ConclusionsIn a 2-D scalar diffraction formalism of closed cavities

it is shown that, by using random spatially distributednoiselike sources as input functions, their quasi-steadystate but stable numerical solutions may be obtainedin the presence of a spatially distributed gain. Thesenoiselike sources are partially coherent in the space-frequency domain and simulate the spatially distributedspontaneous emissions in a real cavity from which thecavity mode evolves to ultimately fill the gain volume.In a particular cavity geometry of interest, it is shownthat a resonator solution is obtained only when thesecond-order correlation function (or the mutual co-herence function) of the input fields (outside a highlycoherent region of calculable width) is no greater than0.2; otherwise, a beam-train solution with periodic(partial) focusing of the input fields is obtained.

In this very large Fresnel number cavity (NF > 250)containing a single internal knife-edge that representsgain generator hardware, it is found that no self-con-sistent field distribution that is invariant from one it-eration to the next is obtained in the presence of a spa-

tially distributed gain. Instead, a quasi-steady stateresults in which the numerical iteration history displayssmall power fluctuations from iteration to iteration,with no overall montonically increasing or decreasingtrends. These fluctuations persist regardless of thenumber of iterations and are interpreted as the redis-tribution of energy from one set of (bare cavity)transverse modes into another. Circulation of thesehigher-order (bare cavity) transverse modes (whoseappropriate linear combination constitutes formally theloaded cavity mode) is permitted by the cavity by virtueof its very large Fresnel number. This quasi-steadystate is regarded as a solution of the closed cavity which,except for small fluctuations, yields a closed-cavitypower that is largely independent of the initial condi-tions.

The variation in the predicted closed-cavity powerwith mode width displays the effect of the upstream-downstream coupling which occurs naturally in thiscavity as a result of the formalism itself in which Fresneldiffraction effects are now included. This variation

CALCULATED

Ye a

ccwUj

A0

8.0

7.01-

6.0

cc

0a.

5.01

4.0

1..u U1.0

XC, CM

EXPERIMENT

1.2 1.4 1.6 1.8 2.0 2.2Xc, CM

Fig. 13. (a) Predicted closed-cavity power variation with X, for twosets of mirror reflectivities; (b) experimental data.

1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS 3345

(b)

0.992/0.998

0.832/0.832

I I I I I

lS

3.01

2.01

shows qualitative agreement with experiments; quan-titative agreement is poor however, and this is attrib-uted to the rudimentary nature of the gain model usedin this paper. The effects of incorporating a more de-tailed gain model are examined in a subsequent paper.In the limit of small Fresnel numbers (NF 1) thisformalism yields the well-known solutions of the stableresonator when appropriate symmetry conditions areapplied to the input function.

Hence, despite the nonunique nature of the solutions,this formalism provides quasi-steady state but stablesolutions for the closed cavity from which useful esti-mates of power and intracavity diffraction losses at anymode width may be obtained.References1. D. J. Spencer, D. A. Durran, and H. A. Bixler, Appl. Phys. Lett.

20, 164 (1972).2. H. Mirels and D. J. Spencer, IEEE J. Quantum Electron. QE-7,

501 (1971).3. D. J. Spencer, H. Mirels, and T. A. Jacobs, Appl. Phys. Lett. 16,

384 (1970).4. L. Forman, AFWL-TR-77-131, Rocketdyne Report RI/RD77-171

(Dec. 1977).5. D. L. Hook et al., AFWL-TR-76-295, TRW Report 27351-

6002-RU-00 (Apr. 1977).6. G. W. Tregay et al., Bell Aerospace Report 9276-928001 (Jan.

1978).7. J. E. Broadwell, Appl. Opt. 13, 962 (1974).8. S. W. Zelazny, R. J. Driscoll, J. W. Raymonda, J. A. Blauer, and

W. C. Solomon, AIAA J. 16, 297 (1978).9. T. T. Yang and R. E. Swanson, AIAA Paper 79-1490, Williams-

burg, Va. (1979).10. J. Theones and A. W. Ratcliff, AIAA Paper 73-644, Palm Springs,

Calif. (1973).11. R. W. F. Gross and J. F. Bott, Eds., Handbook of Chemical Lasers

(Wiley, New York, 1976), p. 110.

12. A. W. Ratcliff and J. Theones, AIAA Paper 74-225, Washington,D.C. (1974).

13. R. J. Hall, IEEE J. Quantum Electon. QE-12, 453 (1976).14. W. L. Rushmore and S. W. Zelazny, AIAA Paper IV-5, Cambridge,

Mass. (1978).15. T. T. Yang, J. Phys. C9, 51 (1980).16. On one occasion a negative branch unstable resonator was used.

See, for example, D. O'Keefe, T. Sugimura, W. Behrens, D. Bul-lock, and D. Dee, Opt. Eng. 18, 363 (1979).

17. R. R. Mikatarian, AIAA Paper 74-547, Palo Alto, Calif. (1974).18. R. Tripodi, L. J. Coulter, B. R. Bronfin, and L. S. Cohen, AIAA

J. 13, 776 (1975).19. A. Bhowmik, T. T. Yang, J. J. Vieceli, and W. D. Chadwick, Appl.

Opt. 22, 3347 (1983), same issue.20. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,

1975), p. 509.21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,

New York, 1968), Secs. 10.4.2 and 10.4.3.22. A. G. Fox and T. Li, Bell Syst. Tech. J. 40, 453 (1961).23. Ref. 21, p. 119.24. E. 0. Brigham, Fast Fourier Transform (Prentice-Hill, Engle-

wood Cliffs, N.J., 1964).25. The propagation Fresnel number NF is defined as NF =

G2 ala 2/\z, where Gal and Ga 2 are the radial extent of the fieldat the input and output stations, respectively, X is the wavelength,and z is the spacing between the two stations.

26. E. A. Sziklas and A. E. Siegman, Appl. Opt. 14, 1874 (1975).27. The periodicity can easily be found by computing the reentrant

condition for the cavity; see, for example, D. R. Herriott, H. Ko-gelnik, ad R. Kompfner, Appl. Opt. 3, 523 (1964).

28. Yu. A. Ananev, Sov. J. Quantum Electron. 5, 615 (1975).29. Ref. 20, p. 509.30. Note that, when K 5d 0, the input function (x) in Eq. (8) is nei-

ther symmetric nor antisymmetric about the optic axis as a resultof the random spatial distribution of the phase function R(xn).

31. L. Mandel and E. Wolf, Opt. Commun. 36, 247 (1981).32. J. K. Cawthra, "Chemical Laser Nozzle Technology," Final Re-

port, (Apr. 1983), in preparation.

Patents continued from page 3310

4,391,525 5 July 1983 (Cl. 356-346)Interferometer.R. A. WOODRUFF. Assigned to Ball Corp. Filed 10 Oct. 1979.

Cat's eye retroreflectors in Michelson interferometer configurations havebeen attributed long ago to Terrien. It is not clear that this interferometer'sraison detre should be based on "the prior art optical configurations are dis-advantageous in that they require the use of a very large and thick beamsplitterand a 45° angle of incidence of the incident beam with respect to the beam-splitter in order to achieve proper splitting of the optical beams." Doublepassage of light is a feature of this interferometer, but no point is made of thisfor the IR spectrum analysis intended. C.F.M.

/,o-'G con 1/.q0a 46es EYE60 ,S'6 TcQzecro 5

4,392,709 12 July 1983 (Cl. 350-3.83)Method of manufacturing holographic elements for fiber andintegrated optic systems.J. L. HORNER and J. E. LUDMAN. Assigned to U.S.A. as repre-sented by Secretary of the Air Force. Filed 29 Oct. 1980.

A method for manufacturing holographic elements for fiber and integratedoptical systems is described. Although this patent is an improvement over theprior arts, the proposed method is the same as the method commonly used inconstructing holograms. The recording material as described is dichromatedgelatin which has a spectral sensitivity in the blue and green region of the visiblespectrum. Problems of the holographic optical elements relating to the changein laser wavelength in the fiber optical system with the use of the diode laserwere not mentioned. W-H. L.

4,392,724 12 July 1983 (Cl. 350-163)Lens system capable of short distance photography.Y. HAMANISHI. Assigned to Nippon Kogaku K.K. Filed 19 Jan.1981. In Japan 31 Jan. 1980.

Six embodiments are given of lenses containing from seven to thirteen ele-ments, with a focal length equal to 105 mm at f/2.8 (except for one in which f

210 mm at f4.5). The designs all contain two positive components and a-r .t 2 negative rear component, and when focusing on a near object the first andsecond components are moved forward while at the same time increasing their

separation. Excellent aberration correction is claimed at any magnificationfrom zero to 11. R.K.

3346 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

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