Parametric generation processes: spectral bandwidthand acceptance angles

Norman P. Barnes and Vincent J. Corcoran

A systematic evaluation of the spectral bandwidth and two acceptance angles of parametric generation pro-cesses is presented. The spectral bandwidth and acceptance angles are determined by expanding the wavevector mismatch in a Taylor series and retaining terms through second order. This allows a determinationof these parameters even when the first order term vanishes. Conditions where the first order term van-ishes are presented and compared with similar cases where the first order term does not vanish.

1. Introduction

The efficiency of parametric generation processesis known to depend on the wavelengths involved inthe interaction and their direction of propagation.This was demonstrated in the first optical parametricgeneration experiments.1 2 It has been noted, how-ever, that under certain conditions the interactioncan be made relatively insensitive to changes inwavelength3 and changes in direction of propagationin the optic plane.3-7 Under these conditions a sig-nificantly larger variation can be tolerated before theefficiency decreases appreciably.

Here, a systematic evaluation of the allowable vari-ation of the wavelength and direction of propagationis presented. This evaluation, unlike most evalua-tions, retains terms through the second order. Thisallows an evaluation of the allowable variation to beperformed even under conditions where the firstorder term vanishes or where the interaction is rela-tively insensitive to changes in the wavelength anddirection of propagation. Since two angles are re-quired to specify the direction of propagation withrespect to the crystallographic axes, the allowablevariation is evaluated for two angular parameters.In most cases the efficiency is relatively insensitive tovariations of one of the parameters. This may ex-plain the general neglect of a two parameter analysisof the angular variation of the efficiency. Conditionsunder which the interaction is made insensitive tovariations in wavelength and direction of propagationare presented. An evaluation of the allowable varia-

N. P. Barnes is with Los Alamos Scientific Laboratory, Los Ala-mos, New Mexico 87545; V. J. Corcoran is with the Institute forDefense Analyses, Arlington, Virginia 22202.

Received 23 June 1975.

tion of the wavelength and direction of propagationunder these conditions has been made. These arecompared to evaluations of the allowable variation incases where the insensitivity condition does not pre-vail.

II. Parametric Generation

Parametric generation is a nonlinear optical effectin which two optical beams, the ir and the pump, aremixed in a nonlinear crystal to produce either a sumor a difference frequency. Second harmonic genera-tion is a special case of sum frequency generation.Optical parametric oscillation is difference frequencygeneration with feedback providing the ir radiation.This mixing process is efficient only if the phasematching conditions are approximately satisfied, thatis,

W3= WI ± 2k 3 = ± k2,

where wi and k are the ith radian frequency andwave vector, respectively. The subscript 1 denotesthe pump, 2 denotes the ir, and 3 denotes the signal.The plus or minus signs are associated with sum anddifference frequency parametric generation, respec-tively. When these relations are satisfied exactly,the condition is described as ideal phase matching.If this phase matching relation is not satisfied exact-ly, the interaction may still occur. However, the effi-ciency of the interaction will be reduced.

The efficiency of the parametric generation pro-cess is dependent on the wave vector mismatch Akwhere

Ak = k k2 -k 3 l

= k COs4k3 ± k2 CoS(p2 - 03) - k 3 .

The wave vector diagram is illustrated in Fig. 1. Theangles 2 and ,63 are the angles that the wave vectors

696 APPLIED OPTICS / Vol. 15, No. 3 / March 1976

(I)

k, 13 k3

0

Fig. 1. General wave vector diagram.

k2 and k3 make with the wave vector k1, respectively.As conditions vary from ideal phase matching, eitherbecause the wavelength or the direction of propaga-tion of the ir radiation varies from the ideal phasematching case, the efficiency will decrease. The con-version efficiency, defined as the ratio of power den-sity in the signal to power density in the ir, is ap-proximately 8 ' 9

= S3/S2 = % sin 2 (Akl/2)/(Ak1/2) 2 ,

where S3 is the signal power density, S2 is the irpower density, i70 is the conversion efficiency withzero mismatch, and is the length of the nonlinearcrystal. One criterion for the maximum allowablewave vector mismatch is given by

Ak = ± 7/l.

In this case, the conversion efficiency drops to ap-proximately 0.4, that of the peak conversion efficien-cy. This criterion for the maximum allowable mis-match will be used throughout the following discus-sion.

To determine the parametric dependence of thewave vector mismatch, Ak will be expanded in a Tay-lor series through second order. The resulting qua-dratic equation

Ak=A0 + Ak AX+ 2,AkAX Ak = o +ax +2 x2AX

can be solved for AX, where AX is maximum allow-able variation of the parameter of interest. This pro-cedure will be performed for the wavelength and twoangular parameters.

Ill. Spectral Bandwidth

The spectral bandwidth is defined as the wave-length interval over which the magnitude of the wave

vector mismatch for the parametric generation pro-cess is not greater than -x/l. The spectral bandwidthover which parametric generation can occur in a 1.0-cm nonlinear crystal is usually only a few Angstromswide. There are certain conditions, however, forwhich the Taylor series expansion as a function ofwavelength has a point of inflection. At that angleand center wavelength, the spectral bandwidth willincrease significantly and can approach a spectralbandwidth 1.0 gm wide in a 1.0-cm crystal.

The spectral bandwidth can be determined by ex-panding the wave vector mismatch in a Taylor seriesas a function of wavelength and solving for AX2 . Theterms in the Taylor series are

ak= 2Ir a

2 x~ 2 227 2r[ an' XOS(2 F 3) -TT n2 COS(l2COS( r ±3)

a2 Ak a 2

n2 lk) -~ X 3

2____ 2 x 2F3 aO82 nax2 L L1\ 2 23 2

± XX3 82 X 2COS(k2 1F0)

±2n 2 COS(02 :: 3 1 F 2n3

AX2 = 2 - 20

where X20 is the wavelength about which the expan-sion occurs. The plus and minus signs are associatedwith sum and difference frequency processes. Thefull spectral bandwidth is twice AX2. The parameterAX2 can be determined by evaluating the partial de-rivatives and using the quadratic formula. The par-tial derivatives obviously depend on the wavelengthabout which the expansion occurs. By evaluatingthe partial derivatives for various wavelengths andrepeatedly applying the quadratic formula, a plot ofAX2 vs X2 can be generated.

In most cases, the first order term in the expansionis large with respect to the second, and consequentlythe spectral bandwidth is small. If the first orderterm dominates, the spectral bandwidth depends in-versely on the length of the nonlinear crystal. Spec-tral bandwidths have been evaluated using only thefirst order expansion in the literature.3-5 1011 Aslong as the first order term is large, this is a valid ap-proximation. However, for certain wavelength com-binations the first order term may become small. Ifthe first order term vanishes, that point is called a Xinflection. In this case, the spectral bandwidth de-pends inversely on the square root of the length ofthe nonlinear crystal. This usually leads to greatlyenhanced. spectral bandwidths. The enhancementmay be several orders of magnitude.

The process of determining the spectral bandwidthas a function of ir wavelength for several crystals il-lustrates the large increase possible when the firstorder term vanishes. Figure 2 plots the spectralbandwidth vs wavelength for two crystals, one exhib-iting an example of a X inflection. In both cases, sumfrequency generation using a 1.064-gm pump was as-

March 1976 / Vol. 15, No. 3 / APPLIED OPTICS 697

II

z

I-

C-)a-ci,

o-3L2.0 3.0 4.0 E

INFRARED WAVELENGTH (m)Fig. 2. Spectral bandwidth vs ir wavelength.

50

sumed in a 1.0-cm crystal. The double local maxi-mum for the spectral bandwidth is characteristic ofthe X inflection. The significant increase in the spec-tral bandwidth associated with the vanishing of thefirst order partial derivative may be noted. The ex-istence of the X inflection has been noted in the liter-ature.3 However, the evaluation of the spectralbandwidth has not received much detailed attention.

IV. Acceptance Angle

The acceptance angle is defined as the planar angleover which the magnitude of the wave vector mis-match for the parametric generation process is nogreater than 7r/. Since two angles are needed to spe-cify the direction of propagation with respect to thecrystallographic axes, two acceptance angles will bedefined. For sake of simplicity, the wave vector mis-match will be expanded as a function of one angle ata time.

Angle tuned parametric generation is usually char-acterized by a small acceptance angle A/ 2 in the opticplane. When the ir and pump wave vectors propa-gate collinearly at an angle to the optic axis, thephase matching condition is critically dependent onthe angle of propagation in the optic plane. Thusthe angular range of ir wave vectors that can undergoefficient parametric generation, that is, the accep-tance angle, is small. Typical acceptance angles forcollinear phase matching are on the order of 1.0mrad. Collinear phase matching is illustrated in Fig.3.

There are two conditions under which the firstorder term vanishes, and thus relatively large accep-tance angles are attainable. The first is the case inwhich the optical beams are propagated noncollinear-

ly in directions such that the tangential phase match-ing condition holds. Tangential phase matching isdefined by the vanishing of the first order term and isillustrated in Fig. 3 for the noncollinear case. Thesecond is the case where collinear phase matching oc-curs in the direction of propagation perpendicular tothe optic axis. Both cases tend to produce large ac-ceptance angles because the first order term in theseries expansion vanishes. The phase mismatch thatoccurs, therefore, as the direction of propagation ofthe ir radiation varies is much less than in the non-tangential phase match situation. For tangentialphase matching, acceptance angles on the order of10.0 mrad are common for 1.0-cm crystal lengths.

k3 +\ki

k3 -Ak

COLLINEAR PHASE MATCHING

TANGENTIAL PHASE MATCHINGFig. 3. Collinear and tangential phase matching.

698 APPLIED OPTICS / Vol. 15, No. 3 / March 1976

IM 0

-2 TANGENTIAL,, 105 - \ L, Nb 0

Z AX Li Nb 0

W

COLLINEAR Aif L 103 _

COLLINEAR L Nb 03

10-32.0 3.0 4.0 50

INFRARED WAVELENGTH (,um)

Fig. 4. Acceptance angles vs ir wavelength.

The acceptance angle in the optic plane can also bedetermined by means of the Taylor series expansion.In this plane, defined by the pump wave vector andthe crystallographic optic axis, the terms in the Tay-lor series are

O$k = ±7 0 + 82-A2cos(4)2 F y))

:F n2sin(42 F O ~3 Op2 XA3 ..j'

O2Ak ____

02= 21r ao2 cos(02 F 3)

___- sin(?k2 T p3 )(1 T ~j')- 2 A -sin(4) 2 F 43)

A2 '(02 03)( a 0 0 ) 3 0p 2 ) X3

- On3 o 2 1 1O03Ol2 X3 '

a0 b3 cos2(4$2 = 22 X3n2 /

Azp2 = 2 - 20 -

] -1/2- Sin2 y

The second acceptance angle AX2 is measured inthe plane that contains the ideally phase matched irwave vector and is perpendicular to the optic plane.The angle X2 is the angle between the ir wave vectorand the optic plane when the ir wave vector is in theplane defined above. Assuming an uniaxial crystal,the terms in the Taylor series are

Ak = 0OX2

O2 Ak = n2 c 3 )2 ,2n12 An COS COS(02 3)

An COS(02 : 3) ( Ix3 ) _ n3 1 (aX3 )2]

a X3 T3 OX2 X2 n3

AX2 = X2 -

Tangential phase matching increases A42 substan-tially over situations where nontangential phasematching is used, but AX2 is slightly larger than AI\2even in tangential phase matching situations. Byevaluating the partial derivatives for various wave-lengths and repeatedly using the quadratic formula,plots of At 2 and AX2 can be generated. The full ac-ceptance angles are twice AI'2 and AX2. The processof determining both A\4t2 and AX2 as a function of irwavelength has been done for several crystals. Theresults of these calculations for two crystals aie illus-trated in Fig. 4. The plots are for sum frequencygeneration in a 1.0-cm long crystal pumped by 1.064-,um radiation. The specific plots illustrate a generalrule. Tangential phase matching can increase A 2substantially over nontangential phase matching.However, for uniaxial crystals, AX2 is substantiallylarger than A42, unless tangential phase matching isemployed.

V. Conclusions

A method for determining the spectral bandwidthand two acceptance angles has been presented. Thismethod utilizes terms through second order, but re-duces to the usual first order approximation whenthe second order term is small. The spectral band-width was evaluated using this procedure. A X in-flection, that is, a situation where the first order termvanishes, was shown to produce a large increase inthe spectral bandwidth. A double local maximumwas shown to result in the case of a X inflection. Theacceptance angle was evaluated in two dimensionsthrough second order. The acceptance angle in thedirection orthogonal to the optic plane was shown tobe significantly larger than the acceptance angle inthe optic plane, unless tangential phase matching isemployed.

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(1968).5. J. Warner, Appl. Phys. Lett. 12, 222 (1968).6. J. Warner, Appl. Phys. Lett. 13, 360 (1968).7. R. A. Andrews, IEEE J. Quantum Electron. QE-6, 68 (1970).8. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Per-

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