Generic description of second-order nonlinearphenomena in whispering-gallery resonators

Boris Sturman1 and Ingo Breunig2,*1Institute for Automation and Electrometry of the Russian Academy of Sciences, 630090 Novosibirsk, Russia2Laboratory for Optical Systems, Department of Microsystems Engineering—IMTEK, University of Freiburg,

Georges-Köhler-Allee 102, 79110 Freiburg, Germany*Corresponding author: ingo.breunig@imtek.de

Received July 7, 2011; revised August 16, 2011; accepted August 19, 2011;posted September 1, 2011 (Doc. ID 150717); published September 23, 2011

Extending Yariv’s generic approach to the description of optical microresonators, we describe the second-harmonic generation and the optical parametric oscillation in whispering-gallery resonators (WGRs). The outputcharacteristics of these nonlinear processes are expressed in terms of conventional cavity/coupling parametersand nonlinear material coefficients. The found relations are relevant to the description and optimization of ex-periments with nonlinearly active WGRs at ultralow input light powers. © 2011 Optical Society of America

OCIS codes: 190.0190, 230.5750, 190.3970.

1. INTRODUCTIONThe studies of nonlinear-optical phenomena in whispering-gallery resonators (WGRs) are now experiencing a strongupsurge [1–5]. This is caused by several reasons. First, re-markable progress in manufacturing of WGRs has been made:ultrahigh-quality factors, Q≳ 1010, a strong transversal lightconfinement, and controllable mode structures are available.Second, techniques for efficient coupling of light into and outof WGRs have been developed, delivering a huge intensity en-hancement in the cavity. Third, microstructuring of WGRsmade of χð2Þ nonlinear materials, including periodic poling,has become possible, allowing quasi-phase-matched nonlinearprocesses [6,7].

With the progress described, bringing the nonlinear-opticalphenomena, such as second-harmonic generation (SHG) andoptical parametric oscillation (OPO), to the power range ofconventional continuous-wave lasers has become possible. Inparticular, efficient frequency doubling within the microwatt-to-milliwatt power range has been demonstrated [6,8], OPOwith a 7 μW pump threshold has been realized [9], and hightunability of the frequencies of the signal and idler waves from1.8 to 2:5 μm wavelength has been achieved [7]. Potentially, afew photons launched into a WGR are able to result in strongnonlinear effects because of the intensity enhancement.

Surprisingly, the choice of an adequate approach to the de-scription of the WGR-related linear and nonlinear phenomenais still a topical issue. It was shown by Yariv that the use of theso-called generic approach greatly simplifies the treatment ofnumerous coupling issues [10,11]. The number of model para-meters is minimized, numerous secondary circumstances,such as the exact mode structure and particularities of thecoupler, are put aside, and the spatial separation of the inputand output is naturally taken into account. While having manyparallels with the known theoretical approaches, the genericapproach substantially simplifies and unifies the notion of allWRG-related phenomena.

In this paper, we extend the generic approach to describethe SHG and OPO in WGRs. The obtained simple formulas

express the actual nonlinear output characteristics via theknown experimental and material parameters. They possesssubstantial predicting power. Comparison with the relevantliterature data is conducted as well.

2. BASIC RELATIONSFigure 1 gives a schematic representation of coupling in andout of a WGR, regardless of the specific details of the coupler(prism, taper, etc.) and the cavity. The input and output com-plex amplitudes of the coupler are designated by a1 and a2,respectively. They correspond to a field with the angular fre-quency ω ¼ 2πc=λ, where λ is the vacuum wavelength and c isthe speed of light. The input and output complex amplitudesof the single WGR mode of the same frequency are b2 and b1,respectively. Each of the amplitudes is normalized in such away that its squared absolute value gives the modal power.

Assuming that the coupling is lossless and linear, we gen-erally express the output amplitudes by the input ones via theunitary coupling matrix [10,11]

�a2b1

�¼

�t κ

−κ� t�

��a1b2

�; ð1Þ

such that jκj2 þ jtj2 ¼ 1. The off-diagonal elements of the ma-trix refer to the coupling efficiency; in the limit κ → 0, there isno coupling. Generally, κ depends on the distance between thecoupler and the WGR rim, on the wavelength λ, and on spe-cific coupler parameters, see, e.g., [12,13] and referencestherein. For all WGR experiments, we can set jκj ≪ jtj≃ 1.Importantly, the coupling is a local effect; it is uncoupled frominternal WGR phenomena.

Within the generic approach, the wave launched into theWGR propagates inside over the distance L (the circumfer-ence) toward the output. Let z be the corresponding propaga-tion coordinate so that the spatial evolution of the modalamplitude is given by bðzÞ, whereby bð0Þ ¼ b1 and bðLÞ ¼ b2;

B. Sturman and I. Breunig Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B 2465

0740-3224/11/102465-07$15.00/0 © 2011 Optical Society of America

see Fig. 1. The real physical content arises when we linkb2 to b1.

In the linear case, we have bðzÞ ∝ expð−0:5αzþ ikzÞ,where α is the extinction coefficient describing the light ab-sorption and (if necessary) the scattering losses, while k ¼nω=c is the propagation constant with n being the relevantmodal refractive index [3,4]. Correspondingly, we get, forthe case of a high-quality WGR (αL ≪ 1),

b2 ¼ ð1 − 0:5αLÞeikLb1; ð2Þ

where kL ≫ 1 and jb1j≃ jb2j. Certainly, this relation is applic-able to the most important equatorial and near-equatorialWGR modes.

By combining Eqs. (1) and (2), one can readily express theoutput amplitude a2 and the internal amplitude b1 by the inputamplitude a1. In the leading approximation in αL and jκj wehave for the corresponding powers [10,11]

jb1j2ja1j2

¼ jκj2jeiϕ − 1þ ðαLþ jκj2Þ=2j2 ;

ja2j2ja1j2

¼ jeiϕ − 1þ ðαL − jκj2Þ=2j2jeiϕ − 1þ ðαLþ jκj2Þ=2j2 ; ð3Þ

where ϕ ¼ kL − argðtÞ. The resonant behavior of the internalpower jb1j2 in ϕ (i.e., in ω) is evident. The resonances corre-spond to ϕ ¼ 0; 2π;…. Here we have

jb1j2ja1j2

¼ 4jκj2ðαLþ jκj2Þ2 ;

ja2j2ja1j2

¼ ðαL − jκj2Þ2ðαLþ jκj2Þ2 : ð4Þ

The maximum in jκj value of the internal power occurs atjκj2 ¼ αL when jb1j2=ja1j2 ¼ 1=αL ≫ 1 and ja2j2 ¼ 0. This isthe case of critical coupling when the input power ja1j2 is to-tally transferred into the WGR [10,14]. In the limit κ → 0, wenaturally have ja2j → ja1j and jb1j → 0.

The ratio of the angular frequency ω to the full width at half-maximum of the resonance gives the loaded quality factor Q.One can obtain from Eq. (3) for b1 that

Q−1 ¼ Q−1i þ Q−1

c ; ð5Þ

where Qi ¼ 2πn=αλ and Qc ¼ 2πnL=λjκj2 are the quality fac-tors relevant to the internal and coupling losses. In the limitingcases jκj2 ≪ αL and jκj2 ≫ αL, we have Q≃ Qi and Q≃ Qc,respectively. For critical coupling, we have Q ¼ Qi=2.

In addition to the loaded quality factor Q, it is useful to in-troduce the loaded finesse f , which characterizes both the re-lative losses during one circulation event and the powerenhancement in the WGR. It is given by

f −1 ¼ f −1c þ f −1c ¼ ðαLþ jκj2Þ=2π; ð6Þ

where f i ¼ 2π=αL and f c ¼ 2π=jκj2 are the finesses corre-sponding to the internal and coupling losses.

Importantly, the introduced characteristics Qi;c and f i;s arenot only the subject of calculations. They are directly measur-able in WGR experiments via scanning the frequency ω andchanging the distance between the WGR rim and the couplerat low input powers.

The above formulas provide a firm basis for the treatmentof nonlinear phenomena. Moreover, it is sufficient to replacethe coupling coefficients κ, t by their absolute values in orderto treat the resonant excitation, which is the case in the sub-sequent sections.

When considering nonlinear-optical effects, it is necessaryto deal with light amplitudes of different frequencies. In orderto distinguish between pump and second-harmonic ampli-tudes, we will employ additionally the subscripts p and s, re-spectively. For OPO, the subscripts p, s, and i will denote theamplitudes of the pump, signal, and idler fields.

3. SECOND-HARMONIC GENERATIONLet the amplitudes of the pump waves ap1;p2 and bp1;p2 corre-spond to the frequency ω, and the amplitudes of the signalwaves as1;s2 and bs1;s2 correspond to the double frequency2ω. The structure of the matrix equation for the s amplitudesis the same as that of Eq. (1). However, the input amplitude as1is zero in this case [see Fig. 1(b)] and the coupling coefficientsts and κs are quantitatively different from tp and κp. As a result,the matrix equation for the s amplitudes is reduced to trivialrelations:

as2 ¼ κsbs2; bs1 ¼ t�sbs2: ð7Þ

Next, we must find internal circulation relations that gen-eralize Eq. (2) and account for nonlinear changes. Assumingthat ω and 2ω are close enough to two WGR eigenfrequencies,i.e., the phase-matching condition is fulfilled, and introducingprovisionally the slowly varying amplitudes bj ¼bj expð−ikjzÞ,where j ¼ s, p, kj ¼ njωj=c, and nj is the modal refractive in-dex, we have the coupled-wave equations

b0s þ 0:5αsbs ¼ −iνb2p; b0p þ 0:5αpbp ¼ −iνbsb�p: ð8Þ

Here the prime indicates the differentiation in z, αp;s are therelevant extinction coefficients, and ν is a real coupling coef-ficient to be specified in Section 5. According to these equa-tions, the total internal power jbsj2 þ jbpj2 is changing along zonly because of the linear losses, i.e., the nonlinear coupling islossless. The structure of the coupled-wave equations is thusdictated by general reasons.

Fig. 1. (a) Schematic of coupling into and out of a WGR; a1 and b2 arethe input amplitudes, while a2 and b1 are the output amplitudes; L isthe circumference. (b) and (c) Illustration of coupling in and out forSHG and OPO; the indices p, s, and i correspond to the pump, signal,and idler waves.

2466 J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011 B. Sturman and I. Breunig

From Eqs. (8), we get readily in the leading approximationin L the necessary links between bs2 ¼ bsðLÞ, bp2 ¼ bpðLÞ andbs1 ¼ bs1 ¼ bsð0Þ, bp1 ¼ bp1 ¼ bpð0Þ:

bs2 ¼ ð1 − 0:5αsLÞbs1 − iνLb2p1;

bp2 ¼ ð1 − 0:5αpLÞbp1 − iνLbs1b�p1: ð9Þ

Since the difference between bj1 and bj2 (j ¼ p, s) is verysmall, as f −1j ≪ 1, it can be safely neglected in the nonlinearterms.

Combining Eqs. (7) and (9), we get at the resonance:

bs1 ¼ −2iνLb2p1

jκsj2 þ αsL≡ −

iνLf sπ b2p1: ð10Þ

Similarly, using Eqs. (1) and (9), we obtain

0:5ðjκpj2 þ αpLÞbp1 þ iνLb�p1bs1 ¼ −jκpjap1: ð11Þ

Eliminating bs1, we arrive at a remarkable relation:

bp1

�1f p

þ f sν2L2jbp1j2π2

�¼ −

jκpjπ ap1: ð12Þ

It allows us to express jbp1j via jap1j and calculate thenjbs1jðjap1jÞ from Eq. (10). Furthermore, it shows that, not onlycoupling into (out of) the WGR changes its finesse, but thatthe nonlinear losses at the pump frequency ω owing to SHGprovide a similar effect. The expression in the bracketsin Eq. (12) can be viewed as the inverse renormalizedfinesse ~f −1p .

Consider the mentioned dependences in some detail. Let usintroduce the dimensionless quantities

y ¼ πjbp1jjap1jjκpjf p

; x ¼ jκpj2f 3pf sν2L2jap1j2π4 ; ð13Þ

instead of jbp1j and jap1j. In essence, x is the normalized inputpump power. Then we have from Eq. (12)

ð1þ xy2Þy ¼ 1: ð14Þ

The only physical solution of this cubic equation gives a po-sitive decreasing function yðxÞ [see Fig. 2(a)]. Obviously,yð0Þ ¼ 1 and y≃ x−1=3 for x ≫ 1. The initial negative slopeis high, while the further decrease is slow. A cumbersome

analytical solution for y is also available; it is of minor impor-tance. Finally, note that ~f p=f p ¼ yðxÞ, i.e., the renormalizedfinesse ~f p is decreasing in x according to Fig. 2(a).

Let us now introduce the efficiency of SHG: ηs ¼jas2j2=jap1j2; see also Fig. 1. By combining Eqs. (7) and (10)one finds

ηs ¼f sf pjκsj2jκpj2

π2 xy4ðxÞ: ð15Þ

This relation is symmetric with respect to the s, p indices.It is not difficult to analyze the function xy4ðxÞ≡ yðxÞ½1 −

yðxÞ� by using Eq. (14). It has a maximum at x ¼ xmax ¼ 4,where y ¼ 1=2 and xy4 ¼ 1=4; see also Fig. 2(b). Note thatxy4ðxÞ decreases slowly for x > xmax. Therefore, the use oflarger than optimum pump powers is not critical for the SHG.

Using Eq. (15), we arrive at the following simple relation forthe maximum (in the pump power) transformation efficiency:

ηmaxs ¼ jκsj2jκpj2

ðjκsj2 þ αsLÞðjκpj2 þ αpLÞ: ð16Þ

It is symmetric with respect to the s and p characteristics.According to Eq. (16), the value of ηmax

s is always smaller than1, but it can closely approach unity. The coupling must bestrongly overcritical, jκsj2 ≫ αsL, jκpj2 ≫ αpL, in order to doso. Equation (16) possesses also the natural feature ηmax

s →

0 for jκpj, jκsj → 0.Figure 3 quantifies the dependence of ηmax

s on the ratiosrp ¼ jκpj2=αpL and rs ¼ jκsj2=αsL. One sees that reachingthe efficiencies close to unity requires very large values ofrp;s. The case of critical coupling corresponds to rp ¼ rs ¼1 and ηmax

s ¼ 1=4.Consider now the expression for the input pump power

maximizing ηs:

jap1j2max ¼ ðjκpj2 þ αpLÞ3ðjκsj2 þ αsLÞ4ν2L2jκpj2

: ð17Þ

In contrast to Eq. (16), it is not symmetric in s, p. For jκsj2 ≫αsL and jκpj2 ≫ αpL, we have jap1j2max ≃ jκpj4jjκsj2=4ν2L2. Thisvalue rapidly grows with increasing jκsj and, especially, jκpj,and does not depend on the internal WGR characteristics.

One more important characteristic is the efficiency ηp ¼jap2j2=jap1j2 representing the unused part of the pump power;see also Fig. 1. It is easy to find from Eqs. (10), (12), and (13)that

(a) (b)

Fig. 2. Functions (a) yðxÞ and (b) xy4ðxÞ; the arrow in (b) indicates the maximum.

B. Sturman and I. Breunig Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B 2467

ap2ap1

¼ 1 −2jκpj2yðxÞjκpj2 þ αpL

: ð18Þ

The right-hand side depends on the ratio rp ¼ jκpj2=αpL andalso on the normalized pump power x. For x → 0, we havey → 1, i.e., we return to the linear case, where ηp turns to zeroat rp ¼ 1. With increasing x, the critical value of rp (where ηpturns to zero) shifts to the right and becomes infinite for x ¼ 4.For x > 4, the dependence ηpðrpÞ is monotonically decreasing.

An even more important issue is the impact of the choice ofrp on the dependence ηpðxÞ. For rp ≪ 1, we have triviallyηpðxÞ≃ 1. Increasing rp changes the function ηpðxÞ quantita-tively and qualitatively; see Fig. 4. For rp ¼ 0:5, it is monoto-nically growing starting from a finite value. For rp ¼ 1, whichcorresponds to the critical coupling, the initial value of ηpðxÞturns to zero. Further increasing rp shifts the zero point ofηpðxÞ to the right. For rp ≫ 1, it approaches a value of 4.The larger rp, the smaller is ηpðxÞ in the range x > 4. This fea-ture is expected because most of the input pump power trans-forms into the second harmonic.

One can show, furthermore, that the sum ηs þ ηp is alwayssmaller than 1 because of the internal losses. However, ittends to 1 for rp;s → ∞, when the internal losses become neg-ligible. This expresses the energy conservation law.

4. OPTICAL PARAMETRIC OSCILLATIONLet now the indices p, s, and i correspond to the pump, signal,and idler modes, respectively; see also Fig. 1(c). The phase-matching conditions are expected to be fulfilled. In particular,we have ωp ¼ ωs þ ωi. The coupled-mode equations for theslowly varying amplitudes bj ¼ bj expð−ikjzÞ with j ¼ p, s, iread

b0p þ 0:5αpbp ¼ −i~νbsbi;b0s þ 0:5αsbs ¼ −iqs~νbpb�i ;b0i þ 0:5αibi ¼ −iqi~νbpb�s ; ð19Þ

where ~ν is the relevant coupling constant (see Section 5) αp;s;iare the extinction coefficients, and qs;i ¼ ωs;i=ωp. The non-linear coupling is lossless again, so that the loss of the totalpower inside the WGR, jbpj2 þ jbsj2 þ jbij2, is solely due tothe linear processes. Designating again the numbers 1 and2 to z ¼ 0 and L, so that bj1 ¼ bj1, we find in the approximationleading in L:

bp2 ¼ ð1 − 0:5αpLÞbp1 − i~νLbs1bi1;bs2 ¼ ð1 − 0:5αsLÞbs1 − iqs~νLbp1b�i1;bi2 ¼ ð1 − 0:5αiLÞbi1 − iqi~νLbp1b�s1: ð20Þ

Next we write down the relevant relations for the WGRcoupling. It is expected that only the p mode is coupled infrom outside; see Fig. 1(c). Thus, we have

ap2 ¼ tpap1 þ κpbp2; bp1 ¼ −κ�pap1 þ t�pbp2: ð21Þ

The coupling relations for the s, i modes are shorter:

as2 ¼ κsbs2; bs1 ¼ t�sbs2; ai2 ¼ κibi2; bi1 ¼ t�i bi2:

ð22Þ

Assuming the resonant excitation, we can replace again thecoupling coefficients by their absolute values.

Combining the internal and external relations for the s and imodes, we easily get

bs1 ¼ −iqsf s~νL

π bp1b�i1; bi1 ¼ −iqif i~νL

π bp1b�s1: ð23Þ

Using next the internal and external relations for the p mode,we obtain for bp1

bp1 ¼ −f pπ ðjκpjap1 þ i~νLbs1bi1Þ: ð24Þ

By substituting Eq. (24) into Eqs. (23), we arrive at the follow-ing algebraic relations:

bs1ð1þ csjbi1j2Þ ¼ idsap1b�i1; bi1ð1þ cijbs1j2Þ ¼ idiap1b�s1;

ð25Þ

0.1

0.25

0.55

0.8

0.9

0.95

Fig. 3. Contour plot of the function ηmaxs ðrp; rsÞ with rp ¼ jκpj2=αpL

and rs ¼ jκsj2=αsL; the numbers on the lines indicate the valuesof ηmax

s .

Fig. 4. Efficiency ηp as a function of x for different values ofrp ¼ jκpj2=αpL.

2468 J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011 B. Sturman and I. Breunig

where cs;i ¼ qs;if s;if p~ν2L2=π2, ds;i ¼ qs;if s;if p~νLjκpj=π2 are fourconstants such that csdi ¼ cids. From here, one can expressjbs1j2 and jbi1j2 by the input pump power jap1j2. One can checkthat csjbi1j2 ¼ cijbs1j2 and, finally,

jbs1;i1j2 ¼qs;if s;if pjκpj2

π2 jap1j2th� ffiffiffiffiffi

ξpp

− 1�; ð26Þ

where ξp ¼ jap1j2=jap1j2th ≥ 1, and

jap1j2th ¼ π4qsqif sf if 2pjκpj2~ν2L2 ð27Þ

is the threshold input pump power. According to theserelations, OPO occurs above the threshold, ξp > 1, wherethe powers jbs1j2 and jbi1j2 depend on jap1j2 according to thesquare-root law.

The threshold power jap1j2th has a pronounced minimum injκpj2. If, e.g., jκs;ij2 ¼ jκpj2 and αp ¼ αs;p, it takes place at jκpj2 ¼αpL=3 whereby jap1j2th ¼ 16ðαpLÞ3=27qsqi~ν2L2. Generally, thesituation with minimization of jap1j2th is more complicated.For jκj2p;s;i ≫ αp;s;iL, we have jap1j2th ≃ ðκpκsκiÞ2=16qsqi~ν2L2.

The efficiencies of the s, i generation, defined as the ratiosηs;i ¼ jas2;i2j2=jap1j2, are of prime interest. They can be easilycalculated using Eqs. (22) and (26):

ηs;i ¼qs;i

ðr−1s;i þ 1Þðr−1p þ 1Þ ×4� ffiffiffiffiffi

ξpp

− 1�

ξp; ð28Þ

where rp;s;i ¼ jκp;s;ij2=αp;s;iL. The second factor in this relationexpresses the pump-power dependence above the threshold.The maximum value of this factor, equal to 1, occurs at ξp ¼ 4.Therefore, the maximum (in jap1j2) values of ηs;i are given bythe first factor in Eq. (28); they do not exceed qs;i ¼ ωs;i=ωp.Overcritical regimes with rp;s;i ≫ 1 are necessary to approachclosely the ultimate values qs;i. These regimes correspond in-deed to fairly high values of the threshold power jap1j2th.

It is also easy to find the efficiency ηp ¼ jap2j2=jap1j2 thatrepresents the remnant part of the pump power. First, we ob-tain from Eqs. (20) and (21)

ap2 ¼1 − rp1þ rp

ap1 −2irp~νL1þ rp

bs1bi1: ð29Þ

The first term of the right-hand side describes the linear case.The product bs1bi1 can be calculated from Eqs. (25). Finally,we get above the threshold:

ηp ¼�1 −

2ξ−1=2p

r−1p þ 1

�2: ð30Þ

It is close to unity forffiffiffiffiffiξp

p≫ 1. This means indeed that the

fraction of the s, i output powers is very small here. This isin agreement with the above observations on ηs;i.

One can show furthermore that the sum ηp þ ηs þ ηi is al-ways smaller than 1. However, it tends to 1 in the limit of neg-ligible internal losses, rp;s;i → ∞, in agreement with the energyconservation law.

Figure 5 illustrates the above-threshold dependences of thetransformation efficiencies on ξp for rp ¼ rs ¼ ri ¼ 10.

Further increase of rp lifts up curves 1 and 3 and shifts thezero point of curve 2 toward a value of 4.

5. COUPLING COEFFICIENTSUp to this moment, we treated ν and ~ν as arbitrary constants.Here we link them to the conventional WGR and nonlinearcharacteristics. The light extinction can be ignored whiledoing so.

The total electric light field E can be represented as

E ¼Xj

EjðzÞΨjð~ρÞe−iωtþikz þ c:c:; ð31Þ

where the summation occurs over all relevant WGR modes,EjðzÞ is a slowly varying field amplitude for the mode j, ~ρis the transverse-coordinate vector, andΨjð~ρÞ is a real dimen-sionless eigenfunction with a unit maximum value.

Let now the indices p and s refer to ω and 2ω modes. Thenwe have, for the amplitudes Ep;s under the condition of phasematching and negligible light absorption [15],

ΨpE0p ¼ −i

2ωpdeffcnp

E�pEsΨpΨs; ΨsE

0s ¼ −i

2ωpdeffcns

E2pΨ2

p;

ð32Þ

where deff is the known nonlinear coefficient and np;s are therelevant modal refractive indices. It implies that the nonlinearpolarization is given by 2ϵ0deffE2, where ϵ0 is the vacuumpermittivity. In the terms used, the modal power is Pj ¼2njcϵ0jEjj2σj , where j ¼ p, s and

σj ¼Z

Ψ2j ð~ρÞd~ρ ð33Þ

is the transverse mode cross section. The product Vj ¼ σjLgives the corresponding mode volume.

It is not difficult to check from Eqs. (32) that the totalpower, Pp þ Ps, is z independent. Furthermore, it is easy tounderstand how to introduce the slowly varying amplitudesbj such that jbjj2 ¼ Pj . The answer is

Ej ¼ bj=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2njϵ0cσj

q: ð34Þ

Fig. 5. Dependences of ηs þ ηi (curve 1), ηp (curve 2), and the sumηp þ ηs þ ηi (curve 3) on ξp for rp;s;i ¼ 10.

B. Sturman and I. Breunig Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B 2469

New amplitudes obey Eqs. (8) with the coupling coefficient,

ν ¼ffiffiffi2

pωpdeffσspp

cffiffiffiffiffiffiffiϵ0c

pnpσp

ffiffiffiffiffiffiffiffiffinsσs

p ; ð35Þ

and the overlap cross section between modes p and s:

σpps ¼Z

Ψ2pð~ρÞΨsð~ρÞd~ρ: ð36Þ

Using Eq. (35), one can express the coupling constant ν by theknown bulk nonlinear coefficient deff and the conventionalWGR characteristics.

Similarly, one can find that the nonlinear coefficient ~ν,relevant to the OPO case, is given by

~ν ¼ffiffiffi2

pωpdeffσpsi

cffiffiffiffiffiffiffiϵ0c

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinpnsni

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiσpσsσip ; ð37Þ

where the cross section σpsi is given by the integralfrom ΨpΨsΨi.

Equations (35) and (37) canbe trivially rewritten in the termsof mode volumes Vj ¼ σjL, Vpps ¼ σppsL, and Vpsi ¼ σpsiL.

6. EXPLICIT RELATIONS AND NUMERICALESTIMATESBy combining Eqs. (17) and (35), we obtain a useful relationfor the input pump power that maximizes the SHG efficiencyηs:

jap1j2max ¼ Pc ×1

f 2i;pf i;s×ðrp þ 1Þ3ðrs þ 1Þ

rp: ð38Þ

It consists of three different factors. The first factor,

Pc ¼ϵ0π3c3n2

pns

ω2pL2d2eff

σ2pσsσ2pps

; ð39Þ

is a characteristic power that depends neither on the modefinesses nor on the WGR coupling parameters. The secondfactor, including the internal finesses f i;p ¼ π=αpL andf i;s ¼ π=αsL, is dimensionless and very large. It characterizesenhancement of the nonlinear processes in the WGR. Thethird factor is also dimensionless; it expresses the impactof the ratios rj ¼ jκj j2=αjL characterizing the couplingcriticality.

For the representative parameters L ¼ 1 cm, λp ¼ 1 μm,np ¼ ns ¼ 2:2, and deff ¼ 10pm=V [16], and the effective crosssection σ2pσs=σ2pps ¼ 100 μm2 [6], we obtain Pc ≈ 220W. It is arealistic estimate for WGRs made of lithium niobate. The sec-ond factor sharply depends on the extinction coefficients αp;s;its variation range can be as large as several orders of magni-tude. By setting f i;p ¼ f i;s ¼ 2000, which can be regarded asrepresentative for experiments with WGRs made of lithiumniobate [6], we have an estimate of ≈10−10 for the second fac-tor. The value of the third factor substantially depends on thedistance between the WGR and the coupler. For the criticalcoupling, when rp;s ¼ 1 and ηmax

s ¼ 1=4, it is equal to 16. Forthe chosen parameters, we have finally jap1j2max ≈ 0:4 μW.Pump powers that are 3–4 orders of magnitude larger areneeded to achieve ηs ¼ ð0:7–0:8Þ in accordance with Eq. (38)and Fig. 3.

Similarly, we can obtain a useful relation for the pumppower, japj2th, that gives the threshold of the OPO. CombiningEqs. (27) and (37), we have

jap1j2th ¼ ~Pc ×1

f i;pf i;sf i;i×ðrp þ 1Þ2ðrs þ 1Þðri þ 1Þ

rp; ð40Þ

where the characteristic power

~Pc ¼ϵ0π3c3npnsni

4qsqiω2pL2d2eff

σpσsσiσ2psi

: ð41Þ

Numerical estimates of jap1j2th and ~Pc are similar to those ofjap1j2max and Pc, respectively.

7. DISCUSSIONIn essence, this work is an adaptation of the general descrip-tion of the nonlinear processes [15,17] to the case of WGRs.However, it differs substantially from the previous theoreticalstudies [6,18–21] in this area in the following respects.

• An extension of the generic Yariv approach [10,11] hasallowed us to get rid of many unnecessary details relevant tothe description of the complicated modal structure of WGRs.

• Employing the conventional coupled-wave equations,we naturally treat the SHG and the OPO as steady-state pro-blems with spatially separated input and output. Neither thefield quantization nor the temporal approach, as employed in[6,19], is needed within our theory.

• The final relations of our theory express the observablenonlinear characteristics—the transformation efficiencies—by the measurable WGR parameters, the pump power, andthe conventional nonlinear coefficients. They show that a non-trivial trade-off between thebenefits anddrawbacksof increas-ing/decreasingpumppower and the coupling coefficients takesplace, leading to useful predictions for the control and optimi-zation of the output.

• The necessity to use strongly overcritical regimes in or-der to approach closely the limiting values of the transforma-tion efficiencies is a general feature of our formulas. Thedistance between the coupler and the WGR rim is thus acrucial control parameter.

The theoretical studies that are most closely related to ourstudy are presented in [6,19]. They deal with χð2Þ nonlinearprocesses in WGRs made of lithium niobate and employ quan-tization of the electromagnetic field and a temporal approach.The corresponding formulas for the SHG transformation effi-ciencies, given without details in [6], show no zero-couplinglimit κ → 0. Most probably, they are relevant to the limitingcase jκp;sj2=αp;sL → ∞. Taking into account the correction ofEq. (5) of [6], indicated in [8], and the differences in notation,we have made sure that full agreement between our relationsfor ηs;i and the results of [6] takes place in this limiting case.The analysis of OPO made in [19] was restricted to the degen-erate case (ωs ¼ ωi ¼ ωp=2) and to the limit jκp;sj2=αp;sL → ∞.The relations of Section 4, which are free of these restrictions,extend substantially this analysis.

It is necessary also to mention a similarity between theWGR-based nonlinear devices and those employing externalresonators; see [22–24] and references therein. Since the func-tionalities of these devices are different, it is rather useless totry to establish here direct links.

2470 J. Opt. Soc. Am. B / Vol. 28, No. 10 / October 2011 B. Sturman and I. Breunig

Since the line width of a WGR mode is directly linked to itsoverall finesse (its overall quality factor), the use of overcriticalregimes not only increases the transformation efficiencies, italso increases substantially the line width, facilitating in thisway the fulfillment of the resonant and phase-matchingconditions.

The newly developed techniques for periodic poling ofWGRs made of ferroelectric materials, such as lithium nio-bate, also strongly extend the possibilities for quasi-phase-matching and tuning of different nonlinear processes [6,7].Importantly, these possibilities can be combined with theuse of the largest nonlinear coefficients. In particular, fulfill-ment of the phase-matching condition can be ensured at theexpense of a modest decrease of the effective nonlinear coef-ficient deff compared to d33.

8. CONCLUSIONSExtending Yariv’s generic approach, we have described theSHG and OPO in whispering-gallery resonators made of activenonlinear material. The actual transformation efficiencies areexpressed by the measurable resonator and coupling charac-teristics and the conventional nonlinear coefficients of thebulk material. The final expressions include a great deal ofinformation for experimental optimization of the nonlinearcharacteristics. The use of overcritical coupling is neededto achieve large values of the nonlinear transformationefficiencies.

ACKNOWLEDGMENTSFinancial support from the Deutsche Forschungsge-meinschaft (DFG) is acknowledged. We are also grateful toA. Matsko for valuable comments.

REFERENCES1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846

(2003).2. T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang,

and K. Vahala, “Fabrication, coupling and nonlinear optics ofultra-high-Q micro-sphere and chip-based toroid microcavities,”in Optical Microcavities, K. Vahala, ed. (World Scientific,2004), Chap. 5.

3. A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, andL. Maleki, “Parametric optics with whispering-gallery modes,”Proc. SPIE 4969, 173–184 (2003).

4. A. B. Matsko and V. S. Ilchenko, “Optical resonators withwhispering-gallery modes—Part I: Basics,” IEEE J. Sel. Top.Quantum Electron. 12, 3–14 (2006).

5. L. Maleki and A. B. Matsko, “Crystalline whispering gallerymode resonators in optics and photonics,” in Ferroelectric

Crystals for Photonic Applications, P. Ferraro, S. Grilli, andP. De Natale, eds. (Springer, 2009), Chap. 13.

6. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki,“Nonlinear optics and crystalline whispering gallery modecavities,” Phys. Rev. Lett. 92, 043903 (2004).

7. T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D.Haertle, K. Buse, and I. Breunig, “Highly tunable low-thresholdoptical parametric oscillation in radially poled whisperinggallery resonators,” Phys. Rev. Lett. 106, 143903 (2011).

8. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen,Ch. Marquardt, and G. Leuchs, “Naturally phase-matchedsecond-harmonic generation in a whispering-gallery-moderesonator,” Phys. Rev. Lett. 104, 153901 (2010).

9. J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen,Ch. Marquardt, and G. Leuchs, “Low-threshold optical para-metric oscillations in a whispering gallery mode resonator,”Phys. Rev. Lett. 105, 263904 (2010).

10. A. Yariv, “Universal relations for coupling of optical power be-tween microresonators and dielectric waveguides,” Electron.Lett. 36, 321–322 (2000).

11. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).

12. D. R. Rowland and J. D. Love, “Evanescent wave coupling ofwhispering gallery modes of a dielectric cylinder,” IEE Proc.J 140, 177–188 (1993).

13. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere re-sonators: optimal coupling to high-Q whispering-gallery modes,”J. Opt. Soc. Am. B 16, 147–154 (1999).

14. M. Cai, O. Painter, and K. J. Vahala, “Observation of critical cou-pling in a fiber taper to a silica-microsphere whispering-gallerymode system,” Phys. Rev. Lett. 85, 74–77 (2000).

15. R. W. Boyd, Nonlinear Optics (Academic, 2008).16. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute

scale of second-order nonlinear-optical coefficients,” J. Opt.Soc. Am. B 14, 2268–2294 (1997).

17. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).18. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki,

“Highly nondegenerate all-resonant optical parametric oscilla-tor,” Phys. Rev. A 66, 043814 (2002).

19. V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki,“Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B 20,1304–1308 (2003).

20. Y. Dumeige and P. Feron, “Whispering-gallery-mode analysis ofphase-matched doubly resonant second-harmonic generation,”Phys. Rev. A 74, 063804 (2006).

21. G.Kozyreff,J.L.Dominguez-Juarez,andJ.Martorell,“Whispering-gallery-mode phasematching for surface second-order nonlinearoptical processes in spherical microresonators,”Phys. Rev. A 77,043817 (2008).

22. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant opticalsecond harmonic generation and mixing,” IEEE J. QuantumElectron. QE-2, 109–124 (1966).

23. R. G. Smith, “Theory of intracavity optical second-harmonicgeneration,” IEEE J. Quantum Electron. 6, 215–223 (1970).

24. V. Berger, “Second-harmonic generation in monolithic cavities,”J. Opt. Soc. Am. B 14, 1351–1360 (1997).

B. Sturman and I. Breunig Vol. 28, No. 10 / October 2011 / J. Opt. Soc. Am. B 2471