3438 OPTICS LETTERS / Vol. 32, No. 23 / December 1, 2007

Miniaturization of Fresnel phase matching using aside-coupled integrated spaced sequence of

resonators (SCISSOR)

Yannick DumeigeENSSAT-FOTON (CNRS-UMR 6082)—Université de Rennes 1, 6 rue de Kerampont, B.P. 80518, 22300 Lannion,

France*Corresponding author: yannick.dumeige@enssat.fr

Received September 12, 2007; revised October 24, 2007; accepted October 24, 2007;posted October 26, 2007 (Doc. ID 87472); published November 28, 2007

It is proposed that a side-coupled integrated spaced sequence of resonators (SCISSOR) be used to adaptFresnel phase matching to the case of highly confining waveguides. As is the case for bulk media, thismethod of quasi-phase-matching (QPM) allows resonant or nonresonant QPM. This property can be used tocontrol the spectral bandwidth of the phase-matching curve. © 2007 Optical Society of America

OCIS codes: 190.4390, 190.2620, 230.4555.

Optical waveguides are attractive devices for the in-tegration of parametric sources or optical signal pro-cessing functions based on second-order optical non-linearities. They provide large spatial power densityover a few millimeters, which is a key parameter forefficient second-order nonlinear photonics. For thispurpose, the use of type III–V materials has severaladvantages, such as large nonlinear susceptibility,high confinement due to large refractive index, andcompatibility with other optoelectronic devices [1].Unfortunately III–V materials are isotropic, so artifi-cial structuration is necessary to obtain sufficientform birefringence to reach phase matching (PM) [1].As III–V materials are not ferroelectric, the quasi-phase-matching (QPM) [2] condition can be obtainedonly by using complex technological processes [3]. Inthis Letter I put forward a method to adapt FresnelPM [4,5] to the case of highly confining waveguidesby using a side-coupled integrated spaced sequence ofresonators (SCISSOR) [6]. In the case of second-harmonic generation (SHG), Fresnel PM consists ofpropagating the fundamental (F) and second-harmonic (SH) fields by use of total internal reflec-tion inside a nonlinear plane parallel plate. The ac-cumulated phase mismatch between the two fields iscompensated at each bounce, since they do not un-dergo the same Fresnel phase shift at reflection.Here, I consider a sequence of microrings coupledonly by a straight access waveguide (see Fig. 1). Itwill be shown that the plate can be replaced by thehighly confining straight waveguide and that theresonators, equally spaced by an odd multiple of thecoherent length, can play the role of the internalbounces. The distance L between each resonator isthe same: L=zp−zp−1 with p� �1,M� and z0=0. Therings are assumed to be equivalent with a radius Rand optical losses characterized by �i with i� �� ,2��, where � is the circular frequency of the Ffield. I consider an epitaxial growth of AlGaAs alloysalong the y= �001� crystallographic axis (see Figs. 1and 2). Consequently, for a propagation direction z= �110� and an F field polarized along the x axis, anSH field polarized along the y axis is obtained. The

0146-9592/07/233438-3/$15.00 ©

effective index medium method is used to take intoaccount the y confinement by an effective index n�

and to reduce calculations to two dimensions. The ef-fective index medium method is then used once morefor x confinement to find the propagation constant ��

of the F and SH modes in the straight waveguide andin the ring, which are assumed to be single mode [7].I follow the coupled mode theory for SHG in AlGaAswaveguides [8], considering only a planar waveguidein the x direction (and then assuming that the modeextension in the y direction is equal to h). Then, inthe straight waveguide the fields are well describedby their y components, which correspond to a mag-netic field H�=A�H��x�exp�j�t− j��z� /2 for the Ffield and to an electric field E2�

=A2��z�E2��x�exp�2j�t− j�2�z� /2 in the case of the SHfield. H��x� and E2��x� are normalized to obtain amode with a power equal to �A��2�h. The couplingbetween the field A�,p��� (with �� �0,2��) in the ringnumber p and the field A��z� in the straight wave-guide (Fig. 2) is given by the matrix relation [6]

�A�,p�0�

A��zp+� = � �� j��

j�� ���A�,p�2��

A��zp−� . �1�

Using the one round-trip transmission a�

=exp�−���R� and phase �=−2���R, let us writethat A�,p�2��=a� exp�j��A�,p�0�. Then we can calcu-late the amplitude transmission of one ring:

Fig. 1. Description of the proposed implementation ofFresnel PM in AlGaAs waveguides. y corresponds to the

crystallographic axis �001� and z to �110�.

2007 Optical Society of America

December 1, 2007 / Vol. 32, No. 23 / OPTICS LETTERS 3439

t� =A��zp

+�

A��zp−�

=�� − a��

2 exp�j��

1 − ��a� exp�j��, �2�

where ��2 +��

2 =�2. The transmission t� is also given

by its trigonometric form: t�=T� exp�j���. First, themethod will be briefly illustrated in the case of loss-less coupling and propagation �a�=�=T�=1�. If thedistance L between two adjacent resonators is chosento be equal to the effective coherent length, the SHgenerated between the two resonators is in phase op-position with the F field. This can be compensated bychoosing �=� �2�� and 2�=2� �2�� so t�=1 ���

=0 �2��� and t2�=−1 ��2�=� �2���. The opposite signof the transmissions of the two fields is equivalent tothe nonlinear susceptibility sign reversal in the usualQPM. In the following this procedure is called reso-nant QPM [5]. As it is the case for Fresnel PM, wecan choose �� and �2� independently and so compen-sate for any values of phase mismatch accumulatedbetween two resonators. This allows the nonresonantQPM condition to be fulfilled [5]. This condition can-not be obtained in the usual QPM, since in that casethe only phase mismatch that can be compensated is�. Note that when the distance between each bounceis shorter than the coherent length, the nonresonantQPM is more efficient that resonant QPM [9]. Now, Iwill detail my proposition in the undepleted F fieldcase. For z� �zp−1 ,zp� and p� �1,M� we can write

dA2�

dz= KA�

2�zp−1+ �exp�j��z�. �3�

Let us call ��=�2�−2�� the effective wave-vectormismatch. The nonlinear coupling is given by K=−jd14��

2S / �2� 0n�4�; d14 is the nonlinear coefficient of

Al0.3Ga0.7As. We will take d14=108 pm/V for a Fwavelength ��=1550 nm. S characterizes the fields’overlap for well-confined modes [8]:

S = �−�

+�

�H��x��2E2��x�dx =32� 0n�

2

3���

2��0

�2�d. �4�

Fig. 2. Coupling between the access straight waveguideand resonator number p.

After integration we obtain

A2��zp−� = A2��zp−1

+ � +2K

��A�

2�zp−1+ �

�exp�j���p −1

2 Lsin���L

2 . �5�

It is very important to note that the quantity ��Rcan be chosen as a whole number different from 2 or−2; thus there is no SHG in the resonators [10] butonly in the straight waveguide: A2��zp

+�= t2�A2��zp−�.

Let us consider a lossless straight waveguide suchthat A��zp−1

+ �= t�p−1A��0�, since A��zp−1

+ �=A��zp−�; then

we have

A2��zp+� = t2�A2��zp−1

+ � +2K

��

t2�

t�2 exp�−

j��L

2 � sin���L

2 �t�2 exp�j��L��pA�

2�0�, �6�

which can be written as a sequence �up� with up=A2��zp

+�=Xup−1+Yqp. The general term for this se-quence is calculated recursively; since u0=0, we ob-tain

up = Xpu0 + Y�k=1

p

Xp−kqk = YqXp − qp

X − q. �7�

The conversion yield ��p�= �A2��zp+��2 / �A��0��2 is

��p� =4�K�2

��2 T2��A��0��2 sin2���L

2 �

T2�p + T�

2p − 2�T2�T��pcos�p���

T2� + T�2 − 2T2�T� cos ��

, �8�

where ��=��L+2��−�2� is the phase error. Equa-tion (8) is similar to the expression found by Haïdaret al. [5] for Fresnel PM. The dimensions of the wave-guide are h=1 �m and d=500 nm, so the waveguideand the ring are not single mode, but a ring geometrycould be chosen to tailor the evanescent fields tocouple only one mode at F and SH frequencies[10,11]. This optimization is not beyond the scope ofthis Letter. The radius of each resonator is R

Fig. 3. Conversion yield ��200� as a function of the Fwavelength in the resonant and lossless case. Phase shifts

��, �2� and the phase error �� are also shown.

3440 OPTICS LETTERS / Vol. 32, No. 23 / December 1, 2007

=5.39 �m, the F incident intensity is 200 MW/cm2,��=�2�=0.9, and p=200. A distance between eachresonator equal to L=9Lc=6.11 �m is used (Lc is theeffective coherent length defined as Lc=� /��) to ob-tain resonant QPM for ��=1550 nm. As shown in Fig.3, the PM curve displays a very narrow bandwidth�����17 pm� due to the slow variation in �� and thefast variation in �2� around the resonance. This in-teresting property could be used in applications need-ing parametric processes with a narrow spectralbandwidth [12]. For an identical structure with R=5 �m and L=6.29 �m, we can also obtain a nonreso-nant QPM as presented in Fig. 4. Note that in thiscase we are still neglecting A2�,p�2��, since both Fand SH fields are nonresonant in the rings. In thiscase the bandwidth of the QPM curve has increased�����0.65 nm� with respect to the case of resonantQPM. Combined with the possibility of tailoring thegroup-velocity dispersion using linear SCISSOR ef-fects [6], the nonresonant QPM could be used for thefrequency conversion of short pulses. I emphasize thecrucial role played by the intrinsic and couplinglosses of each resonator, which have the same effectas surface roughness in the case of Fresnel PM [5].For not perfectly transparent resonators (T� andT2��1), when T��T2�, the conversion yield reachesa maximum value for an optimal number of resona-tors given by

pmax = E�ln� ln T2�

2 ln T� � ln� T�

T2� , �9�

where E�x� is the whole part of x. In the particularcase where T�=T2�, we obtain pmax=E�−1/ln T��,which illustrates that the effective number of resona-tors decreases when losses increase and T� de-creases. In Fig. 5 the conversion yield is plotted as afunction of p for different values of � and �� in thecase of resonant QPM. This shows that couplinglosses as low as 0.8% (corresponding to �

2 =2�2

=0.992) strongly limit the number of effective resona-tors and the conversion yield. In the case of resonantQPM, since �� does not depend on �� [6], � is not de-

Fig. 4. Conversion yield ��200� as a function of the Fwavelength in the nonresonant and lossless case. Phaseshifts ��, �2�, and the phase error �� are also shown.

pendent on fluctuations in ��. In both QPM schemes,

the resonant frequency of each resonator must bewell controlled; it requires a fabrication accuracy bet-ter than 5 nm for p=50. If an effort is made to in-crease the effective coherent length, p could be re-duced. This would limit the detrimental effects oflosses and structural disorder. In conclusion,SCISSOR could be used to obtain the resonant ornonresonant Fresnel PM. The method that I havepresented in this Letter could be generalized to otherkinds of resonators, such as whispering gallery modemicrodisks, which present very low intrinsic and cou-pling losses [13].

This work was supported by the French “AgenceNationale de la Recherche” through project O2E(NT05-3-45032). The author acknowledges stimulat-ing and helpful discussions with Patrice Féron.

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Fig. 5. Conversion yield � as a function of p for differentvalues of �

2 =2�2 =2 and �=��=�2�, in the resonant case

of Fig. 3 and with ��=1550 nm.

Phys. Lett. 90, 051108 (2007).