Prism coupling into second-order nonlinear waveguides

15 January 1998

Optics Communications 146 Ž1998. 95–98

Prism coupling into second-order nonlinear waveguides Maria Giuseppa Masi ) , Gaetano Assanto Department of Electronic Engineering, Terza UniÕersity of Rome, Via della Vasca NaÕale, 86-00146 Rome, Italyl Received 18 June 1997; accepted 27 August 1997

Abstract We numerically investigate the effects of second-harmonic generation in the coupling of an external laser beam into a thin-film waveguide in the presence of a quadratic susceptibility x Ž2.. A nonlinear phase shift through a cascading mechanism affects the in-coupling of a fundamental frequency beam, modifying the effective detuning. Conversely, such nonlinear behaviour can be exploited for in situ evaluation of the waveguide quadratic characteristics. q 1998 Elsevier Science B.V.

Distributed prism coupling is a well-known and commonly used technique for exciting the modes of a guiding structure with confinement in one transverse dimension w1x. High coupling efficiencies, up to 80.1% for a Gaussian beam in the case of an optimized uniform gap w2x, can be obtained through phase matching, i.e. making the wavevector component k in of the external field along the propagation direction equal to the guided mode wavevector bv . Since phase-matching depends on the refractive indices of the structure, it can be modified in the presence of a Kerr-like cubic nonlinearity if one or more of the media which form the planar waveguide exhibit an intensity-dependence. Such a nonlinear response can be readily understood in terms of a coherent travelling wave interaction between guided and injected field components when their relative phase velocity is modified through the Kerr effect w3x. An effective change in phase-velocity, however, can be also produced in the presence of a quadratic nonlinearity, when successive up- and down-conversion processes take place w4,5x. This mechanism, extensively studied with regard to second-harmonic generation ŽSHG., is often addressed to as ‘‘cascading’’ and can be exploited in the realization of all-optical switching and processing devices w6–9x.

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E-mail: [email protected]

In this Letter, we consider the role of the nonlinear phase shift produced by a cascaded x Ž2. Žy2 v ; v , v .: x Ž2. Žyv ; 2 v , v . interaction when prism-coupling an input beam at the fundamental frequency ŽFF. v into a planar waveguide containing one or more media with a quadratic response near phase-matching for SHG. While previous theoretical studies of distributed couplers in the presence of SHG were able to underline the existence of nonlinear resonances w10x, including plane-wave bistability w11x, we specifically refer to the FF finite-beam excitation of a prism coupler, addressing situations of experimental relevance. In the following, in order to retain the relevant physics without the details of a cumbersome vectorial description, we will assume that TE and TM modes propagate with polarizations parallel to the crystal principal axes, and treat the interaction between FF and second-harmonic ŽSH. waves in the scalar approximation. Moreover, without loss of generality, we will assume that the TE 0 Ž v . ™ TM 0 Ž2 v . SHG process is dominant and nearly phase-matched. This is the case, for instance, of a temperature-tuned x-propagating z-cut lithium niobate crystal, the other processes being negligible because of large mismatches or small overlap integrals. The field of the incident s-polarized beam through the prism of refractive index n p excites the FF TE 0 mode which, along propagation, generates an SH TM 0 field, as sketched in Fig. 1.

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 4 9 3 - 8

M.G. Masi, G. Assantor Optics Communications 146 (1998) 95–98

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Fig. 1. Schematic of the studied prism-waveguide coupler, with self-explanatory notation.

The model is readily derived from Maxwell’s equations using coupled-mode theory w2,3x: d Av dz

s t r A in f in Ž z . exp Ž yiD b in z . y

1 lv

Av

y iK NL Av) A 2 v exp Ž yiD k nl z . , d A2 v

1 sy

dz

l2 v

A 2 v y iK NL A2v exp Ž iD k nl z . ,

Ž1. Ž2.

with Av and A 2 v complex eigenmodel amplitudes normalized such that < A v < 2 and < A 2 v < 2 are guided-wave powers per unit wavefronts Žin Wrm along y, the coordinate of invariance.. A in f inŽ z . is the input field distribution at the prism base Ž A2in is the incident intensity in Wrm2 .; t r is the FF transfer coefficient and lv and l 2 v the re-irradiation distances into the prism at FF and SH, . respectively w2x; D b in s Ž k in z y bv the wavevector detuning along the propagation direction, with k in z s Ž vrc . n p sinŽ u in .; D k nl s b 2 v y 2 bv the phase-mismatch Ž2. < v Ž .< 2 2 v Ž . for SHG; K NL s Ž vr4. Hq` Ex x d x y` ´ 0 x y y x E y x the nonlinear coefficient, including the overlap integral over the modal eigenfunctions. Here x yŽ2.y x is – in general – a function of x and represents the second-order susceptibility distribution of one or more of the media in the guiding structure. Let us define the normalized excitation at FF as Pi s Ž d zr2. pr2 Ž A in K NL d z . 2 , with d z the input beam size along z. Assuming the prism to extend from z s y` to z s z 0 , and a uniform low-index Žair. gap, the coupling efficiency Žat v . and the SH transfer efficiency can be

'

calculated versus Pi by numerical integration of Eqs. Ž1., Ž2.. The results for a coupler optimized at low powers Žlinear limit. with z 0 s 0.367d z and lv s 0.74 d z w2x are shown in Fig. 2 for both linear and nonlinear matching, i.e. D k nl s D b in s 0 Žsince we are dealing with guided-wave fields inside the structure, diffraction in z s z 0 has been neglected.. Clearly, the FF input coupling efficiency decreases with excitation as a result of energy conversion to the SH mode, while the conversion efficiency to SH correspondingly increases. Notice, however, that the SH Žsolid line. experiences a maximum, whereas the FF Ždashed line. is monotonically reduced. This rather non-intuitive result can be explained by considering that, as the excitation increases, power is more and more readily transferred to the SH, reducing the interaction strength between the input field and the FF modal amplitude. The optimum coupling distance becomes longer than in the linear limit, thereby limiting the achievable power transfer until the overall photon budget available for frequency doubling begins to reduce. This is also apparent from the line Ždotted line. depicting the total normalized efficiency. Fig. 3 shows the results for the case of a slight linear detuning with respect to the Žlinear. optimum angular condition. The behavior is complicated by the phase shift taking place through ‘‘cascading’’, even though no SHG phase-mismatch is present. The linear detuning, in fact, causes the FF in-coupled components to acquire a phase mismatch with respect to the SH generated field travelling at its own phase velocity. Otherwise stated, the injected field is always in phase-matching with the locally generated second-harmonic, but out of phase with the guided mode. This is equivalent to having a nonzero wavevector mismatch D k nl in a standard SHG process. The resulting

Fig. 2. Coupling efficiency h1 at v Ždashed line., and transfer efficiency h 2 at 2 v Žsolid line., i.e. ratio of the TM 0 Ž2 v . power with respect to the input beam power, versus normalized excitation. The dotted line is the sum of h1 and h 2 . The incident beam is Gaussian and the coupling geometry is optimized to obtain h1 s80.1% at low power. Here D k nl d z s D b in d z s 0.

M.G. Masi, G. Assantor Optics Communications 146 (1998) 95–98

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Fig. 3. Same as in Fig. 2, but for D k nl d z s 0 and D b in d z s 2.

phase shift versus the FF input excitation Pi is shown in Fig. 4 for this case and the perfectly matched cases. The shift, evaluated at the end of the coupling region Žprism termination., is responsible for successive up- and downconversion cycles in the parametric interplay between FF and SH beneath the coupling region. The nonlinear response is complicated by the presence of an SHG mismatch, without ŽFig. 5a, 5b. or with ŽFig. 6a, 6b. linear detuning. While, as expected, the calculated FF and SH efficiencies do not depend on the sign of D k nl if D b in s 0 ŽFig. 5., an imperfect angular Žinput. tuning will result in an asymmetric response about phase-matching. This is shown in the surface plots of Figs. 6a ŽFF. and 6b ŽSH., displaying coupling and transfer efficiencies versus input excitation and SHG mismatch, for a fixed input detuning. These 3D graphs ŽFigs. 5, 6. clearly show that, in the presence of a quadratic susceptibility in any of the

Fig. 4. Nonlinear phase shift DFnl of FF mode, for D k nl d z s D b in d z s 0 Ždashed line. and for D k nl d z s 0, D b in d z s 2 Žsolid line..

Fig. 5. Ža. Coupling efficiency h1 at v and Žb. transfer efficiency h 2 at 2 v versus excitation and phase-mismatch D k nl d z , for D b in d z s 0.

media forming the waveguide, a high power in-coupling characterization can provide an elegant approach to the in situ evaluation of the structure nonlinear performance. Assuming that the linear parameters are known, indeed, efficiency-versus-power measurements can be used to estimate both nonlinearity and SHG mismatch, the latter in both magnitude and sign. Similar results are obtained in the case of pulsed excitations, considering the energy response to pulses such that the quasi-cw approximation holds. This is the case for pulses as short as several hundreds of femtoseconds, provided temporal walk-off can be neglected. Let us consider a fundamental wavelength l s 1.32 mm, a monomode planar waveguide formed by a polymer film with n oŽ l. s 1.7, n eŽ lr2. s 1.66, thickness h s 0.5 mm, a nonlinearity d eff s 10 pmrV, on a linear substrate with a filmrsubstrate index step D nŽ l. s 0.2, using a high-index rutile prism, an optimum air-gap thickness of 0.52 mm Žoptimized for Gaussian beam in-coupling., a Gaussian elliptic beam of axes d z s 2 mm and d y s d zr50, the nonlinear characteristics of the polymer can be evalu-

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M.G. Masi, G. Assantor Optics Communications 146 (1998) 95–98

herent cascading phase effects due to the presence of either linear or SHG detunings, or both. Since the amount of injected light can be evaluated by a power balance at the input Žincluding the reflected portion of the launched beam. or by out-coupling measurements via end-fire or another prism, the features we have investigated numerically provide a convenient approach for the in situ evaluation of the waveguide nonlinear Žquadratic. characteristics, of paramount importance in the realization of guided-wave nonlinear devices involving parametric interactions. Acknowledgements This work was supported by the Italian Research Council Žgrants no. 96.01844.CT11 and 96.02238.CT07.. References

Fig. 6. Same as in Fig. 5, for D b in d z s 2.

ated using input powers ranging from 0.19 to 1.9 kW Ži.e. Pi s 10–100.. In conclusion, the nonlinear relationship between ŽFF and SH. efficiencies and input power in a waveguide exhibiting a quadratic response is determined not only by straightforward frequency conversion, but also by the in-

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