Fresnel phase matching: Exploring the frontiers between ray and ...

Fresnel phase matching: Exploring the frontiers between ray and guided wave quadratic nonlinear optics Myriam Raybaut1*, Antoine Godard1, Alexis Toulouse1, Clément Lubin1 and Emmanuel Rosencher 1,2 1

Office National d’Etudes et de Recherches Aérospatiales (ONERA), Chemin de la Hunière, F–91761 Palaiseau, France 2 Département de Physique, Ecole Polytechnique F 91 Palaiseau, France *Corresponding author: [email protected]

Abstract: Fresnel phase matching is a convenient and universal way to phase match nonlinear three-wave mixing by total internal reflection in isotropic materials like common semiconductors. This technique makes use of the large relative phase lag between the interacting waves at total internal reflection, and was suggested by the nonlinear optics pioneers in the 70’s; it has been worked out by several teams since then but, quite unexpectedly, has never succeeded in producing enough parametric gain to achieve optical parametric oscillation. We show that this failure stems mostly from a basic law of nonlinear reflection, which leads to a spatial walk-off between the pump and the generated parametric waves, resulting in unexpected destructive interference patterns between the waves while bouncing back and forth between the interfaces. Ray tracing or plane wave analysis gives an incomplete representation of the phenomenon while highly multimodal nonlinear guided wave theory reconciles the different views. Very good agreement between the presented theory and experiments is demonstrated in gallium arsenide samples. ©2008 Optical Society of America OCIS codes: (190.4410) Nonlinear optics, parametric processes; (260.6970) total internal reflection.

References and links 1. 2. 3. 4. 5. 6.

7. 8. 9. 10.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463 - 466 (1998). A. Yariv, “Coupled-mode Theory for Guided-Wave Optics,” IEEE J. of Quant. Electron. QE-9, 919-933 (1973). K.L. Vodopyanov, O. Levi, P. S. Kuo, T. J. Pinguet, J. S. Harris, M. M. Fejer, B. Gerard, L. Becouarn, and E. Lallier, “Optical parametric oscillation in quasi-phase-matched GaAs,” Opt. Lett. 29, 1912-1914 (2004). J.A. Armstrong, N. Bloembergen, J. Ducuing, and P.S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962). G. Boyd and C. Patel, “Enhancement of optical second-harmonic generation by reflection phase matching in ZnSe and GaAs,” Appl. Phys. Lett. 8, 456-459 (1966). H. Komine, W.H. Long, J.W. Tully, and E.A. Stappaerts, “Quasi-phase matched second-harmonic generation by use of a total internal-reflection phase shift in gallium arsenide and zinc selenide plates,” Opt. Lett. 23, 661-663 (1998). R. Haidar, N. Forget, P. Kupececk, and E. Rosencher, “Fresnel phase matching for three-wave mixing in isotropic semiconductors,” J. Opt. Soc. Am. B 21, 1522 (2004). N. Bloembergen and P.S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606622 (1962). M. Born and E. Wolf, in Principles of Optics, Pergamon Press (1964). M. Raybaut, A. Godard, A. Toulouse, C. Lubin, and E. Rosencher, “Non linear effects on Fresnel phase matching,” Appl. Phys. Lett. 92, 121112 (2008).

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Received 25 Aug 2008; revised 22 Sep 2008; accepted 3 Oct 2008; published 24 Oct 2008

27 October 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18457

11. 12.

13. 14. 15.

M.M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, “Quasi-Phase-Matched Second Harmonic Generation: Tuning and Tolerances,” IEEE J. Quantum Electron. 28, 2631-3654 (1992). A.V. Smith, D.J. Armstrong, and W.J. Alford, “Increased acceptance bandwidths in optical frequency conversion by use of multiple walk-off-compensating nonlinear crystals,” J. Opt. Soc. Am. B 15, 122 (1998). D. Marcuse, in Theory of dielectric optical waveguides (Academic press, London, 1974). A. Fiore, V. Berger, E. Rosencher, P. Bravetti, N. Laurent, and J. Nagle, “Phase matched mid-infrared difference frequency generation in GaAs-based waveguides,” Appl. Phys. Lett. 71, 3622 - 3624 (1997). A. W. Snyder and R. De La Rue, “Asymptotic solution of eigenvalue equations for surface waveguide structure,” IEEE Trans. Microwave Theory Tech. 18, 650-651 (1970).

1. Introduction Mid-infrared tunable sources are of considerable interest for applications in environmental monitoring. Indeed, the main toxic agents and pollutants present strong absorbing lines in the spectral region from 3 to 12 µm. These lines are moreover highly differentiated from one species to the other, so that each pollutant has its own spectral signature (or so-called spectral fingerprint). The spectroscopic interest is reinforced by the high transmissivity of the atmosphere in the corresponding spectral band, allowing for instance LIDAR applications. Optical parametric conversion appears as a promising solution to cover this wide spectral region: one possible scheme is the down conversion of near-2 µm wave into the mid-infrared, which could be obtained either by difference frequency generation or by parametric fluorescence, leading to optical parametric oscillation. Semiconductors of the technological mainstream (such as gallium arsenide GaAs, indium phosphide InP or zinc selenide ZnSe) are attractive candidates for optical parametric conversion of mid-infrared waves. Indeed, they are transparent and display very low dispersion in the near- and mid-infrared region; they possess high nonlinear 2nd-order coefficients χ(2), and benefit from a mature technology. Moreover, they can stand a high incident energy flux. However, these materials are optically isotropic, so that no natural birefringence phase-matching scenario is possible. Nevertheless, because of the importance of the subject, various solutions have been proposed to get an efficient conversion, among which: artificial birefringence, modal phase-matching, as well as several quasi-phasematching techniques [1-3]. In their founding paper, Armstrong et al suggested using the relative phase change between the parametric and the fundamental waves upon total internal reflection to compensate their phase mismatch [4]. This principle was demonstrated by Boyd and Patel as well as by Komine et al by phase matching second harmonic generation in isotropic semiconductors (GaAs, ZnSe) [5,6]. This approach was generalized to three-wave mixing by our group [7]. We showed that this technique displayed a certain number of advantages compared to the previous ones given in references [1-3]. Firstly, the technique is poorly demanding on technology; secondly, unmatched spectral tuning can be obtained in the midinfrared, (experimentally from 8 to 13 µm tunability with a single wafer while theory predicts tunability even in the terahertz range). Rather good conversion efficiency was obtained by the different groups though, quite unexpectedly, no one ever succeeded in producing enough parametric gain to achieve optical parametric oscillation. In this paper, we show that this failure stems mostly from a basic law of nonlinear reflection already discovered by Bloembergen and his co-workers [8]. This latter phenomenon leads to a non collinear propagation of the pump and the generated parametric waves. As a consequence, the generated beams walk-off during propagation in the slab, resulting in unexpected destructive interference patterns between the waves while bouncing back and forth between the interfaces. In Section 2, we briefly recall the basic principles of Fresnel phase matching (FPM). We present in Section 3 a second harmonic generation (SHG) experiment that shows that the theoretical models presented so far are not sufficient to describe the FPM interaction. In particular, in Section 4, an intuitive description of the influence of nonlinear total internal

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reflection on the parametric conversion yield is shown. This part will serve as a guideline to the development of the multimodal theory which is given in Section 5. Here, an analytical theory based on the infinitely confining and highly confining waveguide assumptions is first developed. This allows us to show the main physical insights of FPM. Numerical models, based on this guided wave theory, of the SHG experiment are then presented and performed. Finally, the comparison between these theoretical predictions and the experimental results are presented in Section 6. 2. Principles of Fresnel phase matching Fresnel phase-matching makes use of the differential Fresnel phase shift ΔφF experienced by three interacting waves (ω1 > ω2 > ω3) at total internal reflection on the air-semiconductor interface (Fig. 1). This Fresnel phase shift ΔφF may be very large (particularly near the critical angle). It can thus compensate for any dispersion phase-mismatch Δk = k1 − k 2 − k 3 , and allows a quasi-phase-matched growth of the conversion signal: the wave vectors k i at pulsation ωi are given by k i = (ω ni / c) ei where c is the light velocity, ei and ni are the unitary direction vector and the materials optical index at angular frequency ωi respectively. x

ΔΦF

x

y

θ

z

t

φ

z y

L

(a)

(b)

Fig. 1. Scheme of the Fresnel phase-matching configuration for SHG and DFG: geometry of the Fresnel phase-matched plate: (a) side view for geometrical description; and (b) view from above for angle definition.

Let us consider the general three waves interaction scheme: ω1 − ω 2 → ω3 (and correlatively ω3 + ω 2 → ω1 ). Two cases can be analysed: –

–

ω3 = ω 2 = ω and ω1 = 2ω : this represents the second harmonic generation (SHG) case treated by [5,6]. Such case is easier to analyse and will be used here in order to highlight the physics involved in the process. ω1 and ω 2 are, for instance, output waves of a nearly degenerated optical parametric oscillator of pump angular frequency ω 0 ( ω1 ≈ ω 2 ≈ ω 0 / 2 ) and ω3 is the difference frequency generated (DFG) angular frequency. This case is of a more important practical importance but is more complex to analyse.

The two pumping waves are injected into the plate of thickness t by one of the slanted faces (Fig. 1). Due to the high optical index of the material, a total internal reflection situation is easily reached (internal reflection angle θ ). The three waves then experience multiple bounces all through the plate. The global phase shift Δφ between two successive bounces is the sum of the three following contributions: Δφ = ΔkLb + Δφ F + δφ .

(1)

Lb being the distance between two successive bounces given by Lb = t cosθ . Δφ F is given

by Δφ F = φ F1 − φF 2 − φF 3 , where φ F i is the Fresnel phase shift experienced by the wave of angular frequency ωi at total internal reflection at the semiconductor-air interface which #100597 - $15.00 USD

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depends on θ and on the different polarizations of the waves [9]. There is a third possible contribution: the effective nonlinear coefficient d eff may change value after reflection. It may be a mere change of sign, which introduces an additional phase shift δφ = π . The Fresnel phase shift Δφ F depends thus on (i) the reflection angle θ , which must be greater than the critical angle for each wave ωi ( θ ≥ θ c = arcsin(next n(ωi )) , next being the optical index of the outside medium i.e. next = 1 in this paper) ; (ii) the polarization of the three waves. This last dependency is responsible for the so-called Fresnel birefringence. Because of the large variation of Δφ F as a function of θ , Eq. (1) indicates that almost any phase mismatch Δk may be compensated, which makes Fresnel phase matching a very versatile and universal phase matching technique [7]. As demonstrated in reference [7], computation of the interacting process provides the proportionality factor between the incoming pump waves intensities I1in and I2in, and the outgoing wave intensity I3out : I3out

Z ω 2 ( Ndeff )2 ⎛ sin(ΔkLb 2) ⎞ = 0 23 ⎜ ⎟ n1n2 n3 ⎝ sin(Δk 2) ⎠ 2c

2

⎛ sin( N Δφ ⎜ ⎝ N sin( Δφ

2

2) ⎞ in in ⎟ I1 I 2 , 2) ⎠

(2)

where Z 0 is the vacuum impedance ( Z 0 = 377 Ω ). The global parametric process efficiency η is the thus product of two main parts. Firstly, there is the parametric conversion on each path Lb between two bounces, so that the first term of the conversion yield, writes ⎡⎣sin( ΔkLb

2

2) (Δk 2)⎤⎦ . Then, the individual fields generated along all the paths will interfere

with each other. This introduces the term: {sin( N Δφ 2) [ N sin(Δφ 2) ]} , where N is the 2

number of bounces inside the plate. In reference [7], we showed that resonant phase matching ( ΔkLb = (2 p + 1)π ) and non resonant phase matching ( ΔkLb ≠ (2 p + 1)π ), with p integer, were both interesting cases, which increase the versatility of the technique. As shown in Fig. 2, resonant phase matching corresponds to the optimal condition where the distance Lb between two successive bounces is exactly an odd number of coherence lengths Λc (= π / ∆k) of the nonlinear process. Optimal thickness

Phase matching I

I

z

z Δk Lb= (2p+1)π

Δk Lb + ΔΦF + δφ = 0 [2π]

Fig. 2. Resonant Fresnel phase-matching: the distance Lb between two successive bounces is optimized (Lb = (2 p + 1) Λc , where p is an integer and Λc the coherence length of the nonlinear interaction ).

Equation (2) also indicates that, when the Fresnel phase matching condition is met, the signal should grow quadratically with the number of bounces N, so that large conversion efficiencies could be obtained for long samples. The sole condition for Fresnel phasematching (FPM) is:

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Δφ (θ ) = ΔkLb (θ ) + ΔφF (θ ) + δφ = 0 [ 2π ] .

(3)

However, such a large conversion yield has never been achieved in such a way that optical parametric oscillation could be obtained. In Ref. [7], we have investigated some possible limitations of this efficiency: (i) the surface roughness and (ii) the Goos-Hänchen effect which can lead to a spatial walk off between the waves. However, these effects could not explain a lack of a factor greater than 10 between the ray theory and the experimental data that were reported in [10] and are recalled in Sec. 3. We show here that another effect introduces a more sever limitation: the nonlinear reflection law at the interfaces. 3. Second harmonic generation experiment In order to investigate in a more systematic way the discrepancies between the results given by the ray theory presented in Sec. 2 and those obtained experimentally, we carried out several second harmonic generation (SHG) experiments in a fixed FPM configuration varying only the nonlinear material sample length while other parameters were kept constant. PPLN OPO OPO

ω

=ω

1

2

Focalisation optics z φ = 30° θ = 28.5°

ω1=ω2

Translation mount

ω3

Filters IR detector

Rotation stage

t = 400 µm L

ω

3

ω

1 Filters

IR detector

(a)

(b)

(c)

Fig. 3. Experimental set-up: (a) nonlinear material samples geometry, the only varying parameter is the length L, (b) SHG experimental setup, (c) single coherence length intensity calibration experiment.

The experimental setup and the obtained results were recently presented in [10], and they are only briefly recalled in the present paper. The experiments were carried out using the 4 µm idler beam emitted by a PPLN optical parametric oscillator (OPO) as the fundamental radiation of the SHG process. The OPO is pumped by a Nd:YAG laser with a 10 kHz repetition rate, the pulse duration at 4 µm is typically 150 ns and the emitted idler beam is nearly diffraction limited ( M 2 ≤ 1.3 ). The experimental set-up is shown in Fig. 3. The p-polarized pump beam is focalised into a 100 µm waist inside GaAs samples of different lengths. The crystals facets are bevelled with an angle of 28.5° (Fig.3(a)), and mounted on a rotation stage, allowing us to inject the pump beam at the expected resonant Fresnel phase matching angle of 32.5°. The remaining 4 µm pump beam is filtered out using glass filters at the output of the plate. The 2 µm second harmonic (SH) signal is detected using a thermoelectrically cooled HgCdTe detector (Fig. 3(b)). In order to facilitate the comparison with the modelling results (see Sec. 5.2), the measured SH intensity is normalized to the SH intensity that is generated in a single coherence length. This normalising intensity was determined by maximizing the SHG in a #100597 - $15.00 USD

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GaAs prism in a single-pass configuration with the same propagation angle relatively to the crystallographic axes as in the GaAs plates (Fig. 3(c)). The obtained ratio thus represents the square of an efficient number of bounces in the plate. The measured normalized SH intensity is plotted as a function of the propagation angle and the crystal length in Fig. 4, as well as the calculated signal expected from the ray theory. First, let us notice that, apart from a 0.3° error due to both the uncertainty on the bevel angle and on the angle rotation measurement, the FPM condition arises at the expected angle, and with the correct acceptance angle (Fig. 4(a)). In terms of sample length influence, we can observe two regimes: for short crystal length ( TE) Calculation 1.2

Intensity (a.u.)

1.0 0.8 0.6 0.4 0.2 0.0 32.0

32.2

32.4

32.6

32.8

33.0

33.2

33.4

33.6

Internal propagation angle (°)

(a)

Normalized SHG Intensity

TM + TM -> TE SHG experiment Experiment Parabolic calculation from the ray theory 4000 3000 2000 1000 0 0

5

10

15

20

25

30

Crystal Length (mm) (b)

Fig. 4. Experimental results of 4 → 2 µ m FPM SHG experiment: SH intensity as a function of the internal angle (a), SHG intensity normalized to one coherence length intensity as a function of the sample length (b).

In order to rule out diffraction effects as a possible cause of the unusual variation of the SH-wave intensity as a function of the fundamental-wave intensity, we also recorded the

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experimental pump and SHG profiles by use of a SWIR camera. As shown in Fig. 5, we can notice that the beam quality of the pump is not degraded by the propagation inside the plate and that the SHG beam is also of good spatial quality. This is consistent with the fact that the Rayleigh distance of the pump beam in GaAs is larger or at least equal to the length of our samples. x profile

Intensity (a.u,)

x profile y profile

y profile

0 4000 6000 8000 10000 12000 14000 X or Y (µm)

(b)

(a)

Fig. 5. Spatial profiles of the pump beam (using an infrared Pyrocam camera) (a) and SHG beam (using a SWIR camera) (b), at the output of the GaAs crystal. Note that the profiles are only given here for a qualitative appreciation of the beam quality, not to give an actual value of the beams sizes.

4. Nonlinear reflection law As noted by Bloembergen and Pershan, when a beam of light is reflected onto nonlinear materials, the laws of reflections must take into account the conservation of the component of the wave momentum parallel to the boundary [8]. This condition implies that, besides the collinear field generated in the bulk, at the interface the nonlinear polarisation driving term radiates an “homogeneous” SH field that satisfies the following generalized Snell-Descartes law, which is visualized in Fig. 6 for a SHG situation: nω sin θω = n2ω sin θ 2ω .

(4)

This equation shows that, because of optical dispersion, the only condition for the SH wave to be a plane wave is to be in a situation where the pump wave incidence angle θω and the SHG incidence angle θ 2ω are not equal and must satisfy Eq. (4). In this manner, the reflected SH wave is collinear with the “homogeneous” SH wave generated at the interface and a plane wave propagation is preserved. The required small variation in incidence angle δθ is trivially obtained by a first order expansion, so that:

δθ ≈ −(δ n n) tan θω ,

(5)

where δ n = n2ω − nω is the optical dispersion and n ≈ n2ω ≈ nω . k2ω, homogeneous

δθ

kω k2ω, collinear

θ2ω θω Nonlinear material Air

nωsin(θω)=n2ωsin(θ2ω) Fig. 6. Nonlinear total internal reflection law

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A more elaborate approach is to unfold the zig-zag motion of the waves in the plate, as represented in Fig. 7. The incident collinear waves ω1 and ω 2 propagate at an angle θ relatively to the virtual stack. The periodicity of the stack introduces a reciprocal lattice vector which has to be determined. The exact calculation is done in the Appendix A. It leads to a vector conservation law given by: k1 + K F = k 2 + k 3 K F = Δφ F t

,

(6)

where K F is the reciprocal lattice vector due to the periodic Fresnel phase shift at the interfaces. This equation is a generalisation of the vector equation given by Fejer et al in reference [11] and leads to the same expression as (4). Equation (6) shows that, indeed, the nonlinear wave is non collinear relatively to the pump waves and that FPM Eq. (2) is an approximation which does not take the non collinearity of the waves into account.

x ×eiΔΦ ×eiΔΦ ×eiΔΦ ×eiΔΦ ×eiΔΦ

O

δθ

θ

k2 k1

k3

k2

θ

δθ

k3

z

t (a)

k1 KF

(b)

Fig. 7. Unfolding of the zig-zag motion of the waves at total internal reflection (a). Because of the nonlinear law of reflection, the waves suffer from a non collinear propagation due to the periodicity of the structure (b).

Non collinear angles of 1 to 2 degrees are thus expected in our experimental conditions, which is indeed quite important. This non collinearity leads to two effects. The first one is a spatial walk-off which decreases the overlap between the interacting beams. This effect has been evaluated using the same expression as for the birefringence walk-off correction term developed in reference [12] and leads typically to an efficiency decrease of a factor 2. It cannot alone explain the large factor 10 observed with respect to the experiments. Moreover, as shown in Fig. 8, the pump and parametric beams may recombine after a certain number of bounces: the generated beam then fills the plate completely. This recombination may lead to complex destructive interferences which may explain the small efficiency experimentally determined. The number of bounces necessary for such a recombination may be roughly evaluated. As shown in Fig. 8, the recombination occurs when the SH wave generated at the beginning of the plate and the fundamental wave experience numbers of bounces that differ form 2 for a given propagation length, so that N sin θω ≈ ( N + 2) sin(θω + δθ ) . This number Nrec of bounces is then given by: N rec ≈ 2 tan θω δθ ≈ 2 n δ n .

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(7)



This means that, after Nrec ≈ 150 bounces which corresponds typically to 15 mm long samples, the two beams can begin to overlap and destructive interferences are likely to occur. This might partly explain the discrepancy between experimental results and nonlinear plane wave theory. A more detailed analysis is now described.

Fig. 8. Recombination of the pump and SHG waves after a few bounces

5. Guided wave multimode analysis of Fresnel Phase Matching Different physical phenomena point to the inadequacy of a plane wave analysis for Fresnel phase matching. Indeed the spatial extensions of the waves are involved in two different aspects: (i) the non collinearity of the waves due to the nonlinear law of reflection, (ii) the Goos-Hänchen shift between the waves. Both effects induce a spatial walk off and interferences between the waves. It is thus natural to resort to a guided wave analysis which takes both phenomena into account in a built-in way. At first sight, it may seem that the guided wave approach is not useful because of the large number of modes which are involved in the wave propagation in the guide. Typically, the pump wavelengths are in the 0.6 µm to 10 µm range in the materials, while the sample thickness covers the 200 µm to 400 µm range, so that we are envisaging many hundred of modes. In fact, it is shown in Appendix B that only a small number of modes are actually involved (typically a few tens) under usual experimental conditions. For simplicity’s sake, and in order to validate this nonlinear guided wave theory with respect to the plane wave one, let us first describe the waveguide analysis of SHG process in an isotropic sample (DFG process is a mere generalization) for a simple case. In a second paragraph, we briefly describe the generalized modal numerical calculation used for modelling of the Fresnel phase matching experiments. 5.1 Analytical description of second harmonic generation process We shall first recall the main points of this theory which are relevant to our situation. The waves are propagating in the z direction and the polarization vectors are oriented in the y direction for the s (TE) waves and in the (x,z) plane for the p (TM) one (see Fig. 9 for the geometry). In order to highlight the main physical insights of this theory, we first analytically present the simplest TEω − TE2ω or sss case. x z x0

λ

θ

t

waist w0

Fig. 9. Fresnel phase-matching waveguide geometry.

For such a configuration, the pump and SH electric fields distributions functions are respectively given by: ω

ω

Eωy ( x, z , t ) = ∑ l Alω ( z ) E l ( x)e− i (ω t − β l z ) e y + cc ,

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(8)



and 2ω

2ω

E 2yω ( x, z, t ) = ∑ m Am2ω ( z ) E m ( x)e− i (2ωt − β m ω

z)

e y + cc .

(9)

th

In these expressions, El ( x) are the l order mode TE wave eigenfunctions, solutions of the linear Maxwell equations. Their expressions can be found in references [2, 13]. The propagation constant β lω is solution of the implicit equation: tan(α lω t ) = 2κ lωα lω

where t is the waveguide thickness. α lω =

⎛ ⎜ ⎝

(α ω ) − (κ ω ) 2

l

( nω kω )

2⎞

⎟, ⎠

l

2

( )

− β lω

2

(10)

(βω )

and κ lω =

l

2

− ( kω )

2

are

the transverse and the evanescent wavenumbers, respectively. In a ray tracing interpretation, it corresponds to a zigzag angle θ lω given by sin θ lω = β lω ( nω kω ) (see Fig. 6). The Alω ( z ) are the slowly varying amplitude of the modes. Their value at z = 0 is obtained by the projection of the input Gaussian field on the waveguide modes and we note lθ the value of l for which Alω (0) is maximum for a given incidence angle θ . As it is shown in Appendix B, few modes

( l ≈ lθ ) are involved in this projection (from 10 to 20). Let us insist that, by construction, expressions (8) and (9) take into account, in a built-in way, the Goos–Hänchen shift as well as the Fresnel phase shift. In the undepleted pump approximation ( Alω ( z ) ≈ Alω (0) ≈ Alω ), the mode coupling equation which describes the energy transfer between the fundamental and the harmonic waves is easily derived (see Ref. [2]): d − dz Am2ω, y ( z ) = i

ω ε0 p0

∑ ∑ χ y(2), y , y Sl ,l ',m ( Alω, y Alω', y ) ei δβ

l ,l ',m

z

;

(11)

l' l

p0 is a normalizing power constant which is such that the amplitude of the Poynting vector given by Π =

1

μ0

+∞ ∫

−∞

E.Bdx =

β lω 2 ωμ 0

+∞ ∫

−∞

2

Elω, y dx = p0

Π

(12)

corresponds to a power of 1 W/m in the y direction. Sl,l’,m is the overlap integral between the modes: Sl ,l ', m =

+∞ ∫

−∞

Elω, y ( x)Elω', y ( x) Em2ω, y ( x)dx ,

(13)

and δβ l ,l ',m are the mismatch in the propagation constants:

δβl ,l ',m = β lω + β lω' − β m2ω .

(14)

Integration of Eq. (11) is trivial and leads to:

(

) ( ( exp(iδβ

(2) Am2ω, x ( z ) = − ωε 0 p0 χ yyy ∑ ∑ Sl ,l ',m Alω, y Alω', y L l' l

)

l ,l ', m L ) − 1

)

δβ l ,l ',m L .

(15)

The normalization constant in Eq. (12) is such that the total amount of harmonic power generated by unit length along y axis is given by the summation over each mode contribution m: #100597 - $15.00 USD

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P2ω ( z ) = ∑ Am2ω, y ( z ) . 2

(16)

m

A careful examination of Eqs. (15) and (16) shows that the energy transfer is efficient when three conditions are met.

(

)

(i) Firstly, as expected, a sinc δβ l ,l ',m L 2 term appears in the conversion Eq. (15). In order to obtain good conversion efficiency, TE modes at pulsations ω and 2ω must be almost phase matched, i.e.:

δβl ,l ',m = β lω + β lω' − β m2ω ≈ 2kω (nω sin θ lω − n2ω sin θ m2ω ) ≈ 0 ,

(17)

This latter approximation is valid since l ≈ l ' . We can notice here that Eq. (17) is similar to Eq. (4) established in the frame of plane-wave theory. Moreover, it is important to highlight the fact that condition (17) is different in nature to the phase matching condition obtained in single mode waveguides [14]. For the latter, since single modes are involved, Eq. (17) is very stringent, as far as the waveguide thickness and the phase-matched wavelengths are concerned. This is why the tunability of nonlinear single mode waveguide is so restricted. However, in the former case, many sets of modes (l,l’,m) are involved in the parametric process so that Eq. (17) is far less stringent than in the single mode waveguides. We are in fact interested with the sets of modes with large coherence lengths Δ l ,l ', m = π δβ l ,l ',m , i.e. which satisfy the following inequality δβl ,l ',m π L or Δ l ,l ', m L . We will show that, in many situations, FPM can be obtained over a large span of wavelengths and sample thicknesses. (ii) The element of the nonlinear susceptibility tensor χ (2) yyy must not be zero. We must stress the fact that, in this guided wave approach, there is no more such consideration as the change of nonlinear coefficient at total internal reflection. This is taken into account by the value of the effective nonlinear coefficient which, in the general case, involves more than a single nonlinear susceptibility tensor element. (iii) The overlap integral Sl ,l ', m between the modes must be optimized. As we will see below, this condition mostly points to the parity of the interacting waves. In order to derive rules of thumbs which will allow us to further validate this nonlinear guided wave theory with respect to the plane wave one and derive approximate results which will be confirmed by comprehensive numerical calculations, we discuss the guided wave multimode analysis of FPM by use of expressions of the overlap integral, the phase mismatch and the mode amplitudes for highly or infinitely confining waveguides (HCW and ICW respectively). One should note that, since the analytic expressions of mode spatial profiles Elω ( x) inside the nonlinear slab involve only trigonometric functions [2], analytic expressions of overlap integrals Sl ,l ', m can be derived in the general case as well. However, the corresponding expressions are complicated and their values do not differ significantly from those calculated in the ICW case. The expression of overlap integrals derived in this latter case are thus considered in the following discussion to gain some physical insights. It can be written as follows Sl ,l ', m ≈ ( −1)

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m ( −1 )

l +l ' −( m+1) 2

Δ

8 2π

2 t β lω β lω' β m2ω

3/ 2 (ωμ0 p0 ) ,

(18)



where the mode mismatch l + l '− m = Δ is an odd integer (see Appendix C). The overlap integral is thus strongly enhanced for small ∆ values (|∆| = 1) when m ≈ l + l ' ≈ 2l , i.e. nω sin θ lω ≈ n2ω sin θ m2ω , which is a rough expression of the momentum conservation at reflection. Larger ∆ values correspond to higher non-collinearity angle between the waves and thus to lower spatial overlap between the interacting waves. The analytical expression of mode amplitudes Alω can also be derived for ICW (see Appendix B) Alω = 2iI

ωμ 0 βl t

p0

π

sin θ e 2w

− F (l ) ⎡ 2

⎣⎢

(

Erfi F ( l ) − i sin2θ

t w0

) − Erfi ( F (l ) + i

)

sinθ t ⎤ , 2 w0 ⎦ ⎥

(19)

where w0 is the incident Gaussian beam waist and F ( l ) is a function given by F (l ) =

1 w0 ⎛ π ⎜ kω cos θ − l 2 sin θ ⎝ t

⎞ ⎟ ⎠

.

(20)

The expression of the mode amplitude (19) shows that the distribution of modes involved in the Gaussian beam propagation is centred near lθ ≈ (cosθ ) kω t π , which is nothing else than the condition for waveguiding for an infinitely confining waveguide. Moreover, the number of excited modes ∆l is roughly given by Δl = 4t sin θ (π w0 ) , meaning that only few modes (typically 10) are usually involved for typical experimental conditions (see Appendix B). The ICW approximation is not suitable to derive useful analytical expressions of phase mismatch δβ l ,l ',m because it overlooks the effect of the Fresnel phase lag at the waveguide interfaces. We thus evaluate δβ l ,l ',m by use of a more sophisticated approach that relies on the Snyder–De la Rue approximation which is particularly valid in a HCW situation [15]. This yields (see Appendix D):

δβl ,l ',m =

nω kω cos θ ⎡⎢ 2 nω cos θ l t β lω∞

⎢ ⎣

nω 2 − 1

⎛ ⎜ ⎝

t cosθ l ⎤⎥ 1 1 1 ⎞ + − − Δ π − π , ⎟ ⎥ Kl Kl ' K m ⎠ Λc

(21)

⎦

where

β lω∞ =

( nω kω ) − ( l π t ) 2

2

,

(22)

is the propagation constant of mode l in an ICW and for TE polarization ⎧ ⎪1 Kl = ⎨ 2 for TM polarization ⎪ ⎩ nω

(23)

describes the TE and TM index contrast in a HCW, i.e. the effect of Fresnel birefringence. Equation (21) is the main result of this calculation. It shows nicely how FPM is used to enhance tunability, the influence of mode mismatch, birefringence, and the wide angle tuning capability of the technique. For the TEω − TE2ω case, the propagation constant mismatch is then :

δβl ,l ',m ≈

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nω kω cosθ ⎛⎜ 2nω cosθ l t β lω∞

⎜ ⎝

nω2 − 1

− Δπ −

⎞

tπ ⎟. Λ c cosθ l ⎟

(24)

⎠



The first term in bracket represents the Fresnel phase shift, the second term is the modal dispersion and the last one is the phase mismatch between two bounces of the waves. The Fresnel phase matching conditions are fulfilled once δβ l ,l ',m ≈ 0 : 2nω cos θ l nω − 1 2

−

tπ ≈ Δπ , Λ c cos θ l

(25)

with Δ an odd number. It is easily shown that such an equation is an alternative formula to the ray analysis one (3). The possible values of the mode mismatch Δ are given by the condition that Eq. (25) has a solution and the phase matched modes l are then the solutions of Eq. (25). We can also notice here that Δ represents the order of quasi-phase matching in the considered three wave mixing process. Many different terms may appear in the summation of Eq. (15). However, because of the mode overlap dependence on Δ , terms where Δ is minimal are privileged. For any values of nω and Λ c (ω ) , i.e., any frequency, and any large value of thickness t (t > Λ c ), Eq. (25) possesses solutions, which explains the large tunability of Fresnel phase matching. The solutions analytically derived in this section confirm the adequacy of the guided wave approach to our physical problem and their coherence with the results found under the ray analysis assumption. They are nevertheless approximations and a numerical analysis of the three waves mixing in the guided wave approach is required to take into account the different contributions summed in Eq. (15), for all polarization cases, and thus come up with calculated results to hold comparison with the experiment that has been presented in Sec. 3. 5.2 Description of the numerical modelling of SHG in a highly multimode guide. The developed calculation code is a generalization for all states of polarizations of the three-wave mixing process analytically described previously. Firstly, the eigenmodes fields Elω ( x) for TE modes and H lω ( x) for TM modes are calculated, as well as the propagation constants, at the pump and the SH wavelength, given the geometrical parameters considered in the calculations, such as: wavelengths, thickness, incidence angle, pump waist and polarization state. The incident Gaussian pump waves are then numerically decomposed on these eigenmodes. In order to minimize the calculation duration, a filtering process is implemented to keep only the waveguide modes that contain a significant amount of the incident power (around 10–20 modes contain the total incident power for incident waists in the 100 µm range and 400 µm thick plates). Let us recall here that [2, 13]: ω

TE,ω

EωTE ( x, z, t ) = ∑ l AlTE,ω ( z ) E l ( x)e−i (ω t − βl

e y + cc

(26)

e y + cc ,

(27)

z)

for a TE wave and, ω

TM,ω

HωTM ( x, z , t ) = ∑ l K lω ( z ) H l ( x)e−i (ωt − βl

z)

so that EωTM ( x, z, t ) =

−i ω t − β ω ω ω ω ∑ ( AlTM, ( z ) ElTM, ( x ) e x + AlTM, ( z ) ElTM, ( x)ez ) e ( ,x ,x ,z ,z

TM ,ω z l

)

+ cc (28)

l

for a TM wave. In a next step, the coefficients of the eigenmodes of the generated TE and TM second harmonic waves are calculated, under the approximation of undepleted pump. The coupling

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equations to calculate each mode coefficient, which are generalization of Eq. (15), are derived from Maxwell equations and presented in reference [2]. −

d TE,2ω ω Am , y ( z ) = −i dz 4 p0

∞ ∫

−∞

PxNL (ω , x, z ) EmTE,2ω ( x)dx

(29)

for a TE SH wave, −

d TM,2ω 1 K p, y ( z ) = dz 4ε 0 n22ω

∞ ⎡

d

∫ ⎢ −∞ ⎣ dz

PxNL (ω , x, z ) −

d NL ⎤ Pz (ω , x, z ) ⎥ H pTM,2ω ( x)dx dz ⎦

(30)

for a TM SH wave, with P NL the nonlinear driving polarization which involves the fundamental wave. For each case (TE or TM SH wave), coordinates changes have to be operated to calculate the nonlinear polarization in the crystallographic coordinates of the crystal. Indeed, in order to explore our experimental conditions, the mixing in a oriented GaAs plate (as described in Fig. 10) is considered. The overlap integrals Sl,l’,m also involved in the nonlinear polarization term are calculated by use of exact analytical expressions to minimize the calculation time. For each polarization state, we can then derive the coefficients of each SH eigenmode fields [2], and thus obtain the SH power as a function of the guide length as well as the spatial profile of the SH wave propagating along the guide. 6. Numerical modelling results and comparison with experiments 6.1 Numerical results for a 4 µm

→ 2 µm SHG process

In order to hold comparisons with the experiments presented in Sec. 3, the numerical calculations presented here are based on the 4 µm → 2 µm SHG process. As a first step before carrying out nonlinear mixing calculation, we check that the pump wave propagation is properly described using our calculation. In this purpose, we consider the propagation of a TM polarized input Gaussian beam with a 100 µm waist into a 400 µm thick gallium arsenide slab for two different incidence angles: either along the longitudinal direction the plate (90° injection angle) or injected at a 32.5° angle. Figure 10 shows the interference pattern between the excited waveguide modes along the slab which fully describes the propagation of the Gaussian beam in both cases. Indeed, as shown in Fig. 10(a) for an on-axis propagation, the natural diffraction of the Gaussian beam is accurately rendered as well as the zigzag propagation shown in Fig. 10(b) where one can notice the interference fringes occurring at each reflection in the overlap of upward and downward beams. Besides, as shown in Fig. 10(c), it is confirmed that the incident pump power is actually coupled to only few modes as expected with the analytical treatment in the ICW case presented in Sec. 5.1.

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(a)

(b)

(c) Fig. 10. Calculation of the pump wave intensity, (a) along the longitudinal direction of the guide, (b) zigzag propagation, (c) modal distribution (intensity) of the incoming pump for the zigzag case geometry.

As a second step, the consistency of the SHG model is validated in the straightforward case of a pump beam propagating along the plate axis where the expected result is wellknown. For a TM fundamental beam at 4 µm, we calculate the generated beam for both TE and TM polarizations. Figure 11(a) shows the generated TE beam at 2 µm. As expected in the case of phase mismatched SHG, the SH beam intensity displays a sinusoidal modulation along its propagation axis. The corresponding SHG coherence length of 28 µm corroborates the one calculated by use of the well-known textbook plane wave formula: Λ c = λ ⎡⎣ 4 ( n2ω − nω ) ⎤⎦ . For the TM case presented in Fig. 11(b), the SH intensity profile exhibits the same periodicity along the propagation axis. But it also displays a modulation along the transverse axis where it cancels out along the longitudinal wafer axis. We also notice that in this case, the maximal SH intensity is orders of magnitude smaller than for the TE case. These atypical features are actually fully explained by the fact that the effective nonlinear coefficient cancels out for

θ = 90° ( d

eff

(

)

= d14 3cos θ cos ϕ − sin 2 θ + cos 2 θ sin 2 ϕ , according to the angular notations

of Fig. 9).

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(b)

(a)

Fig. 11. Calculation of the SHG TE (a) and TM (b) waves generated with a TM pump wave propagating along a 300 µ m guide axis (n.b. the TM intensity is 7 orders of magnitude lower than the TE one).

Then, we calculate the Fresnel phase matched SHG process that corresponds to the experimental configuration investigated in Sec. 4.2, i.e. a 4-µm TM-polarized fundamental beam generating a 2-µm TE-polarized SH beam at an internal propagation angle of 32.5° in a 400-µm thick GaAs slab. The spatial profile of the calculated SH beam is shown in Fig. 12 in the case of a 4-mm long GaAs sample. As expected, one can observe: (i) a zigzag propagation of the generated beam at the right angle, (ii) a periodic modulation of the SH intensity along each zigzag leg that is consistent with the coherence length of our high-order (precisely 17) quasi-phase matching process, (iii) a step-growth of the SH intensity at each bounce inside the plate, meaning that the Fresnel phase matching condition is satisfied.

Fig. 12. Calculation of a TE SHG wave for a TM pump wave injected at a 32.5° angle inside a 400 µ m thick 4 mm long guide.

Similar calculations are also run for longer crystals in order to evaluate the SHG signal as a function of the sample length. The obtained results are discussed hereafter in Sec. 6.2 where they are compared to experiments on actual GaAs plates. 6.2 Experimental results and comparison

x (µm)

Figure 13 shows the results of this model applied to our experimental case, with a 30 mm long GaAs sample. Clearly the non collinear SHG wave completely fills the guide along its generation. -400 -200 0 0

0.5

1

1.5

2

2.5

z (µm)

x 10

4

Fig. 13. Calculation of a TE SHG wave for a TM pump wave injected at a 32.5° angle inside a 400 µ m thick 30 mm long guide.

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Figure 14 shows the calculation of the SHG efficiency as a function of the crystal length, with and without the 0.5% loss per bounces that were experimentally measured at 4 µm. The calculation clearly show that the conversion efficiency does not grow quadratically with the crystal length above 10 mm and even display a saturation around 20 mm. The agreement with the experimental result is rather fair and shows a strong amelioration compared to the plane wave approach. This confirms the relevance of the guided-wave approach and demonstrates that the saturation of the nonlinear efficiency in the considered FPM scheme can be satisfactorily accounted by such approach that includes the main linear and nonlinear propagation phenomena limiting the nonlinear conversion.

Normalized SHG Intensity

Experiment Waveguide calculation Waveguide calculation with losses Ray calculation with losses 4000

3000

2000

1000

0 0

5

10

15

20

25

30

Crystal length (mm) Fig. 14. Comparison of the measured and calculated SHG intensity as a function of the crystal length. Note that the point dispersion on the waveguide calculation line is an aliasing effect.

7. Conclusion New theoretical concepts are presented, which allow a better understanding of Fresnel phase matching. A guided wave approach is described, which allows us to intrinsically take into account all the physical processes involved, in particular limitations such as walk-off effects, due to Goos-Hänchen shift and nonlinear reflection at the interfaces. Our model was validated by second harmonic generation experiments in the mid-infrared. The guided wave description presented here thus allows us to model and generalize this phase matching technique to any nonlinear material plate thickness, from nearly single mode to highly multimode waveguides, to any optical wavelengths. Moreover, this model can be applied to the optimization of Fresnel birefringence phase matched plates for the infrared, allowing optimum structures to be designed and tested in the near future. We also envision to apply our calculation to the generation of terahertz waves by difference frequency generation of close infrared lines. Our model would be there an essential tool, since the semiconductor would behave as a strongly multimode guide for the pumps and a nearly single mode guide for the terahertz wave.

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Appendix A : Reciprocal lattice vector for Fresnel phase matching −i ω t −k .r Let three waves of amplitude Ei ( r ) e ( i i ) interact in a nonlinear materials in a Fresnel

phase matching configuration. We shall treat the case ω1 − ω 2 → ω3 . The generalization to any kind of second order process (SFG, SHG) is trivial. The geometry is shown in Fig. 7: pump beams ω1 and ω 2 propagate in the (x,z) plane at an angle of incidence θ − δθ and we anticipate a reflection angle of θ for the difference-frequency wave. The z axis is chosen to be collinear to the difference-frequency wave vector k3. The phase of the difference-frequency must thus be constant along x axis to be a plane wave. The nonlinear conversion differential equation thus reads: dE3 ω 2 i ( Δk z + Δk⊥ x ) E1 E2*e = −i 3 χ eff , dz n3 c

(A1)

with Δk = (k1 − k2 ) cos δθ − k1 and Δk⊥ = (k1 − k2 )sin δθ .

ℓ

In the coordinates (O, x, z), the boundary of domain number is given by the equation (see Fig. 7): z = − x tan θ + t cos θ . The difference-frequency field increment along domain number is thus given by the following integral: E3 ( z

− i ( +1)φ +1 ) e

= E3 ( z ) e− i φ3 − ei ΔφF

3

z

+1

∫ z

i

ω3 n3 c

ℓ

( 2 χ eff E1 E2* e

i Δk z + i Δk⊥ x )

dz ,

(A2)

with Δφ F = φ1 − φ2 − φ3 + δϕ NL . In order to conveniently integrate the nonlinear equation, we introduce the variable changing z ′ = z + x tan θ + t cosθ so that the integral can be rewritten: E3 ( z

+1

) exp ( −i (

− exp ( i Δφ F )

+ 1) φ3 ) = E3 ( z ) exp ( −i φ3 )

t cosθ

∫ 0

i

ω3

2 E1 E2* exp ⎜ i ⎜ Δk z ′ + ( Δk⊥ − Δk tan θ ) x + Δk χ eff

⎛ ⎛

n3 c

⎝ ⎝

t cosθ

. ⎞⎞ ⎟ ⎟ dz ′ ⎠⎠

(A3)

Since plane-wave solutions are considered, the difference-frequency field increment must not depend on coordinate x, which leads to the following condition: −Δk tan θ + Δk⊥ = 0 .

(A4)

After integration, Eq. (A3) thus yields : E3 ( z −i

+1

) exp ( −i (

+ 1) φ3 ) = E3 ( z ) exp ( −i φ3 )

⎛ ⎛ Δk t ⎞ ⎛ Δk t ⎞ 2 t * sinc ⎜ ⎟ exp ⎜ i ⎟ χ eff E1 E2 exp ⎜⎜ i 2n3 c cosθ ⎝ 2 cos θ ⎠ ⎝ 2 cos θ ⎠ ⎝

ω3

⎛ Δk t ⎜ ⎝ cos θ

⎞⎞ .

+ ΔφF ⎟ ⎟⎟ ⎠⎠

And the difference-frequency field after propagating through N domains reads: ⎛ Δk t ⎞ ⎛ Δk t ⎞ (2) ω t * E3 ( z N ) exp ( −iN φ3 ) = −i 3 sinc ⎜ ⎟ exp ⎜ i ⎟ χ eff E1 E2 θ θ 2n3c cosθ 2 cos 2 cos ⎝ ⎠ ⎝ ⎠ ⎛ ⎛ Δk t × ∑ exp ⎜ i ⎜ ⎜ l =0 ⎝ ⎝ cos θ N −1

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⎞⎞ + Δφ F ⎟ ⎟ ⎟ ⎠⎠

(A5)

(A6)



One thus obtains : 2

Δk t ω3 t 2 2 (2) ⎞ 2 ⎛ χ eff ⎟ E1 E2 sinc ⎜ n c 2 cos θ 2 ⎝ cos θ 3 ⎝ ⎠ ⎛

E3 ( N ) = ⎜ 2

⎡ N ⎛ Δk t ⎜ ⎣ 2 ⎝ cos θ

+ Δφ F ⎟ ⎥

⎞ ⎟ ⎠

⎛ 1 ⎛ Δk t ⎜ ⎝ 2 ⎝ cos θ

+ Δφ F ⎟ ⎟

sin 2 ⎢

sin 2 ⎜

⎞⎤

⎠⎦

⎞⎞

.

(A7)

⎠⎠

In this way, one recovers the Fresnel quasi-phase matching condition : Δk +

Δφ F cos θ = 0 . t

(A8)

Inserting the above equation in condition (A4), (A4) can be rewritten as Δk ⊥ +

Δφ F sin θ = 0 . t

(A9)

It is thus possible to interpret Fresnel quasi-phase matching with the following vector conservation law : k1 + K F = k 2 + k 3 ,

(A10)

where K F = Δφ F t is the reciprocal lattice vector for Fresnel phase matching. Appendix B : Mode amplitudes of a Gaussian beam in the infinitely confining waveguide (ICW) approximation Let us consider a Gaussian beam propagating in an ICW at an angle of incidence θ and focused at z = 0. We neglect any effects related to the beam inclination relatively to the entrance facet. The waist of the beam is w0 at z = 0 (see Fig. 6). The Gaussian beam at z = 0 is thus described by: Einc ( x ) =

⎛ ⎛ I exp ⎜ − ⎜ ⎜ ⎝ ⎝

2

⎞ ⎞ x sin θ ⎟ − ikω x cos θ ⎟ , ⎟ w0 ⎠

(B1)

⎠

where I is the intensity of the beam. The projection of the Gaussian beam onto the eigenmodes ωμ 0 ⎛ π ⎞ Elω, y ( x ) = 2 p sc l x is given by: β l t 0 ⎜⎝ t ⎟⎠ 2

⎛ ⎛ ⎞ t/2 ⎞ ωμ 0 x ⎛ π ⎞ p0 I ∫ exp ⎜ − ⎜ sin θ ⎟ − ikω x cos θ ⎟ exp ⎜ il x ⎟ dx . ⎜ ⎝ w0 ⎟ βl t ⎝ t ⎠ −t / 2 ⎠

Alω = 2

⎝

(B2)

⎠

In fact, Eq. (B2) is a short expression from which the projection on (i) even modes (l odd and sc=cos) is the real part of (B2) and (ii) on odd modes modes (l even and sc=sin) is the imaginary part. This expression is easily calculated and leads to :

)

ωμ 0 π 2 p0 sin θ exp − F ( l ) βl t 2w

⎡ ⎛ × ⎢ Erfi ⎜ F ⎢ ⎝ ⎣

⎛ sin θ t ⎞ sin θ t ⎞ ⎤ (l ) − i ⎟ − Erfi ⎜ F ( l ) + i ⎟⎥ 2 w0 ⎠ 2 w0 ⎠ ⎦⎥ ⎝

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(

Alω = 2iI

(B3)



where 1 w0 ⎛ π⎞ ⎜ kω cos θ − l ⎟ , 2 sin θ ⎝ t ⎠ and Erfi( z ) = −i Erf(iz ) , Erf(z) being the error function. F (l ) =

(B4)

From expression, (B3), it appears that the mode amplitude is maximal for F ( l ) ≈ 0 i.e. for kω cosθ ≈ l π t which is nothing else than the condition for wave guiding for an infinitely

ω

confining waveguide. Moreover, the e–1 FWHM of the Al distribution in l is given by the condition: F (l ) =

1 w0 ⎛ π⎞ ⎜ kω cos θ − l ⎟ = ±1 , t ⎠ 2 sin θ ⎝

(B5)

so that the number of modes involved in the Gaussian beam propagation is roughly given by: Δl =

4 t sin θ . π w0

(B6)

This latter result shows that, indeed only few modes are expected to participate to the Gaussian beam propagation and nonlinear process. For example, in a typical experimental case where θ ∼ 30° , t ∼ 400 µm, w0 ∼ 100 µm, Δl ≈ 3 . We can thus expect in such cases that around 10-20 modes should be sufficient to describe the injected waves, as confirmed by the numerical decomposition (Fig. 7(c)). Moreover, the smaller the Gaussian beam waist w0, the higher the diffraction effects and the number of modes necessary to describe the beam propagation. Appendix C : Overlap integral in an infinitely confining waveguide As far as the overlap integrals are concerned, the phase between the waves are less important and the infinitely confining waveguide approximation is fair enough. The eigenmode electric field distributions for TE waves are given by :

ωμ0 ⎛ π ⎞ p sc l x . β l∞ t 0 ⎜⎝ t ⎟⎠

Elω, y ( x ) = 2

(C1)

where sc is the sine function if l is even (odd modes) and cosine if l is odd (even modes). The overlap integral for a SHG process is thus given by: ⎛ ωμ0 ⎜ 2ω ⎝ t ω ω β β β

2

Sl ,l ', m = 8

l∞ l '∞

m∞

⎞ p0 ⎟ ⎠

3/ 2 t / 2

∫

−t / 2

⎛

π

⎝

t

sc ⎜ l

⎞

⎛

π

⎠

⎝

t

x ⎟ sc ⎜ l '

⎞

⎛

π

⎠

⎝

t

x ⎟ sc ⎜ m

⎞

x ⎟ dx . ⎠

(C2)

A careful examination of Eq. (C2) shows that the overlap integral is non zero only if l + l '− m = Δ is an odd integer. In that case, for large values of mode number l and m, the overlap integral may be approximated by: Sl ,l ', m ≈ ( −1)

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m

( −1)

l + l '− ( m +1) 2

l + l '− m

8 2π

2 t

ω

ω

β l∞ β l ' ∞ β m2ω∞

(ωμ0 p0 )3/ 2 .

(C3)



Appendix D : Phase mismatch in a highly confining waveguide Equation (15) shows that the energy flows from the mode l to the mode m proportionally to

(

the phase mismatch (PM) term sinc δβ l ,l ',m L 2

)

2

. Let us evaluate the wave-vector mismatch

δβ l ,l ',m for a highly confining waveguide (HCW). We will use the Snyder-De La Rue approximations which are particularly valid in a HCW situation [14]. The field distributions display a sin (α l x ) or cos (α l x ) functional form inside the waveguide, depending on whether the modes are odd or even, with α l related to the propagation constants β l by α l 2 + β l 2 = nω 2 kω 2 . The even (respectively odd) modes correspond to odd (respectively even) values of integer number l or m. For instance, the TE0 correspond to l = 1 and the field distribution is a cos one. In the Snyder-De La Rue approximation, the coefficients α lω are given by:

α lω ≈ l

π t

exp ⎡⎣ −1/ ( K l Vω ) ⎤⎦ ,

(D1)

t nω 2 − 1 2

(D2)

with Vω = kω

and ⎧⎪1 Kl = ⎨ 2 ⎪⎩ nω

for TE polarization for TM polarization

(D3)

The propagation constants for the (ω ,l ) and ( 2ω , m ) modes are then respectively:

( nω kω )

β lω =

2

⎛

π

⎝

t

−⎜l

⎞ ⎟ ⎠

2

exp ( −2 / K lVω )

(D4)

and

β m2ω =

⎛

π

⎝

t

( 2n2ω kω ) − ⎜ m 2

⎞ ⎟ ⎠

2

exp ( −2 / K mV2ω )

(D5)

Noting that V is very large in these highly multimodal waveguides (t large), expression (D4) can be expended to:

β lω ≈ β lω∞ + where β lω∞ =

( nω kω )

2

⎛

π

⎝

t

−⎜l

⎞ ⎟ ⎠

⎛ π ⎞ l ω ⎜ t ⎟ K l Vω β l∞ ⎝ ⎠

1

2

(D6)

2

is the propagation constant in an infinitely confining

waveguide ( V = ∞ ). We introduce now the mode mismatch number Δ = l + l '− m ( Δ

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