Carlos Montes
email: carlos.montes@unice.fr

Pierre Aschieri

Antonio Picozzi

Model for Coherence Transfer in a Backward Optical Parametric Oscillator

Keywords: nonlinear coherence transfer, backward mirrorless OPO, counterpropagating three wave interactions

The mirrorless backward optical parametric oscillator (BOPO), where the signal and idler waves are propagating in opposite directions, will establish a distributed feedback mechanism and thus optical parametric oscillation without the need to apply mirrors or external feedback to the cavity. It has been recently demonstrated experimentally by exploiting the periodic poling technique in second-order nonlinear crystals, that the sub-micrometer structured medium achieves an efficient quasi-phase-matching of the three wave interaction in the backward configuration. A remarkable property of such BOPO is the high degree of coherence of the backward wave component, whose spectrum may be several order of magnitudes narrower than that of the pump, due to the convectioninduced phase-locking mechanism. Experimentally and numerically proved the transfer of coherent phase modulations from the pump wave to the parametrically down-converted waves, we show here that this is also possible for a broad bandwidth spectrally incoherent pump. In order to accurately describe the nonlinear counter-propagation dynamics of the three dispersive waves, we have developed for the first time to our knowledge a new numerical scheme which combines the method of the trajectories usually employed to solve the three-wave interaction and the intraband group velocity dispersion effect is performed in the spectral domain with the help of the Fast Fourier Transform (FFT) technique. The model accurately conserves the number of photons and the Manley-Rowe invariants. This allowed us to predict various configurations of MOPOs in which, thanks to the convection-induced phase-locking mechanism, a highly coherent backward wave is spontaneously generated from a highly incoherent pump wave.

Introduction

It has been theoretically predicted [START_REF] Harris | Proposed backward wave oscillation in the infrared[END_REF] that counter-propagating parametric interactions, where the signal and idler waves are propagating in opposite directions, will establish a distributed feedback mechanism and thus optical parametric oscillation without the need to apply mirrors or external feedback to the cavity. Such a device, the mirrorless optical parametric oscillator (MOPO), has been recently realized by employing engineered second-order nonlinear crystals, namely, periodically poled KTiOPO 4 (PPKTP) with the periodicity of the structure of 800 nm [START_REF] Canalias | Nature Photon[END_REF]. One of the remarkable properties of the MOPO was a strong asymmetry in the spectral bandwidth of the signal and idler pulses: the signal spectral bandwidth was of the same order as the pump, whereas the counterpropagating idler wave was characterized by a spectral bandwidth about two orders of magnitude narrower with respect to those of the signal and pump waves [START_REF] Pasiskevicius | Mirrorless OPO: first steps towards unlocking the potential of counter-propagating three-wave interactions[END_REF]- [START_REF] Strömquist | Middle Infrared Coherent Source in Mirrorless Optical Parametric Oscillator[END_REF]. It has been recently experimentally and numerically investigated that this effect occurs when the pump waves exhibits coherent phase modulations: the phase modulations of the pump are coherently tranferred to the co-propagating parametric wave, while the 1 counter-propagating wave retains a narrow bandwidth and high coherence [START_REF] Strömquist | Coherent phase-modulation transfer in counterpropagating parametric down-conversion[END_REF]. Our aim in this work is to prove that this effect also works when the pump is strongly incoherent characterized by a broad bandwidth stochastic spectrum.

The mechanism underlying such a coherence transfer process is based on a convection-induced phase-locking mechanism [START_REF] Picozzi | Parametric Three-Wave Soliton Generated from Incoherent Light[END_REF]- [START_REF] Montes | Coherent signal from incoherently cw-pumped singly resonant Ti:LiNbO3 integrated optical parametric oscillators[END_REF]. In substance, the random phase modulations of the pump are transferred to the co-propagating wave, which "absorbs" the incoherence of the pump wave and thus allows the counter-propagating wave to evolve toward a highly coherent state. For this purpose we have developed a new numerical scheme for the integration of the counter-propagating three-wave interaction in the presence of group velocity dispersion. The numerical scheme combines the method of the trajectories usually employed to solve the nonlinear three-wave interaction and the intraband group velocity dispersion effect is performed in the spectral domain with the help of the Fast Fourier Transform (FFT) technique. To our knowledge this combined numerical method has been rendered stable for the first time. We use it to provide accurate numerical simulations of the referred experimental results on the dynamics of broad bandwidth pumped MOPOs.

The mirrorless optical parametric oscillator (MOPO) relies on counter-propagating three wave mixing (TWM) for establishing a distributed positive feedback for sustained oscillation above threshold [START_REF] Ding | Backward optical parametric oscillators and amplifiers[END_REF]. It is a general property of TWM that the total interaction phase is fixed. In the case of a down-conversion process, the phases φ j (where j = p, s, i) satisfy the relation:

φ p -φ s -φ i = π/2. ( 1
)
Here we demonstrate that the maximized convective wave separation of the counter-propagating TWM is responsible for the unusual spectral and coherence properties of the generated signal and idler waves. The physical mechanism ensures that, for a broadband incoherent pump, the frequency bandwidth of the backward-propagating wave, either the idler for the already realized experience or the signal for the here proposed experience, will be substantially narrower and given by the backward wave coherence time. As a consequence of the phase relation in Eq.( 1), the phase modulation of the pump is then effectively transferred to the co-propagating parametric wave. The major difficulty in achieving mirrorless parametric oscillation is associated with satisfying the momentum conservation condition, which in scalar form reads k p = k sk i for the backward idler or k p = k ik s for the backward signal. In homogeneous dielectrics, the phase matching condition can possibly be satisfied only for large signal-idler detunings. However, due to the very different diffraction properties of the near-infrared signal and the far-infrared idler, achieving oscillation in homogeneous dielectrics is problematic. By employing ferroelectric nonlinear crystals structured with sub-micrometer periodicity, Λ QPM = 2π/(k pk s + k i ) ≃ 800 nm, quasi-phase matching (QPM) [START_REF] Armstrong | Interaction between light waves in nonlinear dielectric[END_REF] has been utilized to achieve MOPO with counter-propagating idler k i in the mid-infrared spectral range [2] [6]. We show in Fig. 1 the different OPO resonant wave vector configurations: three copropagating waves, backward idler wave, backward signal wave, and backward signal and idler waves. The quasi-phase matching periodicity Λ QPM = 2π/K G runs from several microns for the first to a fourth of micron for the last. The three-wave interaction is known to model the evolution of the slowly-varying envelopes, A j (j = p, s, i), of the three coupled pump, signal and idler waves

(∂ t + v p ∂ x + γ p + iβ p ∂ tt )A p = -σ p A s A i (∂ t + v s ∂ x + γ s + iβ s ∂ tt )A s = σ s A p A * i (2) (∂ t -v i ∂ x + γ p + iβ i ∂ tt )A i = σ i A p A * s
where σ j = 2πd eff v j /λ j n j are the nonlinear coupling coefficients, d eff the efficient quadratic susceptibility, and γ j and β j the damping and dispersion coefficients. The idler A i wave is counter-propagating with respect to the pump. The quasi-phase-matching conditions in the backward configuration read

ω p = ω s + ω i and k p -k s + k i = K G = 2πm/Λ QPM , (3)
where K G is the quasi phase-matching (QPM) grating pitch vector, m is the order of the interaction and

Λ QPM = [n p /λ p - n s /λ s + n i /λ i ] -1
is the modulation period of the quadratic nonlinear coefficient.

The plane wave monochromatic model [START_REF] Ding | Backward optical parametric oscillators and amplifiers[END_REF] satisfactorily estimates the order of magnitude of the MOPO threshold for the counter-propagating interaction, namely I pth = cε 0 n p n s n i λ s λ i /2L 2 d 2 eff . Therefore, considering as in the experimental studies [START_REF] Canalias | Nature Photon[END_REF] [6] a crystal length of L = 0.65cm, one obtains I pth = 1.12 GW/cm 2 . Note that such a MOPOs threshold may increase substantially in the presence of an incoherent pump.

We have investigated theoretically the process of phase modulation transfer in MOPO by numerically solving the coupled wave equations and employing different models for the pump pulse, including chirped pulses and pulses containing random phase modulations. The numerical results support very well the experimental results. Note that we have already considered the convection-induced phase-locking mechanism, and the associated process of coherent wave generation from incoherent excitation, in the usual forward configuration of singly resonant (integrated) optical parametric oscillators (SIOPO) [START_REF] Picozzi | Coherence properties of the parametric three-wave interaction driven from an incoherent pump[END_REF] - [START_REF] Montes | Coherent signal from incoherently cw-pumped singly resonant Ti:LiNbO3 integrated optical parametric oscillators[END_REF]. In particular, it was numerically proved that the waveguide geometry considerably increases the conversion efficiency of the nonlinear device, which thus permits to make use of a low-power cw-pumping [START_REF] Sohler | Digest of the Third International Conference on Integrated Optics and Optical Fiber Communication[END_REF] - [START_REF] Hofmann | Nonlinear Guided Waves and Their Applications[END_REF]. However, the conventional forward resonant OPOs impose severe constraints on the resonant phase-matching conditions: one needs to match the group velocity of the pump wave with one of the daughter waves, while still preserving a large group velocity difference (i.e. large convection) with the other daughter wave [START_REF] Montes | Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator[END_REF] [START_REF] Montes | Coherent signal from incoherently cw-pumped singly resonant Ti:LiNbO3 integrated optical parametric oscillators[END_REF]. We underline that this last condition is automatically satisfied in the backward MOPO, since the counter-propagating waves ensure a maximum convection, so that we just need to match the group velocity of the pump with the other wave.

Convection-induced phase-locking mechanism

The coherence properties of the parametric three-wave interaction driven from an incoherent pump has been the subject of an analytical study where the autocorrelation functions are mathematically evaluated in the presence of dispersion [START_REF] Picozzi | Parametric Three-Wave Soliton Generated from Incoherent Light[END_REF] [8] [START_REF] Picozzi | Influence of dispersion on the resonant interaction between three incoherent waves[END_REF], and the convection-induced phase-locking mechanism has been proposed for the forward OPOs configurations [START_REF] Montes | Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator[END_REF] [10]. Let us present here some simple analytical arguments enlighting this mechanism from equations (2) for the backward idler configuration. Let us consider the dispersionless case (β j = 0), σ s = σ i = σ p /2 = σ , and the linear undepleted pump limit with γ p = 0. The incoherent pump may be modeled as a stationary single-variable stochastic function A p (z) with the following autocorrelation function

A p (z -z ′ ) A p * (z ′ ) |A p (0)| 2 = exp(- |z| λ c ), (4)
λ c = v p τ c being the correlation length of the pump in its own traveling frame,

z = x -v p t.
The correlation time is τ c ≃ 1/π∆ν p , where ∆ν p is the (broad)-bandwidth of the incoherent pump spectrum. The role of convection in the coherence of the daughter waves A s and A i may be analyzed by integrating the second equation ( 2) along the characteristic of the signal wave. Then, the third equation (2) yields

DA i = σ 2 t 0 e -γ s (t-t ′ ) A p (z)A p * (z ′ )A i (x ′ , t ′ )dt ′ (5) 3 where D = ∂ /∂ t -v i ∂ /∂ x + γ i z ′ = z -(v s -v p )(t -t ′ ) ; x ′ = x -v s (t -t ′ )
If v s = v p we have z ′ = z and we can extract the pump amplitude from the integral, i.e. A p (z)A p * (z ′ ) = |A p (z)| 2 , which reveals that the idler evolution is independent of the pump phase fluctuations Φ p (z). This means that the rapid random phase fluctuations of the pump do not affect the idler, which undergoes slow phase variations and thus evolves towards a highly coherent state during its parametric amplification. Let us now consider the evolution of the signal wave from the second equation (2):

A s (x, t) = σ t 0 e -γ s (t-t ′ ) A p (z ′ )A i * (x ′ , t ′ )dt ′ . ( 6
)
When v s = v p we have z ′ = z and A p (z ′ ) becomes independent of t ′ , which leads to a signal amplitude A s that is simply proportional to the pump amplitude A p i.e., the signal field absorbs the noise of the co-moving pump field. Note that this pump-signal phase-locking mechanism does not require an exact matching of the group velocities

v s = v p . It is indeed sufficient that |v s -v p | ≪ λ c γ s = v p τ c γ s , (7)
to remove the pump field from the integral so that the idler field follows the rapid pump phase fluctuations.

Highly incoherent pump pulse

In order to test the new numerical model governing the dispersive three-wave interaction, we have studied the backward MOPO pumped by a highly incoherent pump pulse. The numerical scheme combines the method of the trajectories that proved efficient in solving the large convections involved in the counterpropagating configuration of the three wave interaction through a Runge-Kutta algorithm of 4th order [START_REF] Montes | Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor[END_REF][START_REF] Montes | Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons[END_REF], together with the spectral method of fast Fourier transform (FFT) that integrates the second-order temporal derivatives of Eqs.(2) in the Fourier frequency domain. To our knowledge, this is the first time that these two fundamentally different numerical schemes are combined to integrate the counterpropagating dispersive TWI. The scheme of the model is shown in figure 2. The highly incoherent pump pulse is injected at the entry of the MOPO box (of lenght L) where the nonlinear three wave interaction takes place starting from signal and idler spontaneous noises. For solving the nonlinear time evolution in Eqs.

(2) we use the well known numerical model which follows the trajectories and where the integration is done through the Runge-Kutta algorithm. A larger box (M = L+2d) including the first one, discretized in 2 N (N = 16 to 18) points, is used to perform the FFT for evaluating the dispersion terms in Eqs. [START_REF] Canalias | Nature Photon[END_REF]. Smoothed decreasing curves to zero join the input and output wave amplitudes to the ends of this box for correctly performing the FFT. Let us briefly comment some interesting experiments where the process of incoherent parametric excitation has been investigated. In Refs.

[21]- [START_REF] Dubietis | Combining effect in a multiple-beam-pumped optical parametric amplifier[END_REF] the authors experimentally demonstrate that a coherent amplification may be achieved for a single signal wave through its coupling with two distinct pump beams that are not correlated with each other. In other terms, for specific phase-matching conditions, a single signal wave may be phase matched to a couple of pump waves and to the corresponding set of idler waves, so that the signal mode may be efficiently amplified by taking advantage of the two distinct uncorrelated pump beams simultaneously. Moreover, this process of cumulative pump action has also been observed in the spatial domain owing to conical optical beams by exploiting their specific phase-matching conditions [24] [25]. Although these experiments corroborate the results of the convection-induced phase-locking mechanism, it would be of great interest to observe in a straightforward way the predicted phenomenon of incoherent excitation of a coherent signal in a MOPO. Let us also mention that the convection-induced phase-locking mechanism has been also recently predicted to occur in an optical fiber system by exploiting the peculiar dispersion properties of photonic crystal fibers [START_REF] Yan | Coherent light wave generated from incoherent pump light in nonlinear Kerr medium[END_REF].

In the numerical simulations we consider an incoherent pump pulse of spectral bandwidth ∆ν p = 15 THz, for the experimentally accessible wavelengths: λ p = 861.67 nm, λ s = 1217.94 nm, and λ i = 2945.73 nm. This point of operation is plotted on the group velocity dispersion curves reported in figure 3. The crystal length of L = 0.65 cm in the MOPO provides a photon flight time in the cavity t r = L/v p = 41 ps and a FSR = v p /(2L) = 12 GHz. This configuration corresponds to L = 14Λ 0 , where Λ 0 = v p τ 0 = 0.46 mm is the nonlinear characteristic length. The corresponding KTP refractive index, group velocity and dispersion coefficient for the pump are n p = 1.84044, v p /c = 0.526359, β p /v p = 0.254223ps 2 /m; respectively for the signal are n s = 1.824484, v s /c = 0.537107, β s /v s = 0.142531ps 2 /m; and respectively for the idler are n i = 1.784471, v i /c = 0.537301, β i /v i = -0.328753ps 2 /m. The group velocity difference between the pump and the co-propagating signal is |v pv s |/v s ≃ 0.02. The incoherent pump pulse is characterized by an exponential-shaped correlation function < A p (x = 0,t ′ + t)A p * (x = 0,t) > = A p 2 exp(-|t|/τ c ), where τ c = 1/π∆ν p is the correlation time and ∆ν p the bandwidth of the pump spectrum. The results are shown in Figs. 4(a)-(c) and Figs. 5(a)-(c) for a FWHM pump pulse duration of 100 ps and a pump intensity of I p = 4 GW/cm 2 .

OPO L= M -2 d M = 2 < > < > M N d d
In Fig. 4(a)-(c) we see that the MOPO starts oscillating after an initial delay of 120 ps and then the parametric waves increase up to a relative mean amplitude of < |A s (L)| > / < |A p (L)| > = 0.1308 for the signal and < |A i (0)| > / < |A p (L)| >= 0.07964 for the idler. Let us stress the remarkable gain of coherence of the counter-propagating idler wave with respect to the pump, which exceeds two orders of magnitude, ∆ν i /∆ν p ≃ 1/245, as illustrated in Figs. 5(a)-(c). We have always obtained an excellent transfer of incoherence from the pump to the signal, thus allowing the counter-propagating idler wave to evolve toward a highly coherent state (∆ν i /∆ν p ≃ 10 -2 -10 -3 ). This shows that the convection-induced phase-locking mechanism is efficient for both deterministic phase modulations and random phase fluctuations.

Invariants conservation

Let us make a brief comment concerning the conservation of the number of photons in the three-wave mixing process. Neglecting the dissipation (γ j = 0, j = p, s, i), the total number of photons contained in the initial pump pulse namely,

N ph = ∞ -∞ |A p (x, t = 0)| 2 dx,
must be conserved through the interaction in the crystal. More precisely, the balance equation for the output of the forward pump and signal waves at x = L and of the idler at x = 0, should be written in the following form

t=∞ t=0 |A p (x = 0, t)| 2 dt = t=∞ t=0 |A p (x = L, t)| 2 dt + t=∞ t=0 |A s (x = L, t)| 2 (2σ s /σ p ) dt + t=∞ t=0 |A i (x = 0, t)| 2 (2σ i /σ p ) dt. (8)
The numerical simulations conserve the total number of photons better than 10 -5 for more than 6 × 10 6 time steps and high pump incoherence. The dispersion relations of the backward three-wave interaction (TWI) process read

ω j (k) = v j + 1 2 β j k 2 ( j = p, s) ; ω i (k) = -v i + 1 2 βi k 2 (9) 6
where β j = 2β j v j 2 and retaining only the first terms for the perturbative dispersion terms in Eqs.(2) they read

(∂ t + v p ∂ x + γ p + i 1 2 βp ∂ xx )A p = -σ p A s A i (∂ t + v s ∂ x + γ s + i 1 2 βs ∂ xx )A s = σ s A p A * i ( 10
) (∂ t -v i ∂ x + γ p + i 1 2 βi ∂ xx )A i = σ i A p A * s
These equations conserve the Manley-Rowe invariants

I 1 = N p σ p + N s σ s ; I 1 = N p σ p + N i σ i ( 11
)
where N j = |A j (x, t)| 2 dx (j=p,s,i). These invariants are also conserved in our simulations better than 10 -5 .

3rd order MOPO-3

According to the velocity-locking mechanism, the process of coherence transfer becomes more efficient when the group velocity of the co-propagating waves are matched [START_REF] Picozzi | Coherence properties of the parametric three-wave interaction driven from an incoherent pump[END_REF], but this requires another choice of resonant frequencies for the backward MOPO [START_REF] Montes | Backward three-wave optical solitons[END_REF]. In order to take advantage of a type I configuration in PPKTP, it is more convenient to chose the signal wave as the backward field, so that the group velocities of the co-propagating pump and idler waves may be matched exactly, v p = v i . Indeed, such a group velocity matching may be obtained for the following choice of the three wavelengths: λ p = 1060 nm; λ s = 1534 nm; and λ i = 3429 nm. The corresponding refractive indexes, group velocities and dispersion coefficients are: n p = 1.8300, v p /c = 0.5338, β p /v p = 0.1826ps 2 /m; n s = 1.8163, v s /c = 0.5401, β s /v s = 0.0761ps 2 /m; n i = 1.7709, v i /c = 0.5338, β i /v i = -0.5792ps 2 /m. This point of operation is plotted on the group velocity dispersion curves reported in figure 6. Let us note however that this configuration has a major drawback. The modulation period of the nonlinear coefficient for the first order resonant grating should be as short as 417 nm. In the following, we thus propose to consider the next odd order m = 3 [START_REF] Ding | Backward optical parametric oscillators and amplifiers[END_REF], which thus makes the polarization poled grating of Λ QPM = 1251 nm experimentally accessible. For a 1 cm PPKTP crystal lenght, the oscillation threshold would be now I pth (m = 3) = 9I pth (m = 1) = 4.95GW/cm 2 , which would be lower than the damage threshold for a 100 ps pump pulse. We performed numerical simulations of this configuration by considering an input pump intensity of I p = 5 GW/cm 2 . In Eqs.(2) we must change the sign of the signal and the idler group velocities, because the counter-propagating wave now refers to the signal component, and

Λ QPM = [n p /λ p + n s /λ s -n i /λ i ] -1
is the new modulation period of the quadratic nonlinear coefficient. Now, this configuration, perfectly satisfies the phaselocking condition [START_REF] Picozzi | Parametric Three-Wave Soliton Generated from Incoherent Light[END_REF] where the idler takes the place of the signal, namely |v iv p | ≪ λ c γ i = v p τ c γ i . The results of the numerical simulations are reported in figures 7 and 8. We consider here a highly incoherent pump pulse of 35 THz bandwidth and 100 ps duration. As expected from the convection-induced phase-locking mechanism, almost all the incoherence of the pump wave is transferred to the co-propagating idler wave, which thus leads to the generation of a highly coherent backward signal pulse. As remarkably shown in the figure 8(a)-(c), the gain of coherence of the signal with respect to the pump now exceeds three orders of magnitude.

Conclusion

In summary, we have shown that singly backward MOPOs pumped by an incoherent field are characterized, as a general rule, by the generation of a highly coherent backward wave. This remarkable property finds its origin in the convectioninduced phase-locking mechanism, in which the incoherence of the pump is absorbed by the co-propagating wave, thus allowing the backward wave to evolve toward a highly coherent state [START_REF] Picozzi | Parametric Three-Wave Soliton Generated from Incoherent Light[END_REF]- [START_REF] Montes | Coherent signal from incoherently cw-pumped singly resonant Ti:LiNbO3 integrated optical parametric oscillators[END_REF]. On the basis of the recent experiments [START_REF] Canalias | Nature Photon[END_REF] [6], we propose other realistic experimental conditions that may be implemented with currently available technology and in which backward coherent wave generation from incoherent excitation may be observed and studied. We note that it would be interesting to analyze theoretically these phenomena by making use of the kinetic wave theory [START_REF] Zakharov | Kolmogorov Spectra of Turbulence I[END_REF]- [START_REF] Zakharov | One-dimensional wave turbulence[END_REF], in line with the recent works on optical wave thermalization [START_REF] Pitois | Velocity Locking of Incoherent Nonlinear Wave Packets[END_REF]- [START_REF] Barviau | Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber[END_REF] and condensation [START_REF] Dyachenko | Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation[END_REF]- [START_REF] Picozzi | Coherence absorption and condensation induced by thermalization of incoherent fields[END_REF]. Furthermore, a detailed analysis of the dynamics of the MOPO may reveal some connection with a class of solitary wave solutions spontaneously generated in the backward configuration of the three-wave interaction [START_REF] Montes | Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor[END_REF] [START_REF] Montes | Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons[END_REF].

Fig. 1 : 2 2

 12 Fig. 1: Resonant wave vector configurations for OPOs

Fig. 2 :

 2 Fig. 2: Scheme for the Runge-Kutta-FFT numerical model in the backward mirrorless OPO.

Fig. 3 :

 3 Fig. 3: Group velocity dispersion for KTiOPO 4 at 20 o C (MOPO-1 of type I).

Fig. 4 :

 4 Fig. 4: Temporal amplitude evolution in the MOPO for (a) the forward pump pulse, (b) the forward signal, and (c) the backward idler (c).

Fig. 5 :

 5 Fig. 5: (a) Incoherent pump spectrum of broad-bandwidth ∆ν p = 15 Thz, (b) signal spectrum of ∆ν s = 5.7 Thz absorbing the pump stochasticity, and (c) backward idler spectrum with ∆ν i = 57 Ghz showing a coherence transfer of more than 2 orders of magnitude. [FSR = 12 GHz; Λ 0 = L/14 = 0.46 mm; τ 0 = v p /Λ 0 = 2.95 ps]

Fig. 6 :

 6 Fig. 6: Group velocity dispersion for KTiOPO 4 at 20 o C (MOPO-3 of type I).

Fig. 7 :

 7 Fig. 7: Temporal amplitude evolution in the MOPO3 for (a) the forward pump pulse, (b) the backward signal, and (c) the forward idler (c).

Fig. 8 :

 8 Fig. 8: (a) Incoherent pump spectrum of broad-bandwidth ∆ν p = 35 Thz, (b) backward coherent signal spectrum of ∆ν s = 28 Ghz showing a coherence transfer of more than 3 orders of magnitude, and (c) forward idler spectrum with ∆ν i = 17 Thz absorbing the pump stochasticity. [FSR = 12 GHz; Λ 0 = L/14 = 0.46 mm; τ 0 = v p /Λ 0 = 2.95 ps]

Acknowledgements

The authors thank G. Strömqvist and V. Pasiskevicius for stimutating discussions and the GDR PhoNoMi2 n o 3073 of the CNRS (Centre National de la Recherche Scientifique) devoted to Nonlinear Photonics in Microstructured Materials.


Pierre Aschiéri

Valérie Doya

Ray dispersion strongly modified by a periodic index segmentation

Keywords: Segmented waveguides, Ray dispersion, Chaos, Integrated optics PACS: 42.82.Et, 42.15.Dp, 05.45.Gg

The purpose of this article is to analyze the impact of an index segmentation on the ray dispersion in the geometrical approach. The presented study reveals that a periodic index segmentation of the waveguide along with a transverse gaussian index variation exhibits complex ray dynamics that strongly modify ray dispersion properties.

Introduction

A Periodic Segmented Waveguide (PSW) is characterized by an array of high index region along the direction of propagation and by a transverse index profile of high index segments which depends on the waveguide fabrication process. Those waveguides are well known to be very interesting for many purpose such as mode tapers or mode filters [START_REF] Weissman | Modes of periodically segmented waveguides[END_REF][START_REF] Chou | Adiabatically tapered periodic segmentation of channel waveguides for mode-size transformation and fundamental mode excitation[END_REF][START_REF] Ortega | Analysis of "Quasi Modes" in Periodic Segmented Waveguides[END_REF][START_REF] Castaldini | Soft-Proton-Exchange Tapers for Low Insertion-Loss LiNbO3 Devices[END_REF], they have also been used to achieve efficient nonlinear guided wave interaction using the Quasi Phase Matching (QPM) or Balanced Phase Matching (BPM) schemes [START_REF] Bierlein | Balanced phase matching in KT iOP 4 waveguides[END_REF][START_REF] Weissman | Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides[END_REF][START_REF] Risk | Compact Blue/Green Lasers Conference[END_REF] that take advantage of a periodic reversal of the nonlinear coefficient associated with such waveguides. More recently, it has been shown that multimode PSW can be a practical support for classical chaos investigation in the geometrical limit [START_REF] Aschieri | Complex behaviour of a ray in a Gaussian index profle periodically segmented waveguide[END_REF]. The ray dynamics in the simple and ideal 2D case of a PSW characterized by a transverse gaussian index profile is responsible for the emergence of a genuine chaotic behavior of the ray dynamics which is much more complex than that encountered with the usually considered parabolic index profile [START_REF] Rastogi | Ray analysis of parabolic-index segmented planar waveguide[END_REF][START_REF] Thyagarajan | Propagation characteristics of planar segmented waveguides with parabolic index segments[END_REF][START_REF] Casperson | Beam propagation in periodic quadratic-index waveguides[END_REF]. Such a complex behavior is actually comparable to what has already been studied in a periodically perturbed waveguide [START_REF] Abdullaev | Chaos and dynamics of ray in waveguide media[END_REF][START_REF] Faure | Propagating modes in periodic wave guide in the semiclassical limit[END_REF] or more generally, the analysis shows a dynamic that can be found in usual periodically forced nonlinear pendulum. PSW ray dynamics exhibit resonances, frequency locking, as well as a chaotic behavior and as a consequence of this complex aspect of the system, the transit time of rays (e.g. the ray dispersion) in the waveguide is expected to be different as in non segmented waveguide. As observed in previous work [START_REF] Zaslavsky | Chaotic transmission of waves and "cooling" of signals[END_REF], it is shown, here, that the dispersion is determined by the topology of the phase space.

After a description of the system used, numerical results of ray dispersion value for two waveguides configurations will then be discussed and finally, conclusions and perspectives will be drawn.

Dispersion calculation

A schematic representation of a PSW is sketched in figure 1. The PSW presents a transverse gaussian index profile in the high index segments that is, for example, naturally encountered with the proton exchange technique waveguide fabrication on LiN bO 3 substrate [16]. The duty-cycle of a PSW is usually define as DC = d/Λ and a step index profile is assumed along the propagation direction z as it has been done in previous works [START_REF] Aschieri | Complex behaviour of a ray in a Gaussian index profle periodically segmented waveguide[END_REF][START_REF] Rastogi | Ray analysis of parabolic-index segmented planar waveguide[END_REF][START_REF] Thyagarajan | Propagation characteristics of planar segmented waveguides with parabolic index segments[END_REF]. It can be mentioned that a smoother profile will not change qualitatively the results.

Assuming that light wavelength is very small compared to the dimensions of a highly multimode waveguide, ray optics approximation can be used. A ray is define as the path along which light energy is transmitted from one point to another in the optical system with a velocity equal to c/n(x) where c is the speed of light in the vacuum and n(x) is the transverse index profile. In low index segments, the index profile is constant and n(x) = n 2 whereas the index profile of the high index segments is given by :

n (x) = n 2 + δne -x 2 w 2 (1)
where w is the width of the waveguide, δn = n 1n 2 , n 2 is the substrate index and n 1 is the maximum index induced by the waveguide fabrication process. We consider here dielectric waveguides characterized by a low index contrast δn ≪ 1, which allows us to neglect the reflections of rays at each interfaces.

A ray path in a 2D medium can be described by the following equation in cartesian coordinates [START_REF] Ghatak | Introduction to fiber optics[END_REF]:

d 2 x dz 2 = 1 2β 2 dn 2 (x) dx (2)
where, x and z are the transversal and longitudinal coordinates respectively, n(x) is the transverse index profile, β = n (x i ) cos θ i is the invariant of the ray path, x i being the initial position of the ray and θ i being the incident angle of the ray respect to z axis. The ray path can be calculated by analyzing high and low index segments separately and using the law of refraction at each interface. In the low index segment, the solution is trivial and corresponds to a straight line. In the high index segment characterized by a transverse gaussian index profile, the ray equation ( 2) is now :

d 2 x dz 2 = - 2n 2 δn β 2 w 2 xe -x 2 w 2 (3)
in the limit of usual classical assumption δn ≪ 1. Ray paths of a PSW with a gaussian index profile is plotted on figure 2 for two different initial conditions

(θ i = 0 • , x i = 1.7µm) and (θ i = 0 • , x i = 0). When θ i = 0 • or x i = 0, ray
trajectory oscillate with a period of oscillation Z p whereas the trajectory is a straight line when

θ i = 0 • and x i = 0 [17].
It is important to note that the gaussian shape profile makes equation ( 3) nonlinear with respect to variable x, whereas, for example, for a parabolic index profile, equation

is reduced to a linear equation [START_REF] Aschieri | Complex behaviour of a ray in a Gaussian index profle periodically segmented waveguide[END_REF]. For a parabolic PSW, the presence of a linear term in the RHS of equation (3) makes the system analogous to a parametric system [START_REF] Arnold | Mathematical Methods of classical mechanics[END_REF]. This kind of system can exhibits parametric resonances which are responsible for a ray divergence whatever may be the initial conditions, a feature that has already been study in a previous work [START_REF] Rastogi | Ray analysis of parabolic-index segmented planar waveguide[END_REF].

For a gaussian index PSW, the situation is different because the nonlinear term in the RHS of equation ( 3) saturates the parametric instability and leads to nonlinear resonances which occur when the segmentation period Λ and the period of ray oscillation Z p are in a rational ratio [START_REF] Aschieri | Complex behaviour of a ray in a Gaussian index profle periodically segmented waveguide[END_REF]. The presence of nonlinear resonances makes the ray dispersion behavior more interesting and some results will be shown in the next sections.

Ray dispersion, also often call intermodal dispersion, is caused by the fact that different rays take different times to propagate along different paths from the input to the output of the waveguide. For our purpose, we define the ray dispersion in time ∆t in ps/km as the transit time difference between optical path of the considering ray and the straight ray propagating in the center of the waveguide (see figure 2). ∆t is defined by :

∆t = 1 c n 1 L -n(x)ds ( 4
)
where L is the length of the waveguide and ds ≃ √ dx 2 + dz 2 is an arc length along the ray path. In order to get an deeper insight of the ray dynamics and dispersion properties, Poincaré sections were used. Poincaré section consists on a projection at each period of the ray trajectory on to the phase plane (x, θ) where x is the transverse position of the ray and θ is the angle that makes the ray with the propagation axis. Along with the Poincaré section, a dispersion map were used where ∆t values are reported on a graph for different initial conditions except for diverging rays (e. g. rays that escape from the waveguide, ∆t → ∞). Two different transverse index profiles are considered here, a symmetric and a non symmetric one.

Symmetric PSW

The figure 3 kind of waveguide configuration. So, a complex Poincaré section structure will lead to a complex dispersion map and then to a non classical dispersion behavior which has been pointed out in previous work [START_REF] Zaslavsky | Chaotic transmission of waves and "cooling" of signals[END_REF]. The figure 4 is a slice at θ = 0 • of the dispersion map for the same waveguide and for the corresponding equivalent continuous waveguide. A PSW can be represented by a continuous equivalent waveguide with an index n eq (x) = n 2 +δn(x)×DC [START_REF] Rastogi | Ray analysis of parabolic-index segmented planar waveguide[END_REF][START_REF] Thyagarajan | Propagation characteristics of planar segmented waveguides with parabolic index segments[END_REF][START_REF] Li | Linear propagation characteristics of periodically segmented waveguides[END_REF]. The incident angle is fixed and initials conditions are taken such as only the position of the incident ray varies. For the equivalent continuous waveguide, the absolute value of the dispersion increase monotonically as the position of the incident ray moves away from the center of the waveguide.

This behavior is not the same with the PSW where the dispersion curve is not monotonic. For stable trajectories, |x| ≤ 0.5µm, characterized by a close circle curve on the Poincaré section, dispersion value are approximatively the same as the one of the continuous equivalent waveguide. For |x| > 0.5µm, the dispersion curve exhibits plateau with a an apex on the edge. The width of the plateau corresponds to the width of the resonance on the Poincaré section. Away from the resonance, the dispersion curve of the PSW goes back to the dispersion curve of the equivalent continuous waveguide.

The ray dispersion behavior is related to the presence of resonances.

When Z p and Λ are not commensurable, no resonance occurs and ray dispersion is comparable with that of a classical continuous waveguide. Resonances occur when Z p and Λ are commensurable and then, as long as the ratio Z p /Λ is constant, rays trajectories are synchronized to the segmentation period (this phenomenon is also call frequency locking). The ray is catch by the resonance and a slight modification of the guiding condition does not change ray path and, therefore, the transit time remains almost constant. The range where the initial conditions can be taken without modifying the value of the ray dispersion is determined by the width of the resonance. A strong resonance represented by large islands in the Poincaré section permits a quite large variation of angle or position for the initial conditions without changing ∆t value.

Singularities in the ray dispersion can be clearly identified with the help of the Poincaré section which is fixed for a given waveguide configuration, but they can occur for a given input condition and changing the waveguide parameters. The figure 5 represents ray dispersion values for different segmentation periods Λ as a function of the index difference δn. Dispersion behavior of a PSW deviate from that of the continuous equivalent waveguide model. Ray dispersion is not monotonic as a function of δn and ∆t can decrease whereas δn increase which is not intuitive.

Asymmetric PSW

The introduction of an asymmetric index profile in the transverse direction (figure 6) emphasizes the non classical dispersion behavior. It has to be said that a non symmetrical as well as a symmetrical gaussian index profile can be technologically achieved. Figure 7(a) represents the Poincaré section of a waveguide with an asymmetrical gaussian index profile and shows a more complex topology regarding the previous one (figure 3). The corresponding dispersion map is represented in figure 7(b) and it is similar to the Poincaré section.

Conclusions and perspectives

A simple ray propagation model has been used to provide a numerical study of the dispersion property of a PSW characterized by a gaussian in-dex segment profile. The analysis has shown a non standard behavior of the ray dispersion which is related to the complex topology of the Poincaré section and that can not be found in usual waveguides. The deviation from a classical ray dispersion behavior is enhanced with the introduction of an asymmetric transverse index profile. Beside the academic interest that represent a non standard ray dispersion behavior in a multimodes PSW, it might be interesting to consider devices with a periodic index segmentation and with a transverse gaussian index variation. According to the analysis presented here, these devices may exhibit resonances which could help to reduce pulse dispersion in highly multimode communication components for example. However, deeper analysis has to be performed in order to fully investigate the potential of those waveguide, in particular the extension of the analysis to the optical wave domain has to be done. From a technological point of view, no major difficulty seems to exist to achieve an experimental study of our theoretical predictions considering the fact that the waveguide configuration proposed here is based on well known and widely used fabrication process.

Figure captions

Fig. 1 : Schematic of the investigated longitudinally periodic waveguide with a period Λ and the transverse gaussian index profile of the high index segments. Everywhere else, the index is constant and is equal to the substrate index n 2 .

Fig. 2 : Typical ray path in a transverse gaussian index profile PSW. If θ i = 0 • or x i = 0, the ray path oscillate with a period Z p which depends on initial conditions whereas the ray path is a straight line if θ i = 0 • and x i = 0.

4 .

 4 (a) represents the Poincaré section for initial conditions taken in the range of [-3µm, +3µm] for x i and in the range of [-4 • , +4 • ] for θ i . Poincaré sections reveal the mixed aspect of the system characterized by the coexistence of a regular or a chaotic behavior of the ray trajectory depending on initial condition. Non divergent rays are characterized by a concentric close circle present on the center of the Poincaré section and also around 4 fixed points, these 4 fixed points are resonances of the system. Resonances occur when Z p and Λ are comensurable which is the case here where Z p /Λ ≃ Divergent rays are represented by dots which are mainly present on the peripheral part of the graph. The figure 3(b) represents the corresponding dispersion map for the same PSW, where ∆t values are represented in a grey level. Uniform grey level regions represent equivalent transit time for trajectories. As it has been mentioned previously, ∆t are not reported on the graph for diverging rays trajectories in order to keep a clearer picture. These two graphs are similar, the 4 resonances of the Poincaré section are still visible on the dispersion map where 4 wide uniform grey regions are present at the same location. It means that the transit time is similar for rays that have been launched on these resonances. It has to be mentioned that this correspondence between Poincaré section and dispersion map appears for any

  Figure 8 represents 3 different slices for x = 0, x = 1 and x = 1.5µm of the dispersion map(figure 7(b)). The presence of resonances show themselves through appearance of plateaus as in the symmetrical waveguide profile. Stochastic ray dispersion values can be seen between plateaus or on the edge of the curve. These random dispersion values are due to the fact that resonances lies in a stochastic sea which can clearly be seen with the Poincaré section (figure 7 (a)). Like in the symmetric waveguide, the dispersion value is frozen if initial conditions are taken in a resonance on the Poincaré section but chaotic elsewhere. Figure 9 represents ray dispersion values for two different segmentation periods Λ = 5µm and Λ = 10µm as a function of the index difference δn for a given input condition. Dispersion values seem to be randomly distributed around a mean value and the deviation from the continuous equivalent waveguide model is then much important.

Fig. 3 :

 3 Fig. 3 : Poincaré section of a PSW with the following parameters : DC = 0.5, Λ = 50.7µm, δn = 0.029, w = 3.5µm (a) and the corresponding dispersion map for the same waveguide (b).

Fig. 4 :

 4 Fig. 4 : Dispersion curve as a function of the position of the input ray for θ = 0 • for a PSW (continuous line) and for the equivalent continuous waveguide (doted line).

Fig. 5 :Fig. 6 :Fig. 7 :

 567 Fig. 5 : Dispersion values as a function of the index difference δ n for different periods for a given input condition (θ i = 0 • and x i = 1.5µm), others PSW parameters are DC = 0.5, w = 5µm. Fig. 6 : Schematic of the investigated longitudinally periodic waveguide with an asymmetric gaussian index profile, w 1 = w 2 . Outside of the high index segments, the index is constant and equal to the substrate index n 2 . Fig. 7 : Poincaré section of a PSW with the following parameters : DC = 0.5, Λ = 18µm, δn = 0.02, w 2 = 10 × w 1 = 4.3µm (a) and the corresponding dispersion map (b).

Fig. 8 :

 8 Fig. 8 : Dispersion curve as a function of the position of the input ray for θ = 0 • for a PSW (continuous line) and for the equivalent continuous

Fig. 9 :

 9 Fig. 9 : Dispersion values as a function of the index difference δ n for two different periods for a given input condition (θ i = 0 • and x i = 0.5µm), others PSW parameters are DC = 0.5, w 2 = 6 × w 1 = 5µm

Figure 2

 2 Figure 2:

Figure 3 :

 3 Figure 3:

Figure 5 :

 5 Figure 5:

Figure 6

 6 Figure 6:

Figure 8

 8 Figure 8:

Figure 9 :

 9 Figure 9: