1. Introduction

Reverse-proton-exchanged (RPE) waveguides [1] are of major interest in the realization of highly efficient periodically-poled nonlinear devices in LiNbO₃ crystals [2] due to i) the low losses, ii) the good matching of the modes’ field profiles with the fiber ones, and iii) the high overlap between the interacting fields in the nonlinear processes. In order to assess the poling period and to optimize the design of such devices, the knowledge of the effective indices of guided modes is of great importance. We recall that the polarization of such modes is parallel to the extraordinary axis of the crystal due to the fact that proton-exchange increases the extraordinary index and decreases the ordinary one [3].

Actually, standard characterization techniques, such as the well-known m-lines spectroscopy [4], can hardly be used for the determination of the effective indices of the lowest order extraordinary modes. The field profile of these modes is indeed so confined below the sample surface, due to the buried refractive index profile, that the overlap with the evanescent field generated below the prism by total internal reflection (TIR) is negligible. For this reason it is difficult to determine accurately the extraordinary refractive index profile of the waveguide. To overcome this problem most literature about RPE waveguides has inferred some data about that profile starting from the characterization of the ordinary one, which lies at the surface and allows a direct prism-coupling [1,5]. However, no simple correlation exists between ordinary and extraordinary refractive indices in proton-exchanged waveguides [6], and, moreover, the ordinary profile is waveguiding only in the case of very deeply exchanged samples having tens of extraordinary modes [1,5,7,8], which are not well suited in most applications.

In this paper, a new interesting interaction involving ordinary radiation modes as pump (ω) waves and extraordinary guided modes as second harmonic (2ω) waves is exploited in order to determine the effective indices (n_eff) of all 2ω extraordinary modes, even the lowest order ones’.

 

Fig. 1. Typical dispersion curves of the extraordinary and ordinary refractive indices of the substrate,n_es and n_os, ad of the film, n_ef and n_of, for a RPE lithium niobate waveguide. The effective indices (n_eff) of ordinary radiation modes at λ=1.55µm (continuos range) and of extraordinary guided modes at λ=0.775µm (discrete values) are evidenced with a continuos and a dashed line, respectively, so to evidence the possibility of a birefringence phase matching.

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2. Operation principle

The operation principle of the method can be derived from Fig. 1, which reports the dispersion curves of the extraordinary and ordinary refractive indices of the substrate, n_es and n_os, and of the film, n_ef and n_of, for a typical α-phase [9] RPE waveguide; n_ef and n_of correspond respectively to the maximum value of the extraordinary index and to the minimum value of the ordinary index (which is reduced, indeed, by proton exchange [10]). As shown in the figure, the n_eff of the ordinary radiation modes at wavelengths around λ=1.55µm extend over a continuum range which well superimposes the discrete ensemble of neff of the extraordinary guided modes at half the wavelength. Thus, birefringence phase matching can be obtained between ω ordinary radiation modes and 2ω extraordinary guided modes, according to the relation [11]:

where ${\mathrm{n}}_{\text{eff},\mathrm{m}}^{2\mathrm{\omega }}$ and ${\mathrm{n}}_{\text{eff}}^{\mathrm{\omega }}$ are respectively the n_eff of the m^th order 2ω guided mode and of the ω radiation one. As a consequence of Eq. (1), the n_eff of the 2ω guided modes can be determined by measuring the n_eff of the phase-matched ω radiation modes. The advantage of exploiting radiation modes for measuring the n_eff of the guided modes is that the radiation modes field profile is not buried as that of guided ones, and well superimposes the evanescent field tail generated by TIR below a coupling prism. Thus, radiation modes can be efficiently coupled by means of a prism, and their neff can be determined from the corresponding coupling angles [4].

3. Experimental details

The set-up used in the experiments is schematically reported in Fig. 2. The radiation source is a BBO-based parametric amplifier, pumped by the second harmonic of a mode-locked Nd:YAG laser delivering 10ps pulses at 10 Hz repetition rate. To avoid optical damage, the energy of the pulses is kept below 10µJ at the entrance of the rutile coupling prism. In the incidence plane the beam-width is about 3mm, thus enough wide to have a tight angular spectrum and a high coupling selectivity. In the orthogonal direction, a cylindrical lens (f=100mm) focuses the beam to a 60µm wide spot to increase the energy density.

 

Fig. 2. Experimental set-up. LS: Nd:YAG pumped parametric amplifier; VA: variable attenuator; BS: beam-splitter; PD1: InGaAs photo-diode; D: diaphragm; CL: cylindrical lens (f=100mm); WG: waveguide; RS: rotating stage; PD2: Si photo-diode.

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Starting from the orthogonality condition between the prism and the beam, the prism-waveguide ensemble is rotated until the coupled ordinary radiation mode is phase matched to one of the second harmonic extraordinary guided modes. The phase-matching condition is finely tuned by varying the coupling angle until maximum reading is obtained from the silicon detector. In such conditions the n_eff of the coupled ω radiation mode equals that of the 2ω guided mode, so that the latter one can be derived from the measurement of the coupling angle of the former one. Successive rotations are then performed in order to find the phase matching conditions for all remaining 2ω modes, thus determining the full spectrum of the n_eff of the second harmonic guided modes.

4. Experimental results

The method was tested on two Z-cut lithium niobate RPE waveguides realized by the following three-step process: i) proton exchange at 247°C in benzoic acid diluted with 1% lithium benzoate; ii) annealing in air at T=350°C; iii) reverse exchange at T=320°C in an euthectic melt composed by LiNO₃, KNO₃, NaNO₃. The durations of the processes are respectively 26h, 18h and 17.4h for the first sample (#1), and 12.5h, 11.3h and 18h for the second sample (#2).

The measurements were performed with a 1.55µm pump wavelength by setting a TE-polarization for coupling the ordinary radiation modes. From the coupling angles of those modes giving phase matching, the n_eff of the TM guided modes at λ=0.775µm were determined. Table 1 report such n_eff for the two samples (column [a]). In the Table, a second set of n_eff is also reported (column [b]), which is obtained, whenever possible, through a direct prism-coupling of TM modes at λ=0.775µm with a Ti:sapphire laser. The two sets of n_eff are in excellent agreement, demonstrating that the selectivity of the nonlinear interaction is sufficient for an accurate determination of the effective indices. However, only the present method allows the determination of the n_eff of the lowest order TM₀ and TM₁ modes. Indeed, radiation modes penetrate deeply into the substrate and thus superimpose the fields profile of guided modes, thus favoring the nonlinear interaction, while, as already stated, the extension of the evanescent field below the prism is not enough to enable a direct prism-coupling.

Table 1. Effective indices n_eff at λ=0.775µm for #1 and #2, as determined from the coupling angles of the ordinary radiation modes at λ=1.55µm giving phase-matching [a], and from m-lines spectroscopy at λ=0.775µm with a Ti:sapphire laser [b].

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It is worth noting that such method requires a high power source, due to the low conversion efficiency of the nonlinear process. Indeed, the field overlap between guided and radiation modes is rather poor, the interaction length is limited to the coupling region dimension, and the nonlinear susceptibility term involved in the interaction is the χ₃₁ of lithium niobate, which is more than 5 times lower than χ₃₃. A low field overlap is probably the reason why the TM₄ mode of sample #2 was not detected. However, no other optical methods provide a comparable amount of information about the refractive index profile of a buried nonlinear waveguide. Starting from the set of n_eff of #1, which includes all modes, an accurate determination of the refractive index profile was performed, and the result is reported in Fig. 3. The extraordinary index at the surface is almost equal to n_es, indicating a perfect recovery of the Li concentration, as already evidenced by Korkishko et al. by means of a SIMS analysis. The refractive index shape follows a supergaussian 4^th order curve on the surface side, where reverse exchanged has occurred, and an exponential curve on the substrate side, where the refractive index profile is mostly determined by the annealing process.

 

Fig. 3. Extraordinary refractive index profile of #1 at λ=0.775µm, as calculated from the first set of n_eff reported in Table 1.

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5. Conclusions

A new configuration for the second harmonic generation process involving radiation modes as pump fields and guided modes as second harmonic fields has been proposed and experimentally demonstrated in RPE lithium niobate waveguides, which are nowadays the most efficient guiding structures for all-optical nonlinear processes. By measuring the prism coupling angles of the ordinary radiation modes giving phase matching with the extraordinary guided ones, the n_eff of all 2ω guided modes have been determined, even the lowest order ones’. The method proves to be very useful for the full characterization and modelling of the RPE process, but also for determining the n_eff of the TM₀ mode at λ=0.775µm, whose knowledge is of the uppermost importance when the poling period of waveguiding nonlinear devices in the telecom bandwidth is to be designed.