Phase matching using an isotropic nonlinear optical material

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Frequency conversion in nonlinear optical crystals1,2 is an effective means of generating coherent light at frequencies where lasers perform poorly or are unavailable. For efficient conversion, it is necessary to compensate for optical dispersion, which results in different phase velocities for light of different frequencies. In anisotropic birefringent crystals such as LiNbO3 or KH2PO4 (‘KDP’), phase matching can be achieved between electromagnetic waves having different polarizations. But this is not possible for optically isotropic materials, and as a result, cubic materials such as GaAs (which otherwise have attractive nonlinear optical properties) have been little exploited for frequency conversion applications. Quasi-phase-matching schemes1,3, which have achieved considerable success in LiNbO3 (ref. 4), provide a route to circumventing this problem5,6, but the difficulty of producing the required pattern of nonlinear properties in isotropic materials, particularly semiconductors, has limited the practical utility of such approaches. Here we demonstrate a different route to phase matching — based on a concept proposed by Van der Ziel 22 years ago7 — which exploits the artificial birefringence of multilayer composites of GaAs and oxidised AlAs. As GaAs is the material of choice for semiconductor lasers, such optical sources could be integrated in the core of frequency converters based on these composite structures.


Optical frequency conversion1,2 by second-order nonlinear interaction is a way to obtain coherent light in various spectral regions. The frequency doubling process is used, for instance, to obtain green light from the very efficient near-infrared YAG laser, whereas difference frequency generation (DFG) is the basic process for high-power mid-infrared sources such as optical parameter oscillators. The best known nonlinear materials used for frequency conversion are KDP or LiNbO3 (ref. 2), but several other crystals can be used; for example, KTP (KTiOPO4), AgGaSe2, GaAs, and synthetic materials as organic molecules or semiconductor quantum wells8. Among the different characteristics of nonlinear crystals, two are of paramount importance. (1) The nonlinear coefficient χ(2), which is a measure of the strength of the nonlinear interaction, and is related to the degree of asymmetry of the electronic potential at the microscopic level. (2) The ability to match the phase velocities between the different frequencies1: owing to the optical dispersion of the nonlinear materials, the different waves do not travel at the same velocity in the material. After a distance called the ‘coherence length’, the nonlinear polarization and the generated wave acquire a phase lag of π, which produces a destructive interference. The effective conversion length is thus limited to the coherence length, which is often as small as a few micrometres. In reciprocal space, the phase-matching conditions expresses the photon momentum conservation.

To increase the coherence length, phase matching can be obtained in birefringent crystals as KDP or LiNbO3; the polarizations of the different waves are carefully chosen in order to adjust their different phase velocities. GaAs is a very interesting material for nonlinear frequency conversion, because its huge nonlinear coefficient (χ(2) ≈ 240 pm V−1) gives about one order of magnitude greater efficiency that the commonly used materials. However, as it is an optically isotropic (cubic) semiconductor, it is non-birefringent and phase matching is impossible. To use this highly nonlinear material in spite of the problem of phase matching, quasi-phase-matching1,3 has been used. The sign of the nonlinear interaction is changed periodically by reversing the orientation of the material, in order to compensate for the periodic destructive interference due to phase mismatch. This technique has produced impressive results for LiNbO3 (ref. 4), but although quasi-phase-matching has been shown in GaAs for both second-harmonic5 and difference-frequency6 generations, patterning the nonlinear susceptibilities is much more difficult in semiconductors. The technological steps necessary to obtain the material are difficult and the final materials suffer from detrimental losses.

Here we demonstrate perfect phase matching using an isotropic material as a microscopic nonlinear source. Phase matching is obtained by using a built-in artificial birefringence in a new composite multilayer material: the isotropy of bulk GaAs is broken by inserting thin oxidized AlAs (‘Alox’) layers in GaAs. The resulting composite material is not isotropic, and this is used to get birefringent phase matching, but the only nonlinear material present in the structure is the isotropic GaAs. The use of this so-called form birefringence9,10 for nonlinear frequency conversion was proposed in 1975 by Van der Ziel7. Since then, however, the experimental realization of this proposal has not been achieved, because it has not been possible to find a well suited couple of materials having a high nonlinear coefficient and a high enough refractive index contrast for form birefringence phase matching.

A first, intuitive, way to understand the origin of form birefringence is to consider the macroscopic crystal formed by a GaAs/Alox multilayer system. GaAs is a cubic semiconductor of point group &4macr;3 m, and is therefore non-birefringent. The presence of thin Alox layers grown on a (100) substrate breaks the symmetry of three-fold rotation axes and the point group of the composite material is now &4macr;2 m, the same as KDP. This artificial material has the nonlinear properties of GaAs (in particular, the same tensorial character and roughly the same nonlinear coefficient, if we neglect the small reduction of the average nonlinear susceptibility due to the zero contribution of the thin Alox layers), but the linear optical symmetry of KDP. This is possible by taking advantage of the microscopic nature of the nonlinear polarization in GaAs and the macroscopic engineering of the refractive index on the scale of the extended electromagnetic wavelengths. We note that with (111) orientated GaAs, the introduction of Alox layers switches the point group from &4macr;3 m to 3 m, which is that of another nonlinear material, LiNbO3; the same birefringence properties are of course expected.

Van der Ziel7 calculated form birefringence in the multilayer system directly from Maxwell's equations. Another physical explanation can be given using modal wavefunction considerations. We consider a periodic multilayer material, with light propagating in the plane of the layers. Following Joannopoulos11, for a given wavevector the frequency of an allowed electromagnetic mode in a composite medium increases with the fraction of electric field in the low-index material. The difference between the transverse electric (TE) and transverse magnetic (TM) polarizations then arises from the continuity equations at the boundaries between the two materials, as illustrated in Fig. 1. In the TM polarization, the continuity of the electric displacement forces the electric field to have a large value in the low-index material; this mode therefore has a higher frequency ω for a given wavevector k. For a large wavelength compared to the unit cell, the light experiences an effective medium, that is, a homogeneous-like material, the inhomogeneities of the dielectric constant being averaged on the wavelength scale. This is why the dispersion relation ω(k) in Fig. 1 is linear at the origin. Form birefringence then appears as the difference between the slopes of the dispersion relations for TE and TM waves. In this long-wavelength approximation, the two dielectric constants of the uniaxial composite material are given by:

where αi and εi are the filling factors (α1 + α2 = 1) and the dielectric constant, respectively, of the two constitutive materials 1 and 2. These equations are analogous to electrical series and parallel capacitors. This is not surprising, because the charge equality C1V1 = C2V2 between series capacitors 1 and 2 is nothing but the static limit of the electric displacement continuity relation for TM waves ε1E1 = ε2E2, and the bias equality V1 = V2 for parallel capacitors 1 and 2 is equivalent to the electric-field continuity for TE waves E1 = E2. (Here C represents capacitance, and V voltage.)

Figure 1: Dispersion relation for an in-plane propagation in a periodic composite material of period d.

The material consists of 25% of Alox (refractive index n ≈ 1.6; see text) and 75% of GaAs (n ≈ 3.5), for TM modes (full line) and TE modes (dotted lines). The physical origin of form birefringence appears in the mode wavefunction, pictured on the left for a frequency ω = 0.13(2π/d) (corresponding to the open circles in the dispersion relation). These Bloch waves have been calculated using standard periodic multilayer theory10. The direction of propagation is perpendicular to the plane of the figure. Due to the continuity of the electric displacement ε E normal to the layers, the TM mode (solid line in both panels) has a significant overlap with the low-ε layer (Alox, shown in light grey in the right panel), and a lower average dielectric constant. The continuous TE electric field (dotted line in both panels) has a higher value in GaAs (dark grey, right panel), and a higher average dielectric constant.

It appears from equations (1) and (2) that form birefringence (√εTE − √εTM) increases with the refractive-index contrast between the two materials in the multilayer, as do photonic bandgap effects in photonic crystals11. Although form birefringence in a GaAs/AlGaAs multilayer structure has been proposed for phase matching7, the refractive-index contrast between GaAs (n ≈ 3.5) and AlAs (n ≈ 2.9) is too low to provide the birefringe required to compensate for the dispersion. This is why we have used thin-film layers of Alox (n ≈ 1.6) in GaAs to get sufficient form birefringence. Alox results from selective oxidation (at 400–500 °C in a water-vapour atmosphere) of AlAs layers embedded in GaAs. The technology of AlAs oxidation emerged12 in the early 1990s; since then Alox has led to significant advances in the field of semiconductor lasers13 and Bragg mirrors14 thanks to its refractive index contrast with GaAs.

We have demonstrated phase matching of a DFG process in a waveguide containing four Alox layers. The structure which was used for the DFG is shown in detail in Fig. 2, together with the three modes involved in the nonlinear interaction.

Figure 2: Difference frequency generation (DFG) process in the sample.

Here ω1 and ω2 are the pump frequencies of wavelengths 1.32 µm and 1.058 µm, respectively, and ω3 is the frequency difference of wavelength 5.3 µm. Three periods of the composite material GaAs (325 nm)/Alox (40 nm) constitute the core of the waveguide. The birefringence of the composite material was engineered to compensate for the dispersion arising from both the natural dispersion in bulk GaAs and the optical confinement dispersion in the waveguide. The sample was grown by molecular beam epitaxy on a GaAs (100) substrate and consists of: 2,800 nm Al0.97Ga0.03As; 1,500 nm Al0.70Ga0.30As (waveguide cladding layers), three periods of birefringent composite material (40 nm Alox; 325 nm GaAs) × 3 and 40 nm Alox; 1,500 nm Al0.70Ga0.30As and a final 30-nm GaAs cap layer. The oxidation process is described in detail in ref. 15. The three modes involved in the DFG process are pictured together with their polarization (↑ for TM, for TE). The higher overlap of the TM mode with the low-refractive-index Alox layers is apparent, which is the origin of form birefringence. The arrows indicate the “phase matching” momentum conservation. k1 + k3 = k2, where ki indicates the wavevector of the wavenumber i.

We first characterized the form birefringence of the sample. Thedetails of this experiment, using surface emitting second-harmonic generation, are given in ref. 15. A form birefringence nTEnTM = 0.154 was measured. Even higher birefringences of the order of 0.2 have been obtained with different samples. This birefringence is sufficient to phase-match mid-infrared generation between 3 µm and 10 µm by DFG. We note that by increasing the width of Alox layers (as in the example of Fig. 1), much higher birefringences up to 0.65 could be achieved, but this is not necessary here.

For the DFG experiment, two continuous-wave pump lasers (a Nd:YAG with wavelength λ = 1.32 µm, and a tunable Titanium: Sapphire (Ti : Sa) were end-fire coupled in a 4-µm-wide, 1.2-mm-long channel waveguide. Using the χ(2)xyz element of GaAs, the DFG process was:

as it appears in Fig. 2. The infrared signal is shown in Fig. 3 as a function of the Ti:Sa wavelength. This function has the well known ((sin x)/x)2 shape, which is clear evidence of phase matching. All the polarization selection rules for equation (3)have been checked. These experiments show that perfect phase matching has been achieved with a cubic nonlinear material.

Figure 3: Mid-infrared DFG signal measured by an InSb detector, as a function of the wavelength of the Ti:Sa laser.

The pump powers were 0.2 mW and 1.6 mW for the YAG and Ti:Sa lasers, respectively. This function has the well known shape of a phase matching resonance: it is a ((sin x)/x)2 function, characteristic of momentum conservation in a fixed interaction length3. Inset, signal wavelength tunability with temperature.

A mid-infrared output power of 80 nW was obtained for 0.4 mW and 11.6 mW of Nd:YAG and Ti:Sa pump powers, respectively. This result can easily be pushed into the microwatt range by increasing pump powers and reducing scattering losses originating from processing; such power levels are interesting for mid-infrared spectroscopic applications. We note also that the generated wavelength (5.3 µm) is in the absorption range of LiNbO3, where few nonlinear optical materials exist. Fourier-transform infrared spectroscopic measurements show that absorption losses in Alox start for wavelengths >7.5 µm, opening a very large spectral range for tunable DFG. The efficiency of this frequency converter is limited by the mid-infrared losses which are essentially due to scattering on the ridge inhomogeneities introduced during the technological process. Improving the technological process to reduce these losses will be very important for the success of this composite material in frequency conversion devices. The tunability of the mid-infrared wavelength was demonstrated by varying the temperature. We have measured a linear dependence of the signal wavelength from 5.2 to 5.6 µm with a temperature scan from 0 to 150 °C (Fig. 3 inset). No degradation of the sample was observed during temperature cycles.

The birefringence of the composite structure is sufficient not only to phase match DFG in the mid-infrared, but also for second-harmonic generation around 1.55 µm. We give these two examples because of their interest for applications: continuously tunable mid-infrared compact sources are desirable for pollutant detection in the molecular fingerprint region or for process monitoring, and a 1.55-µm signal can be shifted by mixing it with a 0.75-µm pump. The latter function is required in wavelength division multiplexing systems. Other possible implications go beyond use of the simple passive nonlinear material: GaAs is also the favoured material for quantum-well lasers, and the possibility of integrating quantum wells in the core of the nonlinear waveguide is under study. In a first step, this would enable DFG with only one external source, the other frequency being given by the ‘internal’ quantum-well laser. In a second step, parametric fluorescence from this laser would make a completely monolithic micro-optical parametric oscillator — on a GaAs chip, tunable with temperature — a realistic possibility.


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This work was partially supported by the European Community under the IT “OFCORSE” Programme.

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Correspondence to V. Berger.

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