1. Introduction

There are several applications of silicon-based group IV photonic circuits in which a strong second order susceptibility χ⁽²⁾ is needed within the Si waveguide core material instead of having the nonlinearity provided by an organic material situated in a slot. Monolithic manufacturing is easier with the first approach. Recent experimental studies indicate that a crystalline silicon channel-waveguide structure will possess a large second-order optical nonlinearity when the waveguide core is anisotropically strained by a dielectric cladding layer (especially a silicon nitride film) applied to the top of the structure. Early experiments by Jacobsen et al [1] and Fage-Pedersen at al [2] studied linear electrooptic (LEO) modulation in a silicon-on-insulator (SOI) strip waveguide initially coated with SiO₂ and then covered with a straining Si₃N₄ layer. The modulation was linked to a second-order nonlinearity found to be χ⁽²⁾ = 15 pm/V.

The Si₃N₄ layer was grown directly upon Si in subsequent experiments by other research groups. Bianco et al [3] measured the efficiency of near-infrared second harmonic generation in 10-μm-wide SOI/Si₃N₄ strip waveguides and evaluated χ⁽²⁾ to be 20 ± 10 pm/V. Their estimate requires further clarification because measurements were done with multimode waveguides, so it is not clear whether occasional close-to-phase-matching between several pump and harmonic guided modes contributed to the large measured second harmonic output. While the measured value of χ⁽²⁾ might be somewhat questionable, this must be the first experimental report on second harmonic generation in a uniform silicon waveguide. Hon et al [4,5] employed a periodic arrangement of stressing films on silicon for quasi-phase matched difference frequency generation. Significant experiments were performed by Chmielak et al [6] on 400-nm-wide (100)-SOI/Si₃N₄ rib waveguides whose strain was asymmetric compression. Using Pockels-effect modulation and other tests, they found a record high value χ⁽²⁾ = 122 pm/V, and the second-order susceptibility was largest at the side walls of the rib because those walls were pinned at the bottom. Their χ⁽²⁾ does not apply directly to estimating the efficiency of mixing three lightwaves because one wave in LEO modulation is an RF field; nevertheless χ⁽²⁾ will be very large in both situations. Knowing that, we can proceed with our investigation of phase-matched Si waveguides.

In this paper we propose and analyze strained-silicon waveguides (structures uniform in construction along their length) that provide perfect phase matching of the three lightwaves that give rise to sum frequency generation (SFG) or to second harmonic generation (SHG). With appropriate relabeling of the pump and signal wavelengths, the approach described here is also applicable for analysis of wavelength conversion via difference frequency generation (DFG). The phase matching allows buildup of the nonlinear interaction over a long length of uniform waveguide, yielding efficient conversion without anti-phase domains. Our proposed designs simulated here utilize the different effective indices assigned to TE and TM modes (the mode birefringence) to attain the desired synchrony of waves. In a previous publication [7], we used the birefringence method for optical parametric oscillation in a uniform gallium phosphide waveguide clad by sapphire. We believe that the structures described here will aid the construction of practical Si-based devices for SHG, DFG and SFG wavelength conversion. They should also find use in evaluating the second-order nonlinearity with precision.

2. χ⁽²⁾ and χ⁽³⁾ techniques

The fabrication of the straining layer for χ⁽²⁾ is generally easier than constructing the organic-in-slot for χ⁽²⁾ because the former requires a simple deposition of dielectric while the latter requires e-beam fabrication of a nanoscale slot, followed by insertion of the organic, then followed by DC poling of the polymer in order to orient the molecules in the desired direction [8]. In addition the organic approach may suffer from thermal stability issues that are generally not present in the straining layer method.

Crystalline silicon also possesses a fairly strong third order nonlinear optical susceptibility χ⁽³⁾ that is also present in a non-uniformly strained silicon waveguide, although the strain modifies χ⁽³⁾ somewhat. Thus both χ⁽²⁾ and χ⁽³⁾ are simultaneously present in such a waveguide. We can compare the relative contributions of χ⁽²⁾ and χ⁽³⁾ to the overall nonlinear optical susceptibility. At moderate levels of infrared pump intensity, we can say that the three-wave process of χ⁽²⁾ will generally dominate over the four wave process of χ⁽³⁾. Several research groups have been quite successful in using silicon’s χ⁽³⁾ [9] to attain wavelength conversion and optical parametric amplification via four wave mixing, although a special type of waveguide dispersion engineering was needed to attain phase matching in those cases. However, if a strained Si channel waveguide is optimized for SHG by the birefringence method reported in this paper, then the χ⁽³⁾ phase matching will generally be spoiled (be negligible) since the χ⁽³⁾ phase engineering is markedly different.

3. Phase matching for second harmonic generation

We shall examine the dispersion characteristics of an SOI waveguide in order to show that perfect phase matching in a uniform waveguide structure is both credible and feasible. To begin with, the waveguide is considered to be planar (a slab) and the cladding is assumed to have infinite thickness. Later in the paper, we examine a more a realistic structure in which both the waveguide width and the cladding thickness are finite (this is known as the strip channel waveguide). We note here that the planar waveguide approximation will be reasonably accurate for the strip if the aspect ratio of the strip’s guiding core remains high and if the guiding mode is localized in the core with little penetration into the cladding. In this section of the paper, we determine the optimal thickness of a slab waveguide at which phase synchronism is achieved. For a strip, the optimal thickness found previously will be somewhat modified when the finite core width and finite cladding thickness are taken into account. The strip channel is investigated in Section 5 by analytic formulae that separate the vertical and horizontal mode solutions. As mentioned, the strip response is indeed linked to the slab waveguide behavior.

In this system, the third-order nonlinearity is still present leading to the χ⁽³⁾-induced phenomena such as self-phase modulation and cross-phase modulation. They effectively change the modal indices for the waves involved in the χ⁽²⁾ interactions and thus affect the phase-matching condition. At present, we ignore the corrections to the phase-matching conditions due to self-phase and cross-phase modulation. Most likely, these corrections are tiny and would only be needed for fine tuning of the model when system operates with extremely high power densities.

For purposes of description, we introduce a Cartesian coordinate system in which the Z axis is the direction of lightwave propagation (and where Z is also the longitudinal axis of a strip channel waveguide). The waveguide’s TE modes have their E field polarized along the X axis parallel to the plane of a slab waveguide, with X being the “lateral” direction. The TM modes have E field polarized in the Y direction perpendicular to the slab, with Y being the “vertical” direction. For the strained-silicon core region, TE is polarized typically in the Si [110] crystallographic lattice direction, while TM is along the [001] lattice direction [6]. The waveguide structure in which a straining layer is placed on top of a thick SiO₂ cladding is expected to provide smaller strain and consequently smaller nonlinearity than a structure that has Si₃N₄ directly on top of the Si core. For this reason, the waveguides studied here are Si₃N₄/Si/SiO₂ asymmetric planar waveguides, where the SiO₂ is the buried oxide in SOI.

The refractive indices as a function of wavelength (λ, expressed in micrometers) for Si, SiO₂, and Si₃N₄ are assumed to be as follows:

A convenient instrument to study second order nonlinear processes is second harmonic generation. For the asymmetric material system, a suitable pump (subscript p) wavelength could be close to λ_p = 3μm (for instance, an Er:YAG laser operating at 2.94μm) with second harmonic signal (subscript s) at the wavelength close to λ_s = 1.5μm. Wavelengths longer than 3.5μm would not be appropriate because of the absorption in silica. Another choice, as it is done in [3] is λ_p ≈2.3μm with the second harmonic at the short-wavelength edge of silicon’s transparency region, λ_s = 1.15μm. This defines the wavelength range of interest: 2.3μm < λ_p < 3.5μm.

Figure 1 shows modal indices for the TE and TM fundamental modes at the fundamental (λ_p = 2.94μm) and second harmonic wavelengths (λ_s = 1.47μm) as a function of the silicon core layer thickness d in the Si₃N₄/Si/SiO₂ planar waveguide structure. In the absence of material dispersion, the modal indices would be functions of d/λ, so that the dispersion curves at different wavelengths would be appropriately scaled copies of one another.

 

Fig. 1 The modal indices for TE (red) and TM (blue) fundamental modes at the wavelength of 2.94μm (solid) and 1.47μm (dashed) as a function of the silicon core layer thickness in the Si₃N₄/Si/SiO₂ planar waveguide structure. The inset shows the waveguide structure.

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The dependence of material indices upon wavelength brings some minor correction to the waveguide dispersion under analysis. At a given wavelength, the TE mode always has a higher modal index, so that the dispersion curves never intersect. However, dispersion curves for the TM mode at a shorter wavelength and the TE mode at a longer wavelength may indeed intersect. In this particular case, intersection happens at the value of core thickness of d = 212nm.

In the symmetric SiO₂/Si/SiO₂ waveguide, compared to the Si₃N₄/Si/SiO₂ structure, similar analysis leads to the optimal core thickness being about 16% larger. In another symmetric structure, Si₃N₄/Si/Si₃N₄ with both claddings made of silicon nitride, the optimal core thickness is approximately 17% smaller compared to the Si₃N₄/Si/SiO₂ structure. With both claddings made of silicon nitride, the strain in the core may be larger and the waveguide transparency range is wider because it extends from 1.15 μm to 6.7 μm.

The intersection of the dispersion curves in Fig. 1 indicates the feasibility of perfect phase synchronism for second harmonic generation with the pump wave TE-polarized and the second harmonic wave TM-polarized. The corresponding field profiles are illustrated in Fig. 2 . While the modal fields show penetration into the claddings, a significant fraction of the profiles remains within the Si core. The core confines 46.4% of ${\displaystyle \int E^{2}dy}$in the case of the pump wave and 56.3% in the case of the second harmonic. This gives an idea of the percentage of material’s χ⁽²⁾ that can be effectively accessed in SHG. However, for the process of guided wave SHG, the appropriately defined overlap integral is not dimensionless and its interpretation requires some additional considerations (see, section 5).

 

Fig. 2 Profiles of the modal fields vs Y coordinate for the pump (solid red, λ = 2.94μm, electric field strength of TE₀ mode) and second harmonic (dashed blue, λ = 1.47μm, normal component of the TM₀ mode E-field). The Si core width extends from Y = 0.0 to Y = 0.212 μm.

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If the core/cladding interfaces are smooth and the cladding materials do not have many defects, then the mode tailing we have described does not necessarily imply high propagation losses for the guided waves.

Accounting for the finite width of a strip waveguide and the finite thickness of the nitride layer will modify the dispersion curves. However, it is appealing that widely used SOI structures with core layer (top Si) thickness close to 200nm, when covered by a thick straining Si₃N₄ layer, are suitable for perfectly phase-matched second harmonic generation with pump wavelength around 3.0μm.

4. Phase matching for sum-frequency generation

Another second-order optical nonlinear process is sum frequency generation. The pump wavelengths λ₁ and λ₂ in this case generate the signal at wavelength λ_s to satisfy the energy conservation

As all the wavelengths need to be within the transparency range of the particular SOI material system (between λ_a = 1.15μm and λ_b = 3.5μm), this limits possible values of the parameter a:

In this particular example a_max ≈0.671 and a_min ≈0.329. Limitations for the wavelengths are illustrated schematically in Fig. 3 .

 

Fig. 3 Limitations for the choice of wavelength λ₁ and λ₂ for sum- frequency generation: shadowed area shows allowed wavelengths. The relation between the wavelengths for a given value of the parameter a is illustrated by dashed lines.

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Equation (5) now can be used to find the core layer thickness to provide the phase matching in the sum frequency generation process for a given combination of the parameter a and the signal wavelength λ_s (Fig. 4 ). Here we assume TE polarizations for the pump waves and TM for the signal. Sum-frequency generation includes the above-considered second-harmonic generation as a particular case in which a = ½.

 

Fig. 4 Optimal core thickness d versus SFG signal wavelength λ_s for several representative values of a = 0.35, 0.37, 0.40, 0.44, 0.50. Polarizations: TE for both pump waves and TM for the signal at sum frequency.

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With both pump waves having identical polarizations, there is degeneration between the cases corresponding to values a and (1 – a). For this reason, Fig. 4 shows only the cases of a ≤ ½.

In general, however, there are more flexible parameters in the sum frequency generation process compared to the second harmonic generation. Besides the fact that all the wavelengths are different, there is also some freedom in the choice of polarizations. In principle, if the nonlinearity tensor permits and there is enough freedom in choosing indices for the claddings, the input waves can be either both TE polarized or only one of them could be TE and the other TM, while the sum frequency waves in both cases are TM polarized. Other combinations of polarizations are difficult to achieve because both material and waveguide dispersion as a rule result in lower index for longer wavelengths, and the modal index for a TM guided mode in a slab waveguide is lower than the index for a TE mode of the same order. Some additional efforts in engineering the guided wave dispersion may help overcome this limitation, but this issue is beyond the scope of the article.

5. Corrections for the finite width of the waveguide

Finite width of the Si core along X leads to some corrections for the modal indices and thus for the optimal Y thickness of the core required to achieve the phase matching. Rigorous calculation will require numerical methods that might be computationally intensive. Qualitatively, the corrections due to the finite width of the waveguide can be estimated using the effective index method [10, 11]. In partial justification for the use of the effective index method, we note that the optimal core geometry appears to be the one with a quite high aspect ratio on the order of 12.5:1. As a rule, the higher aspect ratio results in better accuracy of the numerical simulation based upon effective index method. We believe that the accuracy of our computation method is quite adequate to show the important second-order behaviors of the 2D and 3D channels.

We assume that the strip waveguide is formed in the Si₃N₄/Si/SiO₂ material system so that the straining Si₃N₄ layer extends far enough beyond the Si ridge. This nitride arrangement implies – within the effective index model – that in the lateral direction the claddings have the index of silicon nitride. Thus, in the X direction, we assume a Si₃N₄/Effective_Medium/Si₃N₄ system, whereas in the Y direction we have an effective medium whose effective index is that of our previously studied Si₃N₄/Si/SiO₂ slab waveguide. Therefore to evaluate the modal index in the strip channel structure for the mode polarized parallel (or orthogonal) to the plane of the wafer, formulas for TM (or TE) modes are applied to the in-plane Si₃N₄/Effective_Medium/Si₃N₄ system, where the effective medium is assumed to have the refractive index equal to the effective index of the TE (or TM) mode in the Si₃N₄/Si/SiO₂ slab waveguide. We have done this and have investigated SHG for the Si strip by deriving dispersion curves for the finite structure.

Dispersion curves similar to those of Fig. 1, but for the strip structure with finite width, are shown in Fig. 5 . As indicated in this Figure, the intersection of appropriate dispersion curves (the black dot locations) indicates the perfect phase-matching condition for second harmonic generation. For instance, at the ridge width of 2.5μm, the optimal thickness of the core is 196nm, and the modal indices for TE-polarized pump at the wavelength of 2.94μm and TM-polarized second harmonic are both equal to 2.26.

 

Fig. 5 Dispersion curves for the TE₀₀ mode at the pump wavelength, λ_p = 2.94μm (solid, shades of red-pink) and TM₀₀ mode at the second harmonic wavelength, λ_s = 1.47μm (dashed, shades of blue-cyan), as a function of the Si core thickness d, for the values of the strip width w, from top to bottom, w = ∞, 5μm, 2.5μm, 1.25μm. Bold black dots indicate the phase matching condition for SHG. The inset shows the optimal thickness of the core versus the strip width.

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The normalized efficiency of second harmonic generation is given by

where ε₀ is vacuum permittivity, c is speed of light, $n_p^{*}$ and $n_s^{*}$are modal indices of the pump and signal (second harmonic) guided modes,

(11)$$I={\displaystyle {\int }_{core}{E}_{sy}^{}\left(x,y\right)E_{px}^2\left(x,y\right)dxdy}$$

is the overlap integral, x and y are the transverse coordinates in the directions parallel to and orthogonal to the substrate, and the E-fields of harmonic and pump are employed. The overlap integral, by definition, is determined by the field distributions only (11). It takes into account that nonlinearity is present only in the core. The core nonlinearity and some other physical parameters of the system under consideration then define the normalized efficiency of SHG (10). With the E-fields normalized so that

(12)$${\displaystyle \int E^2\left(x,y\right)dxdy}=1,$$

the fields are measured in units inverse to the unit of length, say, 1/μm, and the squared overlap integral I² in Eq. (10) is measured in units of 1/μm². In an unrealistic case of modal fields completely confined within the core and uniformly filling the core cross section, the squared overlap integral I² would be equal to 1/wd. Thus in real-world structures, the product wdI² characterizes the degree of mode confinement in the guided-wave second harmonic generation. The overlap integral squared for the case of second harmonic generation with pump at λ_p = 2.94μm, is shown in Fig. 6 at the left.

 

Fig. 6 Left: The overlap integral squared as a function of the strip width. The core thickness is optimized as shown in the inset in Fig. 5. Right: The modal fields in the ridge waveguide calculated using the effective index approach: TE₀₀ mode at the pump wavelength λ_p = 2.94μm (top) and TM₀₀ mode at the second-harmonic signal wavelength (bottom). The field of view in both cases is 4.5μm × 2.0 μm. False colors from red to purple indicate relative strength of the electric field from maximal to minimal value.

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For the strip width greater than 3μm the squared overlap integral is scaled approximately as 1/w. The overlap reaches a maximal value of I² = 0.193μm⁻² at approximately w = 2.5μm. At this strip width the optimal core thickness is d = 196nm, so that wdI² = 0.095.

The modal fields in a waveguide with w = 2.5μm and d = 196nm are shown in Fig. 6 at the right. Field penetration beyond the core is responsible for the product wdI² being well below unity. Nevertheless, the core cross-section in this case is a small fraction of the pump wavelength squared, wd ≈0.057λ_p², which indicates strong mode confinement. This and high expected valued of the core nonlinearity χ⁽²⁾ = 122 pm/V, in accordance with (10), lead to a quite impressive normalized efficiency of second harmonic generation of 4,850% per Watt⋅cm².

6. Conclusions

In conclusion, we show that a uniform waveguide in the Si₃N₄/Si/SiO₂ material system can be designed to provide the perfect phase matching condition for second harmonic generation and sum frequency generation. In the presence of strain-induced second-order nonlinearity, such waveguides can be used in various on-chip integrated silicon photonic devices. The results here provide a foundation for second-order optical parametric oscillation in silicon.

Acknowledgments

This work is supported in part by the Air Force Office of Scientific Research, Gernot Pomrenke, Program Manager, under Grant Number FA9550-10-1-0417.

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