1. Introduction

Parametric nonlinear optical processes allow for laser-driven sources of coherent radiation that can operate at wavelengths difficult to access with conventional lasers. Typically, these optical parametric oscillators (OPOs) [1–6] use second-order nonlinear optical (NLO) materials, although parametric amplification is also feasible via four-wave mixing in third-order materials [7–10]. Recent work on second-order OPOs, based on bulk isotropic crystals, has focused on quasi-phase-matching (QPM) [11] as a means to achieve phase synchronism between parametrically generated waves with their inherently phase mismatched wavevectors. The QPM device is comprised of a periodic set of short crystal domains with alternating crystallographic orientations which effectively changes the sign of the nonlinear polarization when a light wave is traveling from one domain to the next. Besides QPM, multilayer structures made of isotropic layers can reveal artificial birefringence [12–14] large enough to allow for the birefringent phase matching in nonlinear optical parametric interactions, but this approach, as of now, is applicable only to a narrow class of materials such as (Al)GaAs/AlOx. Thus, in isotropic bulk crystals, QPM is the only technique available to achieve the required phase synchronism for efficient power generation at the desired wavelength. Depending on the nature of the nonlinear crystal, fabrication of such periodic domain-reversed structures requires rather sophisticated growth techniques or post-growth treatment such as periodic poling. Further, the QPM oscillation is generally attained using crystals placed in optical resonators with optical field cross sections that are hundreds of times larger than any of the wavelengths involved. Without tight focusing of light waves, the nonlinear optical interactions are relatively weak which in turn leads to large pump threshold powers. The question arises as to whether it is possible to attain the phase matching condition in a structure of simpler design than that required to implement the QPM technique and do it in such a way as to allow maximization of field interactions in order to achieve the lowest possible threshold power. One such method, investigated in this work, is to utilize a waveguide geometry and exploit the inherent refractive index differences associated with TM-like and TE-like guided modes to achieve both the phase matching condition and the strong field overlap required for efficient parametric amplification without the necessity, or growth complexity, of periodic domain inversion.

This paper presents general OPO phase-matching theory for a three-layer waveguide composed of a gallium phosphide thin-film clad below and above by sapphire. High quality films of GaP can be grown epitaxially for application in nonlinear optical frequency conversion devices [15]. This choice of material can be considered representative as a number of material systems can be used; however, the design space associated with a high index difference waveguide, low propagation loss and a large second-order non-linear coefficient must be considered in light of the desired wavelength of interest. In this work, preference was given to the 3 µm to 5µm wavelength range and materials that are lattice matched to Si which motivated the choice of GaP. Nonlinearity of GaP is almost as large as that of GaAs, and unwanted two-photon absorption is suppressed – even as the pump wavelength approaches 1μm – because this material has a relatively wide bandgap. Sapphire was chosen because of its low loss over this wavelength range. The GaP membrane, (waveguide core), is envisioned being bonded to a sapphire wafer in this representative case. A more advanced structure – the subject of future studies – would be a composite core waveguide, in which the nonlinear layer is sandwiched between high-index linear layers, for instance, made of Si. Over the long term, we are visualizing an epitaxial growth approach to the waveguide core problem. We are looking to an improved OPO composite waveguide consisting of a lattice-matched GaP film grown on a Silicon-On-Sapphire wafer. With a top Si cladding, this creates a Si/GaP/Si composite [16] for OPO applications. Generally, we feel that the waveguides discussed in this paper would fill a vital need for on-chip continuous wave (CW) mid infrared sources in a silicon photonics system-on-a-chip [17]. The OPO pump is visualized as being off chip.

This paper is organized as follows. First, we demonstrate that polarization dispersion in a thin-film waveguide can be used to achieve OPO phase matching, with a CW source, in a uniform GaP-on-sapphire waveguide with an output in the 3μm to 5μm mid infrared wavelength band where the atmospheric transmission is high. Next, we compare the waveguide-based OPO and the QPM OPO in terms of the pump power threshold required for oscillation. We derive the general theory for threshold power in a doubly resonant structure. For 1μm – 2μm sources and 3μm – 5μm outputs, phase matching is predicted for 0.14μm – 0.3μm waveguide cores. Numerical simulations of Al₂O₃/GaP/Al₂O₃ waveguides and bulk orientation-patterned GaP QPM structures show that the OPO threshold is 320 × lower in the thin-film waveguide case. We discuss an alternative OPO strategy for the future: the use of a silicon-based waveguide having a composite Si/GaP/Si core.

2. Phase matching in guided wave OPOs

In order to achieve efficient optical parametric oscillations in a non-linear system all waves subject to the nonlinear interaction must have wavelengths and propagation constants that satisfy the energy conservation and phase matching conditions respectively as well as maintaining a high intensity of the pump wave over a long distance in the nonlinear medium.

In the OPO, the pump wave of vacuum wavelength λ_p is converted into signal and idler waves, λ_s and λ_i. In general, the wave with the longer wavelength is called the idler wave, and wave with the shorter wavelength is called the signal wave. Energy conservation requires that

For continuous energy transfer from the pump wave into the signal and idler waves, the corresponding oscillations of electromagnetic field must stay in phase as the waves propagate through the nonlinear medium. This leads to the additional requirement of phase coherence (or, the momentum conservation law)

Material dispersion typically results in lower refractive index for longer wavelengths, which makes it difficult to satisfy (3) in uniform isotropic materials. In birefringent materials, one can choose polarizations of the waves and direction of propagation in such a way that birefringence-related index change would perfectly compensate for the index change associated with material dispersion, so that the phase synchronism (3) would be satisfied. If birefringence is not strong enough, or, if the waves’ polarizations and propagation direction defined by the phase synchronism in a birefringent crystal happen to be inconsistent with the optimal choice of polarizations and propagation direction based on the nonlinearity tensor, one can rely on poling or growth-defined patterning that periodically orients crystal domains in opposite directions. The phase synchronism condition in this case has an additional term associated with the wave vector of the periodical structure. These two methods – birefringence and periodical reorientation of domains – are, in practice, the tools used for achieving phase synchronism with bulk light beams in non-linear crystals.

With guided modes, on the other hand, there are additional means to achieve the phase synchronism [18]. Feasibility of using waveguide dispersion for achieving phase matching in devices that use difference frequency generation in far infrared (including the THz range, in modern terminology) is known for decades [19]. As Eq. (3) suggests, in case of normal material dispersion, the refractive index at the pump wavelength in the left hand side of the equation should be somehow brought down to match the linear combination of indices at longer wavelengths in the right hand side of the equation. The waveguide geometry provides two tools for doing this. One of them relies on polarization dispersion of guided modes: a TM (transverse magnetic) mode of a planar waveguide has lower index compared to a same-order TE (transverse electric) mode. Another approach is to use higher order modes for the pump wave and lower order modes for signal and idler. Modes of the same symmetry (all even, for instance) are preferable in order to avoid vanishing mode overlapping integrals. A possible configuration is a TM ₂ (third-order) pump and TE ₀ (fundamental) signal-and-idler. However, this requires selective launching of the third-order pump, and the overlap of the three modes is relatively small.

Phase matching using polarization mode dispersion in waveguides is especially appealing as long as it allows for synchronism between strongly confined fundamental modes. A particular case of phase synchronism for degenerate signal/idler wavelengths (a = ½, λ_i = λ_s = 2λ_p), or for a reciprocal process – second harmonic generation – has been studied earlier [20]. With well-defined wavelengths such as λ_p and 2λ_p, it is easy to find the waveguide configuration that leads to the phase synchronism. In that case one would plot the effective indices for the TM-polarized mode at wavelength λ_p and for the TE-polarized mode at 2λ_p, both as a function of the guiding film thickness. The intersection of these dispersion curves defines the film thickness at which synchronism is achieved.

A more general case of a ≠ ½ (that is, λ_i ≠ λ_s), requires some additional arithmetic. It is not evident a priori that the phase synchronism for guided modes is possible when the idler and signal wavelengths are substantially different.

To exploit the polarization mode dispersion, the pump wave has to be TM-polarized, and, in general, at least one of the signal/idler waves must be TE-polarized. Having in mind the particular nonlinearity tensor in GaP [21] and the crystallographic orientation of an epitaxial GaP film, the choice of polarizations is further narrowed: the waveguide must be oriented along the [110] direction within a (100) film, and both signal and idler must be TE-polarized. Thus, the phase synchronism condition (3) is rewritten in terms of effective indices of TE- and TM-polarized guided modes $n_{TE}^{*}$ and $n_{TM}^{*}$:

In (4), we explicitly included the waveguide film thickness t as an independent variable. Also, it is presumed that functional dependence ${n_Q^{*}\left(t\right)|}_{\lambda ={\lambda }_p,{\lambda }_s,{\lambda }_i}$, where Q = TE or TM, implicitly includes material dispersion through wavelength-dependent indices n_g(λ), n_s(λ), and n_c(λ) of the guiding core, substrate, and cladding respectively. In case of a thin-film waveguide, the effective indices are found by numerically solving the equation [22]

A numerical search for the waveguide configuration that provides the phase synchronism can be done as follows. Given the pump wavelength λ_p, select the parameter a in the range 0 < a ≤ ½, which then uniquely defines the idler and signal wavelengths ${\lambda }_i={\lambda }_p/a$and ${\lambda }_s={\lambda }_p/\left(1-a\right)$, and, finally, find numerically the film thickness t to satisfy (4) assuming that the effective indices are defined according to (5). By allowing the parameter a to vary in a certain range, one would find the values of film thickness needed to provide phase synchronism for optical parametric oscillations at various wavelengths. The data on λ_i(a,λ_p), λ_s(a,λ_p), and t(a,λ_p) can then be used to plot important dependences such as the wavelengths of the idler and signal λ_i(t,λ_p) and λ_s(t,λ_p), versus film thickness t for given values of the pump wavelength λ_p.

To illustrate the feasibility of OPO phase synchronism in thin film waveguides, we have chosen GaP as the second order nonlinear film. We assumed that the film is bonded to a substrate with substantially lower refractive index, such as, for example, sapphire. The cladding medium is either the same as substrate or is air. The choice of the substrate/cladding material is not critical, however, we wanted it to be transparent in the wavelength range of interest and to provide substantial index contrast compared to the guiding film. Technological considerations, which are beyond the scope of this paper, may give preference to other material(s). In general, as long as the substrate index is well below the film index, the outcomes of further analysis are expected to be qualitatively similar.

The structure under analysis in this paper is a thin-film planar waveguide and the modal indices and fields are calculated assuming this planar geometry. The finite width of the structure in the lateral direction, d, is needed for the estimation of the pump power at the threshold of optical parametric oscillation. An accurate quantitative analysis of how the finite width would affect the phase synchronization and the overlap of the modal fields will be the subject of future studies. In this paper, the width of the waveguides d is considered to be large enough so that all the essential waveguide characteristics are well approximated by the equations for a thin-film planar structure. Symmetric and asymmetric waveguides are schematically shown in Fig. 1 .

 

Fig. 1 (Color) Symmetric (left) and asymmetric (right) waveguides. The width in the lateral direction d is large enough so that modal indices and fields can be estimated assuming planar waveguide geometry.

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In the numerical simulations below, the wavelength dependence of the refractive index of the GaP guiding film was assumed to be given by the Sellmeier equation [23]

Possible birefringence of the substrate is beyond the immediate scope of this study, so we used ordinary index of sapphire and treated the substrate as an isotropic medium. Also, the index difference between the guiding film and substrate in this material system (n_g – n_s ≈1.3), as well as the difference between the effective indices for TE- and TM-polarized waves ($n_{TE}^{*}-n_{TM}^{*}$ ≈0.4, depending on the film thickness and wavelength), is at least an order of magnitude larger than the birefringence of sapphire (n_o – n_e ≈0.008), so, the substrate index was chosen to be [23]

Formula (7) gives accurate values of the refractive index for wavelengths in the range 0.2μm < λ < 5.5μm. The upper cladding index was either equal to that of substrate n_c = n_s, or set be n_c = 1 depending on the waveguide configuration under consideration.

To illustrate how the polarization mode dispersion in a thin-film waveguide allows for compensation of material dispersion in a guided mode OPO, we plot in Fig. 2 the left and right hand parts of Eq. (4) as a function of film thickness for the case of a symmetric waveguide.

 

Fig. 2 (Color) Effective index for fundamental TM ₀ (black solid) and TE ₀ (black long-dash) modes at pump wavelength and linear combination of effective indices for signal and idler TE ₀ modes with values of the parameter a of 0.2 (green dots), 0.3 (green dot-dash), and 0.5 (green short dash), as functions of the film thickness.

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In Fig. 2, the solid black line shows the effective index for the fundamental TM ₀ mode at the pump wavelength λ_p = 1.5μm, which represents the left hand side of Eq. (4). For comparison, the effective index for the TE ₀ mode is shown by long-dash black line, and, as seen, it is always larger than the index of TM ₀ mode. The right hand side of Eq. (4) is a linear combination of the effective indices of the TE ₀ modes at wavelengths λ_i = λ_p/a and λ_s = λ_p/(1 – a) respectively, and is shown in Fig. 2 as a function of various values of a as green dot-, dot-dash, and short-dash lines. The intersection of these lines with the solid black line indicates that the phase matching condition (4) is satisfied. For each value of a (and, respectively, a given value of idler wavelength λ_i = λ_p/a) there are two intersection points at around t ≈0.07μm and between 0.2μm and 0.3μm in Fig. 2. The larger of these two values of the film thickness is preferable as it is expected to provide larger value of the mode-overlap integral suggesting greater parametric conversion efficiency.

Because sapphire has high loss above 5.5µm, the range of appropriate a values is limited. Setting the longest wavelength of the idler wave to be λ_i = 5.5μm, the smallest a is found to be a = λ_p/λ_i ≈0.273. Allowing the parameter a to vary from 0.273 to 0.5, the associated values of idler wavelength, λ_i = λ_p/a, and signal wavelength, λ_s = λ_p/(1 – a), can be plotted as a function of film thickness, t(a), which will allow for phase matching. Results of these calculations are shown in Fig. 3 for pump wavelengths of 1.0µm, 1.5µm and 2.0µm with signal and idler wavelengths shown in blue and red respectively.

 

Fig. 3 (Color) The idler (red) and signal (blue) wavelengths as a function of the film thickness for the pump wavelength of 1.0μm (solid), 1.5μm (long-dash), and 2.0μm (short-dash). The phase synchronism is assumed between the fundamental TM ₀ mode at the pump wavelength and TE ₀ idler and signal modes in a symmetric waveguide.

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The case of the asymmetric waveguide with air as the top cladding give qualitatively similar results as shown in Fig. 4 , though the film thickness required for parametric oscillations at a given wavelength is about 30% larger.

 

Fig. 4 (Color) The idler (red) and signal (blue) wavelengths as a function of the film thickness for the pump wavelength of 1.0μm (solid), 1.5μm (long-dash), and 2.0μm (short-dash). The phase synchronism is assumed between the fundamental TM ₀ mode at the pump wavelength and TE ₀ idler and signal modes in an asymmetric waveguide.

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Although not apparent in Fig. 4 there is, in fact, a qualitative difference between symmetric and asymmetric waveguides. In a symmetric structure, as long as the materials are transparent and the core index is larger than the claddings index, there is no cutoff wavelength for the fundamental modes, either TE or TM. This means that, aside from material properties, there is no fundamental limit on how long the idler wavelength can be. In contrast, in an asymmetric structure, the idler mode will experience cutoff if its wavelength becomes too long. This was not shown in Fig. 4 because the longest wavelength in calculations was set to be at 5.5μm and this long wave cutoff is predicted to be at about 7.1μm (for 0.23μm thick film and transparent claddings with index of 1.7). Although not particularly relevant in the case presented here, cutoff will become important when the index contrast in an asymmetric waveguide is small and/or the idler wavelength is long.

Figures 3 and 4 clearly illustrate the feasibility of phase synchronism in a waveguided OPO which has a GaP core and a lower cladding with refractive index close to that of sapphire. A practical device would have guided modes confined not only in the vertical direction (normal to the film), but also in the lateral direction. As long as the waveguide width in the lateral direction remains much larger than the extent of the modal fields in the vertical direction, the effective indices in such a waveguide would be mainly defined by the waveguide’s refractive index profile in the vertical direction, and thus the procedure used above to select the appropriate film thickness would remain valid. Should the extent of the lateral field become comparable to modal fields in the vertical dimension, the effective indices for the guided modes have to be found using a two-dimensional mode solver rather than the planar waveguide formula (5). In some limiting case, when the waveguide is narrow enough (for instance, a square cross section guide with uniform cladding around it), the difference between TE-like and TM-like modes vanishes and thus the polarization mode dispersion can no longer be used for the purpose of facilitating the phase synchronism. A detailed study of the effects of finite width of the waveguide on the phase matching condition is beyond the scope of this paper.

3. Threshold condition for OPO

In either bulk or thin films, once the phase matching condition is satisfied, the pump, signal and idler waves will exchange energy as they propagate in the second-order NLO medium. Optical parametric oscillations will take place provided the pump wave power exceeds a certain threshold. The determination of this threshold power can be obtained by considering the non-linear properties of the crystal and the means used to achieve the phase matching condition.

The nonlinear polarization of a medium is usually written in a form $P_i^{NL}=2{\epsilon }_0{\displaystyle {\sum }_{i,j,k}{d}_{ijk}{E}_j{E}_k}$ where d_ijk is the second order nonlinearity tensor, ε ₀ is the vacuum permittivity, and E_j and E_k are the time dependent components of the vector of the electric field strength. Here, the tensor terms are expressed in meters per Volt. For a given symmetry of a crystal and given direction of propagation, similar relations hold between the complex amplitudes of polarization and the electric field strength for the pump, signal, and idler waves:

The effective nonlinear polarization term d_eff in (8) depends on the propagation direction and polarizations of the pump, signal, and idler waves. In QPM crystals the relative phase between oscillations of electric field in a wave and corresponding nonlinear polarization produced by the other waves is step-like adjusted at interfaces between the domains. Compared to a case of a perfect phase matching in a uniform crystal, this leads to a reduced effective nonlinearity:

Although the effective nonlinearity is somewhat reduced, the QPM technique is very useful as it is the only viable tool to achieve phase matching in bulk isotropic crystals. Even in the case of birefringent materials such as LiNbO₃, the use of QPM is justified as it allows for selecting the propagation direction and polarizations of waves to maximize d_eff, so that the reduced value of d_QPM = η_QPMd_eff is still larger than the effective nonlinearity of the waves propagating in the direction of phase synchronism defined by the birefringence.

In order to facilitate the exchange of energy as the fields propagate in the non-linear crystal, the signal and idler waves are typically propagated in a resonator with low round trip losses (for example, a doubly resonant Fabry Perot cavity with high Q) and an external optical pump provides parametric amplification large enough to compensate for the cavity losses. For plane waves, the estimation of the amplitude of the electric field strength in the pump wave at the threshold of the onset of parametric amplification $E_{pth}$ is easily found [24] to be

It is noted that formula (11) is not derived for the case of a finite pump beam size but, instead, is obtained by simply relating the electric field amplitude in a plane wave to the power per unit area.

While the case of plane waves presented above is convenient for theoretical analysis, it is not quite consistent with the geometry of a typical experimental setup. In the case of finite cross section beams with Gaussian profiles, for instance, one has to account for the overlap associated with the pump, signal, and idler waves. This is dramatically different than the case of plane waves because in the plane wave case there is always a perfect overlap between the pump, signal and idler waves. For Gaussian beams, this perfect overlap is impossible because all wavelengths cannot have simultaneously the same values for beam radii and confocal parameters. A more significant issue is that product of two Gaussian distributions (such as electric field strengths in the right hand part of (8)) always produces a Gaussian distribution (the nonlinear polarization in (8)) with a radius smaller than that of either of the original Gaussians. In other words, regardless of what beam-waist radii are chosen for the pump, signal, and idler waves, one can never have a perfect match between all the electric field strength distributions on one side and the induced nonlinear polarization distributions on the other side. With an imperfect overlap between electric field distribution in a wave and the corresponding nonlinear polarization distribution produced by the other two waves, the nonlinear interaction is effectively reduced, which eventually leads to the need for greater pump power than would be required for the case of plane wave.

To quantify this, let us note that overlapping of the electric field distribution of, for instance, a laser-beam pump wave $E_p\left(x,y\right)$ with the nonlinear polarization produced by the signal and idler waves $P_p^{NL}\left(x,y\right)$ is expressed through an integral including the product of all three waves. Expressions for the overlapping integrals become especially simple when all the amplitudes are real:

Proper normalization of the field distributions then leads to the overlap integral of the form

The quantity A_norm has the dimension of area and can be considered as a normalized area defined by the overlap of the three waves. Reduced E_p, E_s, and E_i field overlap would result in a larger normalized area, which effectively accounts for a weaker nonlinear interaction of the waves. This normalized area, rather than the pump beam cross section, appears in the formula (11) for the threshold power of the OPO as

To illustrate the effect of finite beam size, let us assume Gaussian distributions for the pump, signal, and idler waves with radii w_p, w_s, and w_i respectively:

Direct integration in accordance with (13) produces following

Note that even for a seemingly perfect overlap between w_p = w_s = w_i, Eq. (16) gives $A_{norm}=\left(9/8\right)\pi w_p^2$ while the cross section of the pump beam is estimated at$A_p={\displaystyle \int E_p^{2}dxdy}/E_{0p}^2=\pi w_p^2/2$.

From (14) and (16), reducing the radii of the pump, signal, and idler field distributions by focusing these waves inside the nonlinear medium is seen as a way to achieve lower pump threshold. Tight focusing, however, would produce a beam with limited focal depth, which may lead to an inefficient use of the available length of the nonlinear medium. To maximize the efficiency of the nonlinear interactions, one would use the longest available crystal (provided that optical losses are negligible) and then adjust focusing of the pumping beam to get the confocal parameter b close to $b=L/\xi$, where ξ is the numerical constant on the order of unity, for Gaussian beams found to be ξ ≈2.84 [25,26]. Overall optimization of the parametric generation with Gaussian beams suggests that all the beams should have same confocal parameter $b=2\pi w_{p,s,i}^2{n}_{p,s,i}/{\lambda }_{p,s,i}$ so that beams radii are $w_{p,s,i}^2=L{\lambda }_{p,s,i}/2\pi \xi n_{p,s,i}$. With a little algebra, the normalized area of Eq. (16) becomes

Taking into account that the wavelengths of the pump, signal, and idler waves are related through the phase matching condition (2), the expression for the normalized area with the beams’ radii optimized for the most efficient parametric interaction is simplified to

Formulas (18) and (14) allow for estimating the threshold pump power in an optical parametric oscillator based on free space Gaussian beam focused on a non-linear crystal.

It is remarkable that the normalized area appears to be relatively large especially when using long nonlinear crystals. With L = 25mm, λ_p = 1.5μm, λ_s = 2.25μm, λ_i = 4.5μm, n_p ≈n_s ≈n_i ≈3.0, and ξ = 2.84, we then find A_norm ≈5⋅10³μm².

Based on the above argument it is clear that pump power can be reduced provided the overlap between pump, signal and idler beams can be increased. The advantage of using guided modes in terms of lowering the required pump power becomes apparent as guided modes effectively eliminate diffraction induced beam divergence. The pump, signal, and idler waves can be confined to a small cross sectional area defined by the geometry of the waveguide structure and refractive indices of the core and cladding, regardless of the extent of the nonlinear medium along the waveguide. Before doing more accurate estimation of pump threshold power, let us note that in a perfectly engineered waveguide, the mode cross section can be as small as, perhaps, A_w ≈(λ/n)². Using the longest wavelength in the above numerical example, λ_i = 4.5μm and n ≈3.0, one finds A_w ≈2.25 μm², so that A_norm/A_w ≈2.2⋅10³. This implies that the threshold power in a waveguide device can be thousands of times smaller than that in the case of bulk beams. Even though a non-perfect overlap of the pump, signal, and idler guided modes can reduce the nonlinear interaction, the expected decrease of the required pump power is still substantial. This brings the threshold power down to the tens or hundreds milliwatt level, quite suitable for chip-scale integrated optoelectronic devices and systems.

For guided modes, formula (14) is modified to

In the numerator of (20), the overlap integral is calculated within the nonlinear part of the cross section of the waveguide.

The waveguide configuration may or may not use QPM. If it does, like in [27–30], the effective nonlinearity d_eff, similar to the case of bulk beams, is reduced, which is accounted for by the factor ${\eta }_{GW}=2/\pi$. On the other hand, if the waveguide is engineered to achieve phase synchronism without QPM, then ${\eta }_{GW}=1$, and this value will be used in further estimations.

In waveguides with large aspect ratio, some simplification of (20) is possible if the modal fields E(x,y) can be factorized and presented in a form E(x)F(y). Then one can also factorize the expression for the normalized area $A_{wnorm}=W_x{W}_y$ with

The effective index method can be used to find the fields F(y) and thus rather accurately predict OPO threshold power in wide waveguides. For the purpose of order-of-magnitude estimations, we accept further simplification assuming that all the field distributions in the lateral direction F(y) are approximated by a model function cos(πy/d) for |y| < d/2 and zero everywhere else, where d is the waveguide width. The integration in (22) then leads to $W_y=9{\pi }^{2}d/128\approx 0.69d$.

Provided that the same nonlinear material with the same d_eff is used in a bulk QPM OPO and in a waveguide-based OPO, and assuming that the length of the nonlinear medium is the same in both cases, while taking the resonator loss factors to be identical for both configurations, the ratio of threshold powers for a guided wave device $P_{thGW}$ and a bulk crystal based OPO $P_{thb}$ is found to be

Formula (23), being derived with a number of simplifying assumptions, is not an accurate evaluation of the threshold power in a guided wave OPO, but rather an estimation showing feasibility of drastic threshold reduction offered by the guided wave configuration. When a more accurate evaluation is needed, one should perform calculations in accordance with (19), (20).

It worth mentioning that the W_x parameter in (23) is not an effective extent of the modal fields in the x-direction, but rather a normalization factor that takes into account both the extent of the modal fields and fields’ overlap within the nonlinear part of the waveguide’s cross section. In particular, all modal fields may be strongly confined, but, if the overlap integral vanishes, or, if the nonlinear material occupies only a tiny fraction of the waveguide’s core, the W_x parameter would diverge. The fact that only the core or, maybe, only a part of the core in a composite waveguide is nonlinear is already accounted for in (20) – (22) as long as the overlap of the E_p, E_s, and E_i fields is calculated within the nonlinear part of the waveguide’s cross section. The overlap of pump, signal, and idler waves in a bulk crystal and in a waveguide is schematically illustrated in Fig. 5 .

 

Fig. 5 Overlap of pump, signal, and idler waves in a bulk crystal (left) and in a waveguide (right). In the case of bulk crystals the cross section of each beam at its waist as well as the normalized area are scaled directly proportional to the nonlinear medium length. The guided modes remain tightly confined regardless of the length of the nonlinear medium.

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In the case of a thin-film waveguide, once the material parameters are chosen and the pump and idler wavelengths are specified, the waveguide layer thickness has to be chosen to satisfy the phase matching condition. With the planar geometry, this really leaves no room for further optimization. Here we estimate the threshold power reduction factor for the representative case of a symmetric waveguide. The pump wavelength was assumed to be λ_p = 1.5μm. With waveguide thickness rounded to t = 0.236μm, the signal and idler wavelengths are λ_s ≈2.25μm and λ_i ≈4.48μm. The strength of electric field for the pump, signal, and idler modes, as a function of a coordinate across the waveguide, is shown in Fig. 6 . The pump wave is a TM ₀ mode, while signal and idler are TE ₀ modes.

 

Fig. 6 (Color) Strength of electric field as a function of a coordinate across the waveguide for pump (black solid), signal (red short-dash), and idler (blue long-dash) guided modes. The pump wave is a TM ₀ mode, while signal and idler waves are TE ₀ modes.

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Evidently, all mode profiles have substantial penetration into the claddings. The TM-polarized pump wave shows a large field discontinuity at the film boundaries so that the field maximum is reached at the interfaces rather than in the middle of the film. TE-polarized modes, in general, are confined more tightly. In this example, the signal wave, even though its wavelength is somewhat longer than the pump wavelength, shows comparable penetration into the claddings, but the mode profile is smooth and the field maximum is reached in the middle of the film, so that this mode is confined somewhat better. The idler wave though, due to its much longer wavelength, shows very long wings expanding far into the claddings. Even without accurate arithmetic it is evident that the guided modes at the pump, signal, and idler wavelengths are not really strongly confined in the guiding film.

A single-film structure, according to the above algorithm for choosing the phase-matching film thickness, has no free parameters for optimization of the field confinement. In a composite waveguide structure with other layers added to the core (for instance, Si/GaP/Si), there will be additional degrees of freedom so that the confinement could be improved while still preserving phase matching.

By performing calculations according to (21), one finds W_x ≈5.5μm. Setting, rather conservatively, d = 10μm (smaller values might well be acceptable as long as d is still several times more than the extent of each of the modal fields in the vertical direction at the 1/e level), one gets $A_{wnorm}$ = 0.69dW_x ≈38μm². Compared to the normalized area (18) estimated for bulk beams A_norm ≈5⋅10³μm², this waveguide configuration provides about 130 times smaller normalized area. Also, with no need for the QPM, the effective nonlinearity in the waveguide’s core is 1/η_QPM = π/2 times larger than in the bulk QPM crystal. This translates into 320 times smaller threshold pump power required for optical parametric oscillator when compared to Gaussian beams propagating in a QPM crystal.

4. Conclusions

Using the polarization dispersion of the fundamental TE and TM modes, we have derived the CW perfect phase matching conditions for a three-layer uniform waveguide in which a thin film of crystalline GaP forms the second-order NLO waveguide core. For Al₂O₃ waveguide claddings and for TM ₀ pump wavelengths in the 1μm to 2μm range, signal outputs in the 3 μm to 5 μm range are predicted for phase matched film thicknesses of 0.14 μm to 0.30 μm. We then used orientation-patterned quasi-phase-matched GaP as a standard of comparison for our waveguide, in terms of the pump power required for OPO threshold, assuming in both cases that the signal mode as well as the idler mode are resonant in a Fabry Perot cavity. Gaussian beam theory and mode overlap theory was used to show that the thin-film waveguide has a threshold 320 × lower than that of QPM GaP case. The present waveguide appears suitable for low-power mid-infrared on-chip optoelectronics networks. Looking to future improvements and alternatives to the GaP/sapphire waveguide, we believe that the uniform OPO waveguide with a Si/GaP/Si core is also capable of perfect mid-infrared phase matching. That waveguide, with its lattice-matched core, has the advantage of being readily integrateable (directly grown, easily manufactured) on a silicon-on-sapphire wafer or on the silicon-on-insulator wafer that has become the standard platform of the silicon photonics industry.