1. Introduction

There is considerable interest in bright and compact sources of correlated and entangled photon pairs for a myriad of applications in quantum information and quantum optics. As such, there has been much effort over the years to improving the generation efficiency and spectral purity of photon pairs, with a steady improvement from many research groups [2, 3, 4, 5]. Most of this work has focused on optimizing the geometry and collection of photon pairs generated from SPDC in first a bulk crystal and then using quasi-phase matching (QPM) techniques to access the entire transparency range of the nonlinear crystal employed. It was found that, for excitation of both bulk and QPM crystals, that the pair generation rate is essentially independent of pump beam size [6, 7]. In a nonlinear crystal with a focused pump beam, the downconverted pairs are emitted into a continuum of spatial transverse modes (forming a cone) with the angular width increasing as the pump focusing increases. The overall pair generation rate is obtained by spatially integrating over this angular distribution, which gives a result that is independent of the pump beam spot size. Thus simply, the number of pairs generated only depend on the incident pump power but are spatially distributed differently depending on how strongly the pump beam is focused. Furthermore, the resultant spectrum is fairly broad due to this spatial walkoff effect.

For waveguide structures, this interpretation does not hold. There is only a finite (and small) number of transverse modes supported by a waveguide for a given wavelength. Furthermore, there is often only a single set of transverse modes that satisfy phase-matching considerations. Thus, essentially all of the SPDC photons are emitted into the same transverse mode, leading to a high density of SPDC photons all propagating along the waveguide, which results in a narrowing of the spectral bandwidth. Furthermore, in this geometry increasing the pump confinement does not result in a corresponding decrease in brightness due to emission into extra transverse modes, suggesting that increased confinement will result in increased pair production.

Recent investigations into SPDC in low-index contrast nonlinear waveguides have shown that the additional confinement does lead to a significant enhancement of the SPDC rate, in agreement with theoretical predictions [1]. This theory predicts that the enhancement scales inversely with confinement, leading to an obvious way of improving SPDC efficiency over low-confinement waveguides by using high-confinement (nanophotonic) waveguides.

In this manuscript we investigate the expected properties of SPDC in a nanophotonic waveguide system. First we modify the theory of SPDC presented previously into a regime valid for high-confinement waveguides. Then we use this theory to calculate the expected pair generation rates in nanophotonic waveguides and compare with theoretical and experimental results for low-confinement waveguides.

2. Theory of spontaneous parametric down conversion in a waveguide

The spontaneous parametric down conversion rate in a waveguide can be calculated in a simple manner by following the approach of Ref [1]. We start by expressing the electric field for the classical pump and the quantized signal and idler fields traveling in the z direction (quantization length L) as:

where the subscripts {P,S,I} describe the pump, signal, and idler, respectively. The classical pump power is P_P, and we assume a single photon in the signal and idler modes. The mode angular frequencies are given by ω _{P,S,I} = ck _{P,S,I} where k _{P,S,I} = 2π/λ_{P,S,I} are the free-space wavenumbers, and the modes are represented by a modal effective refractive index n _e,{P,S,I}. The time dependent creation operator for the signal and idler photons is given by a ^† _{S,I}(t). The transverse electric field distribution is contained in the term ϕ⃗ _{P,S,I}(r⃗_T), with the normalization:

where A_T is the transverse integration area.

The effective interaction Hamiltonian for a χ(2) material is given by:

where we have assumed that the electric fields for the pump, signal, and idler are scalar, with the polarization dependence of the optical modes for the desired interaction incorporated into the effective nonlinear parameter d _eff, and the integration is over the nonlinear interaction volume V. Substituting the expressions for the electric fields of the pump, signal, and idler modes (Eq. 2) into the interaction Hamiltonian, and keeping only the terms which satisfy energy conservation (ω_P = ω_S + ω_I), we obtain:

where the momentum mismatch is given by Δβ = (n_e,pk_P - n_e,Sk_S - n_e,Ik_I).

The downconversion rate is given by employing Fermi’s golden rule, R = ρ_v|〈H_I〉|²/h¯ ², with the density of states in the signal detection bandwidth δv_S = (c/λ_S ²)δλ_S given by ρ_v = (n_e,Sn_e,IL ²/(cλ_S ²))δλ_S. Putting this all together, and integrating over the waveguide interaction length L_C, we get the final expression for the signal generation rate:

where the mode interaction area is defined as

with the integral over the cross-sectional area comprised of the nonlinear material. The down-converted signal power is given by P_signal = (hc/λ_S)R_signal:

Quasi-phase matching (QPM) is often used to obtain a broader wavelength range of nonlinear interactions. In this case, the above formula holds with the substitutions:

where m is an odd integer corresponding to the QPM poling order, and Λ is the QPM period. This expression is similar to that derived for a waveguide in Ref. [1], with the difference that the effective modal index appears explicitly in the downconverted rate/power and in the phase-mismatch term (instead of the material indices and a waveguide dispersion term). Additionally, the effective interaction area is only calculated over the region which experiences nonlinearity. With these modifications, the presented formula is valid for not only the typical low Δn waveguides conventionally fabricated in nonlinear crystals, but also for large index contrast waveguides and waveguiding structures consisting of both nonlinear and linear materials. Note that the expression for the effective area can be easily generalized for an arbitrary distribution of nonlinear material by including the nonlinear material coefficient distribution d _eff(r⃗_T) in the transverse spatial integral for A_I.

Investigating Eq. 8, we see that the downconverted signal power scales with waveguide geometry as ∝ L_C ²/ (n_e,Pn_e,Sn_e,IA_I). Therefore, for a fixed crystal length, the downconverted signal power can be increased by reducing the product of the effective mode indices and the nonlinear interaction area. High-confinement waveguides can often accomplish this feat because of two factors. First, for a higher index contrast between the core and cladding the effective index can be reduced over that of a small Δn material (given that the core refractive index is the same and appropriate dimensions are chosen), as the confinement condition n_cladding < n_e,wav < n_core allows waveguiding with a lower effective index for cases where n_cladding is reduced. Secondly, the higher index contrast can result in a more tightly confined mode, reducing the interaction area. However, we must point out that this is counteracted in part by the reduced fraction of the optical mode which lies in the nonlinear core, although the fact that confinement can be increased approximately an order of magnitude with a less than 50% drop in nonlinear overlap indicates a large benefit occurs for high-contrast waveguides.

It is also instructive to consider the spectral bandwidth of the downconverted photons. This is given by the sinc² term in Eq. 8, which has a FWHM given by:

where all terms are evaluated at the phase-matched center wavelengths. The second and third terms in the denominator result from dispersion, incorporating both material and waveguide dispersion. For the case of a typical low-index waveguide, the waveguide dispersion can be neglected, whereas for high-confinement waveguides the waveguide dispersion can be dramatically larger and must be included. The overall downconverted signal power can be calculated by integrating Eq. 8 over all wavelengths, resulting in a linear dependence of total downconverted signal power on crystal length L_C.

3. Spontaneous parametric down conversion in a PPKTP waveguide

In order to quantify the benefit of using nanophotonic waveguides for SPDC sources, we will numerically investigate the enhancement in a sample system. We chose to investigate PPKTP due to its suitability for down conversion from a visible pump to the near IR (where high performance single photon counters are readily available), and to compare with recent experimental results. We assume a Type II degenerate SPDC geometry, where the pump (λ = 405 nm, Y-polarized) and signal/idler (λ= 810 nm, Y/Z polarized, respectively) beams copropagate along the X-direction. For this choice of beam polarizations (where the pump electric field X- and Z-components are small and can be neglected), the relevant nonlinear coefficient for first-order QPM is d _eff = 2d _{24,bulk}/π, where d _{24,bulk} = 3.92 pm/V [8]. The physical geometry consists of a PPKTP ridge waveguide core formed from Z-cut PPKTP, surrounded on all sides by a linear low-refractive index dielectric, with refractive index 1.45 (such as silica). As KTP is highly birefringent, in order to accurately model this system we employ the Sellmeier equations, given by [9]:

 

Fig. 1. Electric field profile for Type II SPDC in a low index contrast Rb ion-exchanged PPKTP waveguide for both the pump mode (λ= 405 nm, Y-polarized, left panel) and idler mode (λ = 810 nm, Z-polarized, right panel), after [1]. Inset shows the electric field profile for a highly-confined nanophotonic ridge PPKTP waveguide (w=450 nm, h=500 nm) for the same pump and signal wavelengths.

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where n_x,y,z denote the refractive indices in the x,y,z directions, respectively, and λ is the optical wavelength expressed in microns. We neglect the refractive index dispersion of the surrounding dielectric, as for nearly all optically transparent cladding dielectrics the change is negligible (< 0.01) for the wavelengths of interest.

The above system was modeled using a commercial finite element electromagnetic eigen-mode solver (COMSOL), for a variety of different ridge waveguide heights and widths. Figure 1 shows the electric field distribution in a low-index conventional KTP waveguide (modeled after Ref. [1]), with the inset showing the corresponding field profile in a nanophotonic waveguide of core dimensions 450 nm by 500 nm, for both the pump and signal modes. We see that the nanophotonic modes are much more tightly confined due to the higher core/cladding refractive index contrast. Furthermore, the location of the peak electric fields for the modes in the nanophotonic waveguide coincide, in contrast to the low-index case. Both of these points suggest that nanophotonic waveguides enhance SPDC, however the fact that the optical mode in the nanophotonic waveguide is not exclusively located in the nonlinear material will reduce the nonlinear interaction.

The calculated effective indices for the pump, signal, and idler modes are shown in Fig. 2 for waveguide widths of {300,400,500,600} nm versus waveguide height. We see that the effective index decreases from the material core value towards the cladding value as the fraction of the mode in the core decreases (by reducing waveguide width and/or height), as expected. For the waveguide dimensions considered here, the reduction over the core material index is in the range 5 – 10% for each of the effective indices. Figure 3 shows the corresponding period for first order QPM using the calculated effective indices and Eq. 10. The period ranges from 2 – 2.8μm for the waveguide core geometry considered here. For comparison, the first order QPM period is ~ 8.36μm for a low-index waveguide [1].

 

Fig. 2. Effective index for pump mode (top left panel), signal mode (top center panel), and idler mode (top right panel), for various fixed waveguide widths {300,400,500,600} nm and various heights. The effective index increases as waveguide width and/or height increases, as expected. For comparison, the material index is {1.840,1.758,1.843} for the pump, signal, and idler, respectively. Bottom panels show (from left to right) the magnitude of the electric field distribution for the pump, signal, and idler modes for a 500 nm by 500 nm waveguide.

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The effective nonlinear interaction area is determined by Eq. 7, where the integral is over the waveguide core. Figure 4 shows the calculated effective area for waveguide widths of {300,400,500,600} nm and various heights. We see that for a given waveguide width the effective area decreases as waveguide height is increased. While this suggests that further reductions may be obtained by increasing height even further, note that the optical confinement for each of the pump, signal, and idler modes has a minimum versus waveguide height (fixed width). This results in a minimum in the effective area versus height which lies in the 500 – 700 nm range for the waveguide widths considered here. For a given waveguide height, the figure indicates the effective area has a minimum for waveguide widths of 500 – 600 nm. We see that the interaction area has a minimum value of approximately 0.4μm² with a variation of only ~ 10% over a waveguide width and height range of 500 – 600 nm and 450 – 550 nm, respectively. This effective area is ~ 35 × smaller than the low-index waveguide shown in Fig. 1 (16μm²).

 

Fig. 3. First order quasi-phase matching period for Type II SPDC in PPKTP versus waveguide height for waveguide widths of {300,400,500,600} nm.

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Once the effective indices and the effective interaction areas are calculated, the downconverted power spectrum is readily determined from Eq. 8. Figure 5 shows the calculated signal photon peak generation rate and peak signal power spectral density assuming a first-order QPM crystal of length 10 mm, and a pump power of 1 mW. We see that signal photon peak generation rates exceeding 10⁹ photons/sec are achieved, with corresponding peak signal power spectral densities exceeding 1 nW/nm over a wide range of waveguide dimensions. The data trends primarily follow that of the effective area, due to its much stronger dependence on geometry. However, the counteracting effect of the effective indices shifts the location of the maximum in the rate and power plots towards smaller core geometry. The calculations show that as the waveguide width increases, the corresponding maximum shifts monotonically towards smaller waveguide heights. The highest signal photon generation rate occurs for a waveguide dimension of 500 nm by 500 nm, with a value of 6.55 × 10⁹ photons/sec. The corresponding signal peak power spectral density is 1606 pW/nm. These high rates/powers are fairly tolerant to waveguide dimensions, with errors in waveguide dimensions of 100 nm leading to only slight < 10% changes for dimensions near the optimal value. For the optimal waveguide dimension of 500 nm by 500 nm, the spectral width determined by Eq. 11 is 0.47 nm, which is narrower than that calculated for the low-index waveguide shown in Fig. 1 (0.57 nm) due to the strong dispersion in the nanophotonic waveguide.

 

Fig. 4. Effective nonlinear interaction area for a nanophotonic KTP ridge waveguide surrounded by a linear dielectric (index 1.45) for Type II SPDC (405 nm pump Y-polarized, 810 nm signal Y-polarized, 810 nm idler Z-polarized). The area is shown for waveguide widths of {300,400,500,600} nm versus waveguide heights ranging from 200 to 550 nm. The calculations show that the interaction area has a minimum value of approximately 0.4 μm² for waveguide widths around 500 – 600 nm and waveguide heights of 450 – 550 nm.

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Table 1 compares the predicted performance of a nanophotonic waveguide to both a low-index waveguide and a bulk crystal. We have chosen a 500 nm by 500 nm ridge waveguide, which has been shown to yield approximately the maximum pair generation rate for the system of interest (Type II SPDC, 405 nm pump, 810 nm signal/idler). The calculations assume an input pump power of 1 mW, and a first order QPM crystal of length 10 mm. The low-index waveguide is assumed to be formed by Rb ion-exchange with a waveguide width of 4 microns, a 1/e depth of 8 microns (the electric field profile is shown in Fig. 1), and an index step of ≃ 0.02, following Ref. [1]. The calculations show the predicted peak pair generation rate, peak down converted signal power, and peak conversion efficiency (down-converted signal photons per pump photon per nm), with all calculations performed for perfectly phase-matched excitation (Δβ = 0). We see that the nanophotonic waveguides have much higher predicted pair generation rates than both low-index waveguides and bulk crystals. The enhancement factor is approximately 45 × that of low-index waveguides and more than 6500 × that of bulk crystals. This large enhancement over low-index waveguides is primarily attributed to the much smaller effective interaction area in a highly-confining nanophotonic waveguide (factor of ~35), with the remaining difference contributed by the slightly lower effective refractive indices for the pump, signal, and idler modes. The extremely large improvement over the bulk crystal is due to both the strong transverse confinement plus the increased interaction length in a waveguide.

We have recently experimentally and theoretically investigated SPDC in a low-index contrast PPKTP waveguide for Type II SPDC and compared the pair generation rate with that using a bulk PPKTP crystal [1]. These waveguides were formed using Rb ion-exchange to locally enhance the refractive index by a small amount (Δn ≃ 0.02), over an area of a couple microns. The experimental measurements showed very good agreement with the predicted values, with an error < 5% for waveguides and < 20% for bulk crystals. We also note that the theory presented in this work is in agreement for a low-index contrast waveguide.

 

Fig. 5. Signal photon generation rate and peak power spectral density (Δβ = 0) for Type II SPDC (405 nm pump Y-polarized, 810 nm signal Y-polarized, 810 nm idler Z-polarized) in a nanophotonic KTP ridge waveguide surrounded by a linear dielectric (index 1.45). The calculations assume a first-order QPM crystal of length 10 mm, and a pump power of 1 mW, with a detection bandwidth of 1 nm. The calculations are shown for waveguide core widths of {300,400,500,600} nm versus waveguide heights ranging from 200 to 550 nm. The calculations show that the highest photon generation rates occur for a waveguide core dimension of 500 nm by 500 nm, with rates exceeding 6 × 10⁹ photons/sec. The corresponding spectral density exceeds 1600 pW/nm.

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4. Heralded sources of single photons in a PPLN waveguide

Heralded sources are particularly useful for practical quantum information devices [10]. Here we consider the generation of heralded sources of single photons in two wavelengths of particular interest for practical systems: 1310 and 1550 nm, chosen to correspond to the minimum of dispersion and loss, respectively in single-mode optical fiber. We will consider Type I SPDC in first-order periodically-poled lithium niobate (PPLN), which allows access to the d ₃₃ = 31 pm/V nonlinear coefficient which is significantly larger than that possible in KTP. The calculations are performed as described above, with the waveguide core consisting of Z-cut PPLN surrounded by silica (n = 1.45). The PPLN has a refractive index given by the Sellmeier equations [11]:

Table 1. Comparison of nanophotonic waveguides, low-index waveguides, and bulk crystals of PPKTP for Type II SPDC from 405 nm to 810 nm. All calculations assume a crystal length of 10 mm. Nanophotonic waveguides are predicted to have efficiency improvements of 45× over low-index waveguides and more than 6500× that of bulk crystals.

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where n_e denotes the extraordinary index (n_e = n_z), n_o is the ordinary index (n_o = n_x = n_y), and λ is the optical wavelength expressed in microns.

We consider the two cases of SPDC in which a 405 nm pump is downconverted into {1550,548} nm and {1310,586} nm signal and idler pairs. This pump wavelength was chosen to correspond to readily available GaN diode lasers, where the heralding photons at 548 and 586 nm can be detected with efficient Si single-photon detectors. All optical beams are Z-polarized. Table 2 shows the peak downconverted signal photon rate, power, and efficiency for a nanophotonic waveguide of core dimensions 500 nm by 500 nm, for a first-order QPM crystal of length 10 mm. This size was chosen as it lies close to the absolute maximum signal photon generation rate for both cases considered. We see that peak signal photon generation rates exceeding 10¹⁰ photons/sec/nm/mW are possible.

Table 2. Predicted peak signal photon rates, powers, and efficiencies for Type I SPDC from a 405 nm pump to {1550,548} nm and {1310,586} nm signal and idler pairs in a nanophotonic PPLN waveguide. All calculations assume a crystal length of 10 mm.

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5. Conclusion

In this manuscript we presented a theory of spontaneous parametric down conversion valid for nanophotonic waveguides incorporating both linear and nonlinear materials. This theory was applied to Type II SPDC in a PPKTP nanophotonic waveguide to indicate that a dramatic enhancement in downconverted photons is possible, with improvements over 40× that of the more conventional ion-exchanged low-index PPKTP waveguides. We found that for first-order QPM crystals of length 10 mm peak signal generation rates in excess of 10⁹ photons/sec/mW/nm are predicted over a wide range of waveguide core geometries, with only a slight dependence on geometry near the optimal dimensions of 500 nm by 500 nm. For the optimal waveguide core dimension the calculated spectral width (0.47 nm) is found to be narrower than that calculated for a low-index waveguide (0.57 nm) due to the additional dispersion caused by the waveguide. We have also shown that peak pair generation rates exceeding 10¹⁰ photons/sec/nm/mW are possible in heralded sources based on Type I SPDC in a 10 mm long first-order QPM PPLN nanophotonic waveguide.

We must point out that the actual demonstration of SPDC in a nanophotonic waveguide is not trivial. To our knowledge the fabrication of these highly-confined periodically-poled waveguides has not been demonstrated, although most of the challenging process steps have already been demonstrated (e.g. thin film transfer [12], anisotropic etching [13], small period QPM [14]) in LiNbO₃. Furthermore, fabrication imperfections (such as surface roughness and side-wall tilt) will have a stronger effect on the overall device performance, however these can be minimized through optimization of the fabrication procedure.

While we have only analyzed a couple of systems, we expect this enhancement to occur for many other SPDC systems of interest. The very high rates predicted here will allow useful pair generation at pump powers in the nW level. More generally, nanophotonic structures can also be integrated to form high-efficiency chip-scale sources of photon pairs, and can be combined with other planar structures to create additional functionality such as completely-integrated entangled photon sources. Additionally, we expect this enhancement of nonlinearities in strong-confinement nanophotonic waveguides to also allow dramatic improvements in the efficiency of other nonlinear waveguide devices, such as OPO’s.

This work was supported in part by the Disruptive Technology Office under contract NBCHC060076.