1. Introduction

Photon-pair sources based on spontaneous parametric downconversion (SPDC) in a nonlinear crystal waveguide have been shown to be significantly more efficient than those in a bulk crystal [1, 2, 3, 4, 5, 6]. To utilize waveguide sources in quantum information processing (QIP) applications, it is highly desirable to integrate additional functionality such as pump sources and modulators at the waveguide-chip level for compactness, reliability, and ease of operation. As a first step we have developed a waveguide SPDC source with integrated single-mode polarization-maintaining (PM) fibers with frequency-degenerate, orthogonally polarized signal and idler outputs at 1316 nm. The fiber-coupled waveguide source is particularly suitable for long-distance quantum communication protocols such as fiber-based quantum key distribution (QKD) [7, 8, 9]. The operating wavelength of 1316 nm allows the QKD network to coexist with the standard 1.55 µm telecommunication fiber-optic infrastructure with minimal effects due to cross-talk and nonlinear spurious signals.

Two key performance metrics for a photon-pair source are the spectral brightness and the degree of indistinguishability for the photon pairs. Source spectral brightness of a nonlinear crystal waveguide in units of generated pairs/s per mW of pump power per nanometer of bandwidth has been shown to greatly exceed that of its bulk crystal counterpart. Fiorentino et al. reported [5] a 50-fold enhancement in the generation rate of a periodically poled KTiOPO₄ (PPKTP) waveguide over that of a bulk PPKTP crystal and attributed it to a much larger density of states for the waveguide. We investigate the origin of this significant increase in the density of states using a different theoretical model of waveguide SPDC generation. We modify the standard phase matching function of a nonlinear medium by including the transverse wave vector imposed by the cross-sectional index profile of the waveguide and arrive at the same analytical result as that found in [5]. Our model therefore gives a simple physical picture of the waveguide generation rate and may lead to improved design of waveguide sources.

An important usage of a photon-pair source is the generation of entangled photons. For example, one can separate orthogonally polarized frequency-degenerate signal and idler photon pairs and use a 50–50 non-polarizing beam splitter to generate polarization-entangled photons postselectively [10]. The entanglement quality of such a SPDC source can be estimated by how indistinguishable the signal and idler photons are, as measured by their Hong-Ou-Mandel (HOM) quantum interference [11]. Previous HOM measurements of waveguide sources have not shown a high quantum-interference visibility and high-quality polarization entanglement has not been obtained from these waveguide sources. In this work, we have measured a HOM visibility of 98.2% after subtraction of accidental coincidences caused mostly by the high dark count rates of InGaAs single-photon detectors. The high HOM visibility of our waveguide device suggests that it is a suitable source for many QIP applications that require compactness, high spectral brightness, and a high degree of indistinguishability.

In Sec. 2 we develop our theoretical model of SPDC generation in a waveguide based on the phase matching function that takes into account of the waveguide’s index profile. The fabrication of the waveguide in PPKTP is described in Sec. 3. We present our flux and bandwidth characterization of the SPDC waveguide source in Sec. 4 and the HOM quantum interference in Sec. 5, before concluding our work in Sec. 6.

2. Theory of SPDC in Waveguides

SPDC generation efficiency in bulk crystals is typically in the range of 10^-12 to 10^-8, depending on the type of crystal, the crystal length, collection angle and bandwidth. Moreover, in a bulk crystal the total output flux from a bulk crystal is linearly proportional to the pump power and is not dependent on pump focusing. On the other hand, several groups have demonstrated that a nonlinear waveguide yields a significantly higher SPDC efficiency [1, 2, 3, 5, 6, 12]. Fiorentino et al. made a direct comparison between the outputs from a waveguide on PPKTP and a bulk PPKTP crystal, showing a 50-fold enhancement in the case of the waveguide and in agreement with a semiclassical model based on the density of states of guided mode fields [5]. The model suggests that the waveguide supports a much larger density of states than its bulk crystal counterpart. According to this model, the downconverted signal power dP_s within a bandwidth dλ_s is given by [5]

where L is the crystal length, P_p is the pump power, n_k is the waveguide modal index at wavelength λ_k for subscript k being signal s, idler i, or pump p, and A_I is the mode-overlap area of the three interacting fields. For type-II first-order quasi-phase matching (QPM) in PPKTP with a grating period Λ, the effective second-order nonlinear coefficient is d _eff=(2/π)d ₂₄ and the momentum mismatch is

where k_jz is the wave vector k⃗_j in the material projected along the propagation axis z.

Equation (1) shows that the spectral brightness is inversely proportional to the mode-overlap area A_I, and therefore field confinement in a waveguide leads to an enhanced pair generation rate. The difference in the output flux between a waveguide and a bulk crystal is attributed to the larger density of states (excitation modes) in a waveguide [5]. However, it is not obvious how to relate the density-of-states model for a waveguide to the standard model of SPDC in a bulk crystal that has no dependence on the pump beam area [13, 14]. To provide a more intuitive understanding of the physical origin of the enhanced output of a waveguide, we develop an alternative theoretical model that incorporates a natural extension of the conventional phase-matching function by taking into account the transverse momentum imposed by the waveguide.

In a bulk crystal with a propagation geometry shown in Fig. 1 the SPDC signal output power within a dλ_s bandwidth is [15]

where ϕ_s is the angle between the signal wave vector k⃗_s and pump wave vector k⃗_p. Most of the signal output comes from a narrow signal cone, beyond which the phase-matching function f(λ_s,ϕ_s)=sinc²(Δk_zL/2) is negligibly small to contribute. Consider the momentum mismatch for the longitudinal (z) component, as given by Eq. (2), and for the transverse component

where, without loss of generality, we assume that the pump propagates along the z principal axis so that there is no walk-off due to double refraction and that k_pz=k_p. For signal and idler propagating at small angles relative to the pump, ϕ_s,ϕ_i≪1, the z-component of the signal and idler wave vectors can be expressed in terms of their transverse wave numbers k_jt to second order:

and the longitudinal momentum mismatch Eq. (2) becomes

The first term on the right side of Eq. (6) is the standard phase-matching condition for collinear propagation, whereas the second term is the additional contribution from the transverse signal and idler components for noncollinear propagation.

 

Fig. 1. Geometry of noncollinear propagation of pump k⃗_p, signal k⃗_s, and idler k⃗_i.

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The transverse momentum matching of Eq. (4) dictates that the signal and idler transverse components are equal and opposite, k_st=-k_it. For the case of frequency-degenerate SPDC in type-II phase-matched PPKTP, λ_s=λ_i and n_s≈n_i, and therefore k_s≈k_i and ϕ_s≈-ϕ_i. The standard phase matching condition for collinear propagation is Δk_z=0 with k_st=-k_it so that Eq. (6) is simplified to

From the phase-matching function f(λ_s,ϕ_s), we can obtain the phase-matching angular bandwidth by setting Δk_zL=π which yields a divergence angle for SPDC in a bulk crystal of

Consider an ideal rectangular waveguide with dimensions w_x×w_y and a uniform index Δn higher than the surrounding nonlinear material. The transverse index profile of the material, including both the waveguide and its surrounding, is a boxcar function with width w_x (w_y) along the x (y) dimension. The 2D index profile induces a transverse momentum vector k⃗_gt that must be included in the transversal momentum mismatch of Eq. (4). Similar to the longitudinal grating momentum added by periodic poling in nonlinear crystals, we obtain the transversal grating vector k⃗_g from the Fourier transform of the 2D index profile. For the ideal case of a uniform rectangular waveguide, k_gx (k_gy) is simply a sinc function centered at k_gx=0 (k_gy=0) with a half width of π/w_x (π/w_y). We note that the transverse momentum of the signal field in its fundamental waveguide propagating mode is also bounded by |k _sx,sy|≤π/w _x,y.

and the vectorial form allows the possibility of an asymmetric waveguide profile. The longitudinal phase-matching condition Eq. (6) can be written as

Consider the case that C is a constant over the range of possible k_st, which is bounded by |k_st|≤π/w_t for waveguide propagating modes. At the maximum value of k_st=±π/w_t, we have C≥2π ²/w ² _t. On the other hand, at the minimum value k_st=0, C=k ² _gt. Given that k_gt is a sinc function that has the first zero at ±2π/w_t, it is always possible to find a k_gt for any value of k_st such that C is a constant. In this case, the waveguide phase-matching function of Eq. (10) behaves similar to the normal bulk-crystal phase matching.

Noting that the phase-matched output has no transverse wave number dependence within the k_st waveguide-propagating range of ±π/w_st, we can now evaluate the spectral brightness of the output signal. Rewriting the angular integration over ϕ_s in Cartesian coordinates, we have

The spectral brightness of the signal output is then given by

where A _wg=w_xw_y is the cross-sectional area of the waveguide, and Δk_z is given by Eq. (10). The combination of the constant C/2ks and the longitudinal grating momentum 2π/Λ in Eq. (10) yields an effective grating momentum in the waveguide 2π/Λ′=2π/Λ+C/2k_s so that the sinc function dependence of the waveguide output remains the same as in the bulk crystal. The enhanced waveguide output is due to the factor 1/A _wg resulting from a much larger phase-matched k_st range. The SPDC interaction remains phase matched within a large range of effective divergence of the signal field because such divergence is always compensated by the transverse grating momentum k_gt imposed by the waveguide.

Comparing Eq. (13) with Eq. (1), our theory gives a result that is almost identical to that of [5], except we use the waveguide cross-sectional area A _wg whereas [5] uses the field interaction area A_I. In a waveguide with moderate confinement, the majority of the interacting fields lie within the rectangular cross section, and A _wg and A_I are approximately the same. Applying the result of Eq. (13) to our PPKTP waveguide with an approximately 4 µm×8 µm rectangular cross section, we have A _wg=32 µm² and estimate a pair generation rate of 2.1×10⁷ pairs/s/mW of pump power over a bandwidth of 1.1 nm.

3. Fabrication of Fiber-Coupled PPKTP Waveguide

Our waveguide device was fabricated on a 16-mm long, flux-grown KTP crystal. We applied direct contact mask lithography to pattern Al onto the +Z crystal surface to define the width of the waveguides, and the channels were formed by diffusion exchange of 100% Rb+ ion through the +Z surface at 400°C for 120 minutes. The resultant transverse index profile had a 4-µm wide index step of 0.02 in the lateral direction, and a diffusion profile along the Z direction n(z)=n _KTP+0.02exp(-z/d), with d=8 µm. In a type-II phase-matched process, the pump field is polarized along the crystallographic Y axis, while the signal and idler fields are polarized along the crystallographic Y and Z axes, respectively. All fields propagate along the X axis of the crystal. We applied periodic poling to the KTP waveguide crystal with a grating period of 227 µm, designed for type-II QPM with frequency-degenerate outputs at 1316 nm near room temperature. We used the Sellmeier equation for bulk KTP [16] and the finite element method to calculate the eigen-modes of the waveguide. The modal indices for the pump, signal and idler fields are calculated to be n_p=1.783, n_s=1.749, n_i=1.830, respectively.

Waveguide devices are often difficult to use because of tricky input and output coupling, so it is most useful to pre-connect the devices to single-mode optical fibers. For our PPKTP waveguide we attached polarization maintaining (PM) single-mode fibers to the waveguide’s optically polished facets for the pump input and downconversion output. The fast and slow axes of the PM fibers were aligned with the Y and Z axes of the PPKTP crystal. We measured at room temperature a fiber to waveguide coupling efficiency of ~49% for the input fiber (Coastal Connections PM630) and ~50% for the output fiber (Coastal Connections PM1310). We have characterized the PPKTP waveguide device by measuring its type-II phase-matched second harmonic generation (SHG) process. A 1310-nm diode laser served as the pump and we obtained SHG outputs over a range of waveguide temperatures. From the temperature tuning curve we estimate that frequency degenerate operation at the desired 1316 nm would occur in the room temperature range. Also, absolute-power SHG measurements yielded a nonlinear coefficient d _eff=(2/π)d ₂₄ of ~2.1 pm/V.

4. Flux and Bandwidth Characterization

Our theoretical model of SPDC in a waveguide and the measured d _eff using SHG allow a straightforward comparison with the brightness characterization of our waveguide SPDC source. Pumped by a cw diode laser at 658.0 nm, the fiber-coupled SPDC outputs were collimated and sent through a long-pass filter to block the pump, followed by a 10-nm band-pass filter centered at 1316.0 nm. We used a polarizing beam splitter (PBS) to separate the orthogonally polarized signal and idler beams and coupled them into their respective SMF-28 single-mode optical fibers. The signal-idler coincidences were measured using a pair of fiber-coupled InGaAs avalanche photodiode (APD) single-photon counters with a coincidence window of 2.5 ns. The InGaAs APDs operated in the Geiger mode with a gating frequency of 50 kHz, and a 20-ns duty in each cycle. Detector efficiencies were calibrated using a laser source at 1316 nm and a fiber variable attenuator to be 15.4% and 20.5% with corresponding dark counts of ~32.1 kHz and ~17.2 kHz, respectively. Taking into account the waveguide-to-fiber coupling, transmission efficiencies of optical components, the overall signal and idler detection efficiencies were estimated at η_s≃1.8% and η_i≃2.8%.

Figure 2 shows the detected singles and coincidences rates at various pump powers P_p in the waveguide. At low pump powers, the singles rate (after subtraction of dark counts) has a linear power dependence, as expected. At higher pump powers the saturation of the InGaAs APD detectors caused an undercount of the singles rate, as shown in Fig. 2 for pump powers greater than ~1 mW. The shaded region in Fig. 2 shows the singles rates and pump power levels with which there was no detector saturation. Using the standard time-delay technique [17] we also measured the accidental coincidences that can be subtracted from the raw coincidence data to yield the net detected coincidence rate R_c in Fig. 2. Following the method in [2], the pair generation rate can be calculated according to R_gen=R_c/η_sη_i P_p, and we obtain a pair generation rate of ~2.0×10⁷ pairs/s/mW of pump, in excellent agreement with our theoretical estimate of ~2.1×10⁷ pairs/s/mW. Note that the theoretical and measured generation rates are total-flux values because we used a 10-nm bandwidth that was much larger than the expected phase-matching bandwidth of 1.1 nm. Excluding the dark counts of the detectors, the expected singles rate obtained from the net coincidence rate R̃_s,i=R_c/η_i,s is smaller than the measured rate because the measured singles included background photons that were primarily fluorescence photons. Over the 10-nm measurement bandwidth, which invariably included background photons outside of the phase-matching bandwidth, the ratio of total fluorescence photons to the total downconverted photons was 15±5%.

 

Fig. 2. Measured singles counts (dark counts subtracted) for signal, idler, and coincidence counts (accidentals subtracted) versus pump powers. Shaded area is the region of interest in which detector saturation is negligibly small.

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We characterized the SPDC spectrum by using a tunable narrowband filter with a polarization-independent Gaussian transmission spectral bandwidth of 0.79 nm and a 50% peak transmission efficiency. The fiber filter was connected to the waveguide output PM fiber and collimated for free-space output. We scanned the filter center wavelength from 1313.0 nm to 1319.0 nm, and recorded the corresponding singles counts. A typical spectral histogram for signal photons is shown in Fig. 3. After data deconvolution to remove the effect of the filter’s finite bandwidth and fitting it using a sinc-squared function, we obtain an effective signal-photon bandwidth of ~1.29 nm. Expressed in terms of generated pairs per second per GHz of bandwidth per mW of pump, our waveguide source emits ~10⁵ pairs/s/GHz/mW, which is of the same order of magnitude as the pair generation rate of a 3.6-cm-long type-II phase-matched periodically poled lithium niobate waveguide reported in [6]. In our narrowband measurements we have observed a weak pedestal in the spectrum near 1313.5 nm, which has also been reported elsewhere [5]. A similar satellite peak centered at 1318.5 nm was also observed in the idler output spectrum. The satellite peaks, located 2.5 nm from the peak wavelength, may be caused by nonuniformity in the crystal’s periodic grating structure.

 

Fig. 3. Spectral histogram of signal photons. Theoretical curve is obtained by a convolution of the Gaussian transmission spectrum of the filter and a sinc-squared phase-matching function.

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In Fig. 3 there is a uniform band of background counts (~10 counts per bin) that we attribute to fluorescence photons. At the spectral peak, the ratio of fluorescence photons polarized along the Y crystal axis to the downconverted signal photons is ~10%. A similar measurement for the other polarization and the idler photons shows a lower value. However, if we integrate the background counts over a bandwidth of 10 nm we obtain a ratio that is higher than the 15% we measured using only a 10-nm filter. This apparent discrepancy resulted from our arrangement of the filters. In this case, the narrowband fiber filter was connected directly to the waveguide output fiber, thus the strong residual pump produced additional background photons within the narrowband filter. To accurately measure the amount of fluorescence photons, we placed the narrowband filter after the pump-blocking long-pass filter. With the narrowband filter center wavelength fixed at 1316 nm, we changed the waveguide temperature to detune the signal and idler until their spectra were outside of the filter bandwidth. We then observed that the true ratio of total generated fluorescence photons to total downconverted photons within the phase-matching bandwidth was ~2%, which is consistent with the previously obtained ratio of 15±5% over the entire 10 nm bandwidth. We note that the amount of fluorescence photons in our source was much lower than those reported in previous PPKTP waveguides that were pumped at shorter wavelengths [1, 3, 5, 18].

5. Hong-Ou-Mandel Interference Measurements

One of the simplest ways to generate polarization-entangled photons is to send a pair of orthogonally polarized photons as the two inputs to a 50–50 beam splitter [10], which requires that the two photons be completely indistinguishable in their spatial, spectral, and temporal characteristics. The HOM setup shown in Fig. 4 is ideal for measuring the distinguishability of two individual photons. We used the tunable narrowband filter to remove the satellite peaks from the output spectra. The filtering also provided a detection bandwidth that was slightly smaller than the phase-matching bandwidth and ensured a high degree of signal-idler spectral overlap. For maximum spectral overlap between signal and idler we repetitively scanned the signal and idler spectra while we fine-tuned the waveguide temperature. At a pump wavelength λ_p=658.0 nm, we found the signal and idler outputs were wavelength degenerate at a temperature of 19.5° C. Additional free-space filtering of the residual pump and fluorescence photons were provided by a long-pass filter and a 10-nm band-pass filter.

 

Fig. 4. Experimental setup of HOM quantum interference measurement.

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After the signal and idler photons were separated by a PBS, we sent the horizontally polarized signal through a prism delay stage that was controlled by a motorized translation stage. We also made sure that the overall transmission of the signal arm was minimally perturbed over the entire scanning range of the motorized stage. The signal was then coupled into one of the input ports of a 50–50 fiber coupler. We coupled the vertically polarized idler into the other port of the 50–50 fiber coupler and used a paddle polarization controller to match the polarization of the idler to that of the signal. The total path lengths of the fiber and free space were carefully matched for the two arms. The outputs of the 50–50 fiber beam splitter were sent to InGaAs single-photon counters for coincidence measurements, as described in Sec. 4. We note that the splitting ratio of the fiber beam splitter was 49.5:50.5, which had the effect of reducing the HOM visibility by 0.2%.

Figure 5 shows the HOM measurement result at a pump power P_p=57 µW, in which the raw coincidence counts (solid blue diamonds) and the separately measured accidental coincidences (open red squares) are plotted as a function of the path length difference between the two arms. After subtracting the accidental coincidences from the raw data in Fig. 5, we obtain a HOM quantum-interference visibility V=(C _max-C _min)/(C _max+C _min)=98.2±1.0%, where C _max and C _min are the maximum and minimum coincidence counts with accidentals subtracted, respectively. To our knowledge, this is the highest HOM visibility ever reported for waveguide-based photon-pair sources. The 1% uncertainty of the measured visibility is mainly due to the uncertainty of the accidental coincidence rates caused by the high dark count rates of our InGaAs detectors and the long averaging times. The base-to-base width of the HOM dip is 2.3±0.13 mm, corresponding to a two-photon coherence time of 3.83±0.21 ps, or equivalently a two-photon bandwidth of 0.76±0.04 nm, as expected from the narrowband filter bandwidth of 0.79 nm.

 

Fig. 5. Measured HOM coincidences and accidentals counts in 300-s time intervals as function of the optical path difference between the signal and idler arms at waveguide temperature of 19.5 °C and with 57 µW pump power. HOM quantum-interference visibility is 98.2% with accidentals subtracted from the raw data.

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The high HOM visibility was the result of our use of a single-mode fiber beam splitter to ensure spatial mode overlap and our careful adjustment of the waveguide temperature to optimize the spectral overlap. We believe that the visibility did not reach even higher because of multi-pair generation in the downconversion process. For a highly efficient SPDC waveguide source, the probability of double-pair generation within a coincidence window is not negligible even at a low pump power level. Continuous-wave SPDC output is multi-temporal mode and the pair generation probability follows Poisson statistics. Assuming a system detection efficiency η for photon-number non-resolving detectors, one can easily find that the HOM visibility is given by

to first order in the mean number of generated pairs α within a coincidence time window for small α. With our system efficiency estimated at η ~1.1% for the HOM measurements, V _HOM is dominantly determined by the pumping level. In particular, low coincidence detection rates do not imply low pair generation rates. We verify this relationship by repeating the HOM quantum interference measurements at various pump powers. Figure 6 plots the measured HOM visibility (with accidentals subtracted) as a function of the mean pair number. As α increased from 0.3% to 4.0%, the HOM visibility dropped from 98.2% to 85.5%, in good agreement with the prediction of Eq. (14) (solid line) that takes into account the effect of double-pair generation. In Fig. 6, we also plot the expected HOM visibility (dashed curve) by including contributions from all multi-pair events. We note that as α increases, the experimental data drifts away from the staright line of Eq. (14) towards the more exact theoretical prediction. More importantly, Eq. (14) reveals a fundamental trade-off between the brightness and the quantum interference visibility of a photon-pair source, which allows the waveguide SPDC source to be operated according to the needs of specific applications. With single spatial-mode operation and low pumping levels, we have achieved a HOM quantum-interference visibility that is significantly higher than previous high-brightness waveguide sources [5, 6].

For benchmark purposes, we have performed a HOM measurement without the 0.79-nm narrowband filter. For a detection bandwidth much larger than the phase-matching bandwidth, any asymmetric feature in the signal and idler spectra such as the satellite peak in Fig. 5 would cause the HOM visibility to degrade because the signal and idler photons were no longer completely indistinguishable. Under the same operating conditions with a pump power of 57 µW, we measured a HOM visibility of V _broadband=84.2%. Assuming the difference between the broadband and narrowband visibilities was due to spectral indistinguishability, we estimate that the spectral overlap between signal and idler photons was ~92% without the narrowband filter. For the broadband case we expect the two-photon bandwidth to be determined by the SPDC phase-matching bandwidth. The observed base-to-base width of the HOM dip of 1.6 mm yields a two-photon coherence time of 2.67 ps, or a two-photon bandwidth of 1.08 nm, which is in excellent agreement with the phase-matching bandwidth calculated from Eq. (13).

 

Fig. 6. Experimental HOM quantum-interference visibilities versus the mean pair number within a coincidence window of 2.5 ns. Solid line represents theoretical calculation of Eq. (14) that considers only double-pair events and first order in α. Dashed curve is the theoretical prediction that includes contributions from all multi-pair events.

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6. Conclusions

In this work, we have experimentally demonstrated the production of narrowband correlated photon pairs at 1316 nm using a fiber-coupled type-II phase-matched PPKTP waveguide, yielding a source brightness of 2.0×10⁷ pairs/s/mW of the pump over the measured phase-matching bandwidth of 1.08 nm. The output flux of our waveguide source was measured to show excellent agreement with a new theoretical model of SPDC in a nonlinear waveguide. The model considers the transverse index profile of a nonlinear crystal waveguide that imposes an effective transverse momentum on the phase-matching conditions. We find that the effective grating leads to a broader transverse bandwidth of the signal and idler outputs, which in turn explains the much higher spectral brightness of a waveguide SPDC source compared with a bulk-crystal SPDC source.

Our PPKTP waveguide source is particularly useful for applications in the important telecommunication band. The source was fabricated with single-mode fiber coupling at both the input and output ends to facilitate connection to a standard fiber-optic network. We have measured a ratio of fluorescence photons to downconverted photons of 1:50 for our waveguide source, which is much lower than those observed in previous waveguide sources and is comparable to or better than that of a bulk-crystal source. Furthermore, at a low pump power, our device achieved a HOM quantum-interference visibility of 98.2%, the best ever reported for a waveguide SPDC source. This waveguide SPDC source is well suited for the generation of high-quality entangled photons for fiber-optic applications in quantum information science such as quantum key distribution and entanglement distribution. The possibility of an integrated waveguide device incorporating a SPDC sources, modulators, and directional couplers may lead to simpler and more efficient development in future long distance quantum communication protocols.