1 Introduction

Polarization-entangled photons are very useful for many applications such as quantum teleportation, testing the foundations of quantum mechanics, and various photonic quantum technologies [1–5]. For a decade, many researchers have been interested in a periodically poled KTiOPO4 (PPKTP) crystal because of its high generation efficiency [6–11]. In addition, one can efficiently generate polarization-entangled photons using a polarized frequency-degenerate signal and idler photon pair by a post-selection method.

Although photon pair sources based on a periodically poled crystal have the advantage of efficiency, they tend to be vulnerable to instability of the pump laser because of the narrow operating bandwidth (<1 nm) of the photon pair generated in a type-II PPKTP crystal under a collinear phase matching condition. To obtain high-quality photon pairs, the pump laser for photon pair generation in a collinear PPKTP crystal has used an external cavity diode with a narrow spectral width of several megahertz in a single longitudinal mode (LM) [12, 13]. It seems possible that the frequency instability of the pump laser diode may affect the quality of photon pairs generated via spontaneous parametric down-conversion (SPDC). However, it is not easy to know how photon pairs generated in a collinear PPKTP crystal by pumping with a weakly unstable pump laser change as the temperature of the crystal increases. Kwon et al. [14] studied the coherence properties of entangled photon pairs generated via SPDC in a 3-mm-thick type-I beta barium borate (BBO) crystal pumped by a multimode continuous wave laser. However, as is well known, photon pairs generated via SPDC in a PPKTP crystal are completely different from those generated in a BBO crystal because the operating bandwidth of SPDC in BBO is much broader than that in PPKTP. However, the LM of the laser diode may easily be unstable because of environmental effects. To date, the direct effect of LMs (and their instability or multimodal character) on the quality of photon pairs generated in PPKTP has not been studied.

In this paper, we investigate the properties of photon pairs generated under the type-II collinear condition of PPKTP for different LMs of the pump laser diode. A conventional method of confirming the indistinguishability of the signal and idler photon pair generated by SPDC is measurement of the Hong–Ou–Mandel (HOM) interference [15]. Indistinguishability between the probability amplitudes of the signal and idler photons plays an essential role in observing quantum interference among correlated photons. The dependence of the maximum visibility of the HOM interference signals on the LMs of the pump laser diode can be experimentally and theoretically demonstrated. By comparison with the theory, the variation in the HOM signals could be understood in terms of decomposition of the multi-LM instability of the pump laser diode.

2 Experimental setup

Figure 1 shows the experimental setup for HOM interference measurement for different LMs of the pump laser diode. In our experiment, we used a blue-violet–UV diode laser with a wavelength of 406.4 nm (model DL405-040-S). The laser diode can easily create several LMs because its gain profile depends on the temperature, current, and other environmental factors [16, 17]. A Fabry–Perot interferometer was used to monitor the LMs of the pump laser. The polarization of the incident pump beam was aligned along the y axis at the entrance to the PPKTP to generate type-II SPDC photons. The dimensions of the PPKTP crystal, which had a grating period of 10.00 μm, were x = 10 mm, y = 2 mm, and z = 1 mm. The temperature of the crystal was controlled by a heating oven system stabilized to within about 0.1 °C.

Fig. 1
figure 1

Schematic of experimental setup for measuring the HOM interference. LD laser diode, FP Fabry–Perot interferometer, NDF neutral density filter, L lens, FC fiber coupler, SMF single-mode fiber, DM dichroic mirror, H half-wave plate, Q quarter-wave plate, BS beam splitter, P polarizer, and F interference filter

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The pump laser was focused on a spot 69 μm in radius. A dichroic mirror was used to block the pump beam. For spatial filtering of the signal and idler, the output beam was coupled into a single-mode fiber (SMF). In our system, the number of generated photon pairs from PPKTP is estimated to be 0.28 MHz/mW. We used a Michelson interferometer, which acts as a time compensator, to continuously change the time delay τ between the signal and idler photons. At the entrance of the interferometer, a half-wave plate and quarter-wave plate were used to control the polarization of emitted photons after SMF coupling. We measured the HOM interference using a post-selection method in which both photons propagating at the same output port are ignored by measuring the coincidences between the photons on two arms. The state after the Michelson interferometer is represented as \(\left| H \right\rangle_{0} \left| {V(\tau )} \right\rangle_{0}\), where H and V represent horizontal and vertical polarization, respectively. Considering the coincidences in two output ports of the beam splitter (BS) (τ = 0), the state after the BS is represented as \(\left| H \right\rangle_{1} \left| {V(\tau )} \right\rangle_{2} + \left| V \right\rangle_{1} \left| {H(\tau )} \right\rangle_{2}\). Polarizers (P 1 = −45° and P 2 = 45°) located before the single-photon detectors project the state, and we can see the HOM dip by measuring the coincidences. Because the spectral bandwidth of the signal and idler photons is narrow (0.65 nm) and is calculated as a single-mode bandwidth [18], 3-nm interference filters at 812 nm block unwanted photons from entering the single-photon detector (SPCM-AQR4C, PerkinElmer). The efficiency at a wavelength of 812 nm is around 48 % [19]. The dark count for the single-photon counter was measured to be about 482 Hz. The output pulses of the detectors are sent to the coincidence circuit with a 19.45-ns coincidence time window. The measured average single and coincidence counting rates with the pump power of 1.2 mW were about 15,000 and 400 Hz, respectively. The accidental coincidence under our experiment condition was estimated to be 4 Hz.

3 Experimental results and discussion

To determine how the LMs of the pump laser affect the indistinguishability of the photon pair generated in the type-II collinear PPKTP, we measured the visibility of the HOM interference according to the LM feature of the pump laser diode, as shown in Fig. 3. The visibility of the HOM interference is defined as [20]

$$V = \frac{{\hbox{max} (P_{\text{coin}} (\tau )) - \hbox{min} (P_{\text{coin}} (\tau ))}}{{\hbox{max} (P_{\text{coin}} (\tau )) + \hbox{min} (P_{\text{coin}} (\tau ))}},$$
(1)

where \(\hbox{max} (P_{\text{coin}} (\tau ))\) for \(\tau > > \tau_{\text{coh}}\) and \(\hbox{min} (P_{\text{coin}} (\tau ))\) for \(\tau = 0\) are the probabilities of coincidence. We define the indistinguishability in terms of how similar the SPDC-generated photons are; this can be described as the degree of overlap between their quantum states, which increases perfect bunching behavior at the HOM interferometer. In other words, if the photons do not overlap perfectly, the degree of mismatch decreases the indistinguishability of the two-photon state.

Figure 2 shows the HOM interference in the type-II PPKTP crystal under the single LM of the incident 406.4-nm pump laser. Figure 2a, b shows the experimental and theoretical results, respectively, for the temperature dependence of the HOM interference for the single LM of the pump laser. From the results, we can identify only the effects of the temperature on the HOM interference when the incident pump laser operates in the single LM. Figure 2a shows how the HOM interference changes as the temperature of the crystal increases in the principal mode. The top panel shows the HOM interference for an estimated visibility of 0.97 at a crystal temperature of 88.6 °C, at which the signal and idler photons are almost indistinguishable. Using a heating system, we detuned the center frequencies of the signal and idler photons from the optimal condition of type-II collinear quantum phase matching, i.e., a crystal temperature of 88.6 °C. The detuning temperature ranged from 88.6 to 93.9 °C. The difference in the phase matching condition temperature is closely related to the frequency detuning between the signal and idler photons. Figure 2b shows the HOM interference calculated using the Sellmeier and thermal expansion equation of PPKTP in [21–23], where the frequency difference between the photons is zero.

Fig. 2
figure 2

Temperature dependence of coincidence probabilities for the single LM of the pump laser: a experimental results and b theoretical calculations

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Figure 3 shows experimental results for the effects of four different cases of the pump laser diode on the visibility of the HOM interference. To monitor the LMs of the pump laser, we used a scanning Fabry–Perot interferometer. The Fabry–Perot interferometer consists of two mirrors separated by an air gap, the mirror separation can be controlled by means of a piezoelectric spacer. Although the pump laser transmitted through the Fabry–Perot is measured as a function of the length of the cavity of two mirrors, the transmittance peak of the single LM is periodically repeated whenever the free spectral range of the Fabry–Perot. So, we can analyze the difference of LMs in wavelength of two closely spaced wavelength components of the pump laser. Figure 3a shows the four different cases of the LMs of the pump laser diode that are obtained from the Fabry–Perot interferometer, which has a free spectral range of 10 GHz. The top panel shows the single LM of the pump laser diode corresponding to the principal mode of the pump laser. Under the condition of the single mode (case 1) in Fig. 3a, we obtain a triangular HOM interference pattern [24]. The other three cases (cases 2, 3, and 4) in Fig. 3a show the LMs according to the relative magnitude of the two modes (i.e., the principal and side modes). The HOM interference corresponding to the four different LMs of the pump laser is shown in Fig. 3b; we estimated the values for a dip visibility ranging from 0.97 to 0.35. When the LM of the pump laser diode changed from single to multiple, the visibility of the HOM interference was degraded. Although the side mode made only a small contribution, as shown for case 2 in Fig. 3, the dip visibility of the HOM interference was degraded. From the experimental results in Fig. 3, we confirmed that the LM of the pump laser diode significantly affects the indistinguishability of the photon pair generated in type-II collinear PPKTP.

Fig. 3
figure 3

a Four different cases of the pump modes from the Fabry–Perot interferometer and b HOM interference fringes corresponding to the LMs of the pump laser diode. The visibilities V of the modes are 0.97, 0.86, 0.73, and 0.35

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To intuitively understand the effect of the LMs of the pump laser, we theoretically analyzed the HOM interference. We considered the two LM components of the pump laser diode, as shown in Fig. 3a. The two LMs of the pump laser were decomposed into a principal mode and a side mode. For the side mode, \(\lambda_{\text{pump}}\) includes a mode spacing \(\Delta \lambda_{\text{mode}}\) of 0.03 nm, which is calculated to be \(\Delta \lambda_{\text{mode}} = \lambda^{2}_{\text{pump}} /L_{\text{p}}\), where \(L_{\text{p}}\) is a peak separation as in [25]. Under our experimental condition, the wavelengths \(\lambda_{\text{pump}}\) of the principal and side modes of the pump laser are 406.40 and 406.43 nm, respectively.

From the two-photon state of a type-II collinear PPKTP crystal [24], a coincidence probability for a single mode can be calculated [9]. In the case of the principal mode, the interference pattern has triangular shape [24] and the value of coincidence is zero at time delay of zero, where the generated two-photon is indistinguishable. In the case of the side mode, interference pattern has different shape and the value of coincidence is not zero at time delay of zero because of frequency entanglement [9]. The generalized coincidence probability of the LMs \({\text{P}}_{\text{LMC}} (\tau )\), considering the two LM components of the pump laser diode can be described as follows

$$P_{\text{LMC}} (\tau ) = \left\{ {\begin{array}{*{20}l} {\frac{1}{2}( {1 - \sum\nolimits_{i} {m_{i} \frac{{\Delta \omega_{i} }}{{\pi d_{i} }}\sin( {\frac{{\pi d_{i} }}{{\Delta \omega_{i} }}}( {1 - \frac{{\Delta \omega_{i}}}{\pi }\left| \tau \right|})} } ))} \hfill & {{\text{for }}\left| \tau \right| < \frac{\pi }{{\Delta \omega_{i} }}} \hfill \\ {\frac{1}{2}} \hfill & {{\text{otherwise}}.} \hfill \\ \end{array} } \right.$$
(2)

where \(d_{i}\) is the difference between the signal and idler photon center frequencies, \(\Delta \omega_{i}\) is spectral single-photon bandwidth of 0.03 nm, \(\tau\) is time delay, and m i is the weight factor of individual modes of the pump laser diode.

In our experimental condition, the weight factor m i is obtained from values of peak intensity of individual mode in Fig. 3a. Through comparing of magnitude for the principal and the side mode, the weight factors are calculated. When we put the peak intensity of the principal mode and side mode as A and B, the weight factors are considered as A/(A + B) and B/(A + B), respectively. When the pump laser diode has a single operating mode, we can consider that the only principal mode of the pump laser corresponds to m 1 = 1. The pump laser is operated in two modes, one is the principal mode (i = 1) and the other is the side mode (i = 2), where the sum of the weights of the two modes is m 1 + m 2 = 1.

Figure 4a, b shows the experimentally and theoretically obtained HOM interference patterns, respectively, according to the ratios of the two modes of the pump laser diode shown in Fig. 3a. The measured HOM interference patterns in Fig. 4a are in excellent agreement with the patterns calculated according to the LMs of the pump laser in Fig. 4b. We considered the Sellmeier and thermal expansion equation of PPKTP in [21–23] to calculate the HOM interference. For the principal mode, there is a subtle difference in the pattern shape in the region around zero time delay because the transmittance shape of the interference filter we used is not rectangular but Gaussian. Even though the bandwidth of the signal and idler is 0.65 nm, the generated photon pair can be influenced by the interference filter.

Fig. 4
figure 4

Comparison of a experimental results and b calculated HOM interference for four different cases of the pump laser

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From the results in Fig. 4, we confirmed that the sensitivity of the dip visibility of the HOM interference can be affected by the LMs of the pump laser. In other words, the effect of the LMs can significantly decrease the indistinguishability of the photon pair even though we start with HOM interference that has a visibility of 0.97. From the numerical results, the variation in the HOM signals could be understood by decomposition of the LMs of the pump laser diode into the principal and side modes caused by the LM instability of the pump laser diode.

For PPKTP with multimode pumping, is there a phase matching condition for perfect visibility of the HOM signal? The phase matching condition of the PPKTP crystal can be adjusted by varying both the pump laser wavelength and the temperature of the PPKTP crystal. Thus, to find the condition for perfect visibility of the HOM signal with weakly multimode pumping, we investigated the temperature dependence of the HOM interference patterns under case 2 in Fig. 3, as shown in Fig. 5. The result in Fig. 5 is similar to that in Fig. 2 for pumping by the single-mode laser. However, the maximum visibility of the HOM signal is not improved as the temperature changes. We failed to find the condition for perfect visibility of the HOM signal with weakly multimode pumping. The maximum visibility of HOM interference in Fig. 5 was estimated to be 0.86 at a crystal temperature of near 88.6 °C. The degradation of the visibility can be affected by the pump modes, and the modes of the pump laser limit the visibility of the HOM.

Fig. 5
figure 5

Temperature dependence of HOM interference patterns for the multiple-LM pump laser under the case 2 in Fig. 2

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To explicitly confirm the indistinguishability limit of the photon pair generated by the multi-LM pump laser in the type-II collinear PPKTP crystal, we calculated the HOM interference patterns as a function of the temperature of the PPKTP according to the ratio (m 2/m 1) of the two modes of the pump laser diode, as shown in Fig. 6. The ratios correspond to the four experimental conditions in Fig. 3. For m 2/m 1 = 0 (case 1 in Fig. 3), the minimum coincidence probability is 0, corresponding to a HOM visibility of 1.0 at a crystal temperature of near 88.6 °C (gray dashed line). The result for m 2/m 1 = 0.03 (case 2 in Fig. 3) is considered for weakly multimode pumping. The minimum coincidence probability was calculated to be 0.015, corresponding to a HOM visibility of 0.94. The phase matching condition for m 2/m 1 = 0 is nearly the same as that for m 2/m 1 = 0.03. However, as the magnitude of m 1/m 2 increases, the minimum coincidence probability increases, and the HOM visibility decreases. From the theoretical result in Fig. 6, we showed that the LMs of the pump laser diode affect the indistinguishability limit of the SPDC source in PPKTP. Therefore, the indistinguishability of the photon pair generated in the type-II collinear PPKTP crystal is significantly affected by the degree of coexistence of the principal and side modes. Our results confirm the intrinsic effect of the LM of the pump laser diode on the dip visibility of the HOM interference.

Fig. 6
figure 6

Temperature dependence of HOM interference patterns according to the ratio (m 2/m 1) of the two modes of the pump laser diode

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4 Conclusion

We experimentally and theoretically investigated the indistinguishability of a photon pair generated in a collinear PPKTP crystal according to the LMs of the pump laser diode using the visibility of the HOM interference. When the LM of the pump laser diode was changed from single to multiple, the maximum visibility of the HOM interference decreased. The non-ideal interference signal in a HOM interferometer can be understood in terms of a superposition of different pump modes. Considering the multimode effect of the pump laser diode, the calculated HOM interference for each LM is in excellent agreement with the experimental result. Furthermore, we calculated the HOM interference patterns as a function of the temperature of the PPKTP crystal according to the ratio of the two modes of the pump laser diode and showed that the LMs of the pump laser diode affect the indistinguishability limit of the SPDC source in PPKTP. We believe that our results are certainly interesting to the optics and quantum information community working with these types of SPDC sources in type-II PPKTP.