## Abstract

In this paper, we reveal that some kinds of optical nonlinearities are further enhanced when incoherent light, instead of a laser, is used as a pump light. This idea was confirmed both theoretically and experimentally in the case of sum-frequency generation (SFG) using the optical second nonlinearity. The conversion efficiency of the SFG with incoherent light pumping increased as the bandwidth of the incoherent pump light decreased, finally reaching twice the conversion efficiency of conventional second harmonic generation (SHG) by laser pumping. This method dramatically relaxes the severe requirements of phase matching in the nonlinear optical process. The conversion efficiency became less sensitive to misalignment of the wavelength of pump light and also of device operation temperature when the bandwidth of the incoherent pump light was sufficiently broad, although the improvement of the conversion efficiency had an inverse relationship with the insensitivity to the phase-matching condition. The temperature tuning range was enhanced by more than two orders of magnitude in comparison with the conventional SHG method. As an example of a promising application of this new idea, we performed the generation of quantum entangled photon-pairs using cascaded optical nonlinearities (SFG and the subsequent spontaneous parametric down conversion) in a single periodically poled LiNbO_{3} waveguide device, in which the incoherent light was used as the pump source for both the parametric processes. We have achieved high fidelity exceeding 99% in quantum-state tomography experiments.

© 2014 Optical Society of America

## 1. Introduction

Wavelength conversion is a typical and practical application of optical nonlinearities [1, 2]. Up and down conversions using the optical second- and higher-order nonlinearities have been used to realize coherent light sources at wavelengths existing laser technologies cannot cover. Intraband and interband wavelength conversions are promising in future large-capacity optical communication systems based on all-optical schemes [3–5].

As is shown in numerous works concerning optical nonlinearities, the pump source that is commonly used to induce optical nonlinearity is a laser. This is because the high spatial and temporal coherence of lasers has been the most suitable for generating optical nonlinearity at sufficiently high conversion efficiency.

In this paper, we reveal that in some kinds of optical nonlinearities, an incoherent light source
is more preferable as the pump light source to induce optical nonlinearities more efficiently.
Nonlinear mixing between coherent and incoherent beams has been studied in many literatures
including [6–10], and it has been applied to spectroscopy, imaging, etc. Different from such
preceding works, we deal with the case that the nonlinear mixing is generated only by an
incoherent beam in this paper. We investigated this new idea both theoretically and
experimentally in sum-frequency generation (SFG) of the optical second nonlinear
(χ^{(2)}) device [Fig. 1]. The best conversion efficiency that could be obtained using this approach is twice the
conversion efficiency of conventional second-harmonic generation (SHG) by laser pumping. The
experimental results agreed well with the theoretical predictions given in this study. Moreover,
this new method relaxes the severe requirements of phase matching in the nonlinear optical
process, which is often very sensitive and severe in the case of the conventional SHG method.
The temperature tuning range was enhanced by more than two orders of magnitude in comparison
with the conventional SHG method. As an example of a promising application of this new approach,
we also discuss the generation of quantum-entangled photon pairs using cascaded optical
nonlinearities (the SFG and subsequent spontaneous parametric down conversion (SPDC)) with
incoherent light pumping. The quantum state achieved here was highly efficient, pure and
coherent even though the nonlinear processes were driven by an incoherent light source with poor
temporal coherence. High fidelity exceeding 99% in quantum-state tomography experiments was
obtained in this work.

## 2. Theoretical studies

Let us consider the SHG and the SFG at an optical frequency of *f _{0}*
generated from incoherent light, the electric fields of which are given as the dashed curve in
Fig. 2.Here, we assume that the spectrum of the incoherent pump light is

*continuous*, rather than discrete as that of a laser. This is generally expected in an incoherent light source. We also assume that they are spread over an optical frequency of

*f*. The 3-dB bandwidth in the intensity profile is defined as

_{0}/2*Δf*.

_{pump}The SHG at the frequency *f _{0}* is generated only from a frequency component at

*f*. Therefore, it can be neglected because this frequency component occupies a negligibly small portion in the whole spectrum. On the other hand, the SFG at the frequency

_{0}/2*f*can be generated from numerous combinations of symmetric frequency components at ${f}_{0}/2\pm \delta $ in the continuous spectrum. Therefore, the SHG process can be neglected and we can consider only the SFG process in our model.

_{0}First, for simplicity, let us divide the entire spectrum into 2*N* sections every Δ*f* frequency. Each combination of ${e}_{l}$ and ${e}_{-l}$ ($l=1,2,\cdots N$) generates the SFG at *f _{0}*, where ${e}_{l}$ is the amplitude of the electric field of the

*l*-th frequency mode per unit frequency. Nonlinear polarization at

*f*per unit frequency, ${p}^{\left({f}_{o}\right)}$, now yields

_{0}*l-th*frequency mode (${e}_{l}$), respectively.

The coupled mode equation for the SFG is then given by [1, 2],

*f*per unit frequency. In this equation, losses and pump depletion are neglected for simplicity. The phase mismatching parameter for the

_{0}*l*-th mode is given bywhere

*K*is a parameter corresponding to the quasi phase-matching (QPM) structure, and $A\equiv \pi {f}_{0}/{n}_{SFG}\sqrt{{\mu}_{0}/{\epsilon}_{0}}$ (

*n*is the refractive index of the SFG light).

_{SFG}Again for simplicity, we consider the case in which all the combinations of ${e}_{l}$and ${e}_{-l}$ can satisfy the phase-matching or the QPM condition, *i e.,* $\Delta {k}_{l}\equiv 0$ (validity of this assumption is discussed later). In this case, the ${e}_{{f}_{0}}$is simply given by

*L*is the device length.

The SFG intensity at *f _{0}* per unit frequency, ${i}_{SFG}\left({f}_{0}\right)$, is finally given by the following equation:

Because of the incoherent nature of input pump light, there is no phase correlation among different frequency modes. Therefore $\u3008{e}_{l}{e}_{-l}{e}_{m}{e}_{-m}\cdots \u3009=0$ for $l\ne m$, and it has a non-zero value when *l = m*. Consequently, ${i}_{SFG}\left({f}_{0}\right)$ is given by

Because ${\left|{e}_{l}\right|}^{2}$is proportional to the spectral intensity of input light per unit frequency, ${i}_{l}$, Eq. (6) can be rewritten as

*C*is a coefficient that includes

*A, B, L*, and so on.

Next, let us consider a concrete example. Let us assume that input light has a Gaussian spectrum centered at *f _{0}/2* and the 3-dB bandwidth is

*Δf*

_{pump}, i.e.,*I*is the peak spectral intensity per unit frequency.

_{pump}When *Δf* is sufficiently small

The total (averaged) pump power is given by

Substituting Eq. (9) and Eq. (10) into Eq. (7), we can obtain,

As is discussed later in detail, the generated SFG spectrum is also continuous, and the profile is basically identical to the SHG curve obtained in the case of a conventional SHG with single-mode laser pumping. The total SFG power is obtained by integrating the SFG spectrum. Because the SHG curve is basically a *sinc ^{2}* function [1, 2], the total SFG power, ${P}_{SFG}$, is related to the peak SFG power, ${i}_{SFG}\left({f}_{0}\right)$, as

*Δf*is the 3-dB bandwidth of the SHG curve.

_{SHG}Finally, we obtain

When a similar discussion is applied to the case of a conventional SHG with single-mode laser pumping, the SHG power (under phase matching conditions), ${P}_{SHG}$, is expressed as

Although the model above is still inaccurate, its results exhibit some general trends for our new idea. Equation (13) implies that the SFG power with incoherent light pumping is inversely proportional to the spectral bandwidth of the pump light, *Δf _{pump}*. When we compare Eq. (13) with Eq. (14), it is clear that the SFG power in this case finally exceeds the SHG power of conventional laser pumping when the spectral bandwidth is sufficiently narrow (less than 1.5 times that of the SHG bandwidth,

*Δf*) in the case of Eq. (13). Such general tendencies are still valid in the more accurate model discussed below.

_{SHG}When we consider a more general model including the phase-mismatching effect, the spectral intensity of SFG light at frequency *f* should be given by

*f*(SFG).

The total SFG power is given by

First, we discuss the effect of the phase-mismatching parameter, *Δ*, which is, in general, a function of *x* and *f*, and it varies when the frequencies of the two pump lights and SFG light change. However, when the refractive-index profile within the spectrum of the pump light yields a linear relationship, *i.e.*, the refractive index *n* is simply given by $a+b\lambda $, *Δ* can be expressed as,

Equation (18) implies that *Δ* is *x*-independent and has a constant value when the SFG frequency (*f*) is fixed (here, we used the energy conservation law of the SFG process, $1/{\lambda}_{f}=1/{\lambda}_{f/2+x}+1/{\lambda}_{f/2-x}$).

We can also express *Δ* as

*Δ*is also equal to the phase mismatching of the SHG process in the case of conventional laser pumping.

Consequently, when the linear approximation of the refractive index is valid, Eq. (15) can be rewritten as

The *sinc ^{2}*-function in Eq. (20) is equal to that of the SHG curve of conventional laser pumping.

The validity of the linear approximation of the refractive index profile above can be evaluated experimentally on the basis of the bandwidth of differential frequency generation (DFG), as the phase mismatching in the DFG process is also given by *Δ*. According to previous reports, including our report [3–5,11,12], the bandwidth of DFG is as wide as 60 nm, even for a 6-cm-long periodically poled LiNbO_{3} (PPLN) waveguide device. Therefore, when the bandwidth of the input incoherent light is several tens of nanometers at most, the assumption above is valid. In this case, all the spectral components of the input incoherent light, rather than only a part of it, can contribute to the generation of the SFG. As a result, the intensity of the generated SFG light can be very high, even though optical nonlinearity is induced by the incoherent light.

Let us consider again that the spectrum of input incoherent light is Gaussian-shaped. The *sinc ^{2}* function in Eq. (20) can be expressed using the 3-dB bandwidth of the SHG curve with conventional laser pumping,

*Δf*, as follows:

_{SHG}Consequently, the spectral intensity profile of SFG light is given by

*δ*is the deviation of the center frequency of input light’s spectrum from the optimized (phase-matching) condition

_{p}*f*.

_{0}/2In the same manner, the SHG power with the conventional laser pumping including the phase mismatching effect can be expressed as

Comparing Eq. (11) and Eq. (22), the last exponential term in Eq. (22) implies a spectral narrowing effect when the bandwidth of the pump light is narrowed. This term also relates to the tolerance to phase mismatching.

Using Eq. (22) and Eq. (17), we calculated the total SFG power (*P _{SFG}*) and the bandwidth of the SFG spectrum (

*Δf*) as functions of the bandwidth of the pump light (

_{SFG}*Δf*). In these calculations, we consider the ideal case of

_{pump}*δ*= 0.

_{p}The results are shown as solid curves in Fig. 3.
Here, *P _{SFG}* and

*Δf*are normalized to the SHG power (

_{SFG}*P*) and the SHG bandwidth (

_{SHG}*Δf*), respectively, in the case of conventional SHG wih laser pumping at the same pump power. The bandwidth of the pump light is also normalized to

_{SHG}*Δf*.

_{SHG}*P _{SFG}* increased and

*Δf*decreased as

_{SFG}*Δf*decreased.

_{pump}*P*exceeded

_{SFG}*P*when

_{SHG}*Δf*was approximately 1.2 times that of

_{pump}*Δf*in this calculation. This value was slightly less than the value obtained through a rough estimation using Eq. (13), most likely because of spectral narrowing in the SFG spectrum.

_{SHG}*P _{SFG}* finally reached twice the

*P*when

_{SHG}*Δf*was further narrowed. This can be confirmed as follows.

_{pump}When *Δf _{pump}* becomes much less than

*Δf*, the

_{SHG}*sinc*function can be treated as 1, and Eq. (22) is therefore approximately

^{2}This implies that the SFG spectrum was nearly Gaussian-shaped with the 3-dB bandwidth of $\sqrt{2}\Delta {f}_{pump}$. The total SFG power is then given by

Consequently, our model predicts that the SFG with incoherent light pumping improves the conversion efficiency by two times in comparison with the SHG with conventional laser pumping.

When we define the critical bandwidth, *Δf _{cri}*, as the bandwidth of pump light at which

*P*is equal to

_{SFG}*P*, this value depends on the spectral shape of the pump light. In Fig. 3, we also showed the calculated results when the spectral shape of the pump light was rectangular (dashed) and Lorentzian (dotted). We estimated

_{SHG}*Δf*to be approximately 1.6x, 1.2x, and 0.45x

_{cri}*Δf*for rectangular, Gaussian, and Lorentzian shapes, respectively.

_{SHG}Next we consider the tolerance to phase mismatching. The tolerance to phase mismatching implies that to how large a *δ _{p}* is permitted to suppress the degradation of the SFG power well. Large tolerance to the

*δ*implies that the SFG power is less sensitive to changes in operation temperature of a nonlinear device because the change in operation temperature generally changes the phase-matching wavelength (

_{p}*f*) owing to the changes in refractive indices, which gives the

_{0}/2*δ*.

_{p}When we define the tolerable frequency offset *δ _{p,max}* as the frequency offset in which the degradation of SFG power is less than 3 dB of the best (phase-matched) case,

*δ*can be roughly estimated from Eq. (22) as

_{p,max}On the other hand, from Eq. (23), *δ _{p,max}* for the conventional SHG with laser pumping is

Because generally $\Delta {f}_{pump}>>\Delta {f}_{SHG}$, our method is thought to have quite a large tolerance to phase mismatching in comparison with the conventional SHG method.

As a summary of this section, we can point out some advantages of our new idea with incoherent light pumping over the conventional method with laser pumping. First, our method has higher conversion efficiency than conventional laser pumping when the bandwidth of incoherent pump light is sufficiently narrow. Second, our method has a much greater tolerance to phase mismatching than conventional laser pumping when the bandwidth of the input light is sufficiently large (conversely, the conversion efficiency decreases).

Our idea here can be realized using a variety of configurations of incoherent light
sources. Some examples are illustrated in Fig. 4. As
shown in Fig. 4(a), a combination of the incoherent light
source with adequate optical amplifier(s) (and with optical bandpass filter(s) for more
efficient pumping) is a typical setup used to increase the SFG power. The use of plural light
sources, such as an LED array, is also promising for achieving high output power [Figs. 4(b) and 4(c)].
In such a setup, the SFG power could be simply increased by *N ^{2}* times
if we use

*N*light sources with no special consideration of interference among the light sources due to the incoherent nature of such light sources.

Similar setups can be used for the conventional SHG method with laser pumping. In our case, however, we need no severe consideration of the pump wavelength and the operation temperature of the nonlinear device owing to the large tolerance of phase mismatching. This should lead to the achievement of low cost and simple wavelength convertors.

Our idea described here can be expanded to higher-order harmonic generation (third-order harmonic generation and higher) and perhaps to other optical nonlinear phenomena.

## 3. Experimental results

We performed a series of experiments to experimentally evaluate our method. Figure 5 shows the experimental setup. The nonlinear optical
devices used here were custom-made periodically poled LiNbO_{3} waveguide (PPLN-WG)
devices with ridge waveguide structures. Details of the device structure and fabrication process
are available in [11].

We prepared two samples with different interaction lengths. One device was approximately 6-cm long and showed an SHG conversion efficiency of approximately 650%/W under the QPM condition. The QPM wavelength was 1551.05 nm at an operation temperature of 41.5°C. The bandwidth of the SHG curve (*Δf _{SHG}*) was approximately 50 GHz, corresponding to 0.1 nm in the wavelength domain (the measured SHG curve was plotted as red open symbols in Fig. 7). This value is typical for a 6-cm-long PPLN device [11,12]. The other device was approximately 2-cm long, and the maximum SHG efficiency at the QPM condition was 62%/W. The QPM wavelength and

*Δf*were approximately 1548.54 nm (at 25.0°C) and 150 GHz (0.3 nm), respectively.

_{SHG}The PPLN device was packaged in a fiber-pigtailed optical module with a thermistor, a thermoelectric cooler, and two polarization-maintaining optical fibers for standard 1.5-μm telecommunication applications. The insertion losses of the module were approximately 3 dB for the 6-cm-long device and 2 dB for the 2-cm-long device in the 1.5-μm band.

The experimental results described below are the results for the 6-cm-long device. The results for the 2-cm-long devices also exhibited consistency with our theoretical models.

The incoherent light used here was generated by cascading two sets of erbium-doped fiber amplifiers (EDFAs) and optical bandpass filters (OBFs) to tune the power and bandwidth of the pump light. The pump light was coupled to one pigtail fiber of the PPLN module and the SFG or SHG light from another pigtail fiber was measured using an optical spectrum analyzer with a resolution of 0.02 nm. The total SFG (SHG) power was estimated by integrating the measured spectrum. Henceforth, the pump power was taken to be the value just before the input to the pigtail fiber of the module, and the SFG (SHG) power was the value measured in the optical spectrum analyzer after coupling to the pigtail fiber. Therefore, the measured values of the SHG and SFG powers were quite different from the actual ones obtained from the PPLN devices primarily because of large coupling loss to the pigtail fiber (>10 dB). Such a setup, however, was sufficient for confirming the validity of our method.

Figure 6(a) shows the dependence of the SFG power on the pump power, as a function of the bandwidth of the pump light, for the 6-cm-long device. Here, for convenience, the bandwidth of the pump light is expressed in the wavelength domain. In the same figure the results of the SHG with conventional CW laser pumping are also plotted as a dashed curve. The pump laser in the SHG experiments was a standard single-mode semiconductor laser with a linewidth of 100 kHz. In all cases, the center wavelengths of the pump lights were adjusted to coincide with the QPM wavelength (1551.05 nm).

The results showed quadratic dependence on the pump power, as was predicted theoretically. The SFG power with incoherent light pumping showed almost the same value as the SHG power with laser pumping when the bandwidth of the incoherent pump light was 0.45 nm (56 GHz). This bandwidth was approximately 1.13 times that of the bandwidth of the SHG curve (50 GHz). This agreed well with the theoretical prediction in Section II. When we performed similar experiments on the 2-cm-long device, the critical bandwidth (*Δf _{cri}*) was experimentally estimated to be 1.3 nm (163 GHz). This value was approximately 1.1 times that of the SHG bandwidth of the 2-cm-long device (150 GHz) and also agreed well with the theoretical value.

Figure 6(b) shows the relationship between the peak SFG intensity (*i _{SFG}*) and ${\left({I}_{pump}\right)}^{2}\Delta {f}_{pump}$value. These values had a linear relationship, and the slope of the curve was independent of

*Δf*. This can be understood as follows.

_{pump}When we rewrite Eq. (22) using the peak spectral intensity of the input pump light (*I _{pump}*), the peak SFG intensity, ${i}_{SFG}\left({f}_{0}\right)$, can be expressed as

Equation (28) agrees well with the results in Fig. 6(b).

The black solid curve in Fig. 7 is the SFG spectrum generated from the 6-cm-long device. In this experiment the output from the EDFA was directly coupled to the PPLN module without the OBFs. The amplified spontaneous emission from the EDFA had a broad spectral bandwidth exceeding 20 nm, and our theoretical model predicted that the shape of the SFG spectrum should be exactly the same as that of the SHG curve with laser pumping in such a case.

In the same figure, we also plotted the measured SHG curve of the same sample as red open circles. The shapes of the SFG spectra and the SHG curve agree well, as theoretically predicted.

In Fig. 8 we show the summary of the dependence of (a) the SFG power and (b) the SFG bandwidth on the bandwidth of input incoherent light. The averaged pump powers remained at 0 dBm in these experiments. Black circles and red triangles correspond to the results of the 6-cm-long device and the 2-cm-long device, respectively. The SFG powers were normalized to the SHG power with conventional laser pumping at the same pump power (0 dBm).

The experimental results agreed well with the calculated results shown as solid curves in the figures (assuming Gaussian spectra). It was experimentally confirmed that the SFG power was actually enhanced by narrowing the bandwidth of the input pump light, and also that it finally reached twice the SHG power with laser pumping.

Figure 9 shows the results of the temperature tuning as a function of the bandwidth of the pump light; the results can be used to evaluate the tolerance to phase mismatching. As a reference, the results of the SHG with laser pumping are shown as the black dashed curve.

The tolerable temperature range *δT*, defined as the temperature range in which the degradation in the SFG (and SHG) power is less than 3 dB of the best (phase-matched) case, was estimated to be approximately 1.6 °C for the conventional SHG case. Because the temperature dependence of the QPM wavelength was approximately 0.13 nm/°C in our devices, this temperature range corresponded to approximately 26 GHz of the frequency offset. This value was half of the bandwidth of the SHG curve (*Δf _{SHG}*), as predicted theoretically.

On the other hand, in the case of incoherent light pumping, *δT* increased as the bandwidth of the pump light increased. When the bandwidth was 10.8 nm, the SFG power exhibited no significant degradation, even though the operation temperature changed by over 30 °C.

Figure 10 shows a summary of the results of the temperature dependences. The dashed curve shows the calculated result from the theoretical model. The experimental results and calculated results agreed well, verifying the validity of our new method. Such a wide tolerable temperature range is promising for a simple and low-cost wavelength convertor without a temperature control unit.

Such insensitivity to phase-matching condition generally has an inverse relationship with the improvement of the conversion efficiency. However, from Eq. (26), Eq. (27), and Fig. 3, these conditions can be compatible when *Δf _{pump}* satisfies the following condition:

The results in Figs. 9 and 10 seemingly indicate that *δT*, and therefore the tolerance to
phase mismatching, can be similarly increased even in the case of laser pumping when a laser
with a broad spectral bandwidth is used as the pump light. However, this is not correct. Figure 11 shows the temperature tuning
characteristics of the SHG power when we used a multi-longitudinal-mode Fabry-Perot laser diode
as the pump light. The results showed periodic behavior of the SHG power. The period almost
corresponded to half the mode spacing of the Fabry-Perot laser used here (approximately 1.26
nm). This indicates that the SHG power in this case was enhanced only when one of the lasing
modes corresponded to the phase-matching wavelength in the SHG, or when the lasing modes were
located symmetric to the phase-matching wavelength to satisfy the phase-matching condition in
the SFG. These results were quite different from the method in this work, verifying the
advantages of our method.

## 4. Generation of quantum entanglement by cascading χ^{(2)} processes

The results described above indicate that our method has various advantages over the conventional SHG method. Such merits are promising for practical applications including short-wavelength light sources. As a promising application example of an application of this new approach, we performed the generation of quantum-entangled photon pairs using cascaded optical nonlinearities [12–14] (SFG and subsequent spontaneous parametric down conversion (SPDC)) with incoherent light pumping. In this setup, the incoherent pump light first excites the SFG in the PPLN device; the SFG light then works as the pump light in the subsequent SPDC process in the same PPLN device. The PPLN device used here was a 6-cm-long device.

Figure 12 shows the experimental setup for correlated photon-pair generation using the cascaded SFG/SPDC processes. Similar to the experiments in Section III, the incoherent pump light was generated from the two sets of EDFAs and OBFs. The incoherent pump light passed a fiber polarizer (Pol.) and then was coupled to the PPLN module. The center wavelength of the incoherent light was set to coincide with the QPM wavelength (1551.05 nm).

The photon pairs from the PPLN module were then passed through an optical low pass filter (LPF) to eliminate the SFG light and spatially separated to signal photons and idler photons using two optical high-pass filters (HPF#1 and #2) and OBFs. The two stages of the HPFs simultaneously reduced residual 1.5-μm pump light in each output port of the signal photons and the idler photons. The pump suppression ratio was less than −110 dB when the bandwidth of pump light was as narrow as 1 nm, which was sufficiently low for this study. The center wavelengths of the signal photons and the idler photons were 1540.0 nm and 1562.2 nm, respectively. The 3-dB bandwidth, which was determined by the bandwidth of the final OBF, was approximately 0.8 nm.

The signal/idler photons were detected using InGaAs avalanche photodiode-based single photon detectors (Princeton lightwave benchtop receiver PGA-600HSU) (D1, D2). The gate frequency was 40 MHz. The detection efficiencies of both the APDs were approximately 20%. The gate widths were 1 ns. The dark count rates were approximately 2x10^{−6} per gate for both detectors. The single count rate of each single photon detector and the coincidence counts between the two detectors were measured using a time-interval analyzer.

Figure 13(a) shows the dependence of the
coincidence counts per 30 s on the averaged power of the incoherent pump light as a function of
the bandwidth of the incoherent pump light. In this figure, the coincidence counts are the
values obtained after subtracting the accidental coincidence counts at the mismatched time slot
(*R _{um}*) from the coincidence counts at the matched time slot
(

*R*) in the time-interval analyzer. This subtraction was performed because the accidental counts due to residual pump photons ceased to be negligible when the

_{m}*Δf*became large and the HPFs could not suppress the residual pump photons satisfactorily. Theoretically, the

_{pump}*R*value was simply proportional to the mean number of photon pairs [15].

_{m}-R_{um}The results showed quadratic dependence on the pump power, as predicted from the theoretical model of cascaded optical second nonlinearities. In the same figures, the results of conventional laser pumping using cascaded second harmonic generation (SHG) and SPDC [12,13] were plotted as the dashed curve. In the case of incoherent light pumping, the generation rate of photon pairs increased as the bandwidth of input light decreased and finally exceeded the generation rate of that with conventional laser pumping. The critical bandwidth in terms of the photon-pair generation rate was 0.45 nm. This was the same as that of the SFG process described in Section III.

Figure 13(b) shows the relationship between the SFG or the SHG (with laser pumping) power and the coincidence counts. There was a linear relationship, and the slope did not depend on whether the cascaded processes were generated by laser pumping (c-SHG/SPDC) or incoherent light pumping (c-SFG/SPDC). The slope was also independent of the bandwidth of the incoherent pump light. These results indicate that the efficiency of the SPDC process itself was simply determined by the pump power (SFG or SHG power) and was independent of whether the pump source was coherent or incoherent.

Next we generated quantum entanglement using the cascaded SFG/SPDC processes. Figure 14 shows the experimental setup. In addition to the setup depicted in Fig. 12, a Sagnac-loop interferometer composed of a polarization beam splitter/combiner (PBSC) and a WDM filter was used [12,16]. In this setup, the pump light was 45°-polarized by a half-wave plate (HWP). Two counterpropagating pump lights in the Sagnac loop excited the PPLN bidirectionally, and generated two orthogonally polarized biphoton states, ${|H\u3009}_{s}{|H\u3009}_{i}$and ${|V\u3009}_{s}{|V\u3009}_{i}$, respectively [12,16]. The two biphoton states were combined at the PBSC of the Sagnac-loop, generating polarization-entangled state. A Babinet-Soleil compensator (BSC) was used to optimize the phase difference between the two biphoton states. The bandwidth of the pump light was dominated by the transmission bandwidth of the WDM filter in this setup and was approximately 0.95 nm. Two sets of HWPs and quarter-wave plates (QWPs) were used to compensate birefringence of optical fiber lines, and also for quantum-state tomography experiments [17].

A primary advantage of this new quantum entanglement source is that temporal coherence of pump light is not dominated by the bandwidth of the initial incoherent pump light, but rather by the bandwidth of the SFG spectrum. As discussed in earlier sections, the bandwidth of the SFG spectrum is very narrow (approximately 50 GHz in this study) because of the severe quasi-phase matching requirements, although the bandwidth of the initial pump light is very broad. Therefore, the generated photon pairs can still maintain high temporal coherence in our new approach, regardless of the bandwidth of the initial incoherent pump light. This leads to the generation of a highly pure and coherent entanglement state.

Figure 15 shows reconstructed density matrices of the generated entanglement states as a function of pump power. The pump powers were approximately (a) + 3.2 dBm and (b) + 12.9 dBm per end facet of the PPLN module. These pump powers corresponded to mean numbers of approximately 0.0055 and 0.367 photon-pairs per gate, respectively.

In the case of Fig. 15(a) in which the mean number of photon pairs was very low, the values of the diagonal elements and off-diagonal elements were close to 0.5, while those of the other elements were almost zero in both the real and the imaginary parts. This implies that the generated entanglement state was very close to the ideal one, a maximally entangled (Bell) state. The fidelity and purity of the reconstructed density matrix were estimated to be 99.2% and 98.47%, respectively.

With increasing pump power, *i.e*., increasing mean number of photon pairs, the diagonal elements and the offdiagonal elements became smaller while the $|HV\u3009\u3008HV|$and $|VH\u3009\u3008VH|$elements appeared and became larger [see Fig. 15(b)].

This feature closely resembles the features of the Werner state. The Werner state is a mixture of a pure entangled state and fully mixed state [18]. The density matrix of the Werner state, *ρ _{W}*, of a $\left(1-a\right):a$statistical mixture of the $|{\Phi}^{(+)}\u3009$ and the fully mixed state is given by [18],

In the Werner state, the purity (*P*) and fidelity (*F*) in the density matrix are simply related to the visibility (*V*) as [19]

Open symbols in Fig. 16 show the relationships
among the *V, P*, and *F* in the quantum-state tomography
experiments. The solid curves in the figure are the calculated curves from Eq. (31) and Eq. (32). The experimental values agreed very well with the theoretical values. In the
figure, we also show the fidelity of the Werner state (*F _{W}*), defined
as ${F}_{W}\equiv {\left(Tr\sqrt{\sqrt{{\rho}_{W}}\rho \sqrt{{\rho}_{W}}}\right)}^{2}$ [18], as blue open squares.
The

*F*was almost unchanged and showed values higher than 99%, even when

_{W}*V*changed, while the

*F*of the Bell state was monotonically decreased as

*V*decreased. This experimentally verified that our photon-pair source was actually close to the Werner state, similar to the case with conventional laser pumping [19].

Our results for the generation of a high-quality entanglement state were seemingly a result of the narrow bandwidth (0.95 nm) of initial incoherent pump by spectral carving.Considering the physical background of our method, however, the bandwidth of initial incoherent pump is thought to be irrelevant to the quality of the generated quantum state because the temporal coherence of the pump light for the SPDC is dominated by the bandwidth of the SFG, regardless of the bandwidth of initial pump. We used the initial pump with a bandwidth of 0.95-nm in this study merely for experimental convenience. This was first limited by the bandwidth of the WDM filter used here. The other reason for using such an initial pump is as follows. In this paper, we used cascaded nonlinearities in one-chip PPLN for the SFG and the SPDC. This was because the experimental setup could be very simple only using a single PPLN device. However, in this setup, because the initial pump light outputs together with the SPDC in the same wavelength band, we need to eliminate the strong initial pump outside the Sagnac loop. This was the reason why we used a narrow-band pump (0.95 nm) in this study. If we prepare two PPLN samples (one for the SFG and the other for the SPDC), this problem can be solved and no bandpass filters are needed for initial incoherent pump. The incoherent pump light with a broad bandwidth will be used as it is in such cases. The results will be essentially the same as those in this study (the generation rate may change), considering the physical background of this approach.

## 5. Conclusion

In summary, we reported in this paper that some kinds of optical nonlinearities can be generated more efficiently with incoherent pump light, instead of a laser. We applied this approach to the SFG in an optical second nonlinear device and revealed both theoretically and experimentally that this approach enhanced the SFG power by, at most, two times in comparison with the conventional SHG with laser pumping. We also revealed that this method drastically relaxed the tolerance to phase mismatching due to offset of the wavelength of the pumping light source and the operation temperature of the nonlinear device when the bandwidth of the incoherent pump light was sufficiently broad, although the improvement of the conversion efficiency had an inverse relationship with the insensitivity to the phase-matching condition. As an application of this approach, we also discussed the generation of polarization-entangled photon pairs using cascaded SFG/SPDC with incoherent light pumping. The generated quantum states closely resembled the Werner state, similar to the case with conventional laser pumping. We observed high fidelity exceeding 99% in quantum-state tomography experiments, indicating that the quantum state achieved here was highly pure and coherent even though the nonlinear processes were driven by the incoherent light source. The new findings discussed here will have great impacts on related technologies, including displays, optical communications, and quantum communications.

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