© The Author(s), published by EDP Sciences, 2025

1 Introduction

Twin photon generation, also known as spontaneous parametric down-conversion (SPDC), is a second-order nonlinear process in which a pump photon is split into two photons of lower energy. Nowadays, bulk crystals, photonic crystals or waveguides are used as nonlinear media [1]. Photons are emitted in pairs, which radically alters their behavior. They are at the very heart of several non-classical properties, such as for example the squeezing of quantum fluctuations, or quantum entanglement that is the most intriguing [2–4]. These quantum aspects are the key element in a large number of demonstrations such as quantum cryptography, quantum teleportation and quantum information in the broadest sense, or the detection of gravitational waves [5–8]. While the quantum properties of photon pairs are extremely well known, it is still extremely difficult to predict their flux as a function of pump intensity. Major efforts have been made in recent years to develop semi-classical models [9] and quantum models [10–13] of SPDC.

The present work falls within this framework, with the ambition of further improving quantitative predictivity. We have achieved this by modelling a SPDC experiment for which the pump intensity ranges from 2.4 W cm⁻² to 3.7 GW cm⁻², leading to the generation of twin-photons with a flux from 1.1 × 10⁴ Hz to 1.2 × 10²¹ Hz. Our proposal is a semi-classical model based on both the quantum fluctuations of vacuum for the seeding of the process and a propagation step ensured by classical fields, which allowed a satisfying description of these two regimes, with a much better agreement at high intensity than the best quantum models based on the nonlinear momentum or the nonlinear Hamiltonian operators.

2 Material and methods

2.1 Experiment

We designed a SPDC experiment which uses a collinear type II phase-matched 1-cm-long x-cut KTiOPO₄ (KTP) crystal pumped at 532 nm or 527 nm. The energy and momentum conservations write:$$\hslash {\omega }_p-\hslash {\omega }_s-\hslash {\omega }_i=0$$(1)and$$\Delta k\left({\omega }_p,{\omega }_s\right)=\frac{{\omega }_p}{c}{n}_p-\frac{{\omega }_s}{c}{n}_s-\frac{{\omega }_i}{c}{n}_i=0,$$(2)with n_p = n_y (ω_p), n_s = n_z (ω_s) n_i = n_y (ω_i = ω_p − ω_s), y and z referring to the dielectric frame (x, y, z) of KTP. The indices “p”, “s” and “i” refer to the pump, signal and idler, respectively.

In this configuration, the pump field and the down-converted fields are coupled by the nonlinear coefficient ${\chi }_{24}^{\left(2\right)}\left({\lambda }_p\right)=5.3 \mathrm{pm }{\mathrm{V}}^{-1}$ of KTP [14]. Moreover, using the Sellmeier equations, it comes for the phase-matching wavelengths of the twins: (λ_s = 1037 nm, λ_i = 1092 nm) for λ_p = 532 nm and (λ_s = 1013 nm, λ_i = 1099 nm) for λ_p = 517 nm [15]. Figure 1 details the experimental setup that we used for the quantitative measurement of SPDC over a large pump intensity range of 10 orders of magnitude.

Fig. 1 Experimental setup used for performing CW, nanosecond and picosecond Type II SPDC in a x-cut KTP crystal. (ep, es, ei) stand for the polarizations of the pump, signal and idler beams, respectively.

Three different laser sources, Spectra Physics Millenia eV, Coherent Evolution 30 and Continuum, were used interchangeably. Each of them emits a single transverse Gaussian mode beam. Table 1 summarizes their temporal, spatial and energetic characteristics.

The intensities given in Table 1 are calculated from the waist radii given in the same table.

A half-wave plate (HWP) allowed to polarize the pump beam along the y-axis of KTP. The pump beam was focused using an optical lens (L1) into a 1-cm-long x-cut KTP. L1 was chosen so that the Rayleigh length of the pump beam, deduced from the beam waist radius, is greater than the length of the crystal. By this way, the SPDC experiment was performed with parallel beams. The generated beams were further collimated with an optical lens (L2), filtered with a filter-set (F) including notch filters to reject the pump and injected into a single mode fiber (SMF) by use of an aspheric lens (AL). The focal length f₂ of lens L2 was always chosen so that the pump beam size fits well the aspherical lens diameter to maximize the injection efficiency of the SPDC signal in the SMF. Note that the use of single mode fibers associated to the collection lens (L2) with small numerical aperture (0.041 for the CW and ns setups, 0.030 for the ps setup) allows the collinearly propagating component to be spatially selected, and so prevents any non-colinear contributions of SPDC to be collected. For each pump source, the pump intensities were varied using neutral densities.

Highly sensitive MoSi Superconducting Nanowire Single Photon Detectors (SNSPDs) from IDQuantique were used as detectors. The configuration of the detection stage depends on the pump source. For the CW and kHz ns lasers (solid lines in Fig. 1), the signal was separated on a balanced Beam Splitter (BS) and coincidence measurements were performed on channel 2 and 3. In the case of the ps pump laser, the detection on channel 1 was triggered by the signal of a photodiode DET 110 (Thorlabs) installed on a leak of the pump laser and used as external trigger (dotted line in Fig. 1). All the detectors are properly calibrated. For the CW experiment, the absolute twin photon flux as well as the losses were simultaneously deduced by comparing the coincidence to single ratio [16] as described in Appendix D. The nanosecond experiment losses were calibrated from the overlapping intensities with the CW absolute measurement. The losses in the picosecond regime are calibrated from a high energy free space measurement whose collection numerical aperture equals that of the SMF collection set-up, as described in Appendix D.

2.2 Theory

We proposed a semi-classical model, the so-called SCM, where the field motion is governed by the coupled wave amplitudes system coming from Maxwell’s equations, the initial conditions being the pump field at the entrance of the medium and the quantum fluctuations of vacuum at the signal and idler wavelengths. Assuming no spatial and temporal walk-off, the slowly-varying envelope, parallel beams and a collinear interaction, the coupled wave amplitudes system writes [17]:$$\{\begin{array}{c}\frac{\partial E_s\left(Z\right)}{\partial Z}=j{\kappa }_s{\chi }_{\mathrm{eff}}^{\left(2\right)}{E}_p\left(Z\right)E_i^{*}\left(Z\right)\\ \frac{\partial E_i\left(Z\right)}{\partial Z}=j{\kappa }_i{\chi }_{\mathrm{eff}}^{\left(2\right)}{E}_p\left(Z\right)E_s^{*}\left(Z\right)\\ \frac{\partial E_p\left(Z\right)}{\partial Z}=j{\kappa }_p{\chi }_{\mathrm{eff}}^{\left(2\right)}{E}_s\left(Z\right)E_i\left(Z\right)\\ \\ {\mathrm{\kappa }}_{\mathrm{a}}=\frac{{\omega }_a{ϵ}_0{\mu }_{0}c}{2n_a}(a\equiv p,s,i)\end{array}.$$(3)

The spatial coordinate Z is along the direction of propagation, Z = 0 corresponding to the entrance of the crystal, ${\chi }_{\mathrm{eff}}^{\left(2\right)}$ is the effective coefficient that is equal to ${\chi }_{24}^{\left(2\right)}$ defined in the previous section, (E_p, E_s, E_i) are the complex amplitudes of the electric fields of the pump (p), signal (s) and idler (i) waves, n_a (a ≡ p, s, i) is the refractive index at the circular frequency ω_a. The system of equation (3) has analytical solutions that are Jacobi elliptic functions [17]. When the undepleted pump approximation (UPA) is assumed, i.e., $\frac{\partial E_p\left(Z\right)}{\partial Z}=0$ so that E_p(Z) = E_p(0), equation (3) leads to the signal and idler motions described by hyperbolic functions, i.e.:$$\{\begin{array}{c}{E}_s\left(Z\right)=E_s\left(0\right)\cos h\left(\beta Z\right)+j\sqrt{\frac{{\omega }_s{n}_i}{{\omega }_i{n}_s}}\frac{E_p\left(0\right)}{\left|E_p\left(0\right)\right|}{E}_i^{*}\left(0\right)\sin h\left(\beta Z\right),\\ E_i\left(Z\right)=E_i\left(0\right)\cos h\left(\beta Z\right)+j\sqrt{\frac{{\omega }_i{n}_s}{{\omega }_s{n}_i}}\frac{E_p\left(0\right)}{\left|E_p\left(0\right)\right|}{E}_s^{*}\left(0\right)\sin h\left(\beta Z\right).\end{array}$$(4)

The constant β is defined by:$$\beta ={\chi }_{\mathrm{eff}}^{\left(2\right)}\left|E_p\left(0\right)\right|\sqrt{{\kappa }_s{\kappa }_i}.$$(5)

Equation (4) are those describing optical parametric amplification at the signal and idler frequencies, and are usually used to estimate the oscillation threshold pump intensity of an optical parametric oscillator (OPO), which is a resonant SPDC, taking the complex conjugate of the second equation and setting the determinant of the resulting two simultaneous equations equal to zero [18]. Then by doing this, the oscillation threshold condition does not involve the initial fields E_s(0) and E_i(0), which avoids any quantum calculation. In the case of non-resonant SPDC, which is the case of the present study, it is necessary to know the magnitudes of the signal and idler fields. Our approach was then to keep the propagation equation (4) that are relative to classical fields and to take the quantum fluctuations of vacuum as initial fields, i.e.,$E_s\left(0\right)\equiv \Delta E_s^{\mathrm{vacuum}}$ and $E_i\left(0\right)\equiv \Delta E_i^{\mathrm{vacuum}}$. By this way, the quantum fluctuations of vacuum are considered as the stimulation of the scission of the pump photons giving birth to signal and idler twin photons: Then the sinh(βZ) contributions describe the optical parametric amplification of vacuum while the cosh(βZ) contributions describe the generation over the mode not initially present at the entrance of the crystal.

The way to calculate the quantum fluctuations of vacuum is summarized in Appendix A, which leads to [19–21]:$$\Delta E^{\mathrm{vacuum}}=\sqrt{\frac{\hslash \omega }{4\pi c{\mathrm{ϵ}}_{0}n\left(\omega \right)S}\Delta \omega }$$(6)

The parameter S ≈ π $w_0^2$ is the transverse cross section of the interaction volume where w₀ corresponds to the beam radius. The spectral width Δω can be estimated by the calculation of the SPDC spectral acceptance in the considered phase-matching direction, i.e., the x-axis of KTP, taken here at the full-width at half-maximum of the phase-matching peak as shown in Appendix A. An analytical expression can be obtained by approximating the phase-mismatch Δk(ω) by an affine function of slope α that cancels at ${\omega }_s$, i.e., the phase-matching circular frequency of the signal mode, which writes: $\Delta k\left(\omega \right)\approx \alpha \left(\omega -{\omega }_s\right)$. From equation (2) using the dispersion equations of KTP, it comes $\alpha =-3.1\times {10}^{-10}{ \mathrm{m}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}$ [15]. Then the spectral width Δω can be expressed as $\Delta \omega \approx \frac{4}{\left|\alpha \right|L} ={1.29\times 10}^{12} \mathrm{Hz}$ for L = 1 cm. Taking S = 2.5 × 10⁻⁸ m², n(ω_s) = 1.83, n(ω_i) = 1.74, it comes: $\Delta E_s^{\mathrm{vacuum}}=12.66 \mathrm{V}{\mathrm{ }\mathrm{m}}^{-1}\approx \Delta E_i^{\mathrm{vacuum}}=12.67 \mathrm{V}{\mathrm{ }\mathrm{m}}^{-1}.$

Finally the twin photon flux N_s(L) = N_i (L) = N_twin (L) generated over the crystal length L and expressed in [Hz] (twins/s) can be deduced from the generated electrical field $E_s^{\mathrm{generated}}\left(L\right)=E_s\left(L\right)-\Delta E_s^{\mathrm{vacuum}}$ where $E_s\left(L\right)$ is given by equation (4) with $E_s\left(0\right)= \Delta E_s^{\mathrm{vacuum}}$ given by equation (6). Then, using $N_s\left(L\right)=\frac{{\mathrm{ϵ}}_{0}n\left(\omega \right)\mathrm{cS}}{4\hslash \omega }{\left|E_s^{\mathrm{generated}}\left(L\right)\right|}^2$, it comes:$$N_{\mathrm{twin}}\left(L\right)=\frac{{\mathrm{ϵ}}_0{n}_s\mathrm{cS}}{4\hslash {\omega }_s}{\left(\left|\Delta E_s^{\mathrm{vacuum}}\right|\left(\cos h\left(\beta L\right)-1\right)+\sqrt{\frac{{\omega }_s{n}_i}{{\omega }_i{n}_s}}\left|\Delta E_i^{\mathrm{vacuum}}\right|\sin h\left(\beta L\right)\right)}^2$$(7)with$\left|\Delta E_s^{\mathrm{vacuum}}\right| \approx \left|\Delta E_i^{\mathrm{vacuum}}\right|.$

It is important to notice that our calculations are based on a single transverse mode approach, which corresponds to our experiment. In the case of multimodal configurations, it would be necessary to use another semiclassical modeling previously published that gave a reliable description of the spectral and spatial features of the generated twin photons [9]. We can also neglect the temporal walk-off since its amplitude is smaller than the picosecond and nanosecond pulse durations, and it does not exist in the CW regime.

3 Results

3.1 Experiment

At pump intensities higher than 1 GW cm⁻², the twin photons energy was high enough to record a spectrum using a USB 2000 Ocean Optics spectrometer. Two typical spectra, taken with a pump intensity I_p ≈ 2.3 GW cm⁻² at 532 nm are given on Figure 2 together with the intensity beam profile of the twin photons (inset of Fig. 2). They have been registered with or without an iris between the crystal output and the spectrometer, in order to diaphragm the profile. The first one, with the iris, in orange on Figure 2, corresponds to the center of the SPDC beam profile spotted by the orange circle in the inset: The signal and idler wavelengths are in good agreement with the theoretical expectation from equations (1) and (2), i.e., ${\lambda }_s=1037 \mathrm{nm}$ and λ_i = 1092 nm (vertical dashed lines). In contrast, the second spectrum (without the iris), in black, corresponding to the full SPDC beam profile, is broaden because of non-collinear contributions.

Fig. 2 Twin photons spectrum. The dark line is the spectrum obtained when the complete emission cone is collected. The orange line corresponds to the spectrum obtained by diaphragming the non-collinear contributions. On the beam profile in inset, the numerical aperture of the collection lens is drawn as a circle in each case (black and orange circle). The vertical dashed lines correspond to the theoretical expectations.

The selected generated signal and idler beams are collinear and they exhibit a weak divergence. On Figure 3 are plotted the twin photons flux measured as a function of the pump intensity for the three runs: CW, ns and ps. Two regions are distinguishable from either side of an intensity$I_p^{\lim }\approx 40 \mathrm{MW }\mathrm{c}{\mathrm{m}}^{-2}$ the first one, for I_p below $I_p^{\lim }$, corresponds to a linear increase of the twin-photon flux; the second one, above $I_p^{\lim }$, relates to an exponential increase. Note that $I_p^{\lim }$ also corresponds to the intersection of the NMM/NHM curve and the SCM curve.

Fig. 3 Measured and calculated SPDC photon flux generated in a 1-cm long x-cut KTP crystal as a function of the incident pump intensity. $I_p^{\lim }$is the pump intensity delimiting the low and high pump intensity regimes of SPDC. SCM stands for semi-classical model (present work), NMM for nonlinear momentum model [10, 11] and NHM for nonlinear Hamiltonian model [12]. The interpolation plots for NMM and NHM models are obtained by multiplying the second-order nonlinearity ${\chi }^{\left(2\right)}$ by a factor of 3, while the SCM it is obtained by multiplying the vacuum fluctuations amplitude $\Delta E_{s,i}^{\mathrm{vacuum}}$ by a factor of 2.3.

The error bars for the pump intensity shown in Figure 3 are very large when using the 10 Hz-repetition rate picosecond pump laser because of strong spatial beam instabilities leading to large fluctuations of the beam radius. The shot by shot twin photon flux uncertainty is very big due to these pump intensity fluctuations, but an averaging allowed us to reduce the corresponding vertical error bars. Note that we kept the shot by shot horizontal uncertainty on the plot, because as discussed in [12], an averaging with pump intensity fluctuations in this highly nonlinear regime leads to a small overestimation of the twin photon flux.

3.2 Theory

Equation (7) is plotted in Figure 3: As for the experimental data, the theoretical curve exhibits a linear part and an exponential part from either side of $I_p^{\lim }\left(0\right)$. This behavior can be explained by the existence of two different regimes of SPDC according to the relative value of $E_{s,i}$ and $\Delta E_{s,i}^{\mathrm{vacuum}}$ : One below a limit pump intensity $I_p^{\lim }\left(0\right)$, defining the low pump intensity regime, for which$E_{s,i}<\Delta E_{s,i}^{\mathrm{vacuum}}$, and the other one above $I_p^{\lim }\left(0\right)$, i.e., the high pump intensity regime, that corresponds to $E_{s,i}$ > $\Delta E_{s,i}^{\mathrm{vacuum}}$. From equation (7), it is possible to obtain an analytical expression describing these two regimes. For that, $\frac{{\omega }_s{n}_s}{{\omega }_i{n}_i}\approx 1$ and $\left|\Delta E_s^{\mathrm{vacuum}}\right|\approx \left|\Delta E_i^{\mathrm{vacuum}}\right|$ are assumed, and the product βL is taken as the relevant parameter for distinguishing the two regimes since this quantity is the argument of the ch and sh functions of equation (7). Then it comes:$$\{\begin{array}{c}\underset{\beta L\ll 1}{\lim }{N}_{\mathrm{twin}}=\frac{\pi {{\chi }_{\mathrm{eff}}^{\left(2\right)}}^2}{2{ϵ}_{0}c{n}_s{n}_i{n}_p{\lambda }_s^2\left|a\right|}{I}_{p}L,\\ \underset{\beta L\gg 1}{\lim } N_{\mathrm{twin}}=\frac{1}{4\pi \left|a\right|L}\exp \left(2\beta L\right).\end{array}$$(8)

Setting βL = 1 allows us to define the pump intensity $I_p^{\mathrm{limit}}\left(0\right)$ that delimits the two regimes. According to equation (5), it writes:$$I_p^{\lim }\left(0\right)=\frac{n_p}{2{\mu }_{0}c{L}^2{\left({\chi }_{\mathrm{eff}}^{\left(2\right)}\right)}^2{\kappa }_i{\kappa }_s}=30 \mathrm{MW} \mathrm{c}{\mathrm{m}}^{-2}$$(9)

These 30 MW cm⁻² are qualitatively close to the 40 MW cm⁻² found by extrapolating our experimental results, as shown in Figure 3.

To take one step further, the interpolation of the experimental data with the SCM leads to a full agreement in terms of magnitude and behaviour, as shown in Figure 3 (black dotted curve), by taking $\Delta E_{s,i}^{\mathrm{vacuum}}\approx 28 \mathrm{V}{\mathrm{ }\mathrm{m}}^{-1}$ which is very close to the calculated value, i.e., 12.7 V m⁻¹. It is difficult at this step to find a satisfying explanation for this small discrepancy.

Note that the numerical integration of equation (3), where there is no approximation regarding the pump depletion, leads to the same curve than that calculated with equation (7) where the UPA is assumed, which is a validation of equation (7).

4 Discussion

We performed a comparison between our semiclassical model (SCM) with two reliable quantum models previously developed: The nonlinear momentum model (NMM) summarized in Appendix B describes the quantum field in term of space dependent spectral mode operators a(ω, Z) and considers a single collinearly propagating spatial mode [10, 11]. While the nonlinear Hamiltonian model (NHM) summarized in Appendix C describes the quantum field in term of time dependent mode operators a(t) [12]. At first, it appears that these two quantum models provide quite indistinguishable behaviors on the graph of Figure 3 but there is a small difference between them considering the corresponding numerical values as can be seen in Table 2. This proximity between the two models is not surprising, given the identical definition of the quantum field. Furthermore, it appears that NMM and NHM are better than SCM in the linear part, contrary to the exponential part, i.e., above 40 MW cm⁻², where the agreement of the SCM with the experimental data is much better than for the NMM and NHM. Table 2 gives some numerical values that allows us to directly compare the experimental data with the three models.

There is a discrepancy of a factor of about 3 at 550 MW cm⁻² for the SCM while it is two orders of magnitude in the case of the NMM and NHM. And it is a factor 4.6 versus seven orders of magnitude at 3.7 GW cm⁻². The difference between experiment and theory is quantized hereafter using the error quantifier χ² that is the classical error statistical function. Note that we wrote it Δ² in order to avoid any confusion with the second-order electric susceptibility. Its calculation is detailed in Appendix E, and the corresponding numerical values given in Table 3 for SCM and NMM. The better agreement of the SCM at high intensity can be explained by simply considering that in this regime, the generated twin-photons amount is much bigger than the quantum fluctuations of vacuum: from Figure 3 at $I_p^{\lim }$, there are N_twin = 1.5 × 10¹¹ Hz compared to 5.0 × 10¹⁰ Hz of “photons of vacuum”. As a consequence, this range of intensity mainly corresponds to a coupling between classical fields, which is the framework of equation (3) coming from Maxwell’s equations. In other words, the SCM becomes less and less of an approximation as pump intensity increases. It is exactly the reverse for the quantum models NMM and NHM that are a little bit better than the semi-classical model SCM at low pump intensity. Note that the quantum models NMM and NHM can also fit well the experimental data with a multiplicative constant γ = 3 in front of the effective coefficient ${\chi }_{\mathrm{eff}}^{\left(2\right)}$, which is the “Interpolation” for NMM/NHM in Figure 3 (red dotted curve). It is important to notice that there is no upper limit to the validity of the quantum models.

It is important to notice that the fitting parameters have not the same mathematical impact regarding the different equations, which explains that the three interpolations can lead to the same curve. In the case of our model (SCM), the “Interpolation” consists in a global multiplicative constant in front of the quantities$\left|\Delta E_s^{\mathrm{vacuum}}\right| \approx \left|\Delta E_i^{\mathrm{vacuum}}\right|$ in equation (7). As a consequence, the fit does not affect the shape of the curve, which is not the case for the two quantum models (NMM and NHM) where the fitting parameter, at the level of ${\chi }_{\mathrm{eff}}^{\left(2\right)}$ (cf. Appendices B and C) acts outside and inside exponential functions.

Note that small pump intensity ranges were not investigated using the picosecond pump laser with several attenuators. The small level of twin flux N_twin reported in Table 2 ranging between 10⁴ and 10⁷, it then would correspond to an average twin photon number of $N_{\mathrm{twin}}^{\mathrm{pulse}}=\frac{N_{\mathrm{twin}}}{10}$ ranging from 10³ and 10⁶ twins generated per pulse using the 10 Hz picosecond source. But in this case, we were disturbed by a background noise coming from the pump laser synchronized with our measurements that we were unable to avoid despite notch filters.

5 Conclusion

We performed a second-order spontaneous parametric down-conversion experiment in a Type II phase-matched KTP crystal pumped at 532 nm and 527 nm along the x-axis of the crystal. Three pump sources have been used: In the CW, nanosecond and picosecond regimes. That allowed us to cover a pump intensity range from 2.4 W cm⁻² to 3.7 GW cm⁻² leading to the generation of twin photons with a flux N_twin from 1.1 × 10⁴ Hz to 1.2 × 10²¹ Hz, respectively, the corresponding quantum efficiencies N_twin/N_p being 1.8 × 10⁻¹¹ and 9.2 × 10⁻⁴. We identified two SPDC regimes regarding the pump intensity: One below 40 MW cm⁻² for which the flux increases linearly, and the other one above 40 MW.cm⁻² for which it behaves exponentially.

We proposed a semi-classical model under the undepleted pump approximation based on the quantum fluctuations of vacuum for the seeding of the process and a propagation ensured by single spatial mode classical fields, which allowed a satisfying description of these two regimes, with a much better agreement at high intensity than the quantum models based on the nonlinear momentum or the nonlinear Hamiltonian operators [11, 12]: The difference with respect to experiment reaches seven orders of magnitude with the quantum models, whereas it is one order of magnitude with our model. Then our model can be particularly useful for the design of high brightness twin-photon devices in the framework of space quantum telecommunications [22]. The other advantage of the semi-classical model over the quantum models is its simplicity of implementation and its possible numerical resolution.

However, the semi-classical and quantum approaches are complementary since the quantum models give access to a differentiation of these two regimes from the quantum point of view, i.e., a superposition of the vacuum state and the single biphoton Fock state at low pump intensity while multi-biphoton Fock states are involved at high pump intensity [12].

An additional validation of the semi-classical multimode modelling described in reference [9] would be brought by a comparison with the data produced in this article. The following of this work will be first to further improve of our semi-classical model in the high-gain regime, i.e., over the exponential part of the curves of Figure 3, by taking into account the broadening of the spectral bandwidth and the correlations between different modes [13]. In a second step, it will deal with quantum measurements in the two identified regimes, i.e., at low and high pump intensities, as well as the use of our semi-classical model for the twin-photon generation by four-wave-mixing, and also for triple-photon generation [23–24]. It would also be interesting to use our model for the description of optical parametric oscillation (OPO), in particular for the calculation of the oscillation threshold intensity. An additional perspective for this semi-classical modelling is the description of more complex experimental situations, such as simultaneously phase-matched cascaded second-order processes [25], allowed by the possible numerical integration of the equations.

Finally, from the technological point of view, it would be interesting to use OPOs, in particular Backward-wave OPOs, in order to get high intensity SPDC [26]. A comparison with other high gain regime experiments can also be performed [27–28].

Acknowledgments

The authors wish to thank David Jegouso and Corinne Félix for their technical support concerning the laser sources.

Funding

This work was funded in part by the Program QuanTEdu-France n° ANR-22-CMAS-0001 France 2030.

Conflicts of interest

The authors have nothing to disclose.

Data availability statement

Data are available upon reasonable request at the indicated e-mail.

Author contribution statement

J.B. Software, Investigation, Writing – original draft. V.B. Investigation, Validation, Writing – original draft. B.B. Conceptualization, Methodology, Writing – original draft – review & editing.

References

Appendix A: Quantum fluctuations of the vacuum

The electrical field quantum operator is expressed by use of continuous variable, considering a single spatial mode of propagation and taking the longitudinal cavity length as infinite [23]. The quantum field is modelled by space-dependent mode operators a(ω, Z) and writes:$$E\left(t,Z\right)=i\underset{0}{\overset{+\infty }{\int }}\mathrm{d\omega }\sqrt{\frac{\hslash \omega }{4\pi {\mathrm{ϵ}}_{0}n\left(\omega \right)\mathrm{cS}}}a\left(\omega ,Z\right)e^{-\mathrm{i\omega }(t-\frac{Z}{c})}+H.C.=E^{+}\left(t,Z\right)+E^{-}(t,Z)$$(A1)

Z is the longitudinal space coordinate, n(ω) is the refractive index of the nonlinear medium. The parameter S is the surface of the interaction volume. In the initial work from [23], it corresponds to the transverse section of the cavity in which the expression of the electrical field is derived. We chose here to estimate this transverse section S, to define an effective cavity with an equivalent Gaussian beam waist radius $w_0$, i.e., $S=\pi w_0^2$.

Quantum mechanically, the vacuum fluctuations are expressed using 〈vac|Δ²E(t, Z)|vac〉 = 〈vac|E(t, Z)²|vac〉 = 〈vac|E⁺(t, Z) E⁻(t, Z) + E⁻ (t, Z)E⁺(t, Z)|vac〉. The corresponding calculation of the variance of the fluctuations $\Delta E^{\mathrm{vacuum}}$ on a finite spectral range Δω, as shown in Figure A1, centered around a central frequency ω is given by equation (6) [19].

Figure A1 (top) Phase-mismatch angular frequency dependency of type II SPDC in a x-cut KTP. The blue curve corresponds to the calculation of Δk(ω) using Sellmeier’s equations from [1]. The dashed line corresponds to the interpolation using Δk ≈ α(ω − ωs). (bottom). The quantity Δ2Eω is the vacuum fluctuation spectral density.

Δ²E_ω is integrated over the spectral bandwidth $\Delta \omega$ defined by the slope α and the $\mathrm{sinc}$ function, leading to Δ² E^vacuum and finally to $\Delta E_{}^{\mathrm{vacuum}}=\sqrt{{\Delta }^2{E}_{}^{\mathrm{vacuum}}}$ [24].

Appendix B: The nonlinear momentum model

The nonlinear momentum model (NMM) describes the quantum field in term of space dependent spectral mode operators $a\left(\omega , Z\right),$ and considers a single collinearly propagating spatial mode, assuming the UPA [10, 11]. The corresponding calculation of the twin photon flux is detailed in [11]. Two regimes can be identified according to the parameter C⁽²⁾ expressed as:$$C^{\left(2\right)}\left(\omega \right)=2\pi f^{\left(2\right)}{\left(\omega \right)}^2{I}_p\left(0\right){\chi }_{\mathrm{eff}}^{{\left(2\right)}^2}-\frac{\Delta k{\left(\omega \right)}^2}{4},$$(B1)with$$f^{\left(2\right)}\left(\omega \right)=\sqrt{\frac{\omega \left({\omega }_p-\omega \right)}{16\pi {ϵ}_0{c}^3{n}_p\left({\omega }_p\right)n_s\left(\omega \right)n_i({\omega }_p-\omega )}}.$$(B2)

I_p (0) is the pump intensity at the entrance of the nonlinear crystal, ${\chi }_{\mathrm{eff}}^{\left(2\right)}$ is the effective coefficient, and Δk is given by equation (2) where n_p, n_s and n_i are the refractive indices defined in Section 2.1.

Then it comes for the signal photon flux:$$N_s\left(\omega ,Z\right)=\{\begin{array}{c}\frac{I_p\left(0\right){\chi }_{\mathrm{eff}}^{{\left(2\right)}^2}{f}^{\left(2\right)}{\left(\omega \right)}^2}{C^{\left(2\right)}\left(\omega \right)}{\sin h}^2\left(\sqrt{C^{\left(2\right)}\left(\omega \right)}Z\right)\mathrm{if} C^{\left(2\right)}\left(\omega \right)>0\\ I_p\left(0\right){\chi }_{\mathrm{eff}}^{{\left(2\right)}^2}{f}^{\left(2\right)}{\left(\omega \right)}^2{Z}^2{\sin c}^2\left(\sqrt{\left|C^{\left(2\right)}\left(\omega \right)\right|}Z\right)\mathrm{if} C^{\left(2\right)}\left(\omega \right)<0\end{array}.$$(B3)

Z is the space coordinate along the direction of propagation.

Finally, the twin photon flux writes:$$N_{\mathrm{twin}}\left(Z\right)=N_s\left(Z\right)=N_i\left(Z\right)=\underset{0}{\overset{+\infty }{\int }}\mathrm{d}\omega N_s\left(\omega ,Z\right).$$(B4)

For low pump intensities, all the spectral components are verifying $2\pi f^{\left(2\right)}{\left(\omega \right)}^2{I}_p\left(0\right){\chi }_{\mathrm{eff}}^{{\left(2\right)}^2}\ll \frac{\Delta k{\left(\omega \right)}^2}{4}$, and an analytical expression of the generated twin-photon flux can be deduced, i.e.:$$\underset{\sqrt{|C^{\left(2\right)| }}\ll 1 }{\lim }{N}_{\mathrm{twin}}\left(Z\right)\approx {\frac{{\pi }^2{{\chi }_{\mathrm{eff}}^{\left(2\right)}}^2}{2{ϵ}_{0}c{n}_p{n}_s{n}_i{\lambda }_s^2\left|\alpha \right|}I}_{p}Z.$$(B5)

Note that this expression is equal to equation (8) by a multiplicative factor of π, which is an additional validation of our semi-classical model.

Appendix C: The nonlinear Hamiltonian model

The nonlinear Hamiltonian model (NHM) describes the quantum field in term of time dependent field mode operators assuming the undepleted pump approximation [12]. The Heisenberg’s equation of motion propagates those quantum fields through time. A variable change $z=\mathrm{ct}$ gives the spatial dependency. Two intensity regimes can be defined, one being linear, the other one exponential, which gives for the twin photon flux [12]:$$N_{\mathrm{twin}}\left(Z\right)=\{\begin{array}{c}\frac{2{\chi }_{\mathrm{eff}}^{{\left(2\right)}^2}}{{ϵ}_0{n}_s^2{n_i}^2{n}_p{\mathrm{\lambda }}_{\mathrm{s}}^2}\frac{n_{\mathrm{gs}}{n}_{\mathrm{gi}}}{\Delta n_g} {\left|\frac{{\sigma }_p^2}{{\sigma }_s^2+2{\sigma }_p^2}\right|}^2{I}_p\left(0\right)Z, (\mathrm{small} I_p)\\ \frac{c}{2L{n}_s}\exp \left(2{\kappa }_{\mathrm{time}}Z\right), \left(\mathrm{high} I_p\right)\end{array}$$(C1)with$${\kappa }_{\mathrm{time}}^2=\frac{{\pi }^2{{\chi }_{\mathrm{eff}}^{\left(2\right)}}^2}{4{ϵ}_{0}c{n_s}^2 {n_i}^2{n}_p{\lambda }_p^2}{\left|\frac{{\sigma }_p^2}{{\sigma }_s^2+2{\sigma }_p^2}\right|}^2{I}_p.$$(C2)

The quantities ${\sigma }_{p,s }$ are the transverse section of both pump and down-converted fields, $n_{\mathrm{gs},i}$ are the group velocity indices of signal and idler, and $\Delta n_g=|n_{\mathrm{gs}}-n_{\mathrm{gi}}|$.

Appendix D: Detection method

For the CW experiment, coincidence measurement is used between detector 2 and detector 3. The absolute fiber coupling is measured by comparing the coincidence to single ratio of detectors 2 and 3 [12–16]. The transmissions of channel 2 and 3 are written $T_2$ and $T_3$, respectively: they include all common optical losses, the coupling efficiency in the fiber, and the detectors efficiencies. If $N_2$ and $N_3$ are the flux of detection events on detector 2 and detector 3, and $N_{23}$ the flux of joint detection events between detector 2 and detector 3, then straightforward statistical calculations lead to [12–16]:$$T_2=\frac{2N_{23}}{N_3-N_3^{\mathrm{dark}}}, T_3=\frac{2N_{23}}{N_2-N_2^{\mathrm{dark}}},$$(D1) $$N_{\mathrm{twin}}=\frac{\left(N_2-N_2^{\mathrm{dark}}\right)\left(N_3-N_3^{\mathrm{dark}}\right)}{2N_{23}}$$(D2)

The quantities $N_i^{\mathrm{dark}}$ are measured by rotating the pump polarization angle of 90°: they correspond to coincidences generated by parasitic ambient light. With these equations, it is possible through statistical measurements to measure $T_2$, $T_3$ and then N_twin which is the real twin photon flux corrected from the losses. The numerical aperture of the collection device is empirically dimensioned so that the pump beam radius equals the aspherical lens pupil radius, the geometry of the SPDC collected mode then corresponding to the geometry of the pump beam mode.

For the nanosecond experiment, the collection set-up is the same as in the SPDC experiment, the losses being calibrated by comparison with the CW absolute measurement at equal intensities.

For the picosecond experiment, the generated flux, for example when pumping at 1 GW cm⁻², are sufficient to have a macroscopic measurement available with a pico-joulemeter Molectron J3S10. This is used as a quantitative reference: a measurement at the same pump intensity is then realized with SNSPDs and calibrated neutral densities. By doing this, the coupling is easily and very accurately measured. Here again, the collection numerical aperture is dimensioned to collect the same geometrical spatial mode than the pump beam, for both the Molectron and SNSPD measurements. Thus, the pump intensity can be investigated at smaller values, the coupling efficiency being assumed to not depend on the pump intensity. Note that the calibrated neutral densities are always chosen so that the average number of photons per pulse is small versus 1, in order to certify the linearity of the SNSPD response.

Appendix E: Calculation of the Δ² error quantifier

In low gain regime, we chose to neglect the horizontal error which is small compared to the vertical one, so that it comes:$${\Delta }_{\mathrm{lowgain}}^2=\frac{1}{\mathrm{number} \mathrm{of} \mathrm{data}}\sum _i\frac{{\left({N_{\mathrm{twin}}^{\mathrm{measured}}}_i-{N_{\mathrm{twin}}^{\mathrm{theory}}}_i\right)}^2}{{{\sigma }_{\mathrm{vertical}}}_i^2}.$$

σ_vertical i stands for the error on the measurement of the twin photon flux (vertical error bars on Fig. 3).

In high gain regime, the horizontal error bars are predominant, which gives:$${\Delta }_{\mathrm{highgain}}^2=\frac{1}{\mathrm{number} \mathrm{of} \mathrm{data}}\sum _i\frac{{\left({I_p^{\mathrm{injected}}}_i-{I_p^{\mathrm{theory}}}_i\right)}^2}{{{\sigma }_{\mathrm{horizontal}}}_i^2}.$$

σ_{horizontal i} corresponds to the horizontal uncertainty. ${I_p^{\mathrm{injected}}}_i$ and ${I_p^{\mathrm{theory}}}_i$ are the intensities that leads to the same value of twin photon flux $N_{\mathrm{twin}}^i$.

All Tables

All Figures

	Fig. 1 Experimental setup used for performing CW, nanosecond and picosecond Type II SPDC in a x-cut KTP crystal. (ep, es, ei) stand for the polarizations of the pump, signal and idler beams, respectively.
In the text

Fig. 2 Twin photons spectrum. The dark line is the spectrum obtained when the complete emission cone is collected. The orange line corresponds to the spectrum obtained by diaphragming the non-collinear contributions. On the beam profile in inset, the numerical aperture of the collection lens is drawn as a circle in each case (black and orange circle). The vertical dashed lines correspond to the theoretical expectations.

In the text

Fig. 3 Measured and calculated SPDC photon flux generated in a 1-cm long x-cut KTP crystal as a function of the incident pump intensity. $I_p^{\lim }$is the pump intensity delimiting the low and high pump intensity regimes of SPDC. SCM stands for semi-classical model (present work), NMM for nonlinear momentum model [10, 11] and NHM for nonlinear Hamiltonian model [12]. The interpolation plots for NMM and NHM models are obtained by multiplying the second-order nonlinearity ${\chi }^{\left(2\right)}$ by a factor of 3, while the SCM it is obtained by multiplying the vacuum fluctuations amplitude $\Delta E_{s,i}^{\mathrm{vacuum}}$ by a factor of 2.3.

In the text

	Figure A1 (top) Phase-mismatch angular frequency dependency of type II SPDC in a x-cut KTP. The blue curve corresponds to the calculation of Δk(ω) using Sellmeier’s equations from [1]. The dashed line corresponds to the interpolation using Δk ≈ α(ω − ωs). (bottom). The quantity Δ2Eω is the vacuum fluctuation spectral density.
In the text