Counter-propagating spontaneous four wave mixing: photon-pair factorability and ultra-narrowband single photons

In this section we describe the two-photon state for the CP-SFWM process in a ${\chi }^{(3)}$ medium. We will initially write down expressions for the two-photon state which permit each of the four waves to propagate in different transverse and polarization modes, where ${k}_{1}(\omega )$, ${k}_{2}(\omega )$, ${k}_{s}(\omega )$, and ${k}_{i}(\omega )$ represent the frequency-dependent wavenumbers for each of the four waves: pump 1(1), pump 2(2), signal(s), and idler(i). Later in the paper we will concentrate our discussion on the case where all four waves are co-polarized and involve the same transverse mode so that ${k}_{1}(\omega )={k}_{2}(\omega )={k}_{s}(\omega )={k}_{i}(\omega )\equiv k(\omega )$. Throughout this paper, while the pump 1 and the idler waves are forward-propagating, the pump 2 and signal waves are backward propagating; we adopt a sign convention for which all wavenumbers are positive, with explicit signs appearing in accordance to the direction of propagation.

We start from the interaction Hamiltonian governing SFWM processes, given by

$$\begin{eqnarray}&&\hat{H}(t)=\displaystyle \frac{3}{4}{\epsilon }_{o}{\chi }^{(3)}\int {{\rm{d}}}^{3}{\bf{r}}{\hat{E}}_{1}^{(+)}({\bf{r}},t){\hat{E}}_{2}^{(+)}({\bf{r}},t){\hat{E}}_{s}^{(-)}({\bf{r}},t){\hat{E}}_{i}^{(-)}({\bf{r}},t),\end{eqnarray} \tag{ 1 }$$

where the integration is carried out over the portion of the nonlinear medium for which the pump fields are temporally and spatially overlapped, ${\chi }^{(3)}$ is the third-order nonlinear susceptibility, and ${\epsilon }_{o}$ is the vacuum electrical permittivity. In equation (1), the subscripts ⁽⁺⁾/⁽⁻⁾ refer to the positive frequency / negative frequency parts of the electric field operators. In our analysis, we assume that the two pumps can be well-described by classical fields, i.e. no longer operators, of the form

$$\begin{eqnarray}&&{\hat{E}}_{\nu }^{(+)}({\bf{r}},t)\to {A}_{\nu }{f}_{\nu }(x,y)\int {\rm{d}}\omega {\alpha }_{{\nu }_{\pm }}(\omega )\,{\rm{\exp }}[-{\rm{i}}(\omega t\mp k(\omega )z)],\end{eqnarray} \tag{ 2 }$$

with $\nu =1,2$ for the two pumps and ${A}_{\nu }$ represents the field amplitude. ${\alpha }_{{\nu }_{\pm }}(\omega )$ is the spectral envelope (the meaning of the signs ± is defined below), and ${f}_{\nu }(x,y)$ is the transverse spatial field distribution, which is normalized so that $\int \int | {f}_{\nu }(x,y){| }^{2}\,{\rm{d}}{x}{\rm{d}}{y}=1$ and is approximated to be frequency-independent within the pump bandwidth.

The quantized signal and idler fields are expressed as

$$\begin{eqnarray}&&{\hat{E}}_{\mu }^{(+)}({\bf{r}},t)={\rm{i}}\sqrt{\delta k}{f}_{\mu }(x,y)\displaystyle \sum _{k}{\rm{\exp }}[-{\rm{i}}(\omega t\mp k(\omega )z)]{\ell }(\omega ){\hat{a}}_{\mu \pm }(k),\end{eqnarray} \tag{ 3 }$$

with $\mu =s,i$ and $\delta k=2\pi /{L}_{Q}$ the mode spacing, written in terms of the quantization length L_Q. Function ${\ell }(\omega )$ is given as follows

$$\begin{eqnarray}&&{\ell }(\omega )=\sqrt{\displaystyle \frac{{\rm{\hslash }}\omega }{\pi {\epsilon }_{o}{n}^{2}(\omega )}},\end{eqnarray} \tag{ 4 }$$

in terms of the (linear) refractive index of the nonlinear medium $n(\omega )$ and of Planck's constant ${\rm{\hslash }}$. In equation (3), ${\hat{a}}_{\mu \pm }(k)$ is the annihilation operator (the meaning of the signs ± is defined below), and ${f}_{\mu }(x,y)$ represents the transverse spatial distribution of the field, which is also normalized as the corresponding pump functions, and is assumed to be frequency-independent within the bandwidth of signal and idler modes.

Note that in equations (2) and (3) the $-/+$ signs, in front of the propagation constant $k(\omega )$ and the corresponding subscripts $+/-$ in the annihilation operators and the pump spectral envelopes, indicate optical fields propagating along the fiber in the forward/backward directions.

Following a standard perturbative approach [29] and our treatment in reference [30], it can be shown that the two-photon state produced by CP-SFWM can be written as $| {\rm{\Psi }}\rangle ={| 0\rangle }_{s}{| 0\rangle }_{i}+\eta {| {\rm{\Psi }}\rangle }_{2}$, in terms of the two-photon component

$$\begin{eqnarray}&&{| {\rm{\Psi }}\rangle }_{2}=\displaystyle \sum _{{k}_{s}}\displaystyle \sum _{{k}_{i}}{\ell }({k}_{s}){\ell }({k}_{i})F({k}_{s},{k}_{i}){\hat{a}}_{s-}^{\dagger }({k}_{s}){\hat{a}}_{i+}^{\dagger }({k}_{i}){| 0\rangle }_{s}{| 0\rangle }_{i},\end{eqnarray} \tag{ 5 }$$

and the constant η, which is related to the conversion efficiency and is given by

$$\begin{eqnarray}&&\eta ={\rm{i}}(2\pi )\delta k\displaystyle \frac{3{\epsilon }_{o}{\chi }^{(3)}}{4{\rm{\hslash }}}{A}_{1}{A}_{2}{{Lf}}_{{\rm{eff}}},\end{eqnarray} \tag{ 6 }$$

where L is the fiber length and f_eff is the spatial overlap integral between the four fields given by

$$\begin{eqnarray}&&{f}_{{\rm{eff}}}=\int {\rm{d}}{x}\int {{\rm{d}}{\text{yf}}}_{1}(x,y){f}_{2}(x,y){f}_{s}^{* }(x,y){f}_{i}^{* }(x,y).\end{eqnarray} \tag{ 7 }$$

In equation (5) ${k}_{s}\equiv {k}_{s}({\omega }_{s})$ is the propagation constant for the backward-propagating signal mode, and ${k}_{i}\equiv {k}_{i}({\omega }_{i})$ is the propagation constant for the forward-propagating idler mode; ${\hat{a}}_{s-}^{\dagger }(k)$ represents the creation operator for the backward-propagating signal mode, while ${\hat{a}}_{i+}^{\dagger }(k)$ represents the creation operator for the forward-propagating idler mode. $F({k}_{s},{k}_{i})$ is the joint amplitude function, which can be written in terms of frequencies rather than wave numbers, in which case it is referred to as the joint spectral amplitude (JSA), and is expressed as $F({\omega }_{s},{\omega }_{i})$.

In our analysis we first consider a source configuration in which both pumps are pulsed. In this case, the JSA function ${F}_{{\text{}}P}({\omega }_{s},{\omega }_{i})$ can be shown to be given by

$$\begin{eqnarray}&&{F}_{{\text{}}P}({\omega }_{s},{\omega }_{{\rm{i}}})=\int {\rm{d}}\omega {\alpha }_{1+}(\omega ){\alpha }_{2-}({\omega }_{s}+{\omega }_{{\rm{i}}}-\omega ){\rm{sinc}}\left[\displaystyle \frac{L}{2}{\rm{\Delta }}k\right]{{\rm{e}}}^{{\rm{i}}\tfrac{L}{2}\kappa }{{\rm{e}}}^{{\rm{i}}\omega \tau },\end{eqnarray} \tag{ 8 }$$

where ${\alpha }_{1+}(\omega )$ represents the pump spectral envelope for the forward-propagating pump, ${\alpha }_{2-}(\omega )$ represents the pump spectral envelope for the backward-propagating pump, and τ represents the time of arrival difference between the two pump pulses at their respective ends of the fiber. Note that τ can be controlled externally with a relative delay between the two pumps; in particular, $\tau =0$ implies that the pump pulses corresponding to pumps 1 and 2 arrive at the same time at the two ends of the fiber. Equation (8) is expressed in terms of the phase mismatch function ${\rm{\Delta }}k\equiv {\rm{\Delta }}k(\omega ,{\omega }_{s},{\omega }_{i})$, and the function $\kappa \equiv \kappa (\omega ,{\omega }_{s},{\omega }_{i})$ defined as

$$\begin{eqnarray}&&{\rm{\Delta }}k={k}_{1}(\omega )-{k}_{2}({\omega }_{s}+{\omega }_{i}-\omega )-{k}_{s}({\omega }_{s})+{k}_{i}({\omega }_{i})+{\phi }_{{\rm{NL}}},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\kappa ={k}_{1}(\omega )+{k}_{2}({\omega }_{s}+{\omega }_{i}-\omega )+{k}_{s}({\omega }_{s})+{k}_{i}({\omega }_{i}),\end{eqnarray} \tag{ 10 }$$

where ${\phi }_{{\rm{NL}}}$ is a nonlinear phase shift derived from SPM and CPM (see below for further discussion and for expressions). Note that the energy conservation constraint is already included in the resulting joint amplitude.

Let us now consider a pumps configuration defined as the limit where the backward-propagating pump wave becomes monochromatic at frequency ${\omega }_{{cw}}$, in which case the corresponding electric field can be expressed as ${E}_{{cw}}^{(+)}({\bf{r}},t)={{af}}_{2}(x,y){\rm{\exp }}[-{\rm{i}}({\omega }_{{cw}}t+k({\omega }_{{cw}})z)]$ with a the field amplitude, while the forward-propagating pump remains broadband; we refer to this as the mixed pumps configuration. In this case, the JSA function becomes

$$\begin{eqnarray}&&{F}_{{\text{}}M}({\omega }_{s},{\omega }_{{\rm{i}}})={\alpha }_{+}({\omega }_{s}+{\omega }_{{\rm{i}}}-{\omega }_{{cw}}){\rm{sinc}}\left[\displaystyle \frac{L}{2}{\rm{\Delta }}{k}_{{\text{}}M}\right]{{\rm{e}}}^{{\rm{i}}\tfrac{L}{2}{\kappa }_{{\text{}}M}},\end{eqnarray} \tag{ 11 }$$

where ${\rm{\Delta }}{k}_{{\text{}}M}\equiv {\rm{\Delta }}{k}_{M}({\omega }_{s},{\omega }_{i})$ and ${\kappa }_{{\text{}}M}\equiv {\kappa }_{{\text{}}M}({\omega }_{s},{\omega }_{i})$ are defined as

$$\begin{eqnarray}&&{\rm{\Delta }}{k}_{{\text{}}M}={k}_{1}({\omega }_{s}+{\omega }_{i}-{\omega }_{{cw}})-{k}_{2}({\omega }_{{cw}})-{k}_{s}({\omega }_{s})+{k}_{i}({\omega }_{i})+{\phi }_{{\rm{NL}}},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\kappa }_{{\text{}}M}={k}_{1}({\omega }_{s}+{\omega }_{i}-{\omega }_{{cw}})+{k}_{2}({\omega }_{{cw}})+{k}_{s}({\omega }_{s})+{k}_{i}({\omega }_{i}).\end{eqnarray} \tag{ 13 }$$

The nonlinear phase shift ${{\rm{\Phi }}}_{{\rm{NL}}}$, appearing in equation (9) (for the pulsed pumps case), can be been shown to be given as follows [9, 31]

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{NL}}}=({\gamma }_{1}-2{\gamma }_{21}-2{\gamma }_{s1}+2{\gamma }_{i1}){P}_{1}-({\gamma }_{2}-2{\gamma }_{12}+2{\gamma }_{s2}-2{\gamma }_{i2}){P}_{2},\end{eqnarray} \tag{ 14 }$$

where P₁ and P₂ represent the peak powers for pumps 1 and 2 (related, for pumps with Gaussian spectra, as ${P}_{\nu }={p}_{\nu }{\sigma }_{\nu }/[\sqrt{2\pi }R]$ with the average pump powers p_ν, where R is the repetition frequency and ${\sigma }_{\nu }$ is the corresponding bandwidth). The nonlinear phase shift ${{\rm{\Phi }}}_{{\rm{NL}}}$, appearing in equation (12) (for the mixed pumps case) is given by equation (14), with the substitution ${P}_{2}\to {p}_{2}$.

The coefficients ${\gamma }_{1}$ and ${\gamma }_{2}$ result from SPM of the two pumps, and are given, with $\nu =1,2$ by

$$\begin{eqnarray}&&{\gamma }_{\nu }=\displaystyle \frac{3{\chi }^{(3)}{\omega }_{\nu }^{0}{f}_{{\rm{eff}}}^{\nu }}{4{\epsilon }_{0}{c}^{2}{n}_{\nu }^{2}}.\end{eqnarray} \tag{ 15 }$$

In equation (15), the refractive index ${n}_{\nu }\equiv n({\omega }_{\nu }^{0})$ and the spatial overlap integral ${f}_{{\rm{eff}}}^{\nu }\equiv \int \int {\rm{d}}{x}{\rm{d}}{y}| {f}_{\nu }(x,y){| }^{4}$ (where the integral is carried out over the transverse dimensions of the fiber) are defined in terms of the carrier frequency ${\omega }_{\nu }^{0}$ for pump-mode ν [32].

In contrast, coefficients ${\gamma }_{\mu \nu }$ ($\nu =1,2$ and $\mu =1,2,s,i$) correspond to the CPM contributions that result from the dependence of the refractive index experienced by wave μ, with $\mu =1,2,s,i$, on the pump intensities ($\nu =1,2;$ note that CPM of the signal/idler photons on the pumps as well as SPM of the signal and idler waves are small effects which we neglect). These coefficients are given by

$$\begin{eqnarray}&&{\gamma }_{\mu \nu }=\displaystyle \frac{3{\chi }^{(3)}{\omega }_{\mu }^{0}{f}_{{\rm{eff}}}^{\mu \nu }}{4{\epsilon }_{0}{c}^{2}{n}_{\mu }{n}_{\nu }},\end{eqnarray} \tag{ 16 }$$

where ${n}_{\mu ,\nu }\equiv n({\omega }_{\mu ,\nu }^{0})$ is defined in terms of the central frequency ${\omega }_{\mu ,\nu }^{0}$ for each of the four participating fields, and ${f}_{{\rm{eff}}}^{\mu \nu }\equiv \int \int {\rm{d}}{x}{\rm{d}}{y}| {f}_{\mu }(x,y){| }^{2}| {f}_{\nu }(x,y){| }^{2}$ is the two-mode spatial overlap integral (note that ${f}_{{\rm{eff}}}^{\mu \nu }={f}_{{\rm{eff}}}^{\nu \mu }$).

In designing two-photon sources, it is helpful to be able to estimate the source brightness in terms of all relevant experimental parameters. Expressions for the emitted flux for co-propagating and co-polarized SFWM in single-mode fibers have been reported by us previously [30].

The number of photon pairs generated per second, or source brightness, is given by

$$\begin{eqnarray}&&N=R\displaystyle \sum _{k}\langle {\psi }_{2}| {\hat{a}}_{s-}^{\dagger }(k){\hat{a}}_{s-}(k)| {\psi }_{2}\rangle ,\end{eqnarray} \tag{ 17 }$$

where $| {\psi }_{2}\rangle$ is defined in equation (5) and R is the pump repetition rate (for the pulsed pump in the mixed pumps case, and assumed to be equal for both pumps in the pulsed pumps case); note that the brightness in equation (17) can likewise be expressed in terms of the idler annihilation operator. From this equation we have derived expressions for the number of photon pairs generated per second for the two pump configurations described above, which are represented by ${N}_{{\text{}}P}$ and ${N}_{{\text{}}M}$, respectively. In the analysis we assume that the pulsed pump fields have a gaussian spectral envelope

$$\begin{eqnarray}&&{\alpha }_{\mu }(\omega )=\displaystyle \frac{{2}^{1/4}}{{\pi }^{1/4}{\sqrt{\sigma }}_{\mu }}\,\exp \left[-\displaystyle \frac{{(\omega -{\omega }_{\mu }^{0})}^{2}}{{\sigma }_{\mu }^{2}}\right],\end{eqnarray} \tag{ 18 }$$

where ${\omega }_{\mu }^{0}$ and ${\sigma }_{\mu }$ (with $\mu =1,2$) are the central pump frequency and the pump bandwidth for the two pumps, respectively. Note that the function ${\alpha }_{\mu }(\omega )$ has been normalized so that $\int {\rm{d}}\omega | {\alpha }_{\mu }(\omega ){}^{2}=1$.

Substituting the two-photon state, equation (5), into equation (17) while appropriately turning sums into integrals in the limit $\delta k\to 0$, it can be shown that ${N}_{{\text{}}P}$ is given by

$$\begin{eqnarray}&&{N}_{{\text{}}P}=\displaystyle \frac{{2}^{5}{n}_{1}{n}_{2}{c}^{2}{L}^{2}{\gamma }^{2}{p}_{1}{p}_{2}}{{\pi }^{3}{\omega }_{1}^{0}{\omega }_{2}^{0}{\sigma }_{1}{\sigma }_{2}R}\int {\rm{d}}{\omega }_{s}\int {\rm{d}}{\omega }_{i}\,h({\omega }_{s},{\omega }_{i})| {F}_{{\text{}}P}({\omega }_{s},{\omega }_{i}){}^{2},\end{eqnarray} \tag{ 19 }$$

where ${n}_{1}\equiv {n}_{1}({\omega }_{1}^{0})$ (${n}_{2}\equiv {n}_{2}({\omega }_{2}^{0})$), c is the speed of light in the vacuum, L is the fiber length, ${p}_{\nu }$ and ${\sigma }_{\nu }$ (with $\nu =1,2$) are the average power and bandwidth of the two pumps, respectively, ${F}_{{\text{}}P}({\omega }_{s},{\omega }_{i})$ is the JSA given in the equation (8), and $h({\omega }_{s},{\omega }_{i})$ is a function defined as

$$\begin{eqnarray}&&h({\omega }_{s},{\omega }_{i})=\displaystyle \frac{{\omega }_{s}{k}_{s}^{\prime }({\omega }_{s})}{{n}_{s}^{2}({\omega }_{s})}\displaystyle \frac{{\omega }_{i}{k}_{i}^{\prime }({\omega }_{i})}{{n}_{i}^{2}({\omega }_{i})},\end{eqnarray} \tag{ 20 }$$

where ${k}_{\mu }^{\prime }({\omega }_{\mu })$ denotes the frequency derivate of the propagation constant ${k}_{\mu }({\omega }_{\mu })$, and γ is the SFWM nonlinear coefficient given by

$$\begin{eqnarray}&&\gamma =\displaystyle \frac{3{\chi }^{(3)}\sqrt{{\omega }_{1}^{0}{\omega }_{2}^{0}}{f}_{{\rm{eff}}}}{4{\epsilon }_{0}{c}^{2}{n}_{1}{n}_{2}},\end{eqnarray} \tag{ 21 }$$

in terms of the third-order nonlinear susceptibility, ${\chi }^{(3)}$, the electric permittivity of free space, ${\epsilon }_{0}$, and the spatial overlap integral f_eff defined in equation (7). Note that in our analysis we have assumed that the transverse electric field distributions for the various fiber modes depend only weakly on the frequency, so that γ has been regarded as a constant, and taken out of the integral, as in equation (19).

Similarly, it can be shown that for the mixed pumps case the number of photon pairs generated per second, ${N}_{{\text{}}M}$, is given by

$$\begin{eqnarray}&&{N}_{{\text{}}M}=\displaystyle \frac{{2}^{11/2}{n}_{1}{n}_{2}{c}^{2}{L}^{2}{\gamma }^{2}{p}_{1}{p}_{2}}{{\pi }^{3/2}{\omega }_{1}^{0}{\omega }_{2}\sigma }\int {\rm{d}}{\omega }_{s}\int {\rm{d}}{\omega }_{i}\,h({\omega }_{s},{\omega }_{i})| {F}_{{\text{}}M}({\omega }_{s},{\omega }_{i}){}^{2},\end{eqnarray} \tag{ 22 }$$

where p₁ and σ represent the average power and bandwidth of the pulsed pump, respectively; p₂ is the power of the monochromatic pump wave; ${F}_{{\text{}}M}({\omega }_{s},{\omega }_{i})$ is the JSA given in the equation (11); $h({\omega }_{s},{\omega }_{i})$ is given by equation (20) and γ is defined according to equation (21).

In order for the CP-SFWM process to exist, linear momentum must be conserved which is equivalent to a phasematching condition ${\rm{\Delta }}k=0$ for the pulsed pumps case, or ${\rm{\Delta }}{k}_{M}=0$ for the mixed pumps case (note that since in general all four waves are polychromatic, these phasematching conditions are fulfilled exactly for specific frequencies regarded as 'central' for each wave). It is straightforward to verify from equations (9) and (12) that if all four waves propagate in the same transverse and polarization mode, then phasematching is always attained, provided that the nonlinear term ${\phi }_{{\rm{NL}}}$ is negligible, at frequencies satisfying the following relationships: ${\omega }_{1}={\omega }_{s}$ and ${\omega }_{2}={\omega }_{i}$. This is a remarkable property of CP-SFWM: phasematching is fulfilled automatically, for an arbitrary single-mode fiber, using frequency non-degenerate pumps centered at ${\omega }_{1}$ and ${\omega }_{2}$ so as to generate a backward-propagating signal photon with frequency ${\omega }_{s}={\omega }_{1}$ paired with a forward-propagating idler photon with frequency ${\omega }_{i}={\omega }_{2}$. In other words, basic phasematching properties (i.e. the determination of emission SFWM frequencies as a function of pump frequencies), become decoupled from the fiber dispersion and are in fact identical for all conceivable single-mode fibers. Note that a particular case of the scenario above is that for which the pumps are frequency-degenerate which in fact leads to all four waves being frequency degenerate. Also, note that a possible source of noise in CP-SFWM is spontaneous Brillouin scattering of the pump fields, which would appear in the same frequencies and directions of propagation as the generated photons [32].

In figure 1(c) we have plotted the ${\rm{\Delta }}k=0$ contour (solid black straight lines), i.e. the signal and idler frequencies which satisfy perfect phasematching, as a function of the pump frequency ${\omega }_{1}$, while ${\omega }_{2}$ remains fixed at ${\omega }_{2}=2\pi c/0.532\,\mu {\rm{m}}$. Note that this diagram is 'universal', in the sense that it applies to all conceivable single-mode fibers. While a fixed pump 2 frequency leads to an equally fixed idler frequency, since ${\omega }_{i}={\omega }_{2}$, there is a linear dependence between the remaining two frequencies, as ${\omega }_{s}={\omega }_{1}$. Note also that the intersection of the two straight lines corresponds to the degenerate pumps case, for which ${\omega }_{1}={\omega }_{2}=2\pi c/0.532\,\mu {\rm{m}}$. In particular, in the figure 1(c) the red vertical line corresponds to ${\omega }_{1}=2\pi c/0.820\,\mu {\rm{m}}$, so that its intersection with the ${\rm{\Delta }}k=0$ contour indicates that perfect phasematching occurs for ${\omega }_{s}=2\pi c/0.820\,\mu {\rm{m}}$ and ${\omega }_{i}=2\pi c/0.532\,\mu {\rm{m}}$. A different choice of fixed pump 2 frequency would simply lead to a vertically displaced horizontal tuning curve for the idler photon. Let us emphasize that the automatic phasematching observed for CP-SFWM is achromatic in the sense that it is attained for any choice of ${\omega }_{1}$ and ${\omega }_{2}$ (leading to ${\omega }_{s}={\omega }_{1}$ and ${\omega }_{i}={\omega }_{2})$, regardless of the specific underlying fiber dispersion. This opens a wealth of possibilities for the implementation of photon-pair sources in optical fibers.

Note that if the nonlinear term ${\phi }_{{\rm{NL}}}$ is non-zero, the symmetry between each pair of counterpropagating pump and generated SFWM photon is broken; in principle, this could be useful in order to slightly offset the generation frequencies from the pump frequencies so as to simplify the experimental discrimination of signal and idler photons from scattered pump photons. Likewise, this symmetry can be broken with cross-polarized SFWM processes of the kind xyxy or xxyy in brirefringent fibers. However, for experimental conditions regarded as typical (conventional fibers and typical values of pump power and/or typical birefringence values) the resulting offset tends to be insufficient in practice for theeffective discrimination between SFWM photon pairs and pump photons.

Another interesting possibility is for the four waves to propagate in different transverse modes (in a few-mode or multi-mode fiber), likewise leading to an offset of the generation frequencies from the pump frequencies. Thus, let us discuss the case of CP-SFWM implemented in a few-mode optical fiber [33, 34] leading to an intermodal process; this means allowing some of the waves involved in the SFWM process to travel in higher-order transverse modes. As an example, if the forward propagating pump travels in the fundamental fiber mode and the signal photon travels in a certain higher-order mode X, while the backward-propagating pump mode travels in the same higher-order mode X and the idler photon travels in the fundamental mode, then perfect phasematching will occur for signal and idler frequencies that are shifted from those of the pumps. Specifically, this will result in a signal photon with frequency ${\omega }_{1}+\delta$ and in an idler photon with frequency ${\omega }_{2}-\delta$, with a frequency offset δ which depends on the dispersion relations of the two modes; note that while the frequency offsets are equal (opposite in sign), the resulting wavelength offsets will differ between signal and idler. Importantly, as the order of mode X increases, δ also increases. In table 1 we summarize the emission wavelengths and the resulting wavelength offsets that result from intermodal CP-SFWM in a step-index fiber (with numerical aperture ${\rm{NA}}=0.3$ and core radius r = 2 μm) that supports three higher-order modes, for the case in which the pump wavelengths are 820 nm (forward propagating pump) and 532 nm (backward propagating pump).

Table 1.  Emission wavelengths (${\lambda }_{s}$ and ${\lambda }_{i}$) and wavelength offsets (${\rm{\Delta }}{\lambda }_{s}$ and ${\rm{\Delta }}{\lambda }_{i}$) for intermodal CP-SFWM, for different choices of excited mode X, in a few-mode step-index fiber with numerical aperture ${\rm{NA}}=0.3$ and core radius r = 2 μm.

Fiber mode X	${\lambda }_{s}$ (nm)	${\rm{\Delta }}{\lambda }_{s}$(nm)	${\lambda }_{i}$ (nm)	${\rm{\Delta }}{\lambda }_{i}$(nm)
${{LP}}_{11}$	816.1	−3.9	533.7	1.7
${{LP}}_{21}$	811.1	−8.9	535.8	3.8
${{LP}}_{02}$	809.7	−10.3	536.4	4.4

We point out that while we have verified that the conclusions reached in this paper with respect to factorability and ultra-narrowband single photon generation are unaffected by the use of the intermodal CP-SFWM process described above for signal/idler-pumps discrimination, for simplicity in the rest of the paper we concentrate on a CP-SFWM process which utilizes a single transverse mode.

In this section we show that under certain approximations it becomes possible to derive analytical expressions, in closed form, for both the JSA and for the emitted flux. Specifically, these approximations involve: (i) writing the propagation constant $k(\omega )$, for each of the four interacting fields, as a first-order Taylor expansion around the frequencies for which perfect phasematching is obtained, and (ii) assuming that the function $h({\omega }_{s},{\omega }_{i})$ (see equation (20)) varies slowly within the spectral range of interest, so that we can regard it as a constant when evaluating the integrals in equations (19) and (22) in the section 2.2. Note that these approximations are no longer valid for large spectral spreads of the signal and idler frequencies around the frequencies which yield perfect phasematching.

For the pulsed pumps case (assumed to be Gaussian in spectrum, see equation (18)), under the approximations mentioned above, and using the integral form of the ${\rm{\text{sin}}}{\rm{c}}$ function

$$\begin{eqnarray}&&{\rm{sinc}}(x)=\displaystyle \frac{1}{2}{\int }_{-1}^{1}{\rm{d}}\xi {{\rm{e}}}^{{\rm{i}}{x}\xi },\end{eqnarray} \tag{ 23 }$$

the integral in equation (8) can be carried out analytically resulting in the approximate expression for the JSA ${f}_{{\text{}}P}^{{\rm{lin}}}({\nu }_{s},{\nu }_{i})={\alpha }_{{\text{}}P}({\nu }_{s},{\nu }_{i}){\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$, where ${\alpha }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ is determined by the two pump waves and is given by

$$\begin{eqnarray}&&{\alpha }_{{\text{}}P}({\nu }_{s},{\nu }_{i})={\rm{\exp }}\left[-\displaystyle \frac{{({\nu }_{s}+{\nu }_{i})}^{2}}{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}\right],\end{eqnarray} \tag{ 24 }$$

while ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ is determined both by the pump waves and the properties of the fiber, and has the form

$$\begin{eqnarray}&&{\phi }_{{\text{}}P}(x)={\rm{\exp }}[-{B}^{2}{x}^{2}]\left[{\rm{erf}}\left(\displaystyle \frac{1+{\rm{\Lambda }}}{4B}+{\rm{i}}{B}{x}\right)+{\rm{erf}}\left(\displaystyle \frac{1-{\rm{\Lambda }}}{4B}-{\rm{i}}{B}{x}\right)\right],\end{eqnarray} \tag{ 25 }$$

given in terms of the variable x, and parameters B and Λ, defined as

$$\begin{eqnarray}&&x={T}_{s}{\nu }_{s}+{T}_{i}{\nu }_{i},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&B=\displaystyle \frac{\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}}{{t}_{12}{\sigma }_{1}{\sigma }_{2}},\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{\rm{\Lambda }}=\displaystyle \frac{1}{{t}_{12}}(2\tau +{\tau }_{12}),\end{eqnarray} \tag{ 28 }$$

where ${\rm{erf(.)}}$ denotes the error function, ${\nu }_{\mu }={\omega }_{\mu }-{\omega }_{\mu }^{0}$ are detuning variables ($\mu =s,i)$, and T_s, T_i, t₁₂, and ${\tau }_{12}$ are defined in table 2; τ was defined in the context of equation (8). Note that here ${\omega }_{\mu }^{0}$ (with $\mu =1,2,s,i$) represent the central frequencies of the four waves involved . The definitions provided in tables 2 correspond to temporal variables; ${\tau }_{{ij}}$ terms represent transit time differences through the fiber between waves i and j, while t_ij terms represent transit time sums through the fiber between waves i and j. In conventional fibers, with a length of a few cm, t_ij is on the order of tenths of nanoseconds, while ${\tau }_{{ij}}$ is on the order of few picoseconds.

Table 2.  Temporal parameters in analytical expressions for two cases. Left: general case, for which the four waves could involve different polarizations and propagation modes, right: identical polarizations and propagation modes for all four waves; note that the definition ${k}_{\mu }^{\prime }\equiv {k}_{\mu }^{\prime }({\omega }_{\mu }^{0})$ is used throughout.

${T}_{s}={t}_{2s}-\tfrac{{\sigma }_{1}^{2}}{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}{t}_{12}$	${T}_{i}={\tau }_{2i}-\tfrac{{\sigma }_{1}^{2}}{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}{t}_{12}$	${T}_{s}=\tfrac{{\sigma }_{2}^{2}}{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}{t}_{12}$	${T}_{i}=-\tfrac{{\sigma }_{1}^{2}}{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}{t}_{12}$
${t}_{12}=L({k}_{1}^{\prime }+{k}_{2}^{\prime })$	${\tau }_{12}=L({k}_{1}^{\prime }-{k}_{2}^{\prime })$	${t}_{12}=L({k}_{1}^{\prime }-{k}_{2}^{\prime })$	${\tau }_{12}=L({k}_{1}^{\prime }-{k}_{2}^{\prime })$
${t}_{1s}=L({k}_{1}^{\prime }+{k}_{s}^{\prime })$	${\tau }_{1s}=L({k}_{1}^{\prime }-{k}_{s}^{\prime })$	${t}_{1s}=2{{Lk}}_{1}^{\prime }$	${\tau }_{1s}=0$
${t}_{1i}=L({k}_{1}^{\prime }+{k}_{i}^{\prime })$	${\tau }_{1i}=L({k}_{1}^{\prime }-{k}_{i}^{\prime })$	${t}_{1i}={t}_{12}$	${\tau }_{1i}={\tau }_{12}$
${t}_{2s}=L({k}_{2}^{\prime }+{k}_{s}^{\prime })$	${\tau }_{2s}=L({k}_{2}^{\prime }-{k}_{s}^{\prime })$	${t}_{2s}={t}_{12}$	${\tau }_{2s}=-{\tau }_{12}$
${t}_{2i}=L({k}_{2}^{\prime }+{k}_{i}^{\prime })$	${\tau }_{2i}=L({k}_{2}^{\prime }-{k}_{i}^{\prime })$	${t}_{2i}=2{{Lk}}_{2}^{\prime }$	${\tau }_{2i}=0$

 In the left-hand side of table 2 we have shown the various temporal parameters which define the two-photon state in the general case where the four waves may involve different polarizations and transverse modes. In the right-hand side table, we have specialized this to the case for which all four waves involve the same polarization and the same transverse mode.

Note that for a sufficiently large time of arrival difference between the two pump pulses at the opposite fiber ends τ, leading to the condition ${\rm{\Lambda }}\gg 1$, the pump pulses overlap temporally outside the fiber and the process ceases to occur. If this time of arrival difference τ vanishes, the value of Λ tends to be small since it is given by the ratio of the transit time difference through the fiber of the two pumps, divided by transit time sum; thus, often we may approximate ${\rm{\Lambda }}\approx 0$.

Note that the assumptions (see first paragraph of this section) used for the derivation of the approximate expression for the JSA ${f}_{{\text{}}P}^{{\rm{lin}}}({\nu }_{s},{\nu }_{i})$ are no longer valid for sufficiently large signal and idler spectral spreads around the central SFWM frequencies ${\omega }_{s}^{0}$ and ${\omega }_{i}^{0}$. It is worth pointing out that for the specific source designs presented in this paper (see figures 3 and 5, below) these approximations are well justified: plots of the joint spectrum derived from the expression $| {f}_{{\text{}}P}^{{\rm{lin}}}({\nu }_{s},{\nu }_{i}){| }^{2}$ are in excellent agreement with plots derived from direct numerical integration, without resorting to approximations, according to equation (8).

The same assumptions considered in the derivation of ${f}_{{\text{}}P}^{{\rm{lin}}}({\nu }_{s},{\nu }_{i})$ can be applied in equation (19) in order to obtain a closed analytical expression of the emission rate, which leads to

$$\begin{eqnarray}&&{N}_{{\text{}}P}^{{\rm{lin}}}=\displaystyle \frac{{2}^{5}{n}_{1}{n}_{2}{c}^{2}{\gamma }^{2}{p}_{1}{p}_{2}h({\omega }_{s}^{0},{\omega }_{i}^{0})}{R({k}_{1}^{\prime }+{k}_{2}^{\prime })({k}_{s}^{\prime }+{k}_{i}^{\prime }){\omega }_{1}^{0}{\omega }_{2}^{0}}\left[{\rm{erf}}\left(\displaystyle \frac{1+{\rm{\Lambda }}}{2\sqrt{2}B}\right)+{\rm{erf}}\left(\displaystyle \frac{1-{\rm{\Lambda }}}{2\sqrt{2}B}\right)\right].\end{eqnarray} \tag{ 29 }$$

From the above equation, using the property that the ${\rm{erf}}(x)$ function saturates to a value of 1 for $x\gtrsim 2$ (or to a value of −1 for $x\lesssim -2$), we may show that there exists an effective fiber length L_eff given by

$$\begin{eqnarray}&&{L}_{{\rm{eff}}}=\displaystyle \frac{4\sqrt{2}\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}}{(1+{\rm{\Lambda }})({k}_{1}^{\prime }+{k}_{2}^{\prime }){\sigma }_{1}{\sigma }_{2}}=\displaystyle \frac{4\sqrt{2}\sqrt{{\rm{\Delta }}{t}_{1}^{2}+{\rm{\Delta }}{t}_{2}^{2}}}{(1+{\rm{\Lambda }})({k}_{1}^{\prime }+{k}_{2}^{\prime })},\end{eqnarray} \tag{ 30 }$$

where ${\rm{\Delta }}{t}_{1}\equiv 1/{\sigma }_{1}$ and ${\rm{\Delta }}{t}_{2}\equiv 1/{\sigma }_{2}$ represent the temporal durations for pump 1 and pump 2, respectively, with the property that increasing the fiber length beyond $L={L}_{{\rm{eff}}}$ does not lead to any further increase of the source brightness; thus, L_eff corresponds to the maximum interaction length. Physically, L_eff represents the length of fiber over which the two pumps overlap temporally. Note that making one of the two pumps approach the continuous wave (monochromatic) limit implies that the interaction length can increase without limit, which as is described below is helpful for the optimization of the source brightness.

Let us consider a case where both pumps have a non-zero bandwidth; we can then write $B=\sqrt{1+r}/({\sigma }_{1}{t}_{12})$, or $B=\sqrt{1+r}{\rm{\Delta }}{t}_{1}/{t}_{12}$, (with $r\equiv {\sigma }_{1}^{2}/{\sigma }_{2}^{2}$). Thus, if ${\rm{\Delta }}{t}_{1}$ is much smaller than the sum of the transit times through the fiber of the two pump pulses, represented by t₁₂, then B can be a small number. Essentially, a small B implies that since the interaction length is much shorter than the fiber length, the two-photon state is free from any effects related to the air-fiber and fiber-air interfaces. In this $B\to 0$ limit, which can always be reached through a combination of pulsed pumps with a sufficiently small pump duration together with a sufficiently long fiber, the phasematching function becomes ${\phi }_{{\text{}}P}(x)\to 2{\rm{\exp }}(-{B}^{2}{x}^{2})$. This limit is interesting for applications where the suppression of the sinc-function sidelobes is beneficial, as is the case for the generation of very high-quality factorable states.

Let us now consider the case where the pump bandwidths are highly unbalanced. In particular, the condition ${\sigma }_{1}\ll {\sigma }_{2}$ leads to $B\approx 1/({t}_{12}{\sigma }_{1})$, while similarly ${\sigma }_{2}\ll {\sigma }_{1}$ leads to $B\approx 1/({t}_{12}{\sigma }_{2})$. In the limit where the smaller of the two bandwidths becomes very small, the value of B becomes very large, in which case it may be shown that the phasematching function becomes ${\phi }_{{\text{}}P}(x)\to {\rm{sinc}}(x/2)$. In practice, for values $B\gtrsim 1.0$, the phasematching function is already well described by a sinc function; in this regime, unlike for the small B limit, the fiber edges play an essential role.

The behavior of the function $| {\phi }_{{\text{}}P}(x)|$ as parameter B varies is summarized in figure 2. On the one hand, panel (a) illustrates the small B limit (in this case with $B=0.01$), in which the function $| {\phi }_{{\text{}}P}(x)|$ becomes the Gaussian function ${\rm{\exp }}(-{B}^{2}{x}^{2})$. On the other hand, panel (c) illustrates the large B limit (in this case with B = 1) in which the function $| {\phi }_{{\text{}}P}(x)|$ becomes ${\rm{sinc}}(x/2)$. In both of these panels, the $| {\phi }_{{\text{}}P}(x)|$ function as given by equation (25) is plotted with a solid black line, while the Gaussian or sinc limiting behaviors are plotted with a dashed yellow line. It becomes evident that there is an excellent agreement between these. In panel (b) we show an intermediate case with $B=0.2$ for which $| {\phi }_{{\text{}}P}(x)|$ is not well described neither by a Gaussian nor by a sinc function.

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Let us now analyze how the functions ${\alpha }_{P}({\nu }_{s},{\nu }_{i})$ and ${\phi }_{P}({\nu }_{s},{\nu }_{i})$ define the joint spectral intensity of CP-SFWM photon pairs. It is clear from equations (24) and (25) that while ${\alpha }_{P}({\nu }_{s},{\nu }_{i})$ is oriented at $-45^\circ$ in $\{{\omega }_{s},{\omega }_{i}\}$ space with a width $\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}$, the orientation and width of ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ depend, both, on pump and fiber parameters. The orientation angle of the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ is given by

$$\begin{eqnarray}&&{\theta }_{{si}}=\arctan \left(-\displaystyle \frac{{T}_{s}}{{T}_{i}}\right)=\arctan \left(\displaystyle \frac{{\sigma }_{2}^{2}}{{\sigma }_{1}^{2}}\right),\end{eqnarray} \tag{ 31 }$$

with the last equality valid for the case where all four waves have the same polarization and transverse mode. Note that ${\sigma }_{2}^{2}/{\sigma }_{1}^{2}\geqslant 0$, so that ${\theta }_{{si}}$ is constrained as $0\leqslant {\theta }_{{si}}\leqslant 90^\circ$, i.e. the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ has contour curves with non-negative slope, including the two limiting cases of horizontal and vertical orientations. As regards the width of the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$, it is helpful to consider separately the limiting cases for large B where this function is well described by ${\rm{sinc}}(x/2)$, and for small B for which this function becomes $\exp (-{B}^{2}{x}^{2})$. In the first case, the width is inversely proportional to the fiber length L (since both T_s and T_i in $x={T}_{s}{\nu }_{s}+{T}_{i}{\nu }_{i}$ are linear in L). In the second case, the width no longer depends on L (since B is proportional to ${L}^{-1}$ while x is proportional to L). Thus, as the fiber length is increased the width of the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ diminishes, eventually the shape turning Gaussian at which point the width and shape of the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ no longer responds to further increasing L. Thus, increasing L beyond the length defined by non-zero temporal overlap between the two pump pulses, L_eff, has no effect neither on the flux nor on the joint spectral intensity.

The discussion of the previous paragraph is illustrated in figure 3, in which we show the two-photon state obtained for a step-index fiber (with core radius $r=1.5\,\mu {\rm{m}}$ and numerical aperture ${\rm{NA}}=0.13$) with length L = 1 cm and two different pulsed pumps configurations: (i) ${\sigma }_{1}=0.01\,{\rm{THz}}$ and ${\sigma }_{2}=0.03{\rm{THz}}$, panels (a)–(d), for which $B=1.07;$ (ii) ${\sigma }_{1}={\sigma }_{2}=0.01$ THz, panels (e)–(h), for which $B=1.43$. In this block of figures, the function ${\alpha }_{P}({\nu }_{s},{\nu }_{i})$ is shown in the first column, the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ in the second column, the JSI $| {\alpha }_{P}({\nu }_{s},{\nu }_{i}){\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i}){| }^{2}$ in the third column, while the numerically calculated JSI is shown in the fourth column. Note that while the third column corresponds to the analytical joint spectral intensity, defined as $| {f}_{{\text{}}P}^{{\rm{lin}}}({\nu }_{s},{\nu }_{i}){| }^{2}$, the fourth column was obtained by numerical integration of equation (8) without resorting to the linear approximation of the ${\rm{\Delta }}k$ function. It is evident that the approximate analytical results agree extremely well with the numerically calculated ones. Also, consistent with equation (31), while the orientation of the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ is 45^◦ for equal pump bandwidths, it approaches a vertical orientation for unequal pump banwidths ${\sigma }_{2}\gt {\sigma }_{1}$, and becomes fully vertical for ${\sigma }_{2}\gg {\sigma }_{1}$. Similarly, (not shown in the figure), for ${\sigma }_{2}\lt {\sigma }_{1}$ the function ${\phi }_{{\text{}}P}({\nu }_{s},{\nu }_{i})$ approaches a horizontal orientation while it becomes fully horizontal for ${\sigma }_{2}\ll {\sigma }_{1}$.

For highly unbalanced pump bandwidths, and in particular when one of the two pumps approaches the monochromatic limit, the interaction length between the two pump pulses increases, in principle, without limit. Such a mixed pumps configuration could have important implications for the ability to reach high emission rates.

Following a similar treatment as used above for the pulsed pumps case, it can be shown that the JSA function given in equation (11) can be expressed, under the linear ${\rm{\Delta }}k$ approximation, as ${f}_{{\text{}}M}^{{\rm{lin}}}({\nu }_{s},{\nu }_{i})={\alpha }_{{\text{}}M}({\nu }_{s},{\nu }_{i}){\phi }_{{\text{}}M}({\nu }_{s},{\nu }_{i})$, with the pump envelope function ${\alpha }_{{\text{}}M}({\nu }_{s},{\nu }_{i})$ and the phasematching function ${\phi }_{{\text{}}M}({\nu }_{s},{\nu }_{i})$ given by

$$\begin{eqnarray}&&{\alpha }_{{\text{}}M}({\nu }_{s},{\nu }_{i})={\rm{\exp }}\left[-\displaystyle \frac{{({\nu }_{s}+{\nu }_{i})}^{2}}{{\sigma }^{2}}\right],\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&{\phi }_{{\text{}}M}({\nu }_{s},{\nu }_{{i}})={\rm{sinc}}\left[\displaystyle \frac{1}{2}({\tau }_{1s}{\nu }_{s}+{t}_{1{i}}{\nu }_{{i}})\right]{\rm{\exp }}[{{\rm{i}}{t}}_{1s}{\nu }_{s}+{{\rm{i}}{t}}_{1{i}}{\nu }_{{i}}],\end{eqnarray} \tag{ 33 }$$

where σ is the bandwidth of the pulsed pump and ${\tau }_{1s}$, ${t}_{1s}$, and ${t}_{1i}$ are defined in table 2. Likewise, it can be demonstrated by integration of equation (22), and under the linear phasemismatch approximation, that the emitted flux can be expressed as

$$\begin{eqnarray}&&{N}_{{\text{}}M}^{{\rm{lin}}}=\displaystyle \frac{{2}^{6}{n}_{1}{n}_{2}{c}^{2}{\gamma }^{2}{p}_{1}{p}_{2}{Lh}({\omega }_{s}^{0},{\omega }_{i}^{0})}{{\omega }_{1}^{0}{\omega }_{2}^{0}| {k}_{s}^{\prime }+{k}_{i}^{\prime }| },\end{eqnarray} \tag{ 34 }$$

where the dependence on the various experimental parameters of the emission rate appears explicitly. Particularly, it can be seen, as expected, that ${N}_{{\text{}}M}$ increases linearly with the fiber length, indicating that the interaction length is not capped as it is for the pulsed pumps configuration.

In figures 3(i)–(l) we illustrate the two-photon state obtained for a CP-SFWM source in the mixed pumps configuration, for which we have assumed the same parameters as in figures 3(e)–(h), except that pump 2 is now monochromatic (${\sigma }_{2}\to 0$); in this case the contours of the phasematching function become horizontal. Note that for the mixed pumps configuration the dependence of the spectral envelope function and the phasematching function on the parameters of the source become decoupled; i.e., the width of ${\alpha }_{{\text{}}M}({\nu }_{s},{\nu }_{i})$ is proportional to the pulsed pump bandwidth (with an orientation at $-45^\circ$), while the width of ${\phi }_{{\text{}}M}({\nu }_{s},{\nu }_{i})$ depends on fiber properties, including its length L and dispersion (with a horizontal orientation) .

In figure 4 we illustrate the behavior of the source brightness as a function of the fiber length, for both the pulsed and mixed pumps cases. In panels (a)–(c) we show this behavior for three different values of ${\sigma }_{2}$ (1 THz, 0.05 THz, and 0.005 THz, respectively), and for a fixed value ${\sigma }_{1}=1$ THz, where a vertical dashed line indicates the effective length L_eff; it may be appreciated that the brightness reaches a plateau at $L\approx {L}_{{\rm{eff}}}$. In panel (d) we show the corresponding behavior for the mixed pumps scheme, for $\sigma =1$ THz; note that in this case the brightness grows linearly with L without saturating to a fixed value.

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