## 7 Joint Spectrum  
The joint spectral amplitude, \(F(\omega_{s},\omega_{i})\), is an important physical quantity to show the spectral property and directly influence on the amount of entanglement in a photon-pair. Now, it is convenient to apply the integration transformation, \(\Omega=\upsilon_{p}-\frac{\upsilon_{p}+\upsilon_{i}}{2}\), in Equation (23). \(\upsilon_{j}\) (with \(j=p,s,i\)) represents the frequency detuned from the center frequency, \(\tilde{\omega}_{j}\). From the phase-matching condition \(\Delta k\), each \(k_{j}(\omega_{j})\) (with \(j=p,s,i\)) can be extended into a Taylor series around the center frequency which is given by \(\Delta k=k_{p_{1}}\left(\tilde{\omega}_{p_{1}}+\frac{\upsilon_{p}+\upsilon_{i }}{2}+\Omega\right)+k_{p_{2}}\left(\tilde{\omega}_{p_{2}}+\frac{\upsilon_{p}+ \upsilon_{i}}{2}-\Omega\right)-k_{s}\left(\tilde{\omega}_{s}+\upsilon_{s} \right)-k_{i}\left(\tilde{\omega}_{i}+\upsilon_{i}\right)\). For the degenerate pump beams, \(\omega_{p}=\omega_{p_{1}}=\omega_{p_{2}}=\tilde{\omega}_{p}+\upsilon_{p}\). As a result, the joint spectrum of the generated photon pair can be given as  
\[ \begin{split} F(\omega_{s},\omega_{i})&=B\int\int d \omega_{p_{1}}d\omega_{p_{2}}\text{e}^{-\left(\frac{\upsilon_{p_{1}}-\tilde{ \omega}_{p_{1}}}{\sigma_{p_{1}}}\right)^{2}}\text{e}^{-\left(\frac{\upsilon_{p_{ 2}}-\tilde{\omega}_{p_{2}}}{\sigma_{p_{2}}}\right)^{2}}\text{e}^{-i\frac{ \Delta kL}{2}}\text{Sinc}\left(\frac{\Delta kL}{2}\right),\\ F(\upsilon_{s},\upsilon_{i})&=C\text{e}^{-\left( \frac{\upsilon_{p}+\upsilon_{i}}{2}\right)^{2}}\text{Sinc}\left[\frac{\tau_{s} \upsilon_{s}+\tau_{i}\upsilon_{i}}{2}\right]\text{e}^{-i\frac{\tau_{s}\upsilon _{s}+\tau_{i}\upsilon_{i}}{2}},\\ &=C\text{e}^{-i\frac{\tau_{s}\upsilon_{s}+\tau_{i}\upsilon_{i}}{2}} \text{ }\alpha(\upsilon_{s},\upsilon_{i})\Phi(\upsilon_{s},\upsilon_{i}),\end{split}  \tag{24}\]  
where \(B\) and \(C\) are normalized constants which do not contribute to the joint spectrum property, \(\sigma_{p_{1,2}}\) are the bandwidths of pump beams, and \(\tau_{j}=\left(\frac{1}{\upsilon_{p}}-\frac{1}{\upsilon_{j}}\right)L\) (with \(j=s,i\)) represents the group delay between the pump and signal (\(j=s\)) or idler (\(j=i\)). \(\upsilon\) indicates the group velocity. If the group velocity of pump photon is the same as that of signal photon, both exist in the medium at the same time such that \(\tau_{s}=0\). If they have different group velocities, they gradually become temporally separated in the fiber which is called temporal walk-off. \(\alpha(v_{s},v_{i})\) is the pump envelope function which is governed by the energy conservation and depends on the spectral envelope of the pump beam. \(\Phi(v_{s},v_{i})\) is called the phase-matching function which accounts for the momentum conservation and depends on the dispersion property in the fiber. For example, Figure 15 represents a JSA whose formula is given in Equation (24). Here, the pump beam centered at 725 nm and a 12-cm long MNF with 615 nm-diameter are used for numerical calculation. For the degenerate pump beams at 725 nm, temporal walk-off between pump beams is negligible in which they are overlapped for the length of the fiber. In Ref. [43], the signal and idler photons are generated at 632 nm and 850 nm, respectively. There are central and side lobes which are related to the wings of the sinc function. Due to relatively narrow bandwidth of pump beam (\(\Delta\lambda_{p}=1\) nm) and long fiber length, bandwidths of signal and idler photons are about 2 nm and 5 nm, respectively. Another example is shown in Figure 16. For the pump beam centered at \(\lambda_{p}=1064\) nm and \(\Delta\lambda_{p}=3\) nm bandwidth, signal and idler are generated at 800 nm and 1.5 \(\upmu\)m, by using a 5 cm-long and 950 nm-diameter MNF. As a result, relatively broader bandwidth of pump beam and shorter fiber length than the previous case, lead signal and idler photons to have broad bandwidths about 5 nm and 15 nm.