## 3 Theory and Discussions
\(\mid\psi\rangle\) stands for the entangled state,  
\[\mid\psi\rangle=\mid 0\rangle+\sum_{\vec{k}_{s},\vec{k}_{i}}F(\vec{k}_{s},\vec{k }_{i})\hat{a}_{\vec{k}_{s}}^{\dagger}\hat{a}_{\vec{k}_{i}}^{\dagger}\mid 0 \rangle F(\vec{k}_{s},\vec{k}_{i})\propto\delta(\omega_{s}+\omega_{i}-\omega_{p} )\delta_{\vec{q}_{s}+\vec{q}_{i}}\text{sinc}(\frac{\Delta k_{z}L}{2}),\]  
where \(\omega_{s}\), \(\omega_{i}\) and \(\omega_{p}\) are the angular frequencies of the signal photon, idler photon and laser pump, respectively. \(\vec{q}_{s}\) and \(\vec{q}_{i}\) stand for the transverse wave vectors of signal and idler photons, respectively. \(\Delta k_{z}\) is the phase mismatch in the direction of the pump, which equals \(k_{p}-k_{sz}-k_{iz}\), and \(L\) is the length of the nonlinear crystal. Since there is a polarizing beam splitter (PBS) in the alignment, all the signal photons go to detector \(D_{1}\), and all the idler photons go to detector \(D_{2}\). The field operators at the detectors can be described in terms of annihilation operators of signal and idler photons at the surface of the nonlinear crystal, for example:  
\[E_{1}^{(+)}(x_{1},z_{1},t_{1})=\sum_{\vec{k}_{s}}g_{s1}(\vec{k}_{s},x_{1},z_{1 },t_{1})\hat{a}_{s}(\vec{k}_{s}), \tag{3}\]  
where \(\hat{a}_{s}(\vec{k}_{s})\) is the annihilation operator of a mode of the signal light field at the surface of the BBO crystal. \(g_{s1}(\vec{k}_{s},x_{1},z_{1},t_{1})\) is the optical propagation function, which describes the propagation of the signal light from the surface of the crystal to detector \(D_{1}\). The details of the calculation can be found in [13; 15]. Combining Equations (2) and (3), the spatial part of the measured second order correlation function is:  
\[(G_{12}^{(2)})_{spatial}\propto\mid\sum_{\vec{q}_{s}\vec{q}_{i}}g_{s1}(\vec{ q}_{s},x_{1},z_{1})g_{i2}(\vec{q}_{i},x_{2},z_{2})\delta_{\vec{q}_{s}+\vec{q}_{i}} \mid^{2}, \tag{4}\]  
where \(g_{i2}(\vec{k}_{i},x_{2},z_{2})\) is the optical transfer function describing the propagation of the idler photon from the surface of the crystal to detector \(D_{2}\) and \(z_{1}=z_{2}=z\) in our experiment. It is straight forward to show that the measured \((G_{12}^{(2)})_{spatial}\) is the Fourier transform of the aperture function [9; 13]:  
\[(G_{12}^{(2)})_{spatial}\propto\mid\mathcal{F}\{T(x_{s})\}\mid_{(k_{s}x_{1}+k _{i}x_{2})/z}\mid^{2}, \tag{5}\]  
where \(T(x_{s})\) is the aperture function of the object. The experimental results can be easily understood from Equation (5). First of all, the measured \(G^{(2)}\) function can be interpreted as the result of the coherent superposition of two-photon probability amplitudes. In the measurement of Figure 2a, \(D_{1}\) is fixed and \(D_{2}\) is scanned, which means the transformation parameter is \(k_{i}x_{2}/z=\frac{2\pi}{\lambda_{i}}\frac{x_{2}}{z}\) where \(\lambda_{i}=788\) nm. In the measurement of Figure 2b, \(D_{2}\) is fixed, \(D_{1}\) is scanned and the transformation parameter is \(k_{s}x_{1}/z=\frac{2\pi}{\lambda_{s}}\frac{x_{1}}{z}\) where \(\lambda_{s}=633\) nm. In the measurement of Figure 2c, both detectors are scanned (\(x_{1}=x_{2}=x\)), and the transformation parameter is \((k_{s}x_{1}+k_{i}x_{2})/z=2\pi(\frac{1}{\lambda_{s}}+\frac{1}{\lambda_{i}}) \frac{x}{z}=\frac{2\pi}{\lambda_{p}}\frac{x}{z}\) where \(\lambda_{p}=351\) nm. From the above calculations, we can see why the ratio between FWHMs of diffraction patterns in three measurements equals (within experimental error) the ratio between the wavelength of signal, idler and pump. We notice that when two detectors are scanned synchronously, the signal and idler photons behave like one particle, even if they have different wavelengths. The momenta of \"two photons\" are coherently added together due to the internal binding between them, and the momentum of the biphoton equals \(p_{s}+p_{i}\). The photonic de Broglie wavelength of the biphoton is then \(\lambda=\frac{h}{p_{s}+p_{i}}\) where \(h\) is [PERSON]'s constant. In the first experiment, we study the diffraction of non-degenerate biphotons.