## 3 Theory and Discussions  
The purpose of quantum lithography is to increase the resolution of the image by using entangled photon pairs. The experiment reported in the above section is a proof-of-principle quantum lithography experiment with none-degenerate photon pairs. The connection between the experiment and the resolution enhancement is based on the Abbe theory of image formation [11]. As known from Abbe, only a certain portion of the diffracted components generated by a object are intercepted by the aperture \(L\) of an imaging device in a diffraction-limited imaging system. That is the reason why a diffraction-limited imaging system has finite resolution. In quantum lithography, entangled photon pairs are used to illuminate the object, and the interference-diffraction patterns of the object will be narrower compared to that of classical light. In our non-degenerate quantum lithography experiment, we show that the width of the diffraction pattern is narrower than that of classical light (shown in Figure 2), which means that there are more higher diffraction orders collected. The image from the diffraction pattern via Fourier transform will have a higher resolution. The measurements of Figure 2a,b can be easily understood in [PERSON]'s picture [12]. For example, in the measurement of Figure 2a the fiber tip 1 could be treated as a light source emitting photons at 633 nm backward to the BBO crystal. The crystal functions as a non-linear mirror, which turns the 633-nm photons into 788-nm photons and reflects them to the object. The 788-nm photons are diffracted by the single slit and pass through the Fourier lens to form the diffraction pattern in the detection planes. We should emphasize that the diffraction pattern only exists in the second-order correlation regime, and the single photon counting distributions on both \(X_{1}\) and \(X_{2}\) planes are nearly flat. However, [PERSON]'s picture cannot explain the measurement shown in Figure 2c. To understand the measurement qualitatively, we need to use the concept of the diffraction of biphotons: the non-degenerate biphoton, comprising daughter photons with different wavelengths, is one object due to their internal binding (entanglement). It is diffracted by the single slit as a particle with bigger momentum than its constituent parts (signal and idler photons), that is the reason why the FWHM of the diffraction pattern in the third measurement is narrower than those in the first two measurements. To interpret the experiment quantitatively and to find out why the de Broglie wavelength of non-degenerate biphotons is the same as the one of the pump, we need to calculate the second order correlation functions measured in our experiment. According to quantum optical theory, the second order correlation function is:  
\[G_{12}^{(2)}=tr[\hat{\rho}E_{1}^{(-)}E_{2}^{(-)}E_{2}^{(+)}E_{1}^{(+)}], \tag{1}\]  
where \(\hat{\rho}\) describes the quantum-mechanical state of the electromagnetic field on the output face of the crystal, for \(j=1,2\), \(E_{j}^{(+)}=E_{j}^{(+)}(\vec{r}_{j},t_{j})\) is the positive frequency part of the electric field at the point \(\vec{r}_{j}\) evaluated at time \(t_{j}\) and \(E_{j}^{(-)}=(E_{j}^{(+)})^{\dagger}\). In our experiments, the state of entangled light is a pure state, and we assume that the measurements have proceeded at the single photon level. Therefore, the \(G_{12}^{(2)}\) can be written in terms of the effective biphoton probability amplitude [13,14]:  
\[G_{12}^{(2)}=\mid A\mid^{2}=\mid\langle 0\mid E_{2}^{(+)}E_{1}^{(+)}\mid\psi \rangle\mid^{2}, \tag{2}\]  
where \(A\) is the effective two-photon probability amplitude, which describes the joint detection of the biphoton.