# Deriving Expressions for R1, R2, and R12 , Normalized Coincident Detection Probability, and LG Mode Formalism

I am currently reviewing a research paper on orbital angular momentum entanglement between two photons (signal and idler) generated from a pump photon via SPDC. The paper introduces the following expressions:

1. The normalized coincident detection probability is given by:

$$P_{N}(\Phi_{1}, \Phi_{2}) = \frac{\left|\int d x \Phi_{1}^{\star}(x) \Phi_{2}^{\star}(x) \Phi_{0}(x)\right|^{2}}{\sqrt{\left|\int d x \Phi_{1}^{\star}(x) \Phi_{0}(x)\right|^{2}\left|\int d x \Phi_{2}^{\star}(x) \Phi_{0}(x)\right|^{2}}}$$

2. The Laguerre-Gaussian (LG) mode function is defined as:

$$\Phi_{p,l}(r, \varphi) = \sqrt{\frac{2p!}{\pi(|\ell| + p)!}} \sqrt{\frac{1}{w}} \left(\frac{\sqrt{2} r}{\omega(z)}\right)^{\ell} L_{p}^{\ell}\left(\frac{2 r^{2}}{\omega^{2}(z)}\right) \exp \left(-\frac{r^{2}}{\omega^{2}(z)}\right) \times \exp \left(-i \ell \varphi \right)$$

Now, I am trying to derive expressions for the following terms;

1. $$R_{1} = \frac{\left(p_{1} + \left|\ell_{1}\right|\right)!}{p_{1}!} \times \int_{0}^{\infty} d r \, r^{\left(|\ell_{1}| + |\ell_{0}|\right)} e^{-r\left(1 + W^2\right)}\left[L_{p_{1}}^{\left|\ell_{1}\right|}(r)\right]^2\left[L_{p_{0}}^{\left|\ell_{0}\right|}(r W)\right]^2$$

2. $$R_{2} = \frac{\left(p_{2} + \left|\ell_{2}\right|\right)!}{p_{2}!} \times \int_{0}^{\infty} d r \, r^{\left(|\ell_{2}| + |\ell_{0}|\right)} e^{-r\left(1 + W^2\right)}\left[L_{p_{2}}^{\left|\ell_{2}\right|}(r)\right]^2\left[L_{p_{0}}^{\left|\ell_{0}\right|}(r W)\right]^2$$

3. $$R_{12} = \int_{0}^{\infty} d r \, r^{\left(|\ell_{0}| + |\ell_{1}| + |\ell_{2}|\right)/2} e^{-r\left(1 + W^2/2\right)} \times L_{p_{1}}^{\left|\ell_{1}\right|}(r) L_{p_{2}}^{\left|\ell_{2}\right|}(r) L_{p_{0}}^{\left|\ell_{0}\right|}\left(r W^2\right)$$

4. $$P_{N}(\Phi_{1}, \Phi_{2}) = \text{Sinc}^2\left[(\ell_{1} + \ell_{2} - \ell_{0}) \pi\right] \frac{\left|R_{12}\right|^2}{\sqrt{R_{1} R_{2}}}$$

I am seeking assistance in efficiently implementing these calculations using Mathematica. Any guidance, code snippets, or recommendations for a more streamlined approach would be greatly appreciated. Thank you!

• Edit your question to include the code for these expressions (InputForm). If you are having a specific problem with an expression, show what you have tried and explain what specifically you are having a problem with. Commented Dec 20, 2023 at 3:27