# Recent Progress in Multiplexed Single-Photon Sources
## 4 Theory of Multiplexed Single-Photon Sources
As it is shown in the next section, the single-photon probability function depending on the input mean photon number \(\lambda\) always has a well-defined maximum that can be found by applying standard optimization algorithms such as the gradient method. In the case of different input mean photon numbers \(\lambda_{n}\), other optimization methods such as the genetic algorithm can be used. Such optimization algorithms are available in several mathematical software packages. Next, this method is applied for all reasonable values of the number of multiplexed units \(N\), thus obtaining \(P_{1}-N\) pairs forming a discrete function. In the case of SPSs based on certain types of multiplexers, this discrete function has a single maximum that can be selected easily. Examples of such multiplexers are the symmetric multiplexer and the temporal multiplexer based on binary-divided delays [42; 51]. For SPSs based on other types of multiplexers, the discrete function has a well-defined global maximum, but it can contain local maxima as well. In this case, it is still possible to select the global maximum. Input- or output-extended incomplete binary tree multiplexers are representative examples of such multiplexers [53; 54]. Finally, the discrete function might have no global maximum at all; instead, it tends to a finite value less than one as the number of multiplexed units is increased. This behavior is characteristic of SPSs based on asymmetric and stepwise-optimized binary tree multiplexers [52; 55] and for temporal multiplexers based on storage loop delays [37; 39; 47; 49]. Obviously, when choosing well-defined optimal system parameters in this case is not possible, one of the following two approaches can be applied instead. In the first approach, the single-photon probability \(P_{1}\) is maximized for a sufficiently high number of multiplexed units \(N_{\text{high}}\), leading to \(P_{1,N_{\text{high}}}\), and it is considered to be close to the saturated value. Then, the highest single-photon probabilities \(P_{1,N}\) are determined for lower values of the number of multiplexed units, and they are compared to the highest single-photon probability \(P_{1,N_{\text{high}}}\) found for the high number of multiplexed units. In this case, the lowest value of the number of multiplexed units \(N\) for which the difference \(P_{1,N_{\text{high}}}-P_{1,N}\) is smaller than a predefined value is said to be the optimal number of multiplexed units \(N_{\text{opt}}\), the input mean photon numbers that yield \(P_{1,N}\) are called the optimal input mean photon numbers \(\lambda_{n}\), and \(P_{1,N}\) is called the maximal single-photon probability \(P_{1,\text{max}}\). In the second approach, the highest single-photon probabilities obtained for neighboring values \(N\) and \(N-1\) of the number of multiplexed units are compared. In this case, the lowest value \(N\) for which the difference \(P_{1,N}-P_{1,N-1}\) is smaller than a predefined value is called the optimal number of multiplexed units \(N_{\text{opt}}\), the input mean photon numbers that yield \(P_{1,N}\) are called the optimal input mean photon numbers \(\lambda_{n}\), and \(P_{1,N}\) is called the maximal single-photon probability \(P_{1,\text{max}}\).