# Recent Progress in Multiplexed Single-Photon Sources
## 3 Multiplexed Single-Photon Sources
### Spatial Multiplexing
Though the chosen input port of the previous router to which the output of the next router is connected can be arbitrary, it can be proved that to obtain higher single-photon probabilities, the input port with the smaller loss is the right choice. An important advantage of such multiplexers over symmetric multiplexers is that the number of multiplexed units in the resulting structure is not limited to power-of-two numbers. The transmission coefficients of this type of multiplexer can be written as [54]  
\[V_{n}= \begin{cases}V_{b}V_{\min}V_{\max}^{n-1}&\text{if }n<N,\\ V_{b}V_{\max}^{n-1}&\text{if }n=N,\end{cases}  \tag{1}\]  
where \(V_{\min}=\min(V_{r},V_{l})\), and \(V_{\max}=\max(V_{r},V_{l})\). Recently, novel types of multiplexers called incomplete binary tree multiplexers were proposed [53; 54]. These multiplexers are termed incomplete because, as opposed to complete binary-tree multiplexers, the levels of these multiplexers are not complete in general, although for a certain number of photon routers, these multiplexer types include symmetric multiplexers as a special case. Such multiplexers have the same advantage over symmetric  
Figure 1: (**a**) Schematic figure of asymmetric multiplexer-based SPSs. (**b**) Schematic figure of an example of SOBTM-based SPSs. (**c**) Schematic figure of OMAXV multiplexer-based SPSs. MU\({}_{i}\)s are multiplexed units. PR\({}_{i}\)s are photon routers. \(V_{\max}\) and \(V_{\min}\) are the larger and smaller transmission coefficients, respectively, of the photon routers. multiplexers as asymmetric multiplexers, that is, no doubling of the number of multiplexed units is required to form possible multiplexers. Input-extended incomplete binary tree multiplexers can be assembled as follows. Let us start with a complete binary tree formed by a given number of photon routers. This multiplexer is extended by attaching the novel routers to the next level of the tree one by one from left to right (or vice versa) until the next level becomes complete. The formula describing the transmission coefficient of such multiplexers can be found in Ref. [53]. The building strategy of output-extended incomplete binary-tree multiplexers can also be explained by starting from a complete binary tree multiplexer. In this case, the output of the initial complete binary multiplexer is connected to one of the inputs of a novel photon router termed as the base router. The output of the next router is connected to the free input of the base router, thus starting the incomplete branch of the multiplexer. Subsequent routers are added one by one, always to and only to the next level in the incomplete branch of the multiplexer, until the level is completed. Various extension strategies regarding the choice of the arm to which the next router is to be added were proposed in the literature [53; 54]. The most important type of these multiplexers is the one in which the initially symmetric multiplexer is attached to the input of the base router characterized by the smaller transmission coefficient \(V_{\text{min}}\) (minimum-based), and novel routers are added to the arms of the incomplete branch characterized by the largest transmission coefficient \(V_{n}\) (maximum-logic). It was shown that SPSs based on minimum-based, maximum logic, output-extended, incomplete binary tree multiplexers (OMAXV multiplexers) show better performance than SPSs based on any other output-extended incomplete binary tree multiplexers for a wide range of the parameters characterizing such devices [54]. Hence, in the literature, this type of spatially multiplexed SPS was analyzed and compared to other SPSs. The formula describing the transmission coefficients of SPSs based on OMAXV multiplexers has been stated in Ref. [54]. Figure 1c shows an example of the structure of an SPS based on OMAXV multiplexers. The last type of spatial multiplexers is the stepwise-optimized binary tree multiplexer (SOBTM) [55].