## 1 Introduction
### Contributions  
In this paper, we consider a downlink optical IRS-assisted photon-counting communication system in which the signals emitted via multiple LEDs are transmitted through NLOS paths, reflected via optical IRS mirrors, and finally received via multiple photodetectors (PDs). Based on the statistical idea, we use the knowledge of expectation and conditional expectation to derive the MSE of the system under Poisson shot noise. Then, we formulate an optimization problem that aims to minimize the MSE of the system while jointly optimizing the IRS configuration and transmitter precoding. We also propose the corresponding alternating optimization algorithm to solve it. The simulation results demonstrate that our proposed scheme outperforms other baseline schemes in terms of MSE and BER performance. The main contributions of this paper are outlined as follows:  
1. First, we consider the IRS model in the very near field and study the MSE optimization problem of a downlink optical IRS-assisted photon-counting communication system. Taking into account the power constraints in optical communication and the configuration constraints of the optical IRS, we transform it into a joint optimization problem of the optical IRS configuration matrix and the precoder.
2. Next, we use statistical knowledge such as conditional probability and conditional expectation to derive the MSE closed-form expression of the IRS-assisted MU photon-counting system. Then we decompose the original problem into two convex subproblems through a series of convexification methods, and we analyze the convergence and complexity of the proposed alternating optimization (AO) algorithm.
3. Finally, we numerically evaluate the IRS-assisted photon-counting communication system. We select several baseline schemes for comparison, including the IRS random allocation scheme and the ZF precoding scheme. The numerical results show that the proposed A0 algorithm effectively reduces the MSE and BER of the system, outperforms other schemes, and exhibits robustness under different turbulence intensities, background radiation, and imperfect CSI.  
The rest of this paper is organized as follows. In Section 2, we introduce the IRS-assisted MU photon counting system and present the framework of the whole system, especially the detailed introduction of channel modeling. In Section 3, we provide a detailed derivation of the IRS-assisted photon-counting system based on the MMSE principle, and we formulate the optimization problem. In Section 4, we introduce an AO algorithm and analyze its convergence and complexity in detail. We then present our simulation results and discuss their implications in Section 5. Finally, we summarize the paper in Section 6.  
Notations: Scalars are denoted by regular letters, while vectors and matrices are represented by bold letters. The symbol \(\mathbb{R}\) represents the set of real numbers, \(\Pr(\cdot)\) denotes the probability, \(\Pr(\cdot|\cdot)\) denotes the conditional probability, \(\mathbb{E}(\cdot)\) and \(\mathbb{E}(\cdot|\cdot)\) stand for expectation and conditional expectation, respectively, \(\text{Var}(\cdot)\) denotes variance, \((\cdot)^{T}\) represents the transpose, and \(\text{mod}(\cdot)\) represents the modulus operator. The Hadamard product, vectorization operator, and floor operator are represented as \(\odot\), \(\text{vec}(\cdot)\), and \(\lfloor\cdot\rfloor\), respectively.