## 1 Introduction  
In recent years, optomechanical systems have recently been the focus of extensive investigations due to their potential applications in precision measurement of tiny displacements and forces [1]. Such systems enable us to observe quantum behaviors at the macroscale if the mechanical oscillators can be cooled close to the quantum ground state [2; 3; 4; 5]. It has been shown that the radiation pressure acting on the mechanical oscillator exerted by the photons in the cavity can lead to the cooling of the mechanical oscillator [6; 7; 8; 9]. It has been demonstrated theoretically [10; 11] and experimentally [12; 13; 14; 15; 16; 17] that the mechanical oscillator can be cooled close to its quantum ground state in the resolved sideband limit, in which the mechanical frequency is much larger than the cavity decay rate. Moreover, the feedback control has been introduced to the optomechanical systems to effectively cool the mechanical oscillator below the environmental temperature in the unresolved sideband regime [18; 19; 20; 21; 22; 23; 24]. The feedback cooling is based on applying a viscous feedback force on the mechanical oscillator to hinder its thermal motion.  
On the other hand, the normal mode splitting in the mechanical and optical fluctuation spectra is a signature of the strong coupling between the cavity field and the mechanical oscillator [25]. The strong coupling in the optomechanical system would allow for the preparation of mechanical quantum states [2; 3; 4]. Many efforts have been devoted to studying the normal mode splitting of the optomechanical systems theoretically [25; 26; 27; 28; 29] and experimentally [30]. It has been investigated that a degenerate optical parametric amplifier in an optomechanical system can produce stronger coupling between the cavity field and the movable mirror so that larger normal mode splitting is observed [26]. Besides, it has been found that the strong coupling can be achieved between two indirectly coupled mechanical and optical modes in an optomechanical system with one mechanical mode and two optical modes [27]. In addition, the two-level atoms embedded into the optomechanical cavity is helpful to generate the normal mode splitting [28]. Moreover, the normal mode splitting can be observed in a weakly coupled electromechanical system by modulating the spring constant of the mechanical oscillator [29].  
In this paper, we propose a coherent feedback scheme to improve the cooling of a movable mirror in the resolved sideband limit. The feedback scheme is realized by reflecting a part of the output field from the cavity back into the cavity via an asymmetric beam splitter. Moreover, we find that the feedback scheme can lead to the occurrence of the normal mode splitting in the spectra of the movable mirror and the output field when the movable mirror and the cavity field are initially weakly coupled to each other, owing to a decrease of the effective cavity decay rate in the presence of the feedback.  
The outline of the paper is as follows. In Section 2, we present the Hamiltonian for the system, give the equations of motion of the system operators, calculate the expectation values of the system operators, obtain the linearized quantum Langevin equations. In Section 3, we analyze how the cooling of the moving mirror can be enhanced in the presence of the feedback. In Section 4, we discuss the normal mode splitting in the mechanical and output optical fluctuation spectra with the feedback.