Goos-Hänchen Shift in $\mathcal{PT}$-Symmetric and Passive Cavity Optomechanical Systems

Abstract

We theoretically investigate the control of the Goos-Hänchen shift (GHS) of a reflected weak probe field in both parity-time ($\mathcal{PT}$)-symmetric and conventional optomechanical systems. The proposed scheme consists of a single optomechanical platform where a passive optical cavity is coupled to an active mechanical resonator, in contrast to standard passive-passive configurations. Analysis of the eigenfrequency spectrum reveals the emergence of an exceptional point under balanced gain-loss conditions at a tunable effective optomechanical coupling strength. Using the transfer-matrix method combined with stationary-phase analysis, we examine the GHS across broken and unbroken $\mathcal{PT}$ phases and compare it with that in the conventional system. The lateral shift exhibits strong phase dependence: it is markedly enhanced in the unbroken regime relative to both the broken phase and the passive configuration. We further show that the GHS can be actively tuned through the cavity detuning and the intracavity medium length. These results provide a controlled means for manipulating beam shifts in optomechanical systems and suggest pathways toward tunable photonic components and precision optical sensing.

Figures (9)

• Figure 1: Schematic illustration of a $\mathcal{PT}$-symmetric optomechanical system comprising a passive optical cavity ($\hat{a}$, resonance frequency $\omega_{a}$, total decay rate $\kappa = \kappa_i + \kappa_e$) and an active mechanical resonator ($\hat{b}$, resonance frequency $\omega_{b}$, gain rate $\gamma$). The cavity is formed by two nonmagnetic mirrors, $M_1$ (fixed) and $M_2$ (movable). A strong control field (amplitude $\Omega_{c}$, frequency $\omega_{c}$) drives the cavity, which exhibits intrinsic and external coupling decay rates $\kappa_i$ and $\kappa_e$, respectively. A transverse electric (TE) polarized probe field (amplitude $E_{p}$, frequency $\omega_{p}$) is incident on mirror $M_1$ at an angle $\theta_i$. The lateral displacement of the reflected probe field, known as the Goos–Hänchen shift, is denoted by $S_r$.
• Figure 2: Eigenfrequencies of $H_{\text{eff}}$ [Eq. (\ref{['H_eff']})] versus normalized effective optomechanical coupling $G_{ab}/\kappa$. (a) Real part: $\mathrm{Re}(\omega_\pm-\omega_{0})/\kappa$; (b) Imaginary part: $\mathrm{Im}(\omega_\pm-\omega_{0})/\kappa$. Red and green solid curves represent the $\mathcal{PT}$-symmetric case with balanced gain and loss ($\gamma/2\pi=\kappa/2\pi=1.0~\mathrm{MHz}$). For a conventional passive–passive system, blue solid and orange dashed curves indicate unequal losses ($\gamma/2\pi=1.0~\mathrm{kHz}$, $\kappa/2\pi=1.0~\mathrm{MHz}$), while cyan solid and purple dashed curves correspond to equal losses ($\gamma/2\pi=1.0~\mathrm{MHz}$, $\kappa/2\pi=1.0~\mathrm{MHz}$).
• Figure 3: (a–c) Absorption spectra ($\mathrm{Re}[E_T]$) and (d–f) dispersion spectra ($\mathrm{Im}[E_T]$) of the output probe field versus normalized probe–cavity detuning $\tilde{\Delta}/\kappa$. Columns correspond to effective optomechanical coupling strengths $G_{ab}$ for each phase: Broken $\mathcal{PT}$ phase: $G_{ab} = 0.44\,\kappa$ (red), $0.46\,\kappa$ (blue), $0.48\,\kappa$ (green); EP: $G_{ab} = 0.5\,\kappa$ (red); Unbroken $\mathcal{PT}$ phase: $G_{ab} = 0.52\,\kappa$ (red), $0.54\,\kappa$ (blue), $0.56\,\kappa$ (green). Fixed parameters: $\eta = 0.5$, and $\gamma/2\pi=\kappa/2\pi=1.0~\mathrm{MHz}$.
• Figure 4: (a) Absorption spectra ($\mathrm{Re}[E_T]$) and (b) dispersion spectra ($\mathrm{Im}[E_T]$) versus normalized probe–cavity detuning $\tilde{\Delta}/\kappa$ for various effective optomechanical coupling strengths: $G_{ab} = 0.4\,\kappa$ (red), $0.5\,\kappa$ (blue), and $0.6\,\kappa$ (green) in a conventional optomechanical system. Fixed parameters: $\eta=0.5$, $\gamma/2\pi= 1.0~\mathrm{kHz}$, and $\kappa/2\pi=1.0~\mathrm{MHz}$.
• Figure 5: (a) Absolute value of the reflection coefficient $|R(k_{z}, \omega_{p})|$ and (b) the normalized GHS $S_{r}/\lambda$ versus incident angle of the probe field $\theta_{i}$ for three effective optomechanical coupling strengths: (i) $G_{ab} = 0.48\,\kappa$ (blue, broken $\mathcal{PT}$ phase), (ii) $0.52\,\kappa$ (green, unbroken $\mathcal{PT}$ phase), and (iii) $0.52\,\kappa$ (red, conventional case) at resonance ($\tilde{\Delta}=0$). Fixed parameters: $\lambda=1064~\mathrm{nm}$, $\kappa/2\pi=1.0~\mathrm{MHz}$, $\epsilon_{0}=1$, $\epsilon_{1}=2.22$, $d_{1}=1~\mu\mathrm{m}$, $d_{2}=10~\mu\mathrm{m}$, $\gamma/2\pi=\kappa/2\pi=1.0~\mathrm{MHz}$ ($\mathcal{PT}$-symmetric system), $\gamma/2\pi=1.0~\mathrm{kHz}$ and $\kappa/2\pi=1.0~\mathrm{MHz}$ (conventional system), and $\eta=0.5$.