Finite-difference time-domain simulation of thermal noise in open cavities

A numerical model based on the finite-difference time-domain (FDTD) method is developed to simulate thermal noise in open cavities owing to output coupling. The absorbing boundary of the FDTD grid is treated as a blackbody, whose thermal radiation penetrates the cavity in the grid. The calculated amount of thermal noise in a one-dimensional dielectric cavity recovers the standard result of the quantum Langevin equation in the Markovian regime. Our FDTD simulation also demonstrates that in the non-Markovian regime the buildup of the intracavity noise field depends on the ratio of the cavity photon lifetime to the coherence time of thermal radiation. The advantage of our numerical method is that the thermal noise is introduced in the time domain without prior knowledge of cavity modes.

Figure 1

(Color online) Noise source electric field $E_s\left(t_j\right)$ generated for $T=30000\text{K}$ (black dots) and $T=50000\text{K}$ (red crosses). The noise correlation time ${\tau }_c\approx 0.337\text{fs}$ for $T=30000\text{K}$ and ${\tau }_c\approx 0.203\text{fs}$ for $T=50000\text{K}$. $\Delta x=1\text{nm}$, $M=2^{21}$, and ${\tau }_{\text{sim}}=7\text{ps}$. The inset is a schematic showing the noise sources placed next to the grid-ABL interface.

Figure 2

(Color online) FDTD-calculated energy density of blackbody radiation propagating in 1D vacuum vs frequency $\omega$ for temperatures $T=30000\text{K}$ (lower) and $T=50000\text{K}$ (upper). The inset shows the energy density for temperature $T=30000\text{K}$ at higher frequencies. The data are obtained by averaging over 2000 calculations with the resolutions $\Delta x=10\text{nm}$ (red crosses) and $\Delta x=1\text{nm}$ (black dots). The source spectra $D\left(\omega ,T\right)$ are also plotted as solid lines on top of the numerical spectra.

Figure 3

(Color online) Temporal correlation function, $⟨E_s\left(t_1\right)E_s\left(t_2\right)⟩$ vs $t_2-t_1$ for the noise electric field at $T=30000\text{K}$ (black circles) and $T=50000\text{K}$ (red crosses). The noise correlation times are ${\tau }_c\approx 0.337\text{fs}$ for $T=30000\text{K}$ and ${\tau }_c\approx 0.203\text{fs}$ for $T=50000\text{K}$. $\Delta x=1\text{nm}$, $M=2^{21}$, and ${\tau }_{\text{sim}}=7\text{ps}$. The lines represent $⟨E_s\left(t_1\right)E_s\left(t_2\right)⟩$ given by the analytical expression in Eq. (4) for $T=30000\text{K}$ (black) and $T=50000\text{K}$ (red). Every fifth data point is taken from the numerical data in order to better show the agreement with the analytical solution.

Figure 4

(Color online) Spatially averaged EM energy density $U\left(\omega \right)$, calculated by the FDTD method, vs frequency $\omega$ in a dielectric slab cavity with refractive index $n=6$ and length (a) $L=2400\text{nm}$ and (b) $L=20\text{nm}$. The vertical black dashed lines mark the frequencies of cavity modes ${\omega }_m$. The spectrum of impinging blackbody radiation $D\left(\omega ,T\right)$ is also plotted (red solid line). In (a) the cavity decay time $\tau =143\text{ps}$ is much longer than ${\tau }_c$. In (b), $\tau =1.19\text{fs}$, comparable to ${\tau }_c$.

Figure 5

(Color online) Number of thermal photons in individual cavity modes $n_m$, calculated via the FDTD method, for a slab cavity with $n=6$. The cavity length $L$ is varied to change $\tau$. The impinging blackbody radiation has $T=30000\text{K}$ and ${\tau }_c=0.337\text{fs}$. The values of ${\tau }_c/\tau$ are 0.0024, 0.29, 0.43, and 0.56. Lines are drawn to connect the data points at the mode frequencies ${\omega }_m=\pi cm/nL$ to illustrate their frequency dependence. For ${\tau }_c\ll \tau$ (only every fifth mode for $\tau =143\text{fs}$ is shown to improve the visibility), the photon number $n_m$ coincides with the Bose-Einstein distribution $n_T$. When ${\tau }_c\sim \tau$, $n_m$ deviates from $n_T$.

Figure 6

(Color online) Number of thermal photons in individual cavity modes $n_m$, calculated via the FDTD method, for a dielectric slab cavity with $n=6$ and $L=20\text{nm}$. The cavity decay time $\tau =1.19\text{fs}$. The temperature of blackbody radiation is varied to change ${\tau }_c$. The values of ${\tau }_c/\tau$ are 0.29 $\left(T=30000\text{K}\right)$, 0.43 $\left(T=20000\text{K}\right)$, and 0.56 $\left(T=15000\text{K}\right)$. Black dashed lines are drawn to connect the data points at the mode frequencies ${\omega }_m=\pi cm/nL$ to illustrate their frequency dependence. For comparison, the Bose-Einstein distribution $n_T\left(\omega \right)$ is also plotted (red solid lines) for the same temperatures.

Figure 7

(Color online) $B_m\left({\tau }_c,\tau \right)\equiv n_m/n_T\left({\omega }_m\right)$ vs the ratio of noise correlation time to cavity decay time ${\tau }_c/\tau$. $B_m\left({\tau }_c,\tau \right)$ depends only on the ratio ${\tau }_c/\tau$ and not on ${\tau }_c$ or $\tau$ individually. This plot was generated by fixing $\tau$ and varying ${\tau }_c$. Cavity modes of higher $m$ have a larger value of $B_m\left({\tau }_c,\tau \right)$ whether in the regime ${\tau }_c\sim \tau$ (main panel) or ${\tau }_c\ll \tau$ (inset).