Effect of a realistic substrate

In this section we present calculations, performed using the method of Section 8.4, for a practical substrate choice of sodium fluoride (the lowest refractive index common mineral, with $n_s = 1.32$ at a wavelength of 852nm), and investigate how this changes the atom waveguide properties from those presented above. We increased $n_g$ to 1.7 (dense flint glass, e.g. BaSF type) in order to provide sufficient index difference from the substrate.

Fixing the width at $W = 1.00 \,\lambda$, we found that a height $H = 0.34 \,\lambda$ gave the largest $\alpha_L$ of $0.20 \pm 0.01$, and allowed both modes to be sufficiently far from cut-off (greater than half the power being carried inside the guide for both modes). The result is a goodness factor $G$ which is approximately half that of the $n_s = 1$ case, with a corresponding halving of the achievable product of depth and coherence time according to (8.11), and doubling of the heating rate at a given $U_{\rm max}$ and $\Delta$ according to (8.13). The shorter decay lengths of 56nm and 68nm (compared to 93nm and 137nm for $n_s = 1$) cause the typical trapping distance $y_0$ to be reduced by a factor of roughly 1.8.

We found that in order to reproduce the depth of $100\,\mu$K and $y_0 = 0.24\,\mu$m of the upper trap of Figure 8.3 (with $\Delta$ unchanged) we needed $P_{\rm tot} = 22$mW, giving $\tau_{\rm coh} = 9\,$ms. The large power increase over the 1mW required for $n_s = 1$ is explained by the fact that this $y_0$ is now towards the upper limit practically achievable rather than the lower. (If $y_0$ is instead scaled in proportion to the new decay lengths, the required increase in $P_{\rm tot}$ is only a factor 1.7). In this example, we find the transverse oscillation frequencies have increased to $\omega_x / 2\pi = 81$kHz and $\omega_y / 2\pi = 202$kHz, compared to the original $\omega_x / 2\pi = 26$kHz and $\omega_y / 2\pi = 109$kHz. The increase in $\omega_y$ is explained entirely by the shorter decay lengths, and the increase in $\omega_x$ (by a factor of over 3) is attributed to tighter optical mode shapes. It is clear that this latter effect outweighs the decrease in $\tau_{\rm coh}$, implying that the inclusion of the substrate has actually increased $Q$ by 50%.

In summary, the effects of including a realistic substrate limit the maximum trapping distance $y_0$ that can be achieved (because of the reduction in decay lengths), lower the goodness factor, increase the heating rate and the required optical power, but also increase the oscillation frequencies. For our substrate choice, each of these changes was approximately a factor of 2, and we believe that they do not alter the basic practicality of implementing our proposed waveguide.

Figure 8.4: Comparison of our optical guide bound-mode numerical implementation against known analytic solutions, in the case of a free-standing dielectric cylinder of $n_g = 1.56$. We used discretization and box-size identical to the rectangular guide case, and observe typical errors of $\pm 0.5\%$ in propagation constant for the first two modes. The mode naming convention and analytic calculation follow Snitzer [183]; an asterisk indicates a doubly-degenerate mode.