Discussion of optical cut-off and substrate choice

We have shown in Figure 8.2 that parameters optimized for trap depth are near optical cut-off. It is worth gathering together the physical reasons for this. Firstly, on general geometric grounds, the typical available intensities in a guide scale inversely with the effective cross-sectional area of the bound mode, which, far from cut-off (that is, when $n_g - n_{\rm eff} \ll n_g - n_s$) follows very closely the cross-sectional area of the guide. This makes it favorable to shrink guiding structures to areas less than a square wavelength, where they generally become single-mode. Secondly, once we're in the single-mode regime, as we approach cut-off the mode power is carried increasingly outside the guide, increasing the ratio of surface intensity to guide center intensity. Thirdly, the evanescent decay length in the vacuum is longer as we approach cut-off (recall that in the case of the slab, this is exactly expressed by $L^{-1} = 2 k_0 \sqrt{n_{\rm eff}^2 - 1}$ [117], where the factor of 2 arises because we are considering intensity decay length rather than amplitude). In the special case of $n_s = 1$, the decay length diverges to infinity as we approach cut-off. Finally, the ratio $\alpha_L$ becomes larger as we approach cut-off, with corresponding beneficial effects on depth and coherence (due to increasing the `goodness factor' we will introduce in Section 8.3.1).

Unfortunately, these purely theoretical reasons for approaching cut-off are in opposition to more practical ones. The closer to cut-off a guide is, the more sensitive it is to manufacturing variations in cross-section: in our case this will be predominantly a sensitivity to $H$. The result is that small variations in $H$ cause large variations in mode size, or, at worst, complete cut-off. If the mode size change is rapid (nonadiabatic) along the z axis, (for instance if this change is due to surface roughness or refractive index inhomogeneities) the resulting mismatches will be a source of scattering of the guided power. Any coherent scattering back down the guide will set up periodic modulations of the light field over long distances. (One way to reduce the distance over which coherent addition is possible is to use very broad line-width light sources, which would dramatically reduce modulations due to both guided and stray scattered light). The consequence for the atoms would be a z-dependent trap depth and shape, and this could lead to partial reflection or even localization of the matter waves. In general we expect scattering to limit how close to cut-off we can reliably operate.

With regard to the substrate, further practical issues arise. In the above cut-off calculation we chose the simplest case of $n_s = 1$, corresponding to a guide surrounded by vacuum. A real substrate with $n_s > 1$ has the unfortunate effect of limiting the propagation constant $k_z$ of strictly bound modes to be larger than the freely propagating wavevector in the substrate; in other words, $n_{\rm eff} > n_s$ must hold or the light field will rapidly tunnel into the `attractive potential' of the substrate. This in turn limits the decay lengths and $\alpha_L$ that can be achieved. The trap properties quoted in the abstract and in Sections 8.3.1 and 8.3.2 rely on very low $n_{\rm eff}$ values (1.07 for the $E^y_{11}$ mode, 1.18 for $E^x_{11}$) for the reason that a low $n_{\rm eff}$ is the only way to create long evanescent decay lengths in the vacuum. (This is equivalent to Ovchinnikov et al. choosing reflection angles very close to critical[156].) However, in Section 8.3.3 we present preliminary results for a substrate of sodium fluoride (the lowest-index common optical mineral, at $n = 1.32$), and do not believe the substrate alters the basic feasibility of our waveguide.

For completeness, here we list some other possible approaches to the substrate issue. 1) Use an aerogel substrate, which can have exceptionally low refractive indices and low loss (films of several $\mu$m thickness with indices of about 1.1 can be produced[147,150]). 2) Use a dielectric multilayer substrate with an effective index of unity or less (very low loss multilayer mirrors[169] with effective indices less than unity can be created). 3) Investigate if there exist guide shapes which have sufficiently small tunnelling rate into a conventional substrate that the fact that the modes are not strictly bound becomes irrelevant (for instance, a wedge shape with the smallest face in contact with the substrate). 4) Unsupported guiding structures could be produced over short distances[176]. Finally, it is important to note that the idea of replacing the substrate by a metallic reflective layer is not practical because they are too lossy.

Ultimately, the best values of $W$ and $H$, the best guide cross-sectional shape, and the substrate choice will depend on many of the above factors and is an area for further research.

Figure 8.3: Trapping potential contours in the xy plane, with guide dimensions of 0.83$\mu$m by 0.21$\mu$m and an index of 1.56. In each case the maximum depth is $100\,\mu$K and the coherence time of atoms in the ground state is 19ms, achieved with detuning $\pm 15$nm from the cesium D2 line. The outer contour shows a depth of $3\,\mu$K, the cesium MOT temperature. Subsequent contours are spaced by $6\,\mu$K. The plots illustrate the range of trapping distances acheivable: the upper trap ($p=0.4$, using a total guided power of 1mW) has a minimum 0.24$\mu$m from the surface; the lower trap ($p=0.2$, total guided power of 7.5mW) has a minimum 0.44$\mu$m from the surface.