Design of the light fields

Our basic task is to create intense evanescent light fields with a potential minimum sufficiently far from a dielectric surface to make the surface interaction potential and heating mechanisms negligible (discussed in Section 8.5.1). The main difficulty arises because the evanescent fields have a typical exponential decay length $\sim \lambda / 2 \pi$, so if we are to have a trap of useful depth, we are restricted to keep it within roughly $\lambda$ of the surface (less than a micron).

A potential minimum in one dimension can be obtained using a blue (repulsive) light field of higher intensity at the dielectric surface than the red (attractive) light field, and ensuring the decay lengths obey $L_{\rm red} > L_{\rm blue}$, giving a potential of the form

$$U_{\rm dip}(y) = A_{\rm blue} e^{-y/L_{\rm blue}} \,- A_{\rm red} e^{-y/L_{\rm red}}\,.$$ (8.6)

This gives a repulsive force at short range, which becomes attractive at long range (see Figure 8.1b), and is the scheme for the planar trap of Ovchinnikov et al. [156]. A large amount of insight into our proposed trap can be gleaned from this simple one-dimensional model (which we call the exponential approximation), because the squared electric fields above the guide will turn out to approximate exponential forms in the vertical direction quite closely.

If we define a normalized decay length difference $\alpha_L \equiv (L_{\rm red} - L_{\rm blue})/L_{\rm blue}$, then we can give two reasons why increasing $\alpha_L$ is a vital design objective. Firstly, it is easy to show that for small $\alpha_L$ the deepest available trap depth (found by optimizing the ratio of surface intensities $A_{\rm red}/A_{\rm blue}$) scales as $\alpha_L$. Secondly, a larger $\alpha_L$ is beneficial for trap coherence, (giving a smaller spontaneous decay rate at a given trap depth and detuning), because the sum of the intensities can be kept lower (see equation (8.5)) for a given intensity difference (equation (8.4)). We will quantify this latter connection in Section 8.3.1.

Our two key differences from the proposal of Ovchinnikov et al. are as follows. Firstly, we create a non-zero $\alpha_L$ by using two orthogonally-polarized bound modes of a dielectric slab guide, which have different evanescent decay lengths at the same frequency^8.2. This contrasts with Ovchinnikov et al. who suggest varying the decay lengths by varying the reflection angles from the inside surface of a glass prism. Secondly, horizontal confinement is achieved by limiting the width of the slab guide to approximately $\lambda$ (forming what is called a channel guide [117]), which automatically creates a maximum in each light intensity field in the horizontal direction. This results in a tight horizontal confinement in the atomic potential, of similar size to the vertical confinement, and is something very hard to achieve in a prism geometry.

A schematic of our design is shown in Figure 8.1a. The optical guide height $H$ and width $W$ are kept small enough to guarantee that there are exactly two bound modes, differing in polarization but not in nodal structure (in optical terminology this is called single-mode): $E_{11}^x$ has an electric field predominantly in the x direction^8.3, and is to be excited by blue-detuned laser light, and $E_{11}^y$ has electric field predominantly in the y direction and is to be excited by red-detuned laser light. We can see why their vertical decay lengths differ by considering the case of the slab (i.e. taking the width $W \rightarrow \infty$), where these modes are simply the slab TE and TM modes respectively. For both these slab modes the purely transverse field obeys the differential equation

$$\frac{\partial^2 \phi}{\partial y^2} = k_0^2 \, [(n_{\rm eff}^{(i)})^2 - n(y)^2] \, \phi \,,$$ (8.7)

where $\phi = E_x$ ($H_x$) for the TE (TM) mode, the eigenvalue $n_{\rm eff}^{(i)} \equiv k_z^{(i)}/k_0$ is the effective refractive index for the $i^{\rm th}$ mode ($k_z$ being the wavenumber in the propagation direction and $k_0$ the free space wavenumber), and $n(y)$ is the spatially-dependent refractive index [117]. (This is equivalent to a one-dimensional quantum problem in the direction normal to the slab, in a potential $-k_0^2 n(y)^2$ with $\hbar^2/2m = 1$). However the boundary conditions on the slab surfaces differ for the two mode types: $\phi$ is always continuous, but $\partial \phi / \partial y$ is continuous for TE as opposed to $n^{-2} \partial \phi / \partial y$ continuous for TM. This asymmetry exists because the permittivity $\epsilon = n^2$ varies in space but the permeability $\mu$ is assumed to be constant. This discontinuity in the gradient for the TM mode forces it to have a lower $n_{\rm eff}$ than that of the TE mode, which means it is less tightly bound so has a longer evanescent decay length. This effect becomes more pronounced as the slab index increases or as optical cut-off is approached (which happens when $n_{\rm eff}$ decreases until it reaches $n_s$ and the mode becomes unbound). This tendency is preserved even as the width is decreased to only a few times the height, as in our scheme.

Figure 8.2: Numerically solved cut-off curves for a dielectric optical waveguide of $n_g = 1.56$ (with $n_s = 1$) as a function of its width and height (dashed curves) and, on the same axes, contours of the maximum trapping potential depth achievable at fixed total laser power (thin solid curves in rectangular overlayed box region). Also shown is the symmetry line $W=H$ (thin dash-dotted line). Note that the contours show depth increasing as $H$ decreases, almost independent of $W$, near the suggested operating dimensions (shown as a solid ellipse). Cut-off is defined as reaching an effective refractive index $n_{\rm eff} \equiv k_z/k_o = 1.05$, except for $E^x_{21}$ and $E^y_{21}$ (thick dashed lines) which we show cut-off at $n_{\rm eff} = 1.02$.

No analytic solution exists for the general rectangular guide, so we used the finite element method discussed in Section 8.4 to solve for the bound mode $n_{\rm eff}$ values and fields as a function of guide dimensions. Figure 8.2 shows the resulting `cut-off curves', that is, contours of constant $n_{\rm eff}$ in the parameter space $(W,H)$. In this example we chose a guide index $n_g = 1.56$ (typical for a polymer dielectric) and, as a preliminary case, a substrate index $n_s = 1$.

The single-mode region, in which we wish to remain, is bounded below by the $E^y_{11}$ and $E^x_{11}$ curves and above by the $E^y_{21}$ curve. Note that, as in any dielectric guiding structure uniform in the z axis, the lowest two modes ($E^y_{11}$ and $E^x_{11}$ in our case) never truly reach cut-off, rather, they approach it exponentially as the guide cross-section is shrunk to zero. For this reason, we chose the practical definition of cut-off for these modes to be $n_{\rm eff} = 1.05$, which corresponds to only about 20% of the power being carried inside the guide. In contrast, higher modes do have true cut-offs[183,117] (this distinction is illustrated by the dispersion curves of Figure 8.4), and for the $E^y_{21}$ mode our (numerically limited) contour choice of $n_{\rm eff} = 1.02$ falls very close to the true cut-off curve.

Using the numerically calculated electric field strengths of the $E^y_{11}$ and $E^x_{11}$ modes, we found the red and blue guided laser powers which gave the deepest trap, subject to the constraint of fixed total power (keeping the detuning constant ^8.4). We also imposed the restriction that the zero of trapping potential come no closer than 100nm along the vertical line $x=0$, which kept the trap minimum a reasonable distance from the surface (see Section 8.5.1). Performing this optimization over a region of the parameter space covering the single-mode region gave a contour plot of maximum achievable depth for a given total power, shown within the rectangle overlayed on Figure 8.2. This depth increases from negligible values in the top left to the largest depths in the lower right, indicating that choosing $W$ and $H$ to be in this latter corner of the single-mode region is best for depth. The depth shows very little variation with $W$ in this corner, rather it is clear that varying $H$ to stay within our definition of the single-mode region has become the limiting factor on achievable depth. We indicate a practical choice of $W = 0.97 \,\lambda$ and $H = 0.25 \,\lambda$ as a small marker on Figure 8.2. Example trapping potentials shapes possible with these parameters are shown in Figure 8.3; we discuss their properties in Section 8.3.

In Figure 8.1a the direct excitation of the optical guide by the two laser beams is shown only schematically. In a realistic experimental setup this coupling into the guide would happen on the order of a centimeter from the atom guiding region, and could involve tapered or Bragg couplers[117] from beams or from other fibers. At this distance we estimate that isotropic stray light due to an insertion loss of 0.5 would have 8 orders of magnitude less intensity than the EW fields in the guiding region. Assuming the light is coherent, this limits the fractional modulation of the guiding potential to $10^{-4}$. More improvements are possible, including the use of absorbing shields, bending the guide through large angles away from the original coupling direction, and reducing the coherence length.