Formation controllable two-dimensional surface optical lattices

Abstract

The evanescent light field created by total internal reflection at a dielectric-vacuum interface has tight spatial confinement and high local intensity. These properties become more robust when a dual layer of dielectric thin films with different thicknesses is deposited on a planar dielectric surface. Optical construction is promising in nanoscale trapping of absorbed atoms congregating at extreme light field intensity distribution regions on the dielectric surface. Here, we theoretically demonstrate the controllable two-dimensional surface optical lattices based on the evanescent electric field generated by the Hermite-Gaussian light beam. By considering the coupling of lattice to quadrupole-active atomic transitions, we show that the topologies cannot just have high stability, but also the optical potential deep, and thus the dynamic trapping can be governed by changing the design parameters of the optical construction. We expect that this study may contribute to providing another unique method for fabricating spatially complicated two-dimensional nanostructures.

KEYWORDS:

• Evanescent field
• Gaussian light
• quadrupole transition
• optical lattice

1 Introduction

Optical lattices can be described in their simplest form as periodic arrays of traps (micro or nanoscales) for cold atoms created by a set of standing wave lasers [1–3]. The electric field of these lasers interacts with these atoms to generate optical forces that can be divided into two types: scattering and gradient forces [4]. The good functioning of the optical lattices depends on the effectiveness of the gradient force, whereby an optical trapping potential emerges in which the atoms are immersed and then congregate at its bottom [5–7]. On the contrary, the scattering force is considered a destructive factor for both the continuity and stability of the action of optical lattices. The effect of the scattering force strength is significantly reduced by the interference pattern of two counter-propagating laser beams, which is the basis of the setting up. Nevertheless, it remains an obstacle that needs more effective treatment to reach a situation close to full repression [8].

Theoretically and practically, the production of optical lattices in the past three decades has been restricted to the electric-dipole coupling between the atom and the electric field of the laser light, which was accompanied by neglecting the rest of the possible couplings [9]. Many suggestions have been presented to enhance the gradient force and reduce the scattering force. Most of these development attempts have been made on optical lattices generated in unbounded space [1–3]. Mature progress has already occurred in this area and has contributed to several different applications, such as atomic clocks, quantum computing, and quantum simulations [10–12].

Studies have been continued to improve lattice performance further. Remarkable progress has been made when generating a surface optical lattice through the evanescent electric field near the dielectric surfaces (i.e. bounded space). This kind of optical lattice is based on the mechanism suggested by Cook and Hill in 1982 [13]. In this mechanism, localizing features of the evanescent light field within a limited range provides a better solution to many practical problems, particularly those related to continuity and stability [14–18].

The mechanism for creating an evanescent electric field has gone through several developmental stages to achieve a dual aim, increase the magnitude of the evanescent field, and reduce the wave-atom interaction time [19–21]. The first leads to achieving the desirable property, which is the deepening of the optical trapping potential. The possibility of spontaneous emission is made significantly lower by the second goal, minimizing the undesirable process of scattering force [9]. The dielectric waveguide scheme is the best optical structure suggested and practically tested with electric dipole interaction [20,21]. A dual layer of dielectric thin films of different thicknesses is deposited on a planar dielectric surface in this scheme. The succession of total internal reflection between the three dielectric regions made the obtained intensity of the evanescent field much higher than the absence of a deposition process (uncoated or bare surface) or even a single-layer deposition. The last upper layer acts as a planar dielectric slab waveguide and an optical filter, allowing only specific optical modes to propagate within. The change in the thicknesses of the dual layer provides the key to controlling all-optical structure properties. Therefore, it can be considered a design parameter system [20,21].

In the last ten years, remarkable attention has been paid to exploiting the unique property of the electric quadrupole interaction [22–25]. In a quadrupole interaction, the atom engages with a gradient of the electric field, while in a dipole interaction, it engages with the field strength. Thus, the quadrupole interaction can lead to transitions for atoms localized in the dark regions of the light wave, where there is weak light intensity but relatively strong field gradients. Many theoretical and practical studies have been presented to enhance the quadrupole interaction magnitude to the extent that it is exploitable [22–25].

The success of the dielectric waveguide scheme in generating a deep optical potential well with the dipole interaction is expected to make it also promising with quadrupole interaction. We use this optical scheme to display the quadrupole interactions of a two-level atom when dipole transitions is forbidden. Additionally, the original incident light beam is one of the complex Gaussian laser beams known as the Hermite beams. This is a class of laser light distinguished by its distribution (in unbounded or bounded spaces) with an array of peaks and valleys of light intensity resembling an egg carton [7,8].

2. Evanescent electric field

One of the basics of wave optics is that when internal reflection occurs between two dielectric media, evanescent light is created in the second one and propagated along the boundary, as in Figure 1(a). The amplitude of evanescent light gradually decreases as the distance from the boundary increases. The associated intensity in the concentration region is typically minimal but may sometimes be enough to influence the atomic properties by generating two separate optical forces: scattering and gradient forces. The basic formulae of the evanescent electric field can be derived by assuming that the simplest form of Gaussian laser wave coming from $z<0$ is internally reflected at a dielectric-vacuum interface $z=0$. If we assume the plane of incidence is the $x-z$ plane with angle ${\theta }_i>{\theta }_c$, the evanescent field in the vacuum is given by [8]: (1) $\begin{array}{r}E(z)=E_{max}^{0}G(x,y)\exp [-\mathrm{\alpha z}]\exp (i{k}_{x}x)\end{array}$(1) where $G(x,y)$ represents the envelope of a Gaussian laser wave along the surface of $\mathrm{xy}$dimensions and varies according to the class of Gaussian wave (the Hermite or Laguerre or Bessel), while it is equal to one for a pure plane wave. $k_x$ is the $k$vector parallel to the interface, which is given by: (2) $\begin{array}{r}{k}_x=\frac{2\pi }{\lambda }\mathrm{nsin}{\theta }_i\end{array}$(2) where $n>0$ is dielectric substrate refraction index. The quantity $\alpha$ represents the penetration depth inverse as follows: (3) $\begin{array}{r}\alpha =\frac{2\pi }{\lambda }(n^2{\sin }^2{\theta }_i-1{)}^{1/2}\end{array}$(3) Finally, $E_{max}^0$ is the maximum field amplitude, and the zero indicates that this is related to the case of a bare structure (or an uncoated substrate), as follows [20]: (4) $\begin{array}{r}{E}_{max}^0=E_{\mathrm{inc}}^{}\frac{4n^2{\cos }^2{\theta }_i}{n^2-1}{\left[\frac{1}{n^2-(n^2+1){\cos }^2{\theta }_i}\right]}^m\end{array}$(4) where $E_{\mathrm{inc}}$ is the original incident field and $m=0$ or $m=1$, respectively, for transverse electric ($\mathrm{TE}$) or transverse magnetic ($\mathrm{TM}$) fields.

Figure 1. Generating evanescent field in the vacuum region at $z>0$; (a) uncoated interface (b) monolayer deposition (c) multilayer deposition (d) counterpropagating fields.

Practically, the reasonable evanescent intensity to reach the desired influence of the atomic properties may require a very high incident laser intensity in average conditions to overcome the thermal energy of atoms at room temperature. A possible alternative is to subject the atoms to a precooling process. Both methods are associated with some undesirable side effects, so from the beginning, in previous studies, it has been tended to search for other ways to enhance. It has been shown by some measurements that exceeding the threshold may only need to raise the magnitude of evanescent intensity by several orders [9]. The two most prominent methods demonstrated in this regard were through the deposition of metallic layers on the dielectric-vacuum interface. These layers are composed of thin films and can be single (monolayer), as in Figure 1(b), or dual (multilayer), as in Figure 1(c).

A multilayer case is characterized by a solid gap layer located in the middle between the substrate and the waveguide layer. Hence, the evanescent field intensity can be controlled up and down with the stability of the original incident light, thereby changing the thickness of this gap layer. The ease of controllability is a unique property not available in monolayer cases. However, the significant development in nano-manufacturing technology has helped obtain a high enhancement (in both deposition cases), that is not essentially less than two to three orders compared to that obtained with the same incident intensity onto a bare interface case.

3. Dielectric waveguide scheme

A dual-deposition structure is usually called a dielectric waveguide scheme. This scheme has received many theoretical and experimental steadies in electric dipole interaction activity cases in optical mirror production. The primary mechanism of this scheme is simply that the incident wave suffers from successive total internal reflection processes until it tunnels into the waveguide (upper layer) from a solid gap layer. Inside the waveguide, it is subjected to consecutive reflections between its upper and lower interfaces. These combined processes make the evanescent wave output build up more and more on the waveguide-vacuum side.

The basic mathematical formula that describes the formation of the evanescent field is the same as the previous bare interface formula, with some minor modifications due to additional layers. For the waveguide scheme with the same $E_{\mathrm{inc}}$ and ${\theta }_i$, the evanescent electric field $E_4$ (in vacuum above the upper side of the waveguide) can be calculated by introducing an enhancement factor $\gamma$. This factor is defined as the ratio between the evanescent electric field $E_{max}$ above the coated substrate and the evanescent electric field $E_{max}^0$ above an uncoated substrate, thus: (5) $\begin{array}{r}\gamma =\frac{E_{max}}{E_{max}^0}=\frac{\tau E_{\mathrm{inc}}}{E_{max}^0}\end{array}$(5) where $\tau$ is the transmission coefficient (the transmission Fresnel equation) for oblique incidence of the electromagnetic wave at plane boundaries. Generally, it can be evaluated by following the steps demonstrated in essential classical electrodynamics books. This particular scheme has been derived in detail in Ref [20].

4. Optical forces and quadrupole transitions

The total average optical force ${\mathbf{F}}_{\{k\}}^{\mathrm{Opt}.}$ due to the electric quadrupole interaction with a two-level atom moving with velocity $\mathbf{V}=\dot{\mathbf{R}}$ is [26]: (6) $\begin{array}{rl}{\mathbf{F}}_{\{k\}}^{\mathrm{Opt}}(\mathbf{R}\mathbf{,}\mathbf{V})& =2\hslash \left[\frac{\begin{array}{c}{\mathrm{\Gamma }}_Q{|{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})|}^2\mathrm{\nabla }{\theta }_{\{k\}}(\mathbf{R})\\ -\left(\frac{1}{2}\right){\mathrm{\Delta }}_{\{k\}}^{}(\mathbf{R},\mathbf{V})\mathrm{\nabla }{({\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R}))}^2\end{array}}{{\mathrm{\Delta }}_{\{k\}}^2(\mathbf{R},\mathbf{V})+2{|{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})|}^2+{\mathrm{\Gamma }}_Q^2}\right]\end{array}$(6) where (7) $\begin{array}{rl}{\mathbf{F}}_{\{k\}}^{\mathrm{Sct}.}(\mathbf{R},\mathbf{V})& =2\hslash {\mathrm{\Gamma }}_Q|{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R}){|}^2\\ & \times \left(\frac{\mathrm{\nabla }{\theta }_{\{k\}}(\mathbf{R})}{{\mathrm{\Delta }}_{\{k\}}^2(\mathbf{R},\mathbf{V})+2{|{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})|}^2+{\mathrm{\Gamma }}_Q^2}\right)\end{array}$(7) (8) $\begin{array}{rl}{\mathbf{F}}_{\{k\}}^Q(\mathbf{R},\mathbf{V})& =-2\hslash \mathrm{\nabla }|{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R}){|}^2\\ & \times \left(\frac{{\mathrm{\Delta }}_{\{k\}}(\mathbf{R},\mathbf{V})}{{\mathrm{\Delta }}_{\{k\}}^2(\mathbf{R},\mathbf{V})+2{|{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})|}^2+{\mathrm{\Gamma }}_Q^2}\right)\end{array}$(8) where ${\mathbf{F}}_{\{k\}}^{\mathrm{Sct}.}$ is called the scattering quadrupole force and ${\mathbf{F}}_{\{k\}}^Q$is the gradient quadrupole force. Matching to the gradient quadrupole force is a quadrupole potential, which can be given by: (9) $\begin{array}{r}{P}_{\{k\}}^Q(\mathbf{R})=-\frac{\hslash {\mathrm{\Delta }}_{\{k\}}(\mathbf{R})}{2}\ln \left(1+\frac{2{|{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})|}^2}{{\mathrm{\Delta }}_{\{k\}}^2(\mathbf{R})+{\mathrm{\Gamma }}_Q^2}\right)\end{array}$(9) Here, $\mathrm{\nabla }{\theta }_{\{k\}}(\mathbf{R})$ is the gradient of the phase, ${\mathrm{\Gamma }}_Q^{}$ is the spontaneous quadrupole emission, and ${\mathrm{\Delta }}_{\{k\}}(\mathbf{R},\mathbf{V})$ is the dynamic detuning, which is a function of both the position and the velocity vectors of the atom ${\mathrm{\Delta }}_{\{k\}}(\mathbf{R},\mathbf{V})={\mathrm{\Delta }}_0-V\cdot \mathrm{\nabla }{\theta }_{\{k\}}(\mathbf{R})$; where ${\mathrm{\Delta }}_0={\omega }_0-\omega$ is the static detuning, with $\omega$ the frequency of the applied light field. The second term in dynamic detuning ${\mathrm{\Delta }}_{\{k\}}(\mathbf{R},\mathbf{V})$ is written ${\delta }_{\{k\}}=-V\cdot \mathrm{\nabla }{\theta }_{\{k\}}(\mathbf{R})$ and arises because of the Doppler effect due to the atomic motion. Finally, ${\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})$ is the quadrupole position-dependent atomic complex Rabi frequency ${\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})$, which is the main factor in optical trapping processes, is defined as follows: (10) $\begin{array}{r}{\mathrm{\Omega }}_{\{k\}}^Q(\mathbf{R})=\frac{|H_{\{k\}}^Q(\mathbf{R})|}{\hslash }\end{array}$(10) where $H_{\{k\}}^Q(\mathbf{R})$ represents the electric quadrupole interaction, which is one of the possible interaction Hamiltonian $H_{\{k\}}^{\mathrm{int}}(\mathbf{R})$ between atom and light field. The interaction Hamiltonian is a multipolar series about the centre of mass coordinate $\mathbf{R}$ and can be written as [27]: (11) $\begin{array}{r}{H}_{\{k\}}^{\mathrm{int}}(\mathbf{R})=H_{\{k\}}^D(\mathbf{R})+H_{\{k\}}^Q(\mathbf{R})+\dots \dots ..\end{array}$(11) where $H_{\{k\}}^D(\mathbf{R})$ is the electric dipole interaction and is given by: (12) $\begin{array}{r}{H}_{\{k\}}^D(\mathbf{R})=-\mu \cdot {\hat{\mathbf{E}}}_{\{k\}}(\mathbf{R})\end{array}$(12) where $\hat{\mu }=q\mathbf{r}$, $\mathbf{r}$ the internal position vector, is the electric dipole moment vector and ${\hat{\mathbf{E}}}_{\{k\}}\mathbf{(}\mathbf{R})$ is the electric field vector. $H_{\{k\}}^Q(\mathbf{R})$ is the electric quadrupole interaction and is defined as: (13) $\begin{array}{r}{H}_{\{k\}}^Q(\mathbf{R})=-\frac{1}{2}\sum _{\mathrm{ij}}{\hat{Q}}_{\mathrm{ij}}\frac{\mathrm{\partial }{\hat{E}}_{\{k\}}^j(\mathbf{R})}{\mathrm{\partial }{R}_i}\end{array}$(13) This is essentially the coupling between the Cartesian components of the quadrupole moment tensor ${\hat{Q}}_{\mathrm{ij}}=-e{q}_i{q}_j$, representing the components of the internal position vector $q=(X,Y,Z)$ and the gradients of the electric field vector components, evaluated at the center-of-mass coordinate $\mathbf{R}=(r,z)=(x,y,z)$. Without a loss of generality, we assume that the electric field ${\hat{\mathbf{E}}}_{\{k\}}\mathbf{(}\mathbf{r})$ is polarized along the x direction, which yields the following form of $H_{\{k\}}^Q(\mathbf{R})$: (14) $\begin{array}{r}{H}_{\{k\}}^Q(\mathbf{R})=-\frac{1}{2}\sum _i{\hat{Q}}_{\mathrm{iX}}\frac{\mathrm{\partial }{\hat{E}}_{\{k\}}^X(\mathbf{\text{R}})}{\mathrm{\partial }{R}_i}\end{array}$(14) where ${\hat{Q}}_{\mathrm{ij}}=Q_{\mathrm{ij}}(\pi +{\pi }^{†})$ are the quadrupole tensor operator elements, $Q_{\mathrm{ij}}=⟨i|{\hat{Q}}_{\mathrm{ij}}|j⟩$are the quadrupole matrix element, and $\pi ({\pi }^{†})$ are the atomic level lowering (raising) operators.

5. Surface optical lattice parameters

Due to complex Gaussian laser light, surface optical lattices depend on evanescent waves created close to dielectric surfaces. In principle, the generation of lattices with a single laser wave in such a light class is possible, but their generation with standing waves created by counter-propagating laser waves ensures precise control and careful preservation. Because of any tiny fluctuation or imbalance in polarization, phase difference and intensity associated with the laser waves could lead to the instability of the lattice.

With an appropriate atomic transition, in which dipole interaction is prohibited, and quadrupole interaction is permitted, this transition engages with the evanescent field at near-resonance, leading to two classes of optical forces. The effects of these forces are different. Therefore, one atomic optical element must constitute a significant obstacle to another. For example, concerning optical lattices, the subject of this work is required to provide distinct periodical regions that possess tight spatial confinement and high local field intensity. It has been described in some recent studies that these different intensity patterns of the lattice are distinct topologies [7]. In these regions, atoms cool down and congregate at the quadrupole potential extrema (a minimum for negative detuning ${\mathrm{\Delta }}_0>0$ and a maximum for positive detuned light ${\mathrm{\Delta }}_0<0$). These atomic ensembles give an ideal quantum system, so they can sometimes tunnel between lattice topologies according to the rules of quantum mechanics. This occurs even if the potential depth of the lattice points exceeds the kinetic energy of the atoms. Indeed, higher intensity means more depth, reducing the rate of quantum tunnelling processes.

The accurate statement of the desired and undesirable optical forces depends on identifying the main lattice parameters in the optical lattices. There are two critical parameters: the trapped potential depth and the periodicity distribution (or the spacing between the adjacent maximum intensity points). It is well-known that the potential depth increases according to the increase in light intensity (or ${\mathrm{\Omega }}_{\{k\}}^Q$). Still, the fluctuation (due to ${\mathrm{\Gamma }}_Q^{}$) and escape (due to $\mathrm{\nabla }{\theta }_{\{k\}}$) of the trapped atoms increase with increasing scattering forces. The waveguide scheme—as proven in dipole interaction—provides a high-intensity output and, at the same time, provides a short interaction time. Thus, it gives an acceptable solution in which the depth increases and the fluctuation decreases simultaneously, while the escape problem can be eliminated by using two counter-propagating identical Gaussian waves. Hence, the first lattice parameter becomes under actual control, giving high stability. For periodicity distribution, the waveguide scheme also provides a precise mechanism that controls the number of peaks and valleys on the lattice (and thus the periodicity) by controlling the width of the waveguide. Controlling the electric field polarization ($\mathrm{TE}$and $\mathrm{TM}$) and its degree by this technique is classified as a high-pass filter structure. This gives more stability and more extended continuity than the traditional method, in which the periodicity is just tuned by changing the incident light intensity with the accompanying technique difficulties related to the required accuracy.

6. Results and discussion

A perfect Gaussian wave acutely represents the ideal laser output. Therefore, physical studies usually take more practical Gaussian waveforms, such as Hermite-Gaussian, Laguerre-Gaussian, or Bessel-Gaussian. These forms are three complete families of exact and orthogonal solutions to the paraxial wave equation and can be derived by acting with differential operators on the plane wave representation of the fundamental Gaussian mode using a seed function [28]. However, an array of laser waves with different potential regions by which optical lattices may be represented (in both bounded and unbounded spaces) may be created by all three forms. Since the Hermite form does not have an azimuthal phase dependence term, as in the Laguerre and Bessel forms (i.e. they possess an orbital angular momentum), making the lattices generated by Hermite form more stable and continuous. At the same time, it is not possible to produce any class of optical vortex from any order of Hermite form because the basis for the emergence of rotation depends on the presence of orbital angular momentum. Here, the Hermite form is used because it is more compatible with the primary purpose, and accordingly, the Hermite-Gaussian envelope $G(x,y)$ for an uncoated substrate is given by [8]: (15) $\begin{array}{rl}G(x,y)& =\frac{{\left(\frac{2}{{2^{p+\ell }}_{}p{!}^{}\ell {!}^{}\pi }\right)}^{1/2}}{\sqrt{1+{(\mathrm{xsin}{\theta }_i/z_R)}^2}}\\ & \times \exp \left[\frac{\mathrm{ik}}{2}\frac{(x^2{\cos }^2{\theta }_i+y^2)}{[\mathrm{xsin}{\theta }_i+(z_R^2/\mathrm{xsin}{\theta }_i)]}\right]\\ & \times \exp \left[-\frac{(x^2{\cos }^2{\theta }_i+y^2)}{w_0^2[1+{(\mathrm{xsin}{\theta }_i/z_R)}^2]}\right]\\ & \times H_p\left(\frac{\sqrt{2}\mathrm{xcos}{\theta }_i}{w_0\sqrt{1+{(\mathrm{xsin}{\theta }_i/z_R)}^2}}\right)\\ & \times H_{\ell }\left(\frac{\sqrt{2}y}{w_0\sqrt{1+{(\mathrm{xsin}{\theta }_i/z_R)}^2}}\right)\\ & \times \exp \left[i(p+\ell +1){\tan }^{-1}\left(\frac{y}{\mathrm{xcos}{\theta }_i}\right)\right]\end{array}$(15) where $w_0$ is the beam waist of the Hermite-Gaussian light and $z_R$ is Rayleigh ranges while the special functions $H_p$ and $H_{\ell }$ are the Hermite polynomials of order $p$ and $\ell$.

For the specifications of the optical structure, the same design value of the parameters used in Refs. [19,20] listed in Table 1 will be selected where they have given significant theoretical and experimental results under the influence of the dipole interaction. With these parameters $\lambda =785.8\mathrm{nm}$, the maximum transmission coefficient is $\tau =165$ at a resonance angle ${\theta }_i^{\mathrm{res}}={62.6}^{\circ }$. The only allowed resonance corresponds to the lowest transverse electric $T{E}_{01}$ and $T{E}_{10}$ fields, which have the lowest cutoff frequency (longest cutoff wavelength), and the lowest attenuation of all modes in a planar dielectric waveguide. Besides, they are polarized everywhere in one direction. This property is critical, particularly in optical lattices, because the possibility of escaping in other directions is reduced by it, supporting continuity and stability. The availability of a high degree of these two features makes it more possible to control the dynamic action of lattice topologies. Cesium atom with the quadrupolar transition corresponds to $\lambda =675.0\mathrm{nm}$ and ${\mathrm{\Gamma }}_Q=7.8\times {10}^5{s}^{-1}$ is taken to explore the influence of the quadrupolar interaction. The effective positive detuning is ${\mathrm{\Delta }}_0=(\omega -{\omega }_0)={10}^3{\mathrm{\Gamma }}_Q$ assumed to be the electric-quadrupolar moment ${\hat{Q}}_{\mathrm{XX}}=10e{a}_B^2$ (where $a_B$denotes the Bohr radius). Finally, an original Hermit-Gaussian field was prepared with $I={10}^{9}W{m}^{-2}$, $w_0=\lambda /2$ and $z_R=1.1\times {10}^{-3}m$.

Table 1. The Design Values.

LAYER	CHEMICAL COMPOUND	REFRACTIVE INDEX	WIDTH
Substrate	LaSFN18 glass	1.894	–
Solid gap layer	Silicon dioxide $\mathrm{Si}{O}_2$	1.46	$350.0\mathrm{nm}$
Waveguide layer	Titanium dioxide $\mathrm{Ti}{O}_2$.	2.37	$87.0\mathrm{nm}$

The spatial distribution (with its density plot) of the quadrupolar potential $P_{k10}^Q(r)$ due to the lowest transverse electric $T{E}_{01}$ field in the $\mathrm{xy}$plane at a fixed value $z>0$ above the waveguide layer is shown in Figure 2. A similar distribution with the same parameters for $T{E}_{10}$ field is shown in Figure 3.

Figure 2. The spatial distribution of the quadrupolar potential of an atom interacting with a evanescent field at $z>0$ for a negative detuning for $T{E}_{01}$ and the corresponding density plot.

Figure 3. The spatial distribution of the quadrupolar potential of an atom interacting with a evanescent field at $z>0$ for a negative detuning for $T{E}_{10}$ and the corresponding density plot.

The topology of the generated lattices contains only two identical quantum wells that are similar in shape and magnitude but perpendicular in direction. These two situations represent the simplest topology that can be obtained with the Hermite-Gaussian. In both, the minimum depth is seen to be of the following order: (16) $\begin{array}{r}{P}_{min}\approx 12.5\times {10}^{-3}K\end{array}$(16) which is sufficiently deep to trap the Cesium atoms. This depth is sufficient to allow for several quasi-harmonic trapping (vibrational) states [22]. The vibrational frequency of these states can be estimated simply using the parabolic approximation or obtained straightforwardly by the numerical solution of the two-dimensional Schrödinger equation. Besides being more accurate, the second method is also characterized by the possibility of using it to calculate the probability of escape by tunnelling from the first to the second well. This is similar to the problem of the square well of finite depth in quantum mechanics.

A variation of the quadrupolar potential in a central cross-section corresponding to a different solid gap thickness $d_2$ is shown in Figure 4. The minimum depth can be further enhanced by increasing the solid gap thickness $d_2$. It is also possible to obtain a minimal enhancement in a limited range by controlling the angle of incident ${\theta }_i$. Still, it is not specific to a waveguide scheme alone, which has not been shown.

Figure 4. Variation of the of the quadrupolar potential of an atom interacting with an evanescent field at $z>0$ for a negative detuning for $T{E}_{10}$within a central cross-section. Here the different curves correspond to different solid gap thickness $d_2$: $390\mathrm{nm}$ (dash-dots); $370\mathrm{nm}$ (dashes); and $350\mathrm{nm}$ (solid-line). The parameters are the same as those used in the evaluation of Figure 2.

The non-shallow depth provided by the dielectric waveguide scheme provided the desired continuity for the period of the lattice work. Continuity in such a range is not possible in traditional schemes with rapid and transient action. The depth of the two wells can be doubled without any spatial displacement by using two identical opposite fields (two counter-propagating fields), as shown in Figure 1(d). This configuration also gave an additional property represented in the total elimination of the scattering in the direction of propagation, which is a significant source of continuity dissipaters in a single field configuration. On the other hand, the high filtering feature provided by the dielectric waveguide scheme limits the effect to the lowest order fields ($T{E}_{01}$ and $T{E}_{10}$), giving an acceptable degree of spatial stability. Even the losses of quantum tunnelling are limited to two adjacent wells. Finally, it should be noted that a more complex optical distribution of the potential can be obtained by increasing the waveguide width $d_3$, which means that higher-order modes are allowed to propagate, and thus a more significant number of quantum wells of different depths are formed on the waveguide-vacuum interface. With this design key, the potential distributions for any order of fields (either $T{E}_{\mathrm{p\ell }}$ or even $T{M}_{\mathrm{p\ell }}$) can be configured, suggesting that it is in principle possible to create a two-dimensional optical lattice of any size.

7. Conclusions

In conclusion, we have presented a controllable scheme of a two-dimensional surface optical lattice with high continuity and considerable topology and intensity distribution stability. We have shown that any neutral atom with an active electric-quadrupole transition and forbidden electric-dipole transition approaching the vacuum-waveguide interface interacts with this lattice to create an effective periodic optical potential that can be exploited in trapping atoms.

The essential properties of the waveguide scheme, such as the decay length of the evanescent field and hence the potential depth and the type of polarized electric field, depending on easy design parameters, as we have seen. The potential depth can be controlled by changing the thickness of the solid gap layer, while the electric field polarization can be controlled by changing the thickness of the waveguide layer. The notable feature is that both controls are performed without changing the original incident field intensity. The simplest lattice topology contains two identical quantum wells and can be produced with the minimum operating waveguide thickness, where the guide allows only the lowest order field to propagate within. These desirable properties make this scheme promising for exploitation in atomic manipulation in a way that exceeds the rest of the traditional scheme.

The results presented here may differ slightly compared to the predictable experimental results because some expected losses from the absorption and scattering processes are related to the nature and roughness of the surface or even those losses arising from manufacturing defects and deposition accuracy. Thus, the exact selection of the optimal thickness of both the solid gap and waveguide layers depends on the values of the losses. All losses can be collected in one factor and then combined in numerical relations, but the actual estimate for each depends on the availability of experimental data. Modern thin-film technologies have made all those losses reach minimal values considered somewhat ineffective [19–21].

Finally, it may be noted that there is insufficient emphasis on the destructive effect of decay emission, as in most of the optical lattices studies generated in unbounded space, uncoated interface, and monolayer scheme. The dielectric multilayer layers allow the suppression of decay emissions and manipulating atom incidents with higher velocities. This is also due to the unique design parameters property so that the atom-field interaction time, and hence the probability of decay emission, can be controlled to keep it very low [9].

As a natural development of this work, it could be interesting to replace the Hermit light with a light possessing angular momentum properties, such as a Laguerre or Bessel light [29–32]. It is expected that various forms of controllable optical vortices are created. This is the topic of ongoing work and will be part of a future publication.

Disclosure statement

No potential conflict of interest was reported by the author(s).