The Influence of s-wave interactions on focussing of atoms

Atom lithography is a technique where the gradient forces applied by laser fields on a beam of atoms are used to direct the atoms into nanostructures deposited on a plane surfaceTimp et al. (1992); McClelland et al. (1993). It has been the topic of considerable study in the field of atom optics over the last two decades as it provides a scheme of writing nanometer structures directly onto a substrate in parallel process Meschede (2005). The main application in the development of nano-lithography could be the race for an increased density of transistors in the computer chips Balykin and Melentiev (2009). The possibility of using light to deposit feature sizes of a few nanometers was first suggested in 1987 by Balykvin and Letokhov Balykin and Letokhov (1987, 1988). Later in 1991, McClelland and Scheinfein McClelland and Scheinfein (1991) proposed a particle optics approach in which the atom lens created by the laser light can focus an atomic beam down to a surface. The principle was experimentally demonstrated by Timp Timp et al. (1992) in 1992 using Na deposited on Si via a standing light wave, and was similarly followed by McClelland et al. McClelland et al. (1993) to focus Cr in the presence of a 1D lattice. Atom lithography continued to develop to two- and three-dimensional nanostructures in a single process utilizing the selectivity of atom-light interaction (using 2D or 3D lattices) Oberthaler and Pfau (2003). There have also been demonstrations depositing Yb Ohmukai, Urabe, and Watanabe (2003) and Fe Myszkiewicz et al. (2004); Smeets et al. (2010). Almost all the experiments accomplished so far, have used an oven source of atoms in which the beam is collimated with an aperture followed by a transverse laser cooling processMcClelland et al. (1993) before traveling through a focusing potential. The traditional study of focusing in optical lattices uses the classical trajectories model of atomic motion when travelling through the light McClelland and Scheinfein (1991). The principle is based on atom-light interactions, resulting from the dipole force Bjorkholm et al. (1978) which causes neutral atoms to become manipulated with a near resonant laser light Bjorkholm et al. (1978, 1980); Balykin et al. (1988). Ideally, this model predicts almost perfect focusing in the absence of interactions.

Nevertheless, there are some advantages in using a Bose-Einstein condensate (BEC) of neutral atoms as the source of atom deposition. Since the de-Broglie wavelength of a gas of atoms is of the order of the mean field distance between particles, an ultra cold source such as a BEC would bring atoms to wavelengths of $\sim$ nm or pm for nano-Kelvin temperatures Henn et al. (2008) resulting in an excellent collimation of the beam of atoms as well as a high flux density Ziegler and Shukla (1998). Using a BEC source can also reduce effects such as chromatic aberration and angular divergence McClelland et al. (1993), with the longitudinal and transverse velocity distributions typically being much lower when incident on the surface compared to those resulted from thermal sources.

However, for BEC’s, the effect of interactions must be accounted for by introducing the s-wave interaction between atoms. The investigation of the matter wave focusing dynamics of a trapped BEC for both an interacting and non-interacting case was conducted theoretically by D. Muuray and P. Ohberg in 2004 Murray and Öhberg (2005). In their work, they derived the time to focus as a function of the focusing strength for a 3D Bose-condensed cloud. The significance of this research is to take atomic interactions into account when focusing a free propagating BEC and to scale its effect on the broadening of the focal spot sizes and peak densities achievable in realistic nano-lithography experiments.

The principle of atom lithography using a BEC is schematically depicted in Fig 1. The cloud of ⁸⁷Rb is initially confined by a harmonic trap. Turning off the trap, the released condensate expands whilst propagating along the vertical axis. It then encounters the optical lattice being focused by its nodes along the horizontal axis resulting in a large periodic array of nano-structures that are separated by $\lambda/2$ where $\lambda$ is the wavelength of lattice.

In this paper, the focal properties of an optical lattice are derived by the classical equations. Using this model, we estimate the Full Width at Half Maximum (FWHM) as well as peak density of ⁸⁷Rb focused structures, which can be further used to determine the possible contributions of structure broadening such as the spherical and diffraction aberrations. Following this, the Gross-Pitaevskii Equation (GPE) is introduced and its application for the purpose of BEC lithography is studied. Conducting various numerical simulations via the GPE for different ⁸⁷Rb BEC and focusing potential parameters, we estimate resultant focused structure resolutions and peak densities. Not only is the key role of the s-wave interaction within the BEC investigated through the GPE, but also by taking the transverse and longitudinal velocity distributions of the condensate, the classical trajectories model is developed to consider the influence of atomic interactions on focused profiles.

When atoms are exposed to a potential field, an atomic dipole moment is induced by the electric field. The induced dipole moment interacts with the gradient of the field resulting in a dipole force gradient applied towards the nodes or anti-nodes of the periodic field. The resultant dipole potential on stationary atoms is well established . However, when atoms carry a momentum, the created optical dipole potential behaves as a focusing thick lens on neutral moving atoms Gordon and Ashkin (1980):

where $\Delta$ denotes the detuning of the laser frequency from the atomic resonance, $\hbar$ is the Plank constant, $\gamma$ is the natural linewidth of the atomic transition (spontaneous decay rate from the excited level) and $I_{s}$ is the saturation intensity related to the atomic D₂ line transition, $5~{}^{2}S_{1/2}\longrightarrow 5~{}^{2}P_{3/2}$, for ⁸⁷Rb. While a red detuned laser light from resonance, $\Delta<0$, in ⁸⁷Rb D₂ line, directs the falling atoms towards the nodes of the standing potential, a blue detuned light, $\Delta>0$ brings the atoms to the anti-nodes of the focusing potential ($\Delta=\omega_{L}-\omega_{0}$ where $\omega_{L}$ and $\omega_{0}$ are respectively the frequency of incident laser light and resonance between $5~{}^{2}S_{1/2}$ and $5~{}^{2}P_{3/2}$). We consider a geometry for the optical potential such that the laser intensity profile, $I(x,z)$, shapes a mask with a periodic scheme of multiple focusing lenses along the direction of focusing atoms. Hence, a Gaussian standing potential is considered which contains of a sinusoidal behavior of the intensity along the $x$-axis (the focusing direction) and a Gaussian envelope function along the $z$-axis, the direction of falling atoms. This optical lattice potential is represented by

where $I_{0}$ is the maximum intensity of the spatially varying Gaussian profile, $\sigma_{z}$ is the radius of the beam at $1/e^{2}$ value of the maximum intensity and $k=2\pi/\lambda$ is the wavenumber. Since almost no force is applied to atoms along the $y$ direction compared to the other two directions, neglecting this causes the focusing potential, Eq. (1), to become independent of $y$.

In order to scale the parameters involved in an optimal focusing potential, we exploit the classical trajectories approach McClelland et al. (1993). Neglecting the $y$ axis due to the symmetry of problem, the classical equations of motion for atomic trajectories are given by

Using conservation of energy, one can combine Eqs. (3) and (4) solving for $x$ as function of $z$

where $E_{0}$ represents the total energy of each atom and $x^{\prime}=\frac{dx}{dz}$.

To obtain the focal properties of the lattice, we consider the paraxial approximation McClelland (1995) in which the sinusoidal term of the potential consisting of a multi-node wave along the $x$ axis is converted to a single node harmonic part. In addition, the approximation neglects aberrations, i.e. the trajectories are assumed to be perfectly parallel to the $z$ axis when falling towards the lattice. These can be implemented by $U(x,z)\ll E_{0}$, $\frac{dx}{dz}\ll 1$ and $kx\ll 1$. Applying these, Eq. (5) converts to

Now considering $\sin(kx)\sim kx$, $\sin^{2}(kx)\sim 0$ and $\cos(kx)\sim 1$, one obtains the following second order differential equation

According to the relation between the maximum intensity and the corresponding value of power in a standing wave Gaussian beam McClelland (1995), $I_{0}=8P_{0}/\pi\sigma_{z}^{2}$, the required power value of the harmonic potential to focus atoms at any desired spots along the focal axis ($z$-axis) is achieved as function of the potential factors and atoms’ initial kinetic energy,

where $\xi=q^{2}\sigma_{z}^{2}$ is a dimensionless parameter.

We now proceed with a specific example for focusing ultra-cold ⁸⁷Rb atoms using the paraxial approximation displayed in Fig. 2. To meet the requirements of approximation, the potential wavelength is chosen to be 400 times greater than the actual ⁸⁷Rb D₂ line wavelength, $\lambda=400\lambda_{\text{D}_{2}}=312~{}\mu$m where $\lambda_{\text{D}_{2}}=780.027$ nm. This forms a potential with a harmonic distribution along the $x$ axis. While the potential radius is adjusted to $\sigma_{z}=100~{}\mu$m, $\xi=5.37$, $3.37$, $2.37$, and $1.37$ corresponding to the potential power of $P_{0}=43.018$ , $26.996$, $18.986$, and $10.975~{}\mu$W respectively provided that the initial velocity of atoms and the detuning from resonance are selected as $v_{i}=v_{z}=1$ cm/s and $\Delta=200$ GHz. For each value of $\xi$ or $P_{0}$, two ⁸⁷Rb atomic trajectories falling symmetrically from $x_{i}=\pm\lambda/4=\pm 78~{}\mu$m landing at $x_{f}=0$ and a particular $z_{f}$ are plotted. As a case in point, for $\xi=5.37$, solving Eq.(7) yields the focal point optimally placed at $z_{f}=0$ and $x_{f}=0$ (the center of potential). However, reducing power values results in the focal point being located at $z<0$ below the potential center.

We note that this is an ideal scenario in which for a particular value of $P_{0}$, all atoms are focused exactly at one spot and no aberration is considered. However, in practice there always exists a spherical aberration in optics while atoms are entering a thick lens Grivet, Hawkes, and Septier (2013). This arises from the fact that atoms traveling farther to the focal axis will experience a weaker force than those that are closer to this axis. This effect can be considered through the exact numerical solution of Eq.(5). As an illustration, the atomic trajectories for ultra-cold ⁸⁷Rb starting at $z_{i}=20~{}\mu$m whilst moving at a longitudinal velocity of $v_{z}=1$ cm/s through the potential of a radius size of $\sigma_{z}=10~{}\mu$m focused at $z_{f}=0$ ($\xi=5.37$) have been numerically simulated and are depicted in Fig 3. Here, we choose a lattice whose wavelength is 16 times larger than the actual wavelength of ⁸⁷Rb D₂ line, $\lambda=16\lambda_{\text{D}_{2}}=12.48~{}\mu$m. This is practical in realistic experiments through the use of a Spatial Light Modulator (SLM) McGloin et al. (2003); Zhu and Wang (2014). Aligning the laser detuning to $\Delta=200$ GHz, the required optimal value of lattice power and maximum peak intensity to focus the atoms to the center of potential ($z=0$) are calculated as $P_{0}=0.068~{}\mu$W and $I_{0}=1.752\times 10^{3}$ W/m². Unlike the paraxial solution, for each lattice node here, the atoms do not fully land at the same focal point, and this process is identical for all lattice nodes implying the spherical aberration. Hence, one could evaluate the linewidth of the created structure and estimate the broadening contribution arising from this effect inside every lattice slit, $D=\lambda/2=6.24~{}\mu$m. For instance, we have selected the lattice central node, from $x=-\lambda/4=-3.12~{}\mu$m to $x=\lambda/4=3.12~{}\mu$m, and plotted a histogram distribution for 62400 trajectories arriving at the focal plane ($z=0$) in Fig 4. Applying a Kernel fit to the focused profile, the value of FWHM is acquired as $(\Delta x)_{\text{sph}}=0.136~{}\mu$m.

However, the spherical aberration is not the only reason for profile broadening. In a lens like lattice, there are finite adjacent apertures positioned in a distance of $\lambda/2$ apart along the entire lens. Since atoms exhibit wave-like behavior, they interfere together with their de-Broglie wavelengths, $\lambda_{\text{dB}}$, while crossing through the apertures. This causes a limit on the linewidth of structures known as the diffraction effect. The angular resolution produced by the diffraction is estimated by Rayleigh Criterion Hecht (2017); Born and Wolf (2013)

where the factor $\beta=1.22$ accounts for a circular aperture Hecht (2017) while $\beta=0.88$ is dedicated to a rectangular (or cylindrical) aperture Shiono et al. (1987); Rozaliya, Gene et al. (2014); McClelland (1995). The de-Broglie wavelength of atoms is defined as $\lambda_{\text{dB}}={h}/{mv_{z}}$ where $v_{z}$ is the most probable longitudinal velocity. The angular resolution in Eq. (10) can be converted to the diffraction-limited FWHM by

where $\theta$ is considered to be a very small angle. Since each aperture has a rectangular structure to the incident atomic beam, $(\Delta x)_{\text{diff}}$ would be

For the previous example, for the atoms with a mass of $m=1.44\times 10^{25}$ Kg falling at $v_{z}=1$ cm/s through a lattice of wavelength of $\lambda=312~{}\mu$m, the diffraction-limited FWHM is estimated as $(\Delta x)_{\text{diff}}=0.3589~{}\mu$m. In this calculation, the focal length (the difference between the coordinates of the focal point and the principal plane location when focusing at $z=0$), $f=5.531~{}\mu$m, $\sigma_{z}=10~{}\mu$m and $\lambda=12.48~{}\mu$m. According to Eq. (12), this broadening is proportionally related to the focal length and is inversely associated with the velocity of atoms as well as the size of apertures. Hence, to minimize $(\Delta x)_{\text{diff}}$, one needs to either use a relatively shorter focal length, a higher longitudinal velocity or a larger potential wavelength.

Eventually, taking into account the total structure broadening predicted by the classical trajectories model in absence of atomic interactions,

and considering the results for $(\Delta x)_{\text{sph}}$ of the stated example, we can estimate the total value of FWHM via the classical trajectories model for the cases, $z_{f}=0$ (with $\sigma_{z}=10~{}\mu$m), which is evaluated as $(\Delta x)_{\text{Classical}}=0.495~{}\mu$m.

Using a BEC Griffin, Snoke, and Stringari (1996) as a source of ultra-cold atoms brings several advantages to atom deposition as it can significantly reduce the linewidth of longitudinal and transverse velocity distributions providing excellent coherence and collimation for the atomic beam as well as offering relatively small de Broglie wavelengths, high peak densities and quality spatial modes Henn et al. (2008); Ziegler and Shukla (1998); Pethick and Smith (2002). In this section, we take into account the atomic interaction when focusing a free propagating ⁸⁷Rb BEC. The impact of interactions on the broadening of the nano-focal spot sizes and peak densities are estimated, which is accomplished through the use of the GPE Gross (1961); Pitaevskii (1961a).

The three-dimensional time dependent GPE modeling the dynamics of a BEC is represented by Gross (1961); Pitaevskii (1961b); Gross (1957); Ginzburg and Pitaevskii (1958)

where $\psi(\mathbf{r},t)$ indicates the BEC wavefunction at different times of propagation, $V_{\text{ext}}$ is the time-dependent external potential applied on the BEC, $m$ is the atomic mass and $\hbar$ is the Planck constant. The interactions between atoms within the cloud are considered using a non-linear mean field potential, $V_{\text{mean}}$, which estimates the averaged exerted potential on any particular atom by all other atoms given by Dalfovo et al. (1999a); Leggett (2001); Pethick and Smith (2002); Pitaevskii and Stringari (2016)

where

quantifies the atomic interactions, $|\psi(\mathbf{r},t)|^{2}$ describes the atomic density, and $a_{s}$ is the s-wave scattering length. The value $a_{s}$ can be practically tuned from $a_{s}>0$ (repulsive interactions) to $a_{s}<0$ (attractive interactions) utilizing Feshbach resonance McDonald et al. (2014).

Due to the s-wave interactions between the atoms within a condensate, the velocity of atoms does not remain constant over time. Hence, a distribution is formed in both the longitudinal (along the $z$ axis) and transverse (along the $x$ axis) velocity profiles so that each one encompasses an associated peak representing the most probable velocity. Taking a Fast Fourier Transform (FFT) directly from the BEC density profile in position space, one is able to gain the BEC density profile in momentum space at different times of propagation. Since inter atomic interactions cause the condensate to expand (for $a_{s}>0$) over time the density profile in momentum space also spreads.

As an illustration, we have studied separately the BEC transverse and longitudinal velocity distributions at $t=2$ ms in Figs 6(a-c). Here, the BEC is kicked by $v_{z}=1$ cm/s when being released from the trap. Neglecting the gravity acceleration in BEC’s falling, the linewidth for each distribution is derived by applying a Gaussian fit [see Figs 6(b, c)]. The values of FWHM for the transverse and longitudinal profiles at $t=2$ ms are estimated as $\Delta v_{x}=0.0264$ cm/s and $\Delta v_{z}=0.1791$ cm/s respectively. It is clear that the tight trap frequency along the $z$ axis ($\omega_{z}=2\pi\times 70$ Hz) causes a significantly enhanced widening velocity profile along the falling axis compared to the horizontal axis ($\Delta v_{z}\gg\Delta v_{x}$).

In sections IV.1 and IV.2, we explore the possibility of implementing the information from $\Delta v_{z}$ and $\Delta v_{x}$ to the classical trajectories model to predict the resultant structure broadenings.

As stated thus far, the influence of the s-wave interaction between atoms appears as the longitudinal and transverse velocity distributions in the classical calculations. That being said, one can define the following equivalency

where

which can be rewritten as following using Eq. (13),

For the examples discussed in sections IV.1 and IV.2, adding $(\Delta x)_{\text{c}}=0.405~{}\mu$m and $(\Delta x)_{\text{a}}=0.146~{}\mu$m to $(\Delta x)^{\text{non}}_{\text{Classical}}=0.495~{}\mu$m [which comprises $(\Delta x)_{\text{sph}}=0.136~{}\mu$m and $(\Delta x)_{\text{diff}}=0.359~{}\mu$m, see section (II)], we estimate the total FWHM of the focused structures as $(\Delta x)^{\text{int}}_{\text{Classical}}=1.046~{}\mu$m via the classical trajectories model. This is in good agreement with GPE results, $(\Delta x)^{\text{int}}_{\text{GPE}}=1.074~{}\mu$m [see Fig. 5(e)] indicating a reliable mapping between the classical trajectories approach using the BEC transverse and longitudinal velocity profiles from the GPE. Table 2 compares the outcomes resulting from the GPE and classical methods for three various examples for both interacting and non-interacting cases.

We now study in more detail the impact of the focusing potential aperture size (or the potential wavelength) as well as its radius on the deposited BEC structures. We consider a ⁸⁷Rb BEC trapped by $\omega_{x}=2\pi\times 10$ Hz, and $\omega_{y}=\omega_{z}=2\pi\times 70$ Hz, whose center-of-mass is at $z_{0}=500~{}\mu$m from the lattice center at $z=0$. In the results presented, the freely propagating and interacting ($a_{s}=100a_{0}$) condensate is aimed to be focused optimally at $z=0$ ($\xi=5.37$).

We initially consider a lattice radius size of $\sigma_{z}=10~{}\mu$m while three various potential wavelengths are employed, $\lambda=16\lambda_{\text{D}_{2}}$, $8\lambda_{\text{D}_{2}}$, and $4\lambda_{\text{D}_{2}}$. The results are extracted for a range of initial momentum kicks differing from $p=16~{}\hbar k$ to $p=128~{}\hbar k$ (where $k$ is the wavenumber associated with the ⁸⁷Rb D₂ line defined by $k=2\pi/\lambda_{\text{D}_{2}}$) is applied to the BEC. As a result, for an optimal focus, the lattice power and peak intensity are adjusted accordingly based on the BEC kinetic energy for any certain momentum kick. Figs. 10(a, b) display the associated outputs for the focused BEC linewidths and peak densities respectively. The FWHM in each numerical calculation is achieved by taking an average over the linewidth of the structure peaks whose height are greater than $1/e$ of the highest central peak.

Analyzing the results, firstly, one can conclude that imparting higher momentum kicks to the BEC reduces the focal spot sizes and enhances the peak densities for any potential wavelength. However, larger $\lambda$ values necessitate higher potential powers and peak intensities resulting in the superior structure resolutions and heights (see the blue stars in Figs. 10(a, b) compared to the red triangles and green crosses). Furthermore, one can understand that the FWHM and peak density data points for $\lambda=16\lambda_{\text{D}_{2}}$, $8\lambda_{\text{D}_{2}}$, and $4\lambda_{\text{D}_{2}}$ tend to a steady state at relatively large magnitudes of initial momentum kick [i.e. $p\geq 96~{}\hbar k$, see Figs. 10(a, b)]. This implies the characteristic factors of a focused profile (including the resolution and peak density) become independent of the BEC momentum kick at relatively high longitudinal velocities. At this point, the profile resolution also becomes independent of the lattice wavelength while the focused peak density remains strongly dependent on the size of the momentum kick.

In Fig. 11(a, b), a fixed value of $\lambda=16\lambda_{\text{D}_{2}}$ is considered while three different lattice radii, $\sigma_{z}=10$, $20$ and $40~{}\mu$m are selected. Since an expansion in the potential radius size leads to a decline in the power and peak intensity values, it is expected larger FWHMs will be obtained as well as lower peak densities for $\sigma_{z}=40~{}\mu$m (indicated by the green crosses) compared to those of $\sigma_{z}=10~{}\mu$m (indicated by the blue stars). As an instance, choosing $\sigma_{z}=10$, $20$ and $40~{}\mu$m would respectively result in $(\Delta x)_{\text{GPE}}^{\text{int}}=19.68$, $23.27$ and $48.83$ nm for the resolutions, and $4.068\times 10^{4}$, $2.692\times 10^{4}$ and $1.501\times 10^{4}$ atoms/$\mu\text{m}^{2}$ for the peak densities at $p=128~{}\hbar k$. Moreover, the profile characteristic factors tend to a steady state for every potential size at larger momentum kicks [i.e. $p\geq 96~{}\hbar k$, see Figs. 11(a, b)].

Finally, the focused ⁸⁷Rb BEC profile is studied through a focusing lattice of $\lambda=\lambda_{\text{D}_{2}}$. The output FWHM and peak density values are plotted as a function of potential power in Figs. 12(a, b) [for $\sigma_{z}=100~{}\mu$m] and Figs. 13(a, b) [for $\sigma_{z}=200~{}\mu$m]. There are six categories of data points representing the condensate kicked by, $p=0$, $2$, $8$, $32$, $128$ and $256~{}\hbar k$. Unlike the previous case, the lattice power selection here takes a certain range (from $P=0.3$ to $1.83$ mW) regardless of an optimal power value for any momentum kick, $P=P_{\lambda_{\text{D}_{2}}}(n\hbar k)$. In other words, the focused profiles are examined at $z=0$ even when they are not at their optimal focus stage.

According to Figs. 12(a, b), for each value of momentum kick, a rise in the focusing potential power results in an enhanced resolution as well as a higher peak density. For instance, setting $P=1.83$ mW for $p=256~{}\hbar k$ produces a focused profile with $(\Delta x)_{\text{GPE}}^{\text{int}}=51.25$ nm and a highest peak of $2975$ atoms/$\mu\text{m}^{2}$. However, increasing the value of beam radius (from $\sigma_{z}=100$ to $200~{}\mu$m) broadens the corresponding profile resolutions creating shorter peaks, which is indicated in Figs. 13(a, b). In such a case, for $P=1.83$ mW and $p=256~{}\hbar k$, the resolution and highest peak density respectively reduce to $(\Delta x)_{\text{GPE}}^{\text{int}}=85.87$ nm and $1726$ atoms/$\mu\text{m}^{2}$.

It is also noticeable that for any magnitude of momentum kick applied to the BEC, the focused profile characteristic factors become independent of the potential power at larger $P$ values. As a result, in this case, there exists a threshold in power value for improving the profile resolution and peak density meaning that one may not expect to obtain a considerable improvement in focal spot sizes when using $P>2$ mW [see Figs. 12(a, b) and Figs. 13(a, b)]. We note that the threshold range, $P>2$ mW, is only the case for a lattice of $\lambda=\lambda_{\text{D}_{2}}$ as it could be reduced by a factor of ten to $P_{16\lambda_{\text{D}_{2}}}(128~{}\hbar k)=0.39$ mW for a lattice of $\lambda=16\lambda_{\text{D}_{2}}$, which could lead to a focused profile of $(\Delta x)_{\text{GPE}}^{\text{int}}=19.68$ nm [see Fig. 11(a)].

In this paper, we summarized the classical trajectories approach and its applications in focusing ultra-cold ⁸⁷Rb atoms. The structure of an appropriate focusing potential as well as its focal properties for an optimal focus were analyzed using the paraxial solution to the equations of atomic motion. Moreover, we derived a relation between the required value of potential power and the desired focus point on the focal plane along with employing a relevant example. The spherical aberration on broadening of focused structures was explored through numerical solutions to the classical equations. We then calculated the structure linewidth resulting from the diffraction contribution. We inferred that an increase in lattice aperture size and atomic longitudinal velocity or a reduction in the focal length results in a lower FWHM of diffraction as in this case atoms would have less chance to interfere together.

We then moved on to consider the influence of s-wave interactions between the atoms on focusing of ⁸⁷Rb condensate using the GPE. It is found that the resultant structure linewidths are significantly impacted by s-wave interactions. Moreover, we concluded that either reducing the lattice radius size or increasing the BEC initial kinetic energy could improve the characteristic factors (resolutions and peak densities) of a focused ⁸⁷Rb profile. Further to this, the contribution for both the BEC longitudinal and transverse velocity distributions in broadening the structure size was explored via the classical trajectories model. We found a reliable agreement between the GPE approach and a classical trajectories model which includes the interaction contribution.

Finally, conducting a variety of numerical simulations with different lattice radii and wavelengths, we noticed that narrower and higher density structures arise from relatively lower potential radius sizes and greater wavelengths. It was shown that nano-meter structures are achievable in principle. The example of this is a resultant structure of $(\Delta x)_{\text{GPE}}^{\text{int}}=19.68$ nm via a focusing lattice of $\sigma_{z}=10~{}\mu$m and $\lambda=16\lambda_{\text{D}_{2}}$. Furthermore, it was inferred that applying higher momentum kicks necessitating greater optimal potential powers and peak intensities cause superior profile resolutions and peak fluxes. However, for a certain lattice wavelength, there is a threshold point in potential power where for larger values than this point, the focused profile characteristic factors are no longer dependent on power magnitudes tending to a steady state. It was observed that exploiting a relatively smaller lattice wavelength increases the threshold power whilst causing a destructive impact on the structure resolution and peak density. For instance, the approximate threshold power for a lattice of $\lambda=\lambda_{\text{D}_{2}}$ and $\lambda=16\lambda_{\text{D}_{2}}$ is about $P_{\lambda_{\text{D}_{2}}}=1.83$ mW and $P_{16\lambda_{\text{D}_{2}}}=0.39$ mW resulting in $(\Delta x)_{\lambda_{\text{D}_{2}}}^{\text{GPE}}=51.25$ nm and $(\Delta x)_{16\lambda_{\text{D}_{2}}}^{\text{GPE}}=19.68$ nm.

The authors would like to thank Nicholas P. Robins for useful discussions and feedback. Grateful acknowledgement is also extended to Timothy Senden for the financial support of the project. This research was supported by the Research School of Physics, the Australian National University, and the School of Physics, University of Melbourne.