# Optimal binary gratings for multi-wavelength magneto-optical traps

Oliver S. Burrow  1,* Robert J. Fasano  2,3 Wesley Brand  2,3 Michael W.
Wright  1 Wenbo Li  1 Andrew D. Ludlow  2,3 Erling Riis  1 Paul F. Griffin  1
and Aidan S. Arnold1,$\dagger$ 1Department of Physics, SUPA, University of
Strathclyde, Glasgow, G4 0NG, United Kingdom  2National Institute of Standards
and Technology, 325 Broadway, Boulder, Colorado 80305, USA  3University of
Colorado, Department of Physics, Boulder, Colorado 80309, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Grating magneto-optical traps are an enabling quantum technology for portable
metrological devices with ultracold atoms. Gratings are chromatic, with
However, beam diffraction efficiency and angle are affected by wavelength,
creating a single-optic design challenge for laser cooling in two stages at
two distinct wavelengths – as commonly used for loading e.g. Sr or Yb atoms
into optical lattice or tweezer clocks. Here, we optically characterize a wide
variety of binary gratings at different wavelengths to find a simple empirical
fit to experimental grating diffraction efficiency data in terms of
dimensionless etch depth and period for various duty cycles. The model avoids
complex 3D light-grating surface calculations, yet still yields results
accurate to a few percent across a broad range of parameters. Gratings
optimized for two (or more) wavelengths can now be designed in an informed
manner suitable for a wide class of atomic species enabling advanced quantum
technologies.

††journal: oe

https://eqop.phys.strath.ac.uk/atom-optics/grating-mots

##  1 Introduction

Laser cooled atoms are required for a wide range of quantum technologies, and
there is growing demand to create portable, compact and robust devices capable
of leaving controlled laboratory environments [1, 2, 3, 4, 5, 6, 7]. The
grating magneto-optical trap (GMOT) [8, 9, 10, 11, 12, 13, 14] simplifies
laser cooling of thermal atomic vapors or beams when compared to traditional
methods, reducing the optical system requirements to a single input beam and a
planar optic. This reduction in size and complexity enables compact cold-atom
sources [15, 16, 17]. There are a number of advantages to tetrahedral
pyramid[18] and grating-based designs over standard single-input beam
pyramidal geometries [19, 20, 21, 22], namely: the mitigation of absorption-
induced beam shadows particularly for larger MOTs; high optical access; and
the optic can be both mass-produced and used ex-vacuo. This has led to
increasing interest in the GMOT technique for a variety of sensing and physics
applications [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].

The laser wavelength is critical for laser cooling, and must correspond to a
suitable transition in the atomic (or molecular [37, 38, 39]) species being
cooled. Gratings are chromatic by nature, with laser wavelength particularly
affecting both diffraction angle and efficiency – thereby affecting the
function of a GMOT. GMOTs have been realized for the atomic species Rb [9],
Li[26] and Sr [40, 41, 42] with grating optics designed specifically for these
elements. Furthermore, Rb optimized binary gratings [43, 44] have been
demonstrated to have suitable radiation pressure balance over a broad
wavelength range covering all alkali metals (Fig. 6 in [44]), and are
particularly appropriate for e.g. dual-species potassium and rubidium
experiments [45].

The broadband performance of the gratings can be exploited to design GMOT
optics suitable for laser cooling at multiple wavelengths. For example, in Yb
and Sr optical lattice clocks, large atom numbers are typically collected from
a thermal source using the primary cooling transition, at a ‘blue’ wavelength
($\lambda_{\textrm{b}}$) with a broad natural linewidth that connects singlet
states $|g\rangle\leftrightarrow|e_{\textrm{b}}\rangle$. Subsequently, in
order to reach ultracold temperatures and higher densities for loading into an
optical dipole trap or lattice, this is usually followed by cooling on a
secondary transition, at a more ‘red’ wavelength ($\lambda_{\textrm{r}}$) with
a narrow linewidth that connects a singlet to a triplet state
$|g\rangle\leftrightarrow|e_{\textrm{r}}\rangle$. Developing GMOT optics
suitable for these species will aid further development of compact, portable
optical lattice clocks [46, 4, 6].

Figure 1:  Simplified three-state atomic energy level diagram (a), with
species-specific ‘red’ and ‘blue’ transitions with wavelengths and natural
linewidths of $\lambda_{\textrm{r,b}}$ and $\Gamma_{\textrm{r,b}},$
respectively (Table 1). The approximate 2D schematic (b) illustrates how
overlaid ‘red’ and ‘blue’ beams of radius $\rho$ propagate downwards onto a
grating (gray), diffract at different first-order Bragg angles
$\theta_{\textrm{r,b}}=\arcsin(\lambda_{\textrm{r,b}}/d)$, and form beam
overlap volumes (shaded diamonds) with corresponding heights
$h_{\textrm{r,b}}=\rho\cot\theta_{\textrm{r,b}}$, respectively. A more
accurate 3D rendering of the overlap volumes is shown in (c). MOTs can in
principle form and be moved anywhere in the overlap zone corresponding to the
wavelength used, via the central zero of the magnetic quadrupole field
supplied by appropriate coils (not shown). Any ‘handover’ between the MOTs
(unaided by a transport trap) must occur in the intersection of the two
overlap volumes. The grating period is $d$ and etch depth $T$ which is the
same nomenclature as in [44, 43], except here the etched fraction of the
period is $p_{\textrm{e}}$ instead of $1-r$ to avoid confusion with the ‘red’
r labels.

The level scheme and concept of the bi-chromatic grating MOT are shown in Fig.
1 (a)-(c). A summary of some common laser-cooled elements with their ‘red’ and
‘blue’ transitions can be seen in Table 1. The most noticeable chromatic
grating effect is the different red (r) and blue (b) $m^{\textrm{th}}$-order
diffraction angles $\theta_{\textrm{r,b}}$ as governed by Bragg’s law
$\theta_{\textrm{r,b}}=\arcsin(m\lambda_{\textrm{r,b}}/d)$ for wavelengths
$\lambda_{\textrm{r,b}}$ normally incident on a grating of period $d$.
Consequentially the capture volume height
$h_{\textrm{b,r}}=\rho\cot\theta_{\textrm{b,r}}$ is smaller for the ‘red’ MOT
for the same input beam radius $\rho$, as illustrated in Fig. 1 (b,c). Here we
constrain grating periods $\lambda_{\textrm{r}}<d<2\lambda_{\textrm{b}},$ to
simplify the ‘blue’ cooling with only first-order diffraction
($\theta_{\textrm{b}}>30^{\circ}$) and have ‘red’ cooling at a real angle
($\theta_{r}<90^{\circ})$. Precluding second-order diffraction is not strictly
necessary for MOT operation (see [40]), but does mean that no other beam
overlap regions and their radiation pressure balance need to be studied, or
any transitions between these regions.

| Divalent atoms | Alkali metals
---|---|---
Element | Ca | Sr | Yb | Tm | Li | Na | K | Rb | Cs
$|g\rangle$ | (4)${}^{1}\textrm{S}_{0}$ | (5)${}^{1}\textrm{S}_{0}$ | ${}^{1}\textrm{S}_{0}$ | ${}^{2}\textrm{F}_{7/2}$ | 2S1/2 | 3S1/2 | 4S1/2 | 5S1/2 | 6S1/2
$|e_{\textrm{r}}\rangle$ | ${}^{3}\textrm{P}_{1}$ | ${}^{3}\textrm{P}_{1}$ | ${}^{3}\textrm{P}_{1}$ | – | 2P3/2 | 3P3/2 | 4P3/2 | 5P3/2 | 6P3/2
$|e_{\textrm{b}}\rangle$ | ${}^{1}\textrm{P}_{1}$ | ${}^{1}\textrm{P}_{1}$ | ${}^{1}\textrm{P}_{1}$ | – | 3P3/2 | 4P3/2 | 5P3/2 | 6P3/2 | 7P3/2
$\Gamma_{\textrm{r}}\,$(MHz) | 0.0004 | 0.007 | 0.183 | 0.345 | 5.9 | 9.8 | 6.0 | 6.1 | 5.2
$\lambda_{\textrm{r}}\,$(nm) | 657 | 689 | 556 | 531 | 671 | 589 | 767 | 780 | 852
$\lambda_{\textrm{b}}\,$(nm) | 423 | 461 | 399 | 411 | 323 | 330 | 405 | 420 | 455
$\Gamma_{\textrm{b}}\,$(MHz) | 35 | 32 | 31 | 10 | 0.16 | 0.44 | 0.18 | 0.28 | 0.29
$\lambda_{\textrm{b}}/\lambda_{\textrm{r}}$ | 0.644 | 0.669 | 0.718 | 0.774 | 0.481 | 0.560 | 0.528 | 0.538 | 0.534
Table 1: A summary of the ‘red’ and ‘blue’ cooling transition parameters
(subscripts r and b, respectively) for a selection of laser-cooled elements.
The laser wavelength ($\lambda$) and transition linewidth ($\Gamma$) values
are from the NIST database. GMOTs have already been realized for the atomic
species and wavelengths in bold [26, 9, 40, 41, 42].

Alkaline earth and lanthanide[47, 48, 49] elements have a broad primary ‘blue’
$|g\rangle\leftrightarrow|e_{\textrm{b}}\rangle$ and much narrower ‘red’
$|g\rangle\leftrightarrow|e_{\textrm{r}}\rangle$ wavelength cooling transition
(Table 1). Owing to different atomic structure [50] these roles are reversed
for alkali metals, with loading on the broad ‘red’ transition and an option
for ‘blue’ cooling [51, 52, 53, 54, 55] on a 20-40 times narrow transition, as
first suggested in Ref. [50]. Again, the narrower linewidth of the secondary
cooling yields a lower temperature, and moreover the lower cross-section for
reabsorption ($\propto\lambda^{2}$) means higher densities can also be
achieved. High phase-space density can then be optimized, particularly for
species such as Li[51, 52, 53] and K[54, 55] with closely-spaced hyperfine
levels on the ‘red’ line complicating direct sub-Doppler cooling, but also for
Rb [56, 57]. For 6Li a blue MOT leads to 7 times more quantum degenerate atoms
than even gray molasses [58]. Aside from complexity there may be advantages to
two-color cooling for other atomic species too, for higher-yield quantum
degenerate gas or higher-accuracy quantum metrology experiments.

We here consider microfabricated binary grating chips comprising $N$ sectors
of one-dimensional gratings occupying equal angular regions of $2\pi/N\,$rad
on the microfabricated grating surface. The use of two-dimensional gratings
such as checkerboards [9] significantly improves overlap volume size and
thereby MOT atom number. However, to eliminate higher-order diffraction one
requires the more restrictive condition
$\lambda_{\textrm{b}}/\lambda_{\textrm{r}}\geq 1/\sqrt{2}$ for 2D gratings
(cf. $\lambda_{\textrm{b}}/\lambda_{\textrm{r}}\geq 1/2$ for 1D gratings),
which eliminates most elements in Table 1 (and if $\theta_{\textrm{b}}$ is
minimized to $45^{\circ}$, then $\theta_{\textrm{r}}$ would be $80^{\circ}$
for Yb and $66^{\circ}$ for Tm).

Using the central zero of the magnetic quadrupole field supplied by
appropriate coils (not shown in Fig. 1 (b),(c)) MOTs can form and be moved
anywhere in the overlap volume of all laser beams at the wavelength used.
Overlap volume mismatch between the ‘blue’ and ‘red’ wavelengths is a minor
issue, which can be solved by ramping the MOT’s magnetic quadrupole field
centre when transferring from primary to secondary cooling transitions [41],
as long as MOT handover (if unaided by a transport trap) occurs in the overlap
volume intersection. For alkali metals, where the primary cooling is a ‘red’
wavelength, this problem can be mitigated by creating a smaller beam for the
shorter wavelength to overlap the trap centres. There is overlap of the
volumes regardless of beam size and ratio
$\lambda_{\textrm{b}}/\lambda_{\textrm{r}}$, and since atoms are pre-cooled to
a dense ball on the broad-linewidth transition, a smaller overlap volume on
the narrow transition is unlikely to be a problem anyway.

The standard, albeit complex, approach to determine the behaviour of any
specific grating is to solve polarisation-dependent Maxwell equations in 3D
for a given light-surface boundary interaction comprising sub-wavelength
periodic features [59]. However, in the remainder of the paper we show that a
simple empirical dimensionless fit suffices to accurately describe
experimental grating characteristics over a wide range of physical parameters.
This model can then be used, in conjunction with general GMOT characteristics
and metal coating reflectivity, to elucidate the grating properties for any
combination of wavelengths and thus atomic or molecular species. The specific
examples of Sr and Yb are used to illustrate the protocols, as they are
increasingly valuable elements for ultra-precise atomic metrology.

##  2 Dimensionless empirical fit to experimental grating characteristics

Several hundred $2\,\textrm{mm}\times 2\,\textrm{mm}$ binary 1D gratings were
fabricated using electron-beam lithography[43] varying the three physical
grating parameters – etch depth $T$, period $d$ and etched duty cycle fraction
$p_{\textrm{e}}$ – covering the ranges $105\,\textrm{nm}-195\,\textrm{nm}$,
$600\,\textrm{nm}-4000\,\textrm{nm}$ and $40\%-85\%$, respectively. Most
gratings were coated with $100\,$nm or $200\,$nm Al, which have similar
reflectivities [44] and a few gratings had the same thicknesses of Au. All
gratings were fully characterized using the method in Ref. [44] with
wavelengths mainly targeting specific atomic species: i.e. $399\,\textrm{nm}$,
$420\,\textrm{nm}$, $461\,\textrm{nm}$, $532\,\textrm{nm}$,
$556\,\textrm{nm}$, $689\,\textrm{nm}$ and $780\,\textrm{nm}$. The diffractive
properties of over 700 different grating-laser combinations were tested,
particularly the first ($\eta_{1}$) and zeroth ($\eta_{0}$) order diffraction
efficiencies for incident circularly polarized light [44].

In order to find an empirical model to fit our optical grating
characterizations, they were compared as a function of their dimensionless
parameters: etch depth $T/\lambda$ and first-order diffraction angle
$\theta=\arcsin(\lambda/d).$ In our prior work [44], the model used assumed
constant total power in the combined first and zeroth diffracted orders, i.e.
constant $\eta_{\textrm{t}}=2\eta_{1}+\eta_{0}$. Here, to get a better fit to
the data, as described below, an angular fit factor of the form
$(1-C\,\theta^{2})$ is used, with $\theta$ here in degrees and $C$ a fit
constant. All data was corrected for less-than-unit reflectivity, i.e. all
diffraction efficiencies were divided by the reflectivity of the grating’s
metal coating at the measurement wavelength [60].

$p_{\textrm{e}}$ | $A_{1}$ | $B_{1}$ | $C_{1}$ | $\phi_{1}$ | $\Delta\eta_{1}$ | $A_{\textrm{t}}$ | $B_{\textrm{t}}$ | $C_{\textrm{t}}$ | $\phi_{\textrm{t}}$ | $\Delta\eta_{\textrm{t}}$
---|---|---|---|---|---|---|---|---|---|---
(45,55] | 22.6 | -18.1 | 74 | 1.37 | 4.1 | 86.9 | 11.0 | 14 | 0.79 | 6.6
(55,65] | 23.2 | -18.7 | 58 | 1.41 | 3.2 | 89.2 | 8.2 | 27 | 0.91 | 5.6
(65,75] | 25.9 | -12.4 | 55 | 1.44 | 4.2 | 88.7 | 9.4 | 28 | 1.14 | 6.5
Table 2: The fitted surfaces for $\eta_{1}$ (left, subscript 1) and
$\eta_{\textrm{t}}$ (right, subscript t) via Eq. 1 for three etch duty ranges
$p_{\textrm{e}}$ (in interval notation), with corresponding RMS fit errors
$\Delta\eta_{1}$, $\Delta\eta_{\textrm{t}}$. The units of all quantities in
the table are: $\%$ (for $p_{\textrm{e}},$ $A_{i}$, $B_{i}$, $\Delta\eta_{i}$,
i.e. units matching $\eta_{i}$), radians for $\phi_{i}$, and $10^{-6}$ inverse
square degrees for $C_{i}$.

The two levels of the one-dimensional binary reflection grating – separated by
a height $T$ and thereby a relative return phase of $e^{i4\pi T/\lambda}$ –
lead to interference of the electric field. We note that this simple model
doesn’t include the full physics, e.g. phase shifts from the side walls of the
binary grating, but this may be included in the $\phi$ fit parameter below.
The interference is readily modelled using phasors [44, 43] and shows a depth-
dependent intensity behavior in the first and zeroth diffracted orders with a
form $A+B\sin(\phi+4\pi T/\lambda).$ The phase shift $\phi$, is expected to
differ by $\pi$ between first and zeroth orders, with constants $A$ and $B$
approximately equal for strong zeroth-first order efficiency variation. By
combining the diffraction angle and depth behavior, we arrive at a diffraction
efficiency fit of the form:

$\eta_{i}=(1-C_{i}\,\theta^{2})(A_{i}+B_{i}\sin(\phi_{i}+4\pi T/\lambda)),$
(1)

where the subscript $i$ can be 1, t or 0 depending on whether it refers to the
1${}^{\textrm{st}}$, (t)otal or 0${}^{\textrm{th}}$ order efficiency,
respectively. To put this in context with our previous work [44], we used a
similar form of model but with $C=0,$ $\phi=\pi/2$, and for
$\eta_{1}:A_{1}=-B_{1},$ whilst for $\eta_{\textrm{t}}:B_{\textrm{t}}=0.$
Almost all data in Ref. [44] considered only a fixed relative etch depth to
wavelength ratio of $T/\lambda\approx 0.25$.

Figure 2: Contour plots show the reflectivity-corrected fits to
$55\%<p_{\textrm{e}}\leq 65\%$ etch duty cycle data using Eq. 1 for the first-
order $\eta_{1}$ (a) and total diffracted $\eta_{\textrm{t}}$ (b)
efficiencies, plotted vs. etch depth ratio $T/\lambda$ and diffraction angle
$\theta=\arcsin(\lambda/d)$. Zeroth order efficiency can be inferred from
$\eta_{0}=\eta_{\textrm{t}}-2\eta_{1}$. Contours ranges from white to grey are
(0-50)% and (50-100)%, respectively, with 1% contour spacing and a dark green
contour at $\eta_{1}=33\%$ and $\eta_{\textrm{t}}=80\%$. Data point locations
(dots) at test wavelengths of $(399,420,461,532,556,689,780)\,$nm are shown in
purple, blue, cyan, dark green, light green, red and black (diamonds, circles,
left-triangles, right-triangles, squares, down-triangles and up-triangles),
respectively. The side images show the corresponding data point fit function
residues in % (the units of the contour plots), $\Delta\eta_{1}$ (a) and
$\Delta\eta_{\textrm{t}}$ (b), with light gray zones indicating $\pm 1$
standard deviation (from Table 2).

We collated our data into those from three etch duty ranges (denoting etched
area as a fraction of the total): $45\%<p_{\textrm{e}}\leq 55\%$,
$55\%<p_{\textrm{e}}\leq 65\%$, and $65\%<p_{\textrm{e}}\leq 75\%$, consisting
of 174, 203 and 124 different grating+laser combinations, respectively. For
these three datasets we fit the empirical model Eq. 1 was fitted to both
experimental data for $\eta_{1}$ and $\eta_{\textrm{t}}$ (Table 2), yielding
experiment-to-fit root-mean-square (RMS) residues to the $\eta_{i}$ datasets
which are relatively small i.e. $\Delta\eta_{1}<5\%$ and
$\Delta\eta_{\textrm{t}}<7\%$. Fitting to $\eta_{0}$ directly using Eq. 1 led
to slightly worse RMS errors to experiment than inferring the values from the
two other fits via $\eta_{0}=\eta_{\textrm{t}}-2\eta_{1}$. We note that the
simpler Ref. [44] grating model would increase the Table 1 average fit errors
for $\Delta\eta_{1}$, $\Delta\eta_{\textrm{t}}$, and $\Delta\eta_{0},$ by
$0.5\,\%$, $0.5\,\%$, and $1.7\,\%$, respectively – most strongly affecting
the zeroth order. Further details on the residuals are provided in the
Supplementary Material.

From Table 2 the gratings with etch duty cycle range centred at
$p_{\textrm{e}}=60\%$ generally have the best fit, overall diffraction
efficiency, and strong first-order diffraction efficiencies (radial trapping
and cooling forces), which matches our findings in Ref. [44]. For the
remainder of the paper we will therefore only consider this optimal
$p_{\textrm{e}}$ value. We illustrate the $\eta_{1}$ and $\eta_{\textrm{t}}$
fits and their residuals for the duty range $55\%<p_{\textrm{e}}\leq 65\%$ in
Fig. 2, with the graphs pertaining to etch duties $45\%<p_{\textrm{e}}\leq
55\%$ and $65\%<p_{\textrm{e}}\leq 75\%$ in Fig. S1 of Supplement 1 [61].

In addition we can also use Table 2 to infer other key parameters, e.g. the
zeroth order diffraction efficiency $\eta_{0}$ as well as the ‘balance’
parameters. Whilst radial intensity balance in a GMOT is achieved with a well-
centred beam normally incident to the GMOT optic, the axial force of the
incoming beam also needs to be balanced by the axial forces from the
diffracted beams. This is particularly important for a good optical molasses.
For a spatially-uniform input beam, the axial beam-intensity balance due to
$N$ first-order diffracted beams from a grating with $N$ sectors of 1D grating
with wavelength-dependent coating reflectivity $R(\lambda)$ [60] can be
parametrized by:

$\eta_{N\textrm{B}}=\frac{NR\,\eta_{1}}{1-R\,\eta_{0}}\;\;\;\textrm{i.e.}\;\;\;\eta_{3\textrm{B}}=\frac{3R\,\eta_{1}}{1-R\,\eta_{0}}\;\textrm{(`Tri'
grating),}\;\;\;\eta_{4\textrm{B}}=\frac{4R\,\eta_{1}}{1-R\,\eta_{0}}\;\textrm{(`Quad'
grating)},$ (2)

where perfect axial beam-intensity is achieved at unity (100%). We use the
notation ‘Tri’ and ‘Quad’ to describe GMOT optics formed of $N=3$ and $N=4$
sectors of linear binary grating, with the former having a slightly larger
capture volume for more atoms, albeit with potential loading asymmetry.

Gratings containing holes with diameters greater than or equal to the MOT
radius can have no zeroth order ‘reflection’ at the MOT location [8], with the
additional benefits of allowing atomic loading [26] and extraction [24], as
well as optical interrogation [14] and manipulation [62]. The balance
parameters describing gratings with holes (extra subscript H) are:

$\eta_{N\textrm{BH}}=NR\,\eta_{1}\;\;\;\textrm{i.e.}\;\;\;\eta_{3\textrm{BH}}=3R\,\eta_{1}\;\textrm{(`Tri'
with hole),}\;\;\;\eta_{4\textrm{BH}}=4R\,\eta_{1}\;\textrm{(`Quad' with
hole)}.$ (3)

##  3 Optimal bichromatic gratings

The results from section 2 can now be used to design a suitable binary
diffraction grating for a given atomic species. Here, we present GMOT
solutions for cooling the specific examples of Sr and Yb, which are often used
in optical lattice clocks [6, 63]. The first step is to correct the model
developed in section 2 for the metal reflectivity used at both ‘red’ and
‘blue’ wavelengths. Aluminium has high reflectivity $(>86\%)$ across
wavelengths $(100-5000)\,$nm, and a flat response $R=(92.0\pm 0.5)\%$ over the
broad optical wavelength range $(300-620)\,$nm [60]. Al is also inert to many
atomic species, and protected by a thin natural oxide layer. For both Sr and
Yb gratings we therefore mainly consider aluminium coatings, noting that other
metal coatings may also be suitable – we have realized Rb GMOTs with Al, Au,
Pd and Pt as grating coatings.

Using Eq. 1 and the optimal $55\%<p_{e}\leq 65\%$ duty fit results from Table
2, the reflectivity-adjusted balance (Eq. 2) can be calculated for both
wavelengths and for different grating architectures. We considered the balance
parameter for four macroscopic grating designs, namely three-sector (Tri)
optics with and without holes at the grating centre ($\eta_{3\textrm{BH}}$,
$\eta_{3\textrm{B}}$), and the equivalent for Quad optics
($\eta_{4\textrm{BH}}$, $\eta_{4\textrm{B}}$), shown in Fig. 3 left to right,
respectively, with Al, Al, Al, Pd coatings. The results in Fig. 3 are given
for both Sr (upper row) and Yb (lower row). The grating period considered is
restricted to $\lambda_{\textrm{r}}<d<2\lambda_{\textrm{b}}$ to ensure only
zeroth-order and first-order diffraction occurs at both wavelengths (i.e.
$30^{\circ}<\theta_{\textrm{r,b}}<90^{\circ})$. The plots use an x-axis of the
‘red’ diffraction angle $\theta_{\textrm{r}}$ to aid comparison to Fig. 2, and
we restrict to $\theta_{\textrm{r}}<80^{\circ}$ for which experimental grating
data exists.

Figure 3: Grating balance parameters (Eqs. 2,3) for Sr (upper row) and Yb
(lower row) at both red $\lambda_{\textrm{r}}$ and blue $\lambda_{\textrm{b}}$
wavelengths as a function of binary grating parameter space (period $d$ vs.
etch depth $T$). The GMOT geometries considered (left to right) are: Al-coated
Tri-with-hole ($\eta_{3\textrm{BH}}$, Sr black dot is from [40, 42]), Tri
($\eta_{3\textrm{B}}$), Quad-with-hole ($\eta_{4\textrm{BH}}$); as well as Pd-
coated Quad ($\eta_{4\textrm{H}}$). Contours and shaded regions are displayed
in their respective red and blue colors, denoting wavelength. Contours are
drawn at balances $(90,95,100,105,110)\%$ with increasing dash width,
excepting the solid 100 % balance contour. The $(95-105)\%$ and $(90-110)\%$
regions are shaded dark and light, respectively.

In Fig. 3, we now consider the GMOT geometries left-to-right. For the Tri with
hole the model indicates $\theta_{\textrm{b}}<30^{\circ}$ may be suitable at
both wavelengths, matching the Al-coated Sr Tri grating with hole of Ref. [40,
42] (black dot). We note our model is likely still valid a little below
$\theta_{\textrm{b}}=30^{\circ}$, if the large-angle second-order blue
diffraction is minimal. For both atomic species with holeless Al-coated Tri
designs we see a large overlap of good balance in grating parameter space,
suggesting a promising area for future GMOT experiments. In Quad Al-coated
gratings with holes the overlap is much smaller than for Tri gratings, but the
contour intersection still indicates a more specific useful grating region.
This region has experimental data for the red and blue (Fig. 2), albeit at
high red diffraction angle and hence low red GMOT overlap height. To realize
regular Quad grating GMOTs, or to reduce the extremity of the ‘red’
diffraction angle for quads with holes, an alternative is to pick a different
metal coating, here we show Pd instead of Al for the Quads, which greatly
extends their useful parameter space. A functional Sr Quad with hole GMOT has
already been experimentally demonstrated [41], albeit with a multi-layer metal
coating, giving reduced blue reflectivity. This difference prevents direct
comparison, but matches expectations if the balance parameters in Fig. 3 were
adjusted accordingly.

The intersection of $100\%$ balance contours in Fig. 3 would naïvely lead to
the optimal GMOT grating, however we now stress caveats to this approach. The
two experiments with bi-chromatic Sr gratings [41] and [40, 42], have
$\lambda_{\textrm{r}}$ balances of $72\%$ and $111\%$, respectively,
demonstrating that even the narrow-linewidth red cooling works in a large
range of balances, which matches our experience with Rb GMOT balance [9, 17].
This implies that balances constrained to $(100\pm 10)\%$ are in fact overly
restrictive and it is likely a wider range of grating parameter space is
available.

Seo et al.[32] also reported on a six-parameter simulation of GMOTs, optimized
by machine learning, that indicated higher optical balances capture more
atoms. Caution must be used however, as there is a diffraction-angle specific
value corresponding to a level of optical balance which precludes MOT
formation [18, 64, 41, 65]. As a technically simple fix for GMOT under-
balance, a disk of weak neutral density (ND) filter in the center of the GMOT
input laser beam can also be used [11]. Conversely a weak ND filter with a
disk cut from its centre can restore optical balance when there is too much
diffracted power. In principle, bespoke filters could be fabricated to tune
the balance of both wavelengths to a designated level.

Other considerations for GMOT grating design include the input beam-profile
used [11] – so far we have assumed a grating illuminated with a spatially
uniform beam. Without beam-shaping, most input beams have a Gaussian
transverse intensity profile $I_{0}\exp(-2r^{2}/w^{2})$ with beam waist $w$,
leading to a balance parameter that increases with distance $z$ above the
grating centre. Specifically, this modifies any instance of $\eta_{1}$ in Eq.
2 to $\eta_{1}^{\prime}=\eta_{1}\exp(-2(z\tan\theta)^{2}/w^{2})$, where both
the diffraction angle $\theta$ and $w$ depend on the MOT laser wavelength.
Typically with alkali metal atoms, over-expanding the beam simplifies this
consideration, but the primary cooling transitions for alkaline-earth-like
atoms have a large saturation intensity, at wavelengths where relatively high
power laser sources can be complex or expensive.

Another vital design consideration for GMOTs is the trade-off between axial
and radial trapping and cooling forces, as well as their local and spatially
averaged (trap depth) maxima. These compromises strongly depend on grating
diffraction angle [18, 64, 8, 10, 11, 66, 65]. There is evidence that vapor-
loaded Rb GMOTs have largest atom number at diffraction angles around 45∘ [11,
32], which has been corroborated theoretically [32, 65]. It is therefore also
worth considering favoring the primary cooling transition for loading in the
case of Sr and Yb, and $\theta_{\textrm{b}}\approx 45^{\circ}$ corresponds to
the far right of all images in Fig. 3. Furthermore most work to date considers
only simple atomic models and neglects internal state dynamics, which strongly
affect sub-Doppler cooling mechanisms [67, 34].

Our empirical grating model, in conjunction with a wavelength-specific coating
reflectivity, gives a balance parameter that allows one to simply determine
the optical properties of a given bichromatic binary grating. However, several
other factors – including the effects of the multi-level structure of real
atoms on GMOTs [66] – may need consideration to tailor a grating to a given
user’s GMOT design criteria, and we plan to investigate further [65].

##  4 Conclusion

By processing data from a wide variety of diffraction grating-wavelength
combinations we arrived at a simple dimensionless empirical model of
diffraction efficiencies. The model, in conjunction with coating reflectivity,
allows one to tailor grating design to form compromise fabrication solutions
that works for laser-cooling and trapping in GMOTs at two wavelengths. The
grating model could also be used to design single-color GMOTs more accurately,
or extended to an optically simpler three-color GMOT cooling method for group-
III elements, like Indium [68].

The multi-wavelength diffractive optics proposed here will be ideal for any
high-end application where the size, weight and power of the laser cooling
package is critical, particularly for earth- and space-based portable quantum
technologies [69, 45].

Acknowledgments We thank James McGilligan, William McGehee and Roger Brown for
thorough proofreading; and James McGilligan, David Burt and Brendan Casey for
valuable conversations.

Funding InnovateUK+EPSRC projects EP/M013294/1, EP/M50824X/1, EP/R002371/1,
EP/T001046/1.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are
available in the Dataset, Ref. [61]. For the purpose of open access, the
author(s) has applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript (AAM) version arising from this submission.

Supplemental document See Supplement 1 for supporting content, chiefly an
extended model error analysis via figures S1-S4 which determine whether the
duty cycle range, test laser wavelength, diffraction angle, or etch depth show
any biases – with nothing significant observed. A discussion of residual
errors from measuring gratings with nominally the same manufacturing
parameters is also provided.

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