# Exploiting non-orthogonal eigenmodes in a non-Hermitian optical system to
realize sub-100 MHz magneto-optical filters.

Fraser D. Logue  Department of Physics, Durham University, Durham, DH1 3LE,
United Kingdom  Corresponding author<EMAIL_ADDRESS>Jack D.
Briscoe  Department of Physics, Durham University, Durham, DH1 3LE, United
Kingdom  Danielle Pizzey  Department of Physics, Durham University, Durham,
DH1 3LE, United Kingdom  Steven A. Wrathmall  Department of Physics, Durham
University, Durham, DH1 3LE, United Kingdom  Ifan G. Hughes  Department of
Physics, Durham University, Durham, DH1 3LE, United Kingdom

###### Abstract

Non-Hermitian physics is responsible for many of the counter-intuitive effects
observed in optics research opening up new possibilities in sensing,
polarization control and measurement. A hallmark of non-Hermitian matrices is
the possibility of non-orthogonal eigenvectors resulting in coupling between
modes. The advantages of propagation mode coupling have been little explored
in magneto-optical filters and other devices based on birefringence. Magneto-
optical filters select for ultra-narrow transmission regions by passing light
through an atomic medium in the presence of a magnetic field. Passive filter
designs have traditionally been limited by Doppler broadening of thermal
vapors. Even for filter designs incorporating a pump laser, transmissions are
typically less than 15% for sub-Doppler features. Here we exploit our
understanding of non-Hermitian physics to induce non-orthogonal propagation
modes in a vapor and realize better magneto-optical filters. We construct two
new filter designs with ENBWs and maximum transmissions of 181 MHz, 42% and
140 MHz, 17% which are the highest figure of merit and first sub-100 MHz FWHM
passive filters recorded respectively. This work opens up a range of new
filter applications including metrological devices for use outside a lab
setting and commends filtering as a new candidate for deeper exploration of
non-Hermitian physics such as exceptional points of degeneracy.

††journal: optica

##  1 Introduction

Non-Hermitian physics [1, 2, 3, 4, 5, 6] is opening up new possibilities in
photonics from the creation of omnipolarizers [7, 8, 9] to sensing
applications [10, 11, 12] and tailoring laser output [13, 14, 15]. In general,
non-Hermitian matrices do not guarantee orthogonal eigenvectors [16, 17, 18]
and have real eigenvalues (PT-symmetric) [19, 20, 21], imaginary eigenvalues
(anti-PT-symmetric) [22, 23, 24] or complex eigenvalues throughout [25, 26].
Ultra narrowband magneto-optical filters [27], which rely on complex
refractive indices, are perfect candidates for non-Hermitian physics but have
not yet been studied in this domain. Magneto-optical filters already see many
applications in solar weather studies [28, 29, 30], laser frequency
stabilization [31, 32, 33, 34, 35], LIDAR, [36, 37, 38], quantum hybrid
systems [39], underwater optical communications [40] and ghost imaging [41].
Optimizing filters is a balance between maximizing transmission and minimizing
bandwidth. For example, tens of MHz equivalent noise bandwidth at < $20\%$
transmission has been realized in active filters incorporating a pump beam
[42, 43, 44]. On the other hand, several $\rm{GHz}$ equivalent noise bandwidth
has been realized at $>80\%$ transmission in passive designs [45, 46]. A
fundamental limit on linewidths has not yet been found and could have future
implications for metrology [47, 48, 49].
Of equal importance is the platform, particularly when filters are designed
for use outside a laboratory setting. Most magneto-optical filters rely on
thermal vapor cells due to their compact and resilient setups [50]. Thermal
vapor filters can be tuned by varying the magnetic field, temperature and
input polarization [51, 52] and can switch between different D-line operation
[53]. It has been shown that by cascading cells, line-center or wing features
can be selected [54]. However the two setups require the rearrangement of
optical elements and the interchange of $\sim$kG permanent magnets with
$\sim$100 G solenoids. Note that several metrological applications, such as
those that use space, atmospheric or local environmental conditions to detect
fluctuations, require apparatus that is robust to the elements [55, 56, 57].
Therefore there is interest in finding a reconfigurable setup that allows easy
switching between wing and line-center operation.
The core principle of a magneto-optical filter is the magneto-optical rotation
of light [58, 59, 60]. Faraday [61, 62, 63] and Voigt [64, 65, 66] filters,
where the magnetic field is parallel or perpendicular to the $k$-vector of the
light, rely on circular or linear birefringence respectively. These are
special cases where the eigenvectors are orthogonal even though the refractive
indices are complex. By setting the magnetic field at an oblique angle to the
$k$-vector, the system becomes elliptically birefringent and dichroic [67,
68]. While oblique filters have been constructed [66, 69, 51, 70], a treatment
on how elliptical birefringence can improve filter performance has not yet
been presented. In fact, we show that this elliptical birefringence relies on
the eigenmodes of the medium being non-orthogonal. This results in two
improvements on the Faraday and Voigt cases: firstly, there is better
rejection of light outside the birefringent region and secondly, the
birefringent region is much narrower leading to narrower filter peaks and
better figures of merit.
In this paper, we theoretically predict the benefits of oblique filters and
experimentally realize two competitive filter designs. In Section 2, we show
how oblique filters can reject light outside of the birefringent region at
$>90\%$ efficiency using Stokes parameters to characterise their performance.
In Section 3, we demonstrate that this effect is explained by the oblique
geometry having non-orthogonal eigenmodes and introduce the term invariant
polarizations as the natural additional basis to explain oblique magneto-
optical rotation. We calculate the birefringent region showing it to be
narrower than the Faraday and Voigt cases. In Section 4, we realize two
thermal vapor filters with lower equivalent noise bandwidth than previously
published passive designs. Using naturally abundant Rb vapor, we
experimentally realize the first Oblique-Voigt filter, a cascade of an oblique
and a Voigt cell, and the first oblique double pass filter, a single oblique
cell with light passed through twice. We find excellent agreement with data
giving equivalent noise bandwidths and transmissions of 181 MHz, 42 % and 140
MHz, 17 % respectively. The first filter is the highest Figure of Merit (FOM)
[52] passive filter to date. The second filter is the first sub-100 MHz FHWM
passive design and is reconfigurable allowing the selection of line-center or
wing features by solenoid current variation.

##  2 Elliptical Birefringence

Geometry | $\boldsymbol{\theta\hskip 2.84544pt(^{\circ})}$ | D-line | $\boldsymbol{T\hskip 2.84544pt(^{\circ}\rm{C})}$ | $\boldsymbol{B\hskip 2.84544pt(\rm{G})}$ | Inp. Pol.
---|---|---|---|---|---
Faraday | 0 | D2 | 63 | 160 |  Lin. Hor. $\leftrightarrow$
Voigt | 90 | D1 | 100 | 3500 |  L.H. Circ. $\circlearrowleft$

Figure 1: a) and b) : Schematics of single cell Faraday and Voigt filters in
natural abundance Rb. Note the second polarizer is rotated so that the
hypotenuse is not in the plane of the page. As such the polarizers are
crossed. Parameters shown in the table. c) and d): The filter transmissions
after 75 mm plotted against linear detuning with zero detuning being the
weighted centre of the resonance frequencies of the two Rubidium isotopes.
Colored dotted lines indicate the frequencies plotted on the Poincaré sphere.
Polarization animations shown. e): A Poincaré sphere representation showing
the polarization evolution of light moving through the Rb vapor cell. ’x’
marks the evolution after 75 mm. f) and g) are projections on to the $S_{3}$
and $S_{1}$ planes respectively. Stokes parameters are normalized with respect
to intensity of input light. The Faraday maximum transmission frequency
evolves on the equator while the Voigt maximum transmission frequency evolves
along a meridian.

Geometry | $\boldsymbol{\theta\hskip 2.84544pt(^{\circ})}$ | D-line | $\boldsymbol{T\hskip 2.84544pt(^{\circ}\rm{C})}$ | $\boldsymbol{B\hskip 2.84544pt(\rm{G})}$ | Inp. Pol. | % Rejection (Gold Freq.) | % Rejection (Red Freq.)
---|---|---|---|---|---|---|---
Faraday | 0 | D2 | 63 | 160 | Lin. Hor. $\leftrightarrow$ | 49 | 50
Voigt | 90 | D1 | 107 | 514 | L.H. Circ. $\circlearrowleft$ | 54 | 68
Oblique | 86 | D2 | 99 | 226 | Lin Hor. $\leftrightarrow$ | 96 | 92

Figure 2: a), b), c): Theory plots of Faraday, Voigt and oblique filter
transmissions in 75 mm naturally abundant Rb vapor cells (gray) plotted
alongside $S_{0}$ (blue). Parameters given in table. We choose to plot the D2
line for the Faraday and oblique filters for comparison. We plot the D1 line
instead for the Voigt filter as similar line-center D2 filters are not
competitive in terms of FOM with the Faraday and oblique examples. The filter
maximum transmission frequency (purple dashed line) and two frequencies
outside the birefringent region (yellow- and red-dashed line) are marked. Note
the red Voigt frequency remains slightly left handed after propagation leading
to the > 50% rejection. Animations depict the output polarization before the
second polarizer. d), e), f): Poincaré sphere representations of the three
frequencies. Stokes parameters are normalized with respect to input intensity.
g), h), i): Projections of the Poincaré sphere on to different axes. We note
that output elliptical horizontal states either side of the oblique filter
peak result in better filter rejection than the circular and linear horizontal
states in the Faraday and Voigt filters respectively leading to improved
filter performance.

The orientation of the magnetic field defines the relationship between input
polarization and the induced $\pi$ ($m_{J}=m^{\prime}_{J}$) and $\sigma^{+/-}$
($m_{J}=m^{\prime}_{J}\pm 1$) electric dipole allowed transitions [71, 69,
72]. In the Faraday geometry, a magnetic field is exerted parallel to the
$k$-vector of the input light which we take to be parallel with the $z$-axis.
This results in left/right handed circular light inducing $\sigma^{+/-}$
transitions [73]. In the Voigt geometry, the magnetic field is perpendicular
to the $k$-vector and we have a choice of field lines running parallel to the
$y$\- or $x$-axis. Without loss of generality, in this paper, we always place
magnets in the $x-z$ plane (i.e. on the surface of an optical bench). In this
case, horizontally polarized light induces $\pi$ transitions and vertically
polarized light induces a linear combination of $\sigma^{+}$ and $\sigma^{-}$
transitions [74].
Differences in the transition probabilities lead to polarization dependent
absorption (dichroism) and continuous polarization state transformation
(birefringence) as the light propagates through the vapor [75, 76]. The
Faraday geometry exhibits a circular birefringence as differential refraction
of circular handed light causes magneto-optical rotation of the plane of
linear light. In contrast, the Voigt geometry exhibits linear birefringence
where differential refraction of horizontally and vertically polarized light
leads to the handedness cycling through right, linear and left hands. We can
describe polarization evolution using Stokes parameters [77, 78] defined as,

$S_{0}=\frac{I_{\rm{out}}}{I_{\rm{in}}},$ (1)
$S_{1}=\frac{(I_{\leftrightarrow}-I_{\updownarrow})}{N},\hskip
5.69046ptS_{2}=\frac{(I_{\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}}-I_{\mathrel{\text{$\nwarrow$\hbox
to0.0pt{\hss$\searrow$}}}})}{N},\hskip
5.69046ptS_{3}=\frac{(I_{{\circlearrowright}}-I_{\circlearrowleft})}{N}.$ (2)

$S_{0}$ represents the total transmission and is proportional to
$I_{\rm{out}}$, the output intensity. $S_{1}$, $S_{2}$ and $S_{3}$ together
uniquely describe the polarization state. The arrow indices depict the
polarization state of the output light ($\circlearrowright/\circlearrowleft$
denotes right/left hand circular light) and $N$ is a normalization factor.
Throughout when we write $S_{1}$, $S_{2}$ and $S_{3}$, we are using input
normalized stokes vectors $(N=I_{\rm{in}})$. When we write tildes,
$\tilde{S}_{1},\tilde{S}_{2},\tilde{S}_{3}$, we are denoting output normalized
stokes vectors $(N=I_{\rm{out}})$.
Plotting Stokes parameters on a Poincaré sphere provides a useful way of
visualizing polarization [52, 79, 80, 81]. In this paper, we explain filter
performance by plotting the Stokes parameters as a function of the distance
light has propagated through the vapor. A magneto-optical filter only outputs
light that has rotated to be orthogonal to the input polarization. This is
represented on the Poincaré sphere when curves travel to the antipode of the
input (assuming no loss). Fig. 1 plots evolutions of maximum transmission
frequencies for a Faraday and Voigt filter demonstrating circular
birefringence, where the polarization evolves on the equator, and linear
birefringence, where the polarization evolves along a meridian.

If a magnetic field is applied at an angle $\theta$ to the $k$-vector, the
projection operators for the three transitions are:

$\begin{split}&\sigma^{+}:\frac{\widehat{x}\cos\theta+\mathrm{i}\widehat{y}}{{\sqrt{2}}}.\\\
&\sigma^{-}:\frac{\widehat{x}\cos\theta-\mathrm{i}\widehat{y}}{{\sqrt{2}}}.\\\
&\pi:\hskip 5.69046pt\widehat{x}\sin\theta.\end{split}$ (3)

The $\pi$ transition operator is always horizontally linear. If $\theta$ is an
oblique angle, neither $0$ or $90^{\circ}$ to the $k$-vector, the $\sigma$
transition operators are elliptical. Therefore, the oblique geometry exhibits
elliptical birefringence where we have differential refraction between
elliptical polarization states [67, 68]. A derivation of the projection
operators can be found in Appendix A of the Supplementary Material.
In atom-light interactions, there is a birefringent region, where magneto-
optical rotation occurs. In a Faraday/Voigt birefringent region,
circular/linear birefringence results in linear/circular light as an output.
Outside of the birefringent region, circular/linear dichroism of light outputs
circular/linear light. Given that circular and linear light are not orthogonal
polarization states, a filter cannot completely reject the output light
outside the region but filters only $\sim 50\%$. However, by choosing a
suitable value of $\theta$, it is possible to construct an oblique filter
where light outside the birefringent region is suppressed beyond > 90%. Fig. 2
compares the filter transmissions of Faraday, Voigt and oblique filters with
$S_{0}$.
Note that the Faraday filter has characteristic pedestals either side of the
central peak where circular light is rejected at 50%. In the Voigt filter, we
also see $\sim$ 50% rejection for the selected negatively detuned frequency
(gold). The positively detuned frequency (red) is rejected at > 50% but the S0
value is already less than 10%. However, the oblique filter demonstrates
unparalled rejection of light outside the birefringent region significantly
reducing the equivalent noise bandwidth, (ENBW),

$\mathrm{ENBW}=\int\frac{\mathcal{T}(\nu)\mathrm{d}\nu}{\mathcal{T}(\nu_{\rm{s}})},$
(4)

where $\mathcal{T}(\nu)$ is the transmission for a given frequency, $\nu$, or
the maximum transmission frequency, $\nu_{s}$. In the literature, a Figure of
Merit (FOM) [52],

$\mathrm{FOM}=\frac{\mathcal{T}({\nu}_{\rm{s}})}{\mathrm{ENBW}},$ (5)

is often used to compare filter quality with a lower ENBW increasing FOM.

Geometry | $\boldsymbol{\theta(^{\circ})}$ | D-line | $\boldsymbol{T\hskip 2.84544pt(^{\circ}\rm{C})}$ | $\boldsymbol{B\hskip 2.84544pt(\rm{G})}$ | Inp. Pol. | Birefringent Region FWHM (MHz) | Filter FWHM (MHz)
---|---|---|---|---|---|---|---
Faraday | 0 | D2 | 63 | 160 | Lin. Hor.$\leftrightarrow$ | 850 | 790
Voigt | 90 | D1 | 107 | 514 | L.H. Circ. $\circlearrowleft$ | 320 | 310
Oblique | 86 | D2 | 99 | 226 | Lin Hor.$\leftrightarrow$ | 200 | 150

Figure 3: a), b), c): Theory plots of Faraday, Voigt and oblique filter
transmissions in 75 mm naturally abundant Rb vapor cells (purple) plotted
alongside $S_{0}$ (blue). Parameters given in table. The red and yellow
regions are the $S_{0}$ output if the two invariant polarizations are input
(see Table 3). In the Faraday and Voigt cases, these correspond to the
transmission regions left after absorption due to one or more transitions. In
the oblique case, the invariant polarizations are frequency dependent. The
birefringent region is shown in orange as the overlap between the red and
yellow regions. In the Voigt case, one transmission region ($P_{3}$)
completely overlaps with the other ($P_{4}$). d), e), f) depict the
$\rm{\tilde{S}_{1}}$ and $\rm{\tilde{S}_{3}}$ Stokes parameters for the three
filters with polarization animations for the same frequencies as in Fig. 2.
Note the Stokes vectors are normalized with respect to the output intensity
and therefore give information about polarization state but not intensity
loss. The narrowness of this oblique birefringent region results in a FWHM
over two times and five times narrower than the Voigt and Faraday filter
respectively.

##  3 Birefringent Regions with Non-Orthogonal Eigenmodes

We can learn more about the magneto-optical rotation of a vapor by calculating
the birefringent region explicitly. The birefringent region is the $S_{0}$
output where differential refraction occurs. Better filter peaks are formed
within birefringent regions since they are highly selective resulting in
narrower lineshapes and can rotate light with less loss. A filter may have
additional transmission due to absorption both inside and outside of this
region. Dichroic absorption alone evolves the polarization to one particular
polarization state after which point moving through more vapor reduces the
intensity of the light without altering the polarization state. This is a high
loss process and as discussed in Section 2, light in this region is rejected
by the filter at > 50%.

To compute the birefringent region we need to find a basis composed of
polarizations which do not transform as they move through the vapor. We call
these basis vectors invariant polarizations. In the Faraday and Voigt
geometries, the invariant polarizations are the propagation eigenmodes derived
from solving the wave equation [69]. We then calculate two $S_{0}$ regions,
shown in red and yellow in Fig. 3, with the two eigenmodes as input
polarizations. The overlap of the two regions is the birefringent region,
shown in orange. In this detuning range, any input polarization with non-zero
components of both eigenmodes undergoes continuous polarization state
transformation.

However in the oblique geometry, the eigenmode solutions to the wave equation
are non-orthogonal and frequency dependent. As such the eigenmodes are not
invariant polarizations. Instead, we find the polarizations orthogonal to each
eigenmode which do not undergo differential refraction. For more details, see
Appendix B of the Supplementary Material. The full list of invariant
polarizations is shown in Table 3. In particular, note that the oblique
invariant polarizations generally induce all three transitions provided the
three frequency dependent transition probabilities are non-zero.

Transmission Region | Invariant Pol. | Transitions Ind.
---|---|---
$P_{1}$ (Faraday) | L.H.C $\circlearrowleft$ | $\sigma^{+}$
$P_{2}$ (Faraday) | R.H.C. $\circlearrowright$ | $\sigma^{-}$
$P_{3}$ (Voigt) | Lin. Hor. $\leftrightarrow$ | $\pi$
$P_{4}$ (Voigt) | Ver. Hor. $\updownarrow$ | $\sigma^{-}/\sigma^{+}$
$P_{5}$ (Oblique) | Freq. Dep. | $\sigma^{-}/\sigma^{+}/\pi$
$P_{6}$ (Oblique) | Freq. Dep | $\sigma^{-}/\sigma^{+}/\pi$
Table 3: A table of the 6 $S_{0}$ regions shown in Fig. 3. Each transmission
region is created by inputting an invariant polarization to the cell which
does transform polarization state upon propagation i.e. is not magneto-
optically rotated. Note that in the Faraday and Voigt cases, the invariant
polarizations are the eigenmodes of the system and can be read off from their
respective angular dipole matrix elements (See Appendix A of the Supplementary
Material). In the oblique case, the invariant polarizations are frequency
dependent. The invariant polarizations induce certain transitions and these
regions are the transmission left after transition induced absorption

In Fig. 3, we observe that the width of the birefringent region bounds the
width of the filter peak. In the Faraday and oblique cases this is easier to
see. In the Voigt case, one transmission region ($P_{3}$) lies entirely within
the other ($P_{4}$). The larger transmission of $P_{4}$ results in the center
of the peak being displaced and some absorption outside the birefringent
region contributing to the peak’s profile. The plotted oblique filter has the
narrowest birefringent region resulting in the narrowest FWHM peak, two times
and five times narrower than the Voigt and Faraday filters respectively.

Summarizing Sections 2 and 3, we can give a complete description of why the
oblique filter can be tuned to give the best filter. Given that the invariant
polarizations induce all three transitions, the $S_{0}$ regions and hence
their overlap (the birefringent region) is narrower leading to narrower peaks.
Outside of this region, dichroic absorption results in transformation to
polarization states which can be filtered at > 90%. As such, the filter
profile is improved as most light is transmitted within the birefringent
region. The ENBW is also lower for this reason and additionally the FWHM of
the peak is narrower. The detriment to maximum transmission does not outweigh
the reduction in ENBW resulting in higher FOMs.

##  4 Experiment and Results

In order to test the advantages of the oblique geometry we realized two filter
designs; the experimental setup is shown in Fig. 4. In the experimental arm,
light on the Rb-D2 line with a $1/e^{2}$ beamwidth of $100\leavevmode\nobreak\
\upmu\rm{m}$ passes through an ND filter to lower the power to 300 nW. We work
in the weak probe regime [82, 83, 84] as it is readily available to model and
higher intensity filters are outside the scope of this work.
The aim of the two filter designs is to output the desirable central filter
features studied in previous sections while eliminating features away from
center. Fig. 5 shows how the two designs eliminate unwanted features over a 20
GHz detuning range. The first experiment is a two cell cascade with the first
cell in the oblique geometry and the second in the Voigt geometry. Both cells
are between crossed Glan-Taylor polarizers. The two cells play different
roles: Cell 1’s role is to create the narrow feature at center as studied in
the previous sections while the role of Cell 2 is to absorb features away from
the central region. A Voigt cell is particularly appropriate for this role
since the magnetic field and temperature can be tuned to be highly absorptive
away from center leaving a small transmission region at center for the narrow
peak to be transmitted through. The theory behind the Voigt cell has been
studied in [54]. This is the first Oblique-Voigt cascade recorded in the
literature.
The second experiment is a double pass filter where light is passed through an
oblique cell between polarizers and filtered before being reflected back into
the cell and filtered again. The rays from the two passes follow a similar
path and diverge by less than a 1 mm in the $x-y$ plane over the cell length.
As the two beams are weak, neither beam pumps the atoms into higher states.
On the first pass, the light input is horizontally linear polarized and the
magnetic field makes an angle $\alpha$ to the propagation direction while on
its second pass the light is filtered to be vertically linear polarized and
the angle of the magnetic field is $\alpha^{*}=(180-\alpha)^{\circ}$. The two
passes individually give the same filter output. Left/right handed light
propagating at an angle $\alpha$ to the magnetic field is equivalent to
right/left handed light propagating at an angle $\alpha^{*}$, however given we
are inputting linear light the interchange of $\alpha^{*}$ for $\alpha$
results in no change. Additionally, horizontal and vertical input always gives
the same filter output, provided the polarizers are rotated appropriately,
regardless of the magnetic field angle (see Appendix C of the Supplementary
Material). As such, filtering twice results in a squared intensity effect
which suppresses the features away from center much more than the central
features present after the first pass. Double pass filters have been realized
before [85, 86, 87, 88] however never with an oblique cell.
Using an adapted version of the open-source software package ElecSus [89, 90],
we found parameters for natural abundance Rb that would give a large FOM for
the Oblique-Voigt filter and narrow FWHM for the double pass filter. We found
that both of these objectives can be obtained using very similar parameters
for Cell 1 and Cell 3. We chose 75mm cells for Cell 1 and Cell 3 and a 5mm
cell for Cell 2. Choosing cells is an interplay between achieving a
homogeneous magnetic field which is easiest in a small cell and avoiding
collisional broadening effects which occur at higher temperatures. Hence if a
large number density [91] is needed, longer cells may be better [92]. The
combination of a solenoid and permanent magnets allows us to use a longer cell
than in previous oblique investigations [70, 51] since the magnitude of the
field is largely provided by the permanent magnets while its direction is
determined by small adjustments in the solenoid current. This allows easier
control of the field magnitude and direction with the permanent magnets set
square with the cell resulting in less error in homogeneity.
The results and fit parameters are shown in Fig. 6. The data show excellent
agreement with theory [93] apart from a shoulder on the double pass filter.
The Oblique-Voigt filter has the highest FOM recorded to date of
$2.38\pm\leavevmode\nobreak\ 0.01\leavevmode\nobreak\ \rm{GHz}^{-1}$. The
double pass has the highest FOM of a single cell passive filter recorded to
date, $1.20\pm\leavevmode\nobreak\ 0.01\leavevmode\nobreak\ \rm{GHz}^{-1}$,
and has a sub-100 MHz fit FWHM of 93 MHz.
The insets on the left graph show the dependence of the Oblique-Voigt filter
performance on $\theta$, $B_{1}$ and $T_{1}$. We note the high sensitivity to
$\theta$. If $\theta=90^{\circ}$, there is no filter transmission since in the
Voigt geometry linear horizontal light is not rotated. However as $\theta$
decreases, a filter profile quickly emerges showing the radical difference in
the birefringence between the Voigt and oblique cases.
The insets on the right graph show how the double pass filter can be
reconfigured to select for wing features over the line-center feature. This is
achieved by raising the current which increases the axial field reducing the
angle $\theta$ in the resultant field. We experimentally realize a wing filter
using this procedure with the cuboidal magnets placed slightly closer to raise
the magnitude of the magnetic field to $\sim 300\leavevmode\nobreak\ G$. Such
a feature would be very useful in a manufactured device where it may not be
feasible to rearrange optical elements to achieve the same result.

Figure 4: An illustration of the experiment. An external cavity diode (ECD)
laser tuned to the Rb-D2 line emits light towards reference optics and the two
experiments. The reference optics include a Fabry-Pérot etalon to linearize
the scan and a rubidium reference (Rb Ref.) cell for absolute frequency
calibration. A neutral density (ND) filter lowers the intensity of the light
entering the experiment. $\lambda/2$ waveplates rotate the light into the
correct input polarization. The Oblique-Voigt filter consists of two rubidium
vapor cells between two Glan-Taylor polarizers (GTP) which are crossed. Cell 1
has a magnetic field exerted at an angle $\theta$ to the $k$-vector while the
magnetic field is perpendicular to the $k$-vector for Cell 2. The double pass
filter consists of one cell between crossed Glan-Taylor polarizers for each
pass. On the first pass the magnetic field makes an angle $\alpha$ to the
$k$-vector and on the second pass makes an angle
$\alpha^{*}=(180-\alpha)^{\circ}$. Cells 1 and 3 have magnetic fields
generated via a combination of cuboidal plate magnets and solenoids which
exert the appropriate transverse and axial fields respectively. Cell 1 and 3
are heated using adhesive flexible heating elements stuck along the length of
the cells. Cell 2 is placed in a copper heater and has a transverse magnetic
field generated by NdFeB top hat magnets [94, 95]. M- Mirror, OI - Optical
Isolator, (P)BS - (Polarizing) Beamsplitter, PD - Photodetector.  Figure 5: a)
The output of the Oblique-Voigt filter with the second cell removed (purple
line) and with the second cell (purple shading). The transmission of the
second cell (yellow line) with its absorption region (yellow shading) is
shown. b) The output of the oblique double pass filter (purple). A zero field
natural abundance Rb-D2 spectrum at $35^{\circ}\rm{C}$ is plotted on both
panels (dotted gray).

Filter Type | $\boldsymbol{T_{1}\hskip 2.84544pt(^{\circ}\rm{C})}$ | $\boldsymbol{B_{1}\hskip 2.84544pt(\rm{G})}$ | $\boldsymbol{\theta/\alpha\hskip 2.84544pt(^{\circ})}$ | $\boldsymbol{T_{2}\hskip 2.84544pt(^{\circ}\rm{C})}$ | $\boldsymbol{B_{2}\hskip 2.84544pt(\rm{G}})$ |  ENBW $\boldsymbol{(\rm{GHz})}$ |  FWHM $\boldsymbol{(\rm{MHz})}$ |  FOM $\boldsymbol{(\rm{GHz^{-1}})}$
---|---|---|---|---|---|---|---|---
Oblique-Voigt | $96.8\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 0.1$ | $226.8\pm 0.7$ | $86.3\pm 0.3$ | $123.6\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 0.2$ | $3897\pm 3$ | $181\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 1$ | $145\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 1$ | $2.38\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 0.01$
Double Pass | $99.4\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 0.4$ | $226.3\pm 1.2$ | $87.4\pm 0.3$ | $/$ | $/$ | $140\leavevmode\nobreak\ \pm 1$ | $93\leavevmode\nobreak\ \pm 1$ | $1.20\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 0.01$

Figure 6: (Left) Main plot shows data (gold) and theory (purple) plotted for a
natural abundance Rb-D2 two cell cascade with 75 mm and 5 mm cells in the weak
probe regime. The data show excellent agreement with theory with RMS fit error
of 0.5%. The insets show theory spectra where a) $B_{1}$, b) $\theta$ and c)
$T_{1}$ are varied. All other parameters are fixed. (Right) Main plot shows
data (gold) and theory (purple) for a natural abundance Rb-D2 double pass
filter in the weak probe regime. The data show excellent agreement with theory
apart from a shoulder with RMS fit error of 0.3%. d) shows an experimental
wing filter with fit achieved using the same double pass setup by increasing
the current in the solenoid and decreasing the separation of the cuboidal
magnets. This changes $\alpha$ to $69^{\circ}$ and $B$ to $310$ G, the
temperature has also lowered to $92^{\circ}\rm{C}$ e), f), g) show the
theoretical evolution of the filter’s features as the angle $\alpha$ is
varied. The table shows the mean parameter values obtained from fits of five
spectra. The ENBW, FWHM and FOM fit values also account for the systematic
errors involved in data acquisition and linearization.

##  5 Conclusion

In conclusion, we have presented the underlying non-Hermitian physics behind
filters in the oblique geometry. We have shown that elliptical birefringence
and non-orthogonal propagation eigenmodes can create narrower filters with
greater rejection of light outside the birefringent region. We realized two
new filter designs: an Oblique-Voigt cascade and an oblique double pass which
showed excellent agreement with theory. These are the highest FOM and first
sub-100 MHz FWHM passive design filters recorded respectively. These findings
alongside the reconfigurability and compactness of the setup are promising for
future applications in metrology outside a lab context [96, 97]. This is an
initial work into non-Hermitian filter effects; our theory ElecSus predicts
frequencies where the non-orthogonal eigenmodes almost completely coallesce
suggesting the presence of exceptional points of degeneracy [98, 99, 100,
101]. In future work, we want to consider the additional sensing capabilities
a device would have owing to exceptional points’ $\sqrt{\epsilon}$ sensitivity
for small parameter $\epsilon$ [11, 12]. Funding ESPRC Grant EP/R002061/1

Acknowledgments We would like to thank Thomas Robertson-Brown for their
construction and characterization of the magnetic designs and Sharaa Alqarni
for the use of their reference optics.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are
available in Ref. [102].

## References

  * [1] A. Siegman, “Excess spontaneous emission in non-Hermitian optical systems. I. Laser amplifiers,” Physical Review A 39, 1253 (1989).
  * [2] M. V. Berry, “Physics of nonhermitian degeneracies,” Czechoslovak Journal of Physics 54, 1039–1047 (2004).
  * [3] C. M. Bender, “Making sense of non-Hermitian hamiltonians,” Reports on Progress in Physics 70, 947 (2007).
  * [4] M. Berry, “Optical polarization evolution near a non-Hermitian degeneracy,” Journal of Optics 13, 115701 (2011).
  * [5] R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides, and Ş. K. Özdemir, “The dawn of non-Hermitian optics,” Communications Physics 2, 1–5 (2019).
  * [6] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nature Physics 14, 11–19 (2018).
  * [7] A. U. Hassan, G. L. Galmiche, G. Harari, P. LiKamWa, M. Khajavikhan, M. Segev, and D. N. Christodoulides, “Chiral state conversion without encircling an exceptional point,” Physical Review A 96, 052129 (2017).
  * [8] A. U. Hassan, B. Zhen, M. Soljačić, M. Khajavikhan, and D. N. Christodoulides, “Dynamically Encircling Exceptional Points: Exact Evolution and Polarization State Conversion,” Physics Review Letters 118, 093002 (2017).
  * [9] J. B. Khurgin, Y. Sebbag, E. Edrei, R. Zektzer, K. Shastri, U. Levy, and F. Monticone, “Emulating exceptional-point encirclements using imperfect (leaky) photonic components: asymmetric mode-switching and omni-polarizer action,” Optica 8, 563–569 (2021).
  * [10] M. C. Rechtsman, “Optical sensing gets exceptional,” Nature 548, 161–162 (2017).
  * [11] W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
  * [12] J. Wiersig, “Prospects and fundamental limits in exceptional point-based sensing,” Nature Communications 11, 2454 (2020).
  * [13] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time–symmetric microring lasers,” Science 346, 975–978 (2014).
  * [14] P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
  * [15] Z. Gao, S. T. Fryslie, B. J. Thompson, P. S. Carney, and K. D. Choquette, “Parity-time symmetry in coherently coupled vertical cavity laser arrays,” Optica 4, 323–329 (2017).
  * [16] D. C. Brody, “Biorthogonal quantum mechanics,” Journal of Physics A: Mathematical and Theoretical 47, 035305 (2013).
  * [17] M. A. Nowak and W. Tarnowski, “Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach,” Journal of High Energy Physics 2018, 1–34 (2018).
  * [18] H. Xue, Q. Wang, B. Zhang, and Y. Chong, “Non-hermitian Dirac cones,” Physical Review Letters 124, 236403 (2020).
  * [19] R. El-Ganainy, K. Makris, D. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Optics Letters 32, 2632–2634 (2007).
  * [20] A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Physics-Uspekhi 57, 1063 (2014).
  * [21] L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nature Photonics 11, 752–762 (2017).
  * [22] P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, “Anti-parity–time symmetry with flying atoms,” Nature Physics 12, 1139–1145 (2016).
  * [23] Y. Jiang, Y. Mei, Y. Zuo, Y. Zhai, J. Li, J. Wen, and S. Du, “Anti-parity-time symmetric optical four-wave mixing in cold atoms,” Physical Review Letters 123, 193604 (2019).
  * [24] F. Zhang, Y. Feng, X. Chen, L. Ge, and W. Wan, “Synthetic anti-PT symmetry in a single microcavity,” Physical Review Letters 124, 053901 (2020).
  * [25] Q. Wang, J. Wang, H. Shen, S. Hou, and X. Yi, “Exceptional points and dynamics of a non-Hermitian two-level system without PT symmetry,” Europhysics Letters 131, 34001 (2020).
  * [26] T. Luan, H. Shen, and X. Yi, “Shortcuts to adiabaticity with general two-level non-Hermitian systems,” Physical Review A 105, 013714 (2022).
  * [27] J. A. Zielińska, F. A. Beduini, N. Godbout, and M. W. Mitchell, “Ultranarrow Faraday rotation filter at the Rb D1 line,” Optics Letters 37, 524–526 (2012).
  * [28] R. Erdélyi, M. B. Korsós, X. Huang _et al._ , “The Solar Activity Monitor Network - SAMNet,” Journal of Space Weather and Space Climate 12, 2 (2022).
  * [29] L. Giovannelli, F. Berrilli, D. Calchetti _et al._ , “The Tor Vergata Synoptic Solar Telescope (TSST): a robotic, compact facility for solar full disk imaging,” Journal of Space Weather and Space Climate 10, 58 (2020).
  * [30] A. Cacciani and M. Fofi, “The magneto-optical filter,” Solar Physics 59, 179–189 (1978).
  * [31] P. Chang, H. Shi, J. Miao, T. Shi, D. Pan, B. Luo, H. Guo, and J. Chen, “Frequency-stabilized Faraday laser with $10^{-14}$ short-term instability for atomic clocks,” Applied Physics Letters 120, 141102 (2022).
  * [32] H. Tang, H. Zhao, D. Zhang, L. Li, W. Yang, Z. Yang, H. Wang, X. Xu _et al._ , “Polarization insensitive efficient ultra-narrow diode laser strictly locked by a Faraday filter,” Optics Express 30, 29772–29780 (2022).
  * [33] H. Tang, H. Zhao, R. Wang, L. Li, Z. Yang, H. Wang, W. Yang, K. Han, and X. Xu, “18W ultra-narrow diode laser absolutely locked to the Rb D2 line,” Optics Express 29, 38728–38736 (2021).
  * [34] B. Luo, R. Ma, Q. Ji, L. Yin, J. Chen, and H. Guo, “Thermal and temporal characteristics of Faraday anomalous dispersion optical filters based on a hollow cathode lamp,” Optics Letters 46, 5372–5375 (2021).
  * [35] T. Shi, X. Guan, P. Chang, J. Miao, D. Pan, B. Luo, H. Guo, and J. Chen, “A dual-frequency Faraday laser,” IEEE Photonics Journal 12, 1–11 (2020).
  * [36] F. Vargas, A. Liu, G. Swenson _et al._ , “Mesosphere and lower thermosphere changes associated with the July 2, 2019 total eclipse in South America over the Andes Lidar Observatory, Cerro Pachon, Chile.” Journal of Geophysical Research: Atmospheres 127, e2021JD035064 (2022).
  * [37] Y. Xia, X. Cheng, F. Li, Y. Yang, X. Lin, J. Jiao, L. Du, J. Wang, and G. Yang, “Sodium lidar observation over full diurnal cycles in Beijing, China,” Applied Optics 59, 1529–1536 (2020).
  * [38] T. Pan, T. Chen, D. Sun, Y. Han, X. Xue, R. Zhao, and J. Lan, “Metastable helium faraday filter for helium lidar to measure the density of the thermosphere,” Optics Express 29, 4431–4441 (2021).
  * [39] M. Widmann, S. L. Portalupi, P. Michler, J. Wrachtrup, and I. Gerhardt, “Faraday filtering on the Cs-D1-line for quantum hybrid systems,” IEEE Photonics Technology Letters 30, 2083–2086 (2018).
  * [40] J. Zhang, G. Gao, B. Wang, X. Guan, L. Yin, J. Chen, and B. Luo, “Background noise resistant underwater wireless optical communication using Faraday atomic line laser and filter,” Journal of Lightwave Technology 40, 63–73 (2022).
  * [41] L. Yin, D. Qian, Z. Geng, H. Zhan, and G. Wu, “Using FADOF to eliminate the background light influence in ghost imaging,” Optics Express 30, 36297–36306 (2022).
  * [42] W. Zhuang, Y. Zhao, S. Wang, Z. Fang, F. Fang, and T. Li, “Ultranarrow bandwidth Faraday atomic filter approaching natural linewidth based on cold atoms,” Chinese Optics Letters 19, 030201 (2021).
  * [43] G. Bi, J. Kang, J. Fu, L. Ling, and J. Chen, “Ultra-narrow linewidth optical filter based on Faraday effect at isotope 87Rb 420 nm transitions,” Physics Letters A 380, 4022–4026 (2016).
  * [44] X. Guan, W. Zhuang, T. Shi, J. Miao, J. Zhang, J. Chen, and B. Luo, “Cold-atom optical filtering enhanced by optical pumping,” Frontiers in Physics 10, 1090483.
  * [45] R. Speziali, A. Di Paola, M. Centrone, M. Oliviero, D. B. Calia, L. Dal Sasso, M. Faccini, V. Mauriello, and L. Terranegra, “The first light of the Solar Activity MOF Monitor Telescope (SAMM),” Journal of Space Weather and Space Climate 11, 22 (2021).
  * [46] B. Yin, L. S. Alvarez, and T. M. Shay, “The Rb 780-nanometer Faraday anomalous dispersion optical filter: Theory and experiment,” JPL, The Telecommunications and Data Acquisition Report 116, 71–85 (1994).
  * [47] Y. Yan, J. Yuan, L. Wang, L. Xiao, and S. Jia, “Tailoring the pulse train of an optical frequency comb with a magnetized atomic medium,” Laser Physics Letters 19, 105207 (2022).
  * [48] M. Auzinsh, A. Sargsyan, A. Tonoyan, C. Leroy, R. Momier, D. Sarkisyan, and A. Papoyan, “Wide range linear magnetometer based on a sub-microsized K vapor cell,” Applied Optics 61, 5749–5754 (2022).
  * [49] F. S. Ponciano-Ojeda, F. D. Logue, and I. G. Hughes, “Absorption spectroscopy and Stokes polarimetry in a 87Rb vapour in the Voigt geometry with a 1.5 T external magnetic field,” Journal of Physics B: Atomic, Molecular and Optical Physics 54, 015401 (2020).
  * [50] D. Pizzey, J. D. Briscoe, F. D. Logue, F. S. Ponciano Ojeda, S. A. Wrathmall, and I. G. Hughes, “Laser spectroscopy of hot atomic vapours: from ’scope to theoretical fit,” New Journal of Physics 24, 125001 (2022).
  * [51] C. R. Higgins, D. Pizzey, R. S. Mathew, and I. G. Hughes, “Atomic line versus lens cavity filters: a comparison of their merits,” OSA Continuum 3, 961–970 (2020).
  * [52] W. Kiefer, R. Löw, J. Wrachtrup, and I. Gerhardt, “Na-Faraday rotation filtering: the optimal point,” Scientific Reports 4, 1–7 (2014).
  * [53] Y. Yan, J. Yuan, L. Wang, L. Xiao, and S. Jia, “A dual-wavelength bandpass faraday anomalous dispersion optical filter operating on the D1 and D2 lines of rubidium,” Optics Communications 509, 127855 (2022).
  * [54] F. D. Logue, J. D. Briscoe, D. Pizzey, S. A. Wrathmall, and I. G. Hughes, “Better magneto-optical filters with cascaded vapor cells,” Optics Letters 47, 2975–2978 (2022).
  * [55] E. Baumann, J. Chiles, I. Coddington _et al._ , “White paper: Frequency combs for aerospace applications,” Decadal Survey on Biological and Physical Sciences (BPS) Research in Space 2023-2032 .
  * [56] S. D. Leifer, S. A. Diddams _et al._ , “White paper: Optical frequency combs for space applications,” Decadal Survey on Biological and Physical Sciences (BPS) Research in Space 2023-2032 .
  * [57] H. Wu, Z. Qian, H. Zhang, X. Xu, B. Xue, and J. Zhai, “Precise underwater distance measurement by dual acoustic frequency combs,” Annalen der Physik 531, 1900283 (2019).
  * [58] C. Zhu, J. Guan, F. Zhou, E. Zhu, and Y. Li, “Giant magneto-optical rotation effect in rubidium vapor measured with a low-cost detection system,” OSA Continuum 4, 2527–2534 (2021).
  * [59] L. Wu, Y. Liu, H. Guo, H. Wen, Z. Li, J. Tang, Z. Ma, and J. Liu, “A MOEMS accelerometer based on the magneto-optical rotation effect,” IEEE Sensors Journal 22, 3061–3066 (2021).
  * [60] K. J. Carothers, R. A. Norwood, and J. Pyun, “High Verdet constant materials for magneto-optical Faraday rotation: A review,” Chemistry of Materials 34, 2531–2544 (2022).
  * [61] Y. Liu, B. Yang, J. Wang, W. Huang, Z. Gou, and H. Zhou, “Demonstration of Faraday anomalous dispersion optical filter with reflection configuration,” Chinese Physics B 31, 017804 (2022).
  * [62] M. A. Zentile, D. J. Whiting, J. Keaveney, C. S. Adams, and I. G. Hughes, “Atomic Faraday filter with equivalent noise bandwidth less than 1 GHz,” Optics Letters 40, 2000 (2015).
  * [63] I. Gerhardt, “How anomalous is my Faraday filter?” Optics Letters 43, 5295–5298 (2018).
  * [64] M. W. Kudenov, B. Pantalone, and R. Yang, “Dual-beam potassium Voigt filter for atomic line imaging,” Applied Optics 59, 5282–5289 (2020).
  * [65] L. Yin, B. Luo, J. Xiong, and H. Guo, “Tunable rubidium excited state Voigt atomic optical filter,” Optics Express 24, 6088–6093 (2016).
  * [66] J. Menders, P. Searcy, K. Roff, and E. Korevaar, “Blue cesium Faraday and Voigt magneto-optic atomic line filters,” Optics Letters 17, 1388–1390 (1992).
  * [67] E. Palik and J. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Reports on Progress in Physics 33, 1193–1322 (1970).
  * [68] C. H. Papas, _Theory of electromagnetic wave propagation_ (Dover Publications Inc., New York, 1989).
  * [69] M. D. Rotondaro, B. V. Zhdanov, and R. J. Knize, “Generalized treatment of magneto-optical transmission filters,” Journal of the Optical Society of America B Optical Physics 32, 2507 (2015).
  * [70] J. Keaveney, S. A. Wrathmall, C. S. Adams, and I. G. Hughes, “Optimized ultra-narrow atomic bandpass filters via magneto-optic rotation in an unconstrained geometry,” Optics Letters 43, 4272 (2018).
  * [71] F. S. Ponciano-Ojeda, “PhD Thesis,” http://etheses.dur.ac.uk/14065 (2021).
  * [72] C. J. Foot, _Atomic Physics_ , vol. 7 (Oxford University Press Oxford, 2004).
  * [73] C. S. Adams and I. G. Hughes, _Optics f2f: from Fourier to Fresnel_ (Oxford University Press, 2018).
  * [74] J. Keaveney, F. S. Ponciano-Ojeda, S. M. Rieche, M. J. Raine, D. P. Hampshire, and I. G. Hughes, “Quantitative optical spectroscopy of ${87}$Rb vapour in the Voigt geometry in DC magnetic fields up to 0.4 T,” Journal of Physics B: Atomic, Molecular and Optical Physics 52, 055003 (2019).
  * [75] A. Sargsyan, A. Amiryan, A. Tonoyan, E. Klinger, and D. Sarkisyan, “Circular dichroism in atomic vapors: Magnetically induced transitions responsible for two distinct behaviors,” Physics Letters A 390, 127114 (2021).
  * [76] H. Shang, B. Zhou, W. Quan, H. Chi, J. Fang, and S. Zou, “Measurement of rubidium vapor number density based on Faraday modulator,” Journal of Physics D: Applied Physics (2022).
  * [77] B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” American Journal of Physics 75, 163–168 (2007).
  * [78] A. Z. Goldberg, P. De La Hoz, G. Björk, A. B. Klimov, M. Grassl, G. Leuchs, and L. L. Sánchez-Soto, “Quantum concepts in optical polarization,” Advances in Optics and Photonics 13, 1–73 (2021).
  * [79] A. Dantan, J. Cviklinski, E. Giacobino, and M. Pinard, “Spin squeezing and light entanglement in coherent population trapping,” Physical Review Letters 97, 023605 (2006).
  * [80] L. Weller, T. Dalton, P. Siddons, C. S. Adams, and I. G. Hughes, “Measuring the Stokes parameters for light transmitted by a high-density rubidium vapour in large magnetic fields,” Journal of Physics B: Atomic, Molecular and Optical Physics 45, 055001 (2012).
  * [81] Z.-K. Su and J.-H. Wang, “Measuring Stokes parameters by means of unitary operations,” Applied Optics 60, 9852–9857 (2021).
  * [82] B. E. Sherlock and I. G. Hughes, “How weak is a weak probe in laser spectroscopy?” American Journal of Physics 77, 111–115 (2009).
  * [83] R. Bala, J. Ghosh, and V. Venkataraman, “A comprehensive model for doppler spectra in thermal atomic vapour,” Journal of Physics B: Atomic, Molecular and Optical Physics 55, 165003 (2022).
  * [84] B. Luo, L. Yin, J. Xiong, J. Chen, and H. Guo, “Signal intensity influences on the atomic Faraday filter,” Optics Letters 43, 2458–2461 (2018).
  * [85] B. Luo, L. Yin, J. Xiong, J. Chen, and H. Guo, “Induced-dichroism-excited atomic line filter at 1529 nm,” IEEE Photonics Technology Letters 30, 1551–1554 (2018).
  * [86] L. D. Turner, V. Karaganov, P. Teubner, and R. E. Scholten, “Sub-Doppler bandwidth atomic optical filter,” Optics Letters 27, 500–502 (2002).
  * [87] A. Ray, M. S. Ali, and A. Chakrabarti, “Study of the effect of repump laser on atomic line filter,” JOSA B 30, 2436–2442 (2013).
  * [88] Z. Zhu, H. Li, W. Sun, and S. Liu, “Theoretical model of an atomic line filter based on the combination of a double selective pump,” Applied Optics 56, 78–83 (2017).
  * [89] M. A. Zentile, J. Keaveney, L. Weller, D. J. Whiting, C. S. Adams, and I. G. Hughes, “ElecSus: A program to calculate the electric susceptibility of an atomic ensemble,” Computer Physics Communications 189, 162–174 (2015).
  * [90] J. Keaveney, C. S. Adams, and I. G. Hughes, “ElecSus: Extension to arbitrary geometry magneto-optics,” Computer Physics Communications 224, 311–324 (2018).
  * [91] Z. Wu, M. Kitano, W. Happer, M. Hou, and J. Daniels, “Optical determination of alkali metal vapor number density using Faraday rotation,” Applied Optics 25, 4483–4492 (1986).
  * [92] M. A. Zentile, J. Keaveney, R. S. Mathew, D. J. Whiting, C. S. Adams, and I. G. Hughes, “Optimization of atomic Faraday filters in the presence of homogeneous line broadening,” Journal of Physics B: Atomic Molecular Physics 48, 185001 (2015).
  * [93] I. G. Hughes and T. P. A. Hase, _Measurements and their Uncertainties: A Practical Guide to Modern Error Analysis_ (OUP, Oxford, 2010).
  * [94] D. Brown, B.-M. Ma, and Z. Chen, “Developments in the processing and properties of NdFeB-type permanent magnets,” Journal of Magnetism and Magnetic Materials 248, 432–440 (2002).
  * [95] L. Weller, K. S. Kleinbach, M. A. Zentile, S. Knappe, C. S. Adams, and I. G. Hughes, “Absolute absorption and dispersion of a rubidium vapour in the hyperfine Paschen–Back regime,” Journal of Physics B: Atomic, Molecular and Optical Physics 45, 215005 (2012).
  * [96] L. Chang, S. Liu, and J. E. Bowers, “Integrated optical frequency comb technologies,” Nature Photonics 16, 95–108 (2022).
  * [97] N. Picqué and T. W. Hänsch, “Frequency comb spectroscopy,” Nature Photonics 13, 146–157 (2019).
  * [98] W. Heiss, “The physics of exceptional points,” Journal of Physics A: Mathematical and Theoretical 45, 444016 (2012).
  * [99] D. Heiss, “Circling exceptional points,” Nature Physics 12, 823–824 (2016).
  * [100] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nature Materials 18, 783–798 (2019).
  * [101] M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
  * [102] F. D. Logue, “Dataset,” https://collections.durham.ac.uk/collections/r23x816m645.

## Appendix A: Deriving the Transition Projection Operators in the Oblique
Geometry

We define $z$ as the propagation direction of the light and note that the
light is polarized transversely in the $x-y$ plane. In the case where $z$ is
the quantization axis, the angular part of the dipole matrix element is:

$\mathcal{M}=\sqrt{\frac{4\pi}{3}}\langle
l^{\prime}m_{l}^{\prime}|\frac{\widehat{x}+\mathrm{i}\widehat{y}}{\sqrt{2}}Y_{1,1}+\frac{\widehat{x}-\mathrm{i}\widehat{y}}{\sqrt{2}}Y_{1,-1}+\widehat{z}\hskip
2.84544ptY_{1,0}|lm_{l}\rangle,$ (6)

where $l,m_{l},j$ are the orbital, projection and total angular momentum
quantum numbers and $Y_{l,m_{l}}(\theta,\phi)$ are spherical harmonic
functions. However, since a plane wave is polarized transversely, we can omit
$\widehat{z}$ operators. Therefore, in the Faraday geometry, where the
$k$-vector of the light is parallel with the magnetic field along $z$, we
have:

$\mathcal{M}_{\rm{F}}=\sqrt{\frac{4\pi}{3}}\langle
l^{\prime}m_{l}^{\prime}|\frac{\widehat{x}+\mathrm{i}\widehat{y}}{\sqrt{2}}Y_{1,1}+\frac{\widehat{x}-\mathrm{i}\widehat{y}}{\sqrt{2}}Y_{1,-1}|lm_{l}\rangle.$
(7)

Given that $\Delta m_{l}=\Delta m_{J}$ since $\Delta m_{s}=0$, left/right
handed circular light induces $\sigma^{+/-}$ transitions in the Faraday
geometry [73].

In the Voigt geometry, the $k$-vector of the light is perpendicular to the
magnetic field. Without loss of generality, we choose to rotate in the $x-z$
plane. As such, we substitute $\widehat{z}\rightarrow\widehat{x}$,
$\widehat{x}\rightarrow-\widehat{z}$ in eq. 6. This gives the Voigt angular
dipole matrix element:

$\mathcal{M}_{\rm{V}}=\sqrt{\frac{4\pi}{3}}\langle
l^{\prime}m_{l}^{\prime}|\widehat{x}\leavevmode\nobreak\
Y_{1,0}+\frac{\mathrm{i}\widehat{y}}{\sqrt{2}}(Y_{1,1}-Y_{1,-1})|lm_{l}\rangle.$
(8)

Therefore, horizontally polarized light induces $\pi$ transitions and
vertically polarized light induces a linear combination of $\sigma^{+}$ and
$\sigma^{-}$ transitions [74]. The reader can derive the expression if we
instead rotate in the $y-z$ plane.

A general expression for all geometries is achieved by rotating the magnetic
field by an angle $\theta$. This gives the quantization axis transformations
as:

$\begin{split}\widehat{x}\rightarrow\widehat{x}\cos\theta-\widehat{z}\sin\theta,\\\
\widehat{z}\rightarrow\widehat{z}\cos\theta+\widehat{x}\sin\theta.\end{split}$
(9)

We then arrive at the general angular dipole matrix element by applying the
transformations in Eq. 9 to Eq. 6:

$\begin{split}\mathcal{M}_{\rm{O}}=\sqrt{\frac{4\pi}{3}}\langle
l^{\prime}m_{l}^{\prime}|&\frac{\widehat{x}\cos\theta+\mathrm{i}\widehat{y}}{\sqrt{2}}Y_{1,1}+\frac{\widehat{x}\cos\theta-\mathrm{i}\widehat{y}}{\sqrt{2}}Y_{1,-1}\\\
&+\widehat{x}\sin\theta Y_{1,0}|lm_{l}\rangle.\end{split}$ (10)

If the magnetic field is oblique to the $k$-vector, the projection operators
are elliptical for $\sigma$ transitions and horizontally linear for $\pi$
transitions. However, the operators are non-orthogonal. Therefore, at least
two transitions will be induced for any given input polarization (provided the
probability of the transition in question is non-zero at the particular
frequency).

## Appendix B: Calculating Invariant Polarization States in the Oblique
Geometry

Following the Jones Calculus method from [69], the propagation eigenmodes
$\vec{a}$ and $\vec{b}$ form the conjugate transposed (†) rotation matrix,

$\mathbf{R}=\begin{bmatrix}a_{1}&b_{1}\\\ a_{2}&b_{2}\end{bmatrix}^{\dagger},$
(11)

such that the input electric field $\mathbf{E_{\rm{in}}}$ is propagated to the
output electric field $\mathbf{E_{\rm{out}}}$ as

$\mathbf{E_{\rm{out}}=R^{-1}\cdot T\cdot R\cdot E_{\rm{in}}},$ (12)

where $\mathbf{T}$ is a diagonal matrix with first and second diagonal terms
$f(n_{a})$ and $f(n_{b})$, the function $f$ applied to the refractive indices
$n_{a}$ and $n_{b}$. These refractive indices are the eigenvalues associated
with $\vec{a}$ and $\vec{b}$ respectively.

If an invariant polarization $\vec{p}$ or $\vec{q}$ is the input field to eq.
12, the output field will only depend on one of $f(n_{a})$ or $f(n_{b})$ . As
such, no phase difference is induced and the polarization state does not
evolve.

To calculate the invariant polarizations, we can choose a state orthogonal to
an eigenmode. Formally, we say that the invariant polarizations and eigenmodes
together form a biorthogonal system [16].

The need for a biorthogonal system is a direct result of a non-Hermitian
system Hamiltonian [2, 3, 4, 5, 6]. The eigenvalues (refractive indices)
$n_{a}$ and $n_{b}$ are in general complex.

In the Faraday and Voigt cases, the eigenmodes are orthogonal. Therefore, the
invariant polarizations are the eigenmodes and we say the system is
orthonormal provided the eigenmodes are normalized.

We prove this in the Faraday case where the eigenmodes are left hand and right
hand circular light. We input an eigenmode, left hand circular light,

$\begin{split}\mathbf{E_{\rm{out}}}&=\frac{1}{2}\begin{bmatrix}1&1\\\
-\mathrm{i}&\mathrm{i}\end{bmatrix}\cdot\begin{bmatrix}{{f(n_{a})}}&0\\\
0&{f(n_{b})}\end{bmatrix}\cdot\overbrace{\begin{bmatrix}1&-\mathrm{i}\\\
1&\mathrm{i}\end{bmatrix}}^{{{{{\circlearrowleft,\circlearrowright}}}}}\cdot\overbrace{\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}}^{{{{{\circlearrowleft}}}}}\\\
&=\frac{1}{2}\begin{bmatrix}{{f(n_{a})}}+{f(n_{b})}&-\mathrm{i}({f(n_{a})}-{f(n_{b})})\\\
\mathrm{i}({f(n_{a})}-{f(n_{b})})&{{f(n_{a})}}+{f(n_{b})}\end{bmatrix}\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\\\ &=\begin{bmatrix}f(n_{a})\\\ \mathrm{i}\cdot
f(n_{a})\end{bmatrix}\\\ &=f(n_{a})\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\begin{matrix}\vphantom{x}\\\
\vphantom{y}.\end{matrix}\end{split}$ (13)

The final output field is the same as the input field multiplied by a factor
so the eigenmode is also an invariant. Similarly, we prove this in the Voigt
case where the eigenmodes are horizontally and vertically linear light. We
input horizontally linear light,

$\begin{split}\mathbf{E_{\rm{out}}}&=\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\cdot\begin{bmatrix}{{f(n_{a})}}&0\\\
0&{f(n_{b})}\end{bmatrix}\cdot\overbrace{\begin{bmatrix}1&0\\\
0&1\end{bmatrix}}^{\longleftrightarrow,\hskip
2.84544pt\updownarrow}\cdot\overbrace{\begin{bmatrix}1\\\
0\end{bmatrix}}^{\longleftrightarrow}\\\ &=\begin{bmatrix}f(n_{a})&0)\\\
0&f(n_{b})\end{bmatrix}\cdot\begin{bmatrix}1\\\ 0\end{bmatrix}\\\
&=\begin{bmatrix}f(n_{a})\\\ 0\end{bmatrix}\\\
&=f(n_{a})\cdot\begin{bmatrix}1\\\ 0\end{bmatrix}\end{split}$ (14)

Like the Faraday case, the output field is the input field multiplied by a
factor and so the eigenmode is also an invariant polarization.

In the oblique case, the eigenmodes and the invariant polarizations are not
one and the same since the oblique eigenmodes are in general non-orthogonal.
Without loss of generality, we choose the left hand orthogonal vectors to the
eigenmodes as our oblique invariant polarizations,

$\vec{p}=\begin{pmatrix}-a_{2}\\\ a_{1}\end{pmatrix}^{*},\hskip
11.38092pt\vec{q}=\begin{pmatrix}-b_{2}\\\ b_{1}\end{pmatrix}^{*},$ (15)

with ∗ indicating the complex conjugate.

Figure 7: A diagram of the vectors used in eq. 16. In purple, we have the
linear horizontal eigenmode, $\vec{a}$, and the linear diagonal eigenmode at
$45^{\circ}$, $\vec{b}$. In blue, the invariant polarizations to $\vec{a}$ and
$\vec{b}$, $\vec{p}$ and $\vec{q}$ respectively. In red, an arbitrary input
polarization, $\mathbf{E_{\rm{in}}}$, to be propagated through the vapor.

We illustrate this situation using a toy expression. Fig. 7 is a diagram of
the relevant eigenmodes and invariant polarizations. Our eigenmodes will be
two non-orthogonal polarizations: horizontally polarized light and linear
light at 45∘. We input linear horizontally polarized light. Eq. 12 then gives,

$\begin{split}\mathbf{E_{\rm{out}}}&=\begin{bmatrix}1&0\\\ -1&1\\\
\end{bmatrix}\cdot\begin{bmatrix}{{f(n_{a})}}&0\\\
0&{f(n_{b})}\end{bmatrix}\cdot\overbrace{\begin{bmatrix}1&0\\\
1&1\end{bmatrix}}^{\longleftrightarrow,\hskip
2.84544pt\mathrel{\text{$\nearrow$\hbox
to0.0pt{\hss$\swarrow$}}}}\cdot\overbrace{\begin{bmatrix}1\\\
0\end{bmatrix}}^{\longleftrightarrow}\\\ &=\begin{bmatrix}{{f(n_{a})}}&0\\\
{f(n_{b})}-{f(n_{a})}&{f(n_{b})}\end{bmatrix}\cdot\begin{bmatrix}1\\\
0\end{bmatrix}\\\ &=\begin{bmatrix}{f(n_{a})}\\\
{f(n_{b})}-{f(n_{a})}\end{bmatrix}\\\ &={f(n_{b})}\cdot\begin{bmatrix}0\\\
1\end{bmatrix}-{f(n_{a})}\cdot\begin{bmatrix}-1\\\ 1\end{bmatrix}\\\
&={f(n_{b})}\cdot\updownarrow-{f(n_{a})}\cdot\mathrel{\text{$\nwarrow$\hbox
to0.0pt{\hss$\searrow$}}}.\end{split}$ (16)

While we have input an eigenmode, since the eigenmodes are not orthogonal, the
polarization is not invariant. We can decompose the result into two invariant
polarizations, linear diagonal light at $-45^{\circ}$ and vertically polarized
light. Note the relationship between the eigenmodes and the invariant
polarizations is as described in eq. 15.

Into the same system, we input an invariant polarization, vertically linear
light, and prove its invariance,

$\begin{split}\mathbf{E_{\rm{out}}}&=\begin{bmatrix}{{f(n_{a})}}&0\\\
{f(n_{b})}-{f(n_{a})}&{f(n_{b})}\end{bmatrix}\cdot\overbrace{\begin{bmatrix}0\\\
1\end{bmatrix}}^{\updownarrow}\\\ &=\begin{bmatrix}{0}\\\
{f(n_{b})}\end{bmatrix}\\\ &={f(n_{b})}\cdot\begin{bmatrix}0\\\
1\end{bmatrix}\end{split}$ (17)

Of importance to emphasize is this is a toy example. Horizontally linear light
and linear light at $45^{\circ}$ could never jointly form the eigenmodes of a
system. This is because $\mathbf{R}$ is a rotation matrix for some angle
$\beta$ and multiplicative factor $\mathcal{F}$ implying a constraint,

$\begin{split}\mathbf{R}=\begin{bmatrix}a_{1}&b_{1}\\\
a_{2}&b_{2}\end{bmatrix}^{\dagger}=\mathcal{F}\cdot\begin{bmatrix}\cos\beta&-\sin\beta\\\
\sin\beta&\cos\beta\end{bmatrix}\\\ \implies\lvert a_{1}\cdot
b_{1}\rvert=\lvert a_{2}\cdot b_{2}\rvert.\end{split}$ (18)

The eigenmodes of the Faraday and Voigt geometry respectively adhere to this
constraint. The constraint implies elliptical eigenmodes in the oblique
geometry.

## Appendix C: Equivalent Propagation of both passes in the Double Pass Filter

We consider first the different angles between magnetic field and propagation
direction of both passes, $\alpha$ and $\alpha^{*}=(180-\alpha)^{\circ}$.
Equivalently, we can say in the second pass the magnetic field makes an angle
$\alpha$ with a vector pointing in the opposite direction to $\vec{k}$.

In this picture, the relevant axis for identifying polarizations has flipped.
Instead of looking at the electric field towards the source, we should view it
looking from the source. Therefore, left/right handed polarizations now appear
to be right/left handed.

Given that everything else remains identical, we can say that the evolution of
a left/right handed wave with magnetic field at angle $\alpha$ to the
$k$-vector is the same as the evolution of a right/left handed wave with
magnetic field at angle $\alpha^{*}$ to the $k$-vector. If linearly polarized
light is input, the filter transmission will not differ in the two cases.

The filter output from a horizontally ($\mathbf{E_{\leftrightarrow}}$) or
vertically ($\mathbf{E_{\updownarrow}}$) polarized input with crossed
polarizers is always the same regardless of geometry,

$||\mathbf{P_{\updownarrow}\cdot R^{-1}\cdot T\cdot R\cdot
E_{\leftrightarrow}}||=||\mathbf{P_{\leftrightarrow}\cdot R^{-1}\cdot T\cdot
R\cdot E_{\updownarrow}}||,$ (19)

where $\mathbf{P_{\updownarrow}}$ and $\mathbf{P_{\leftrightarrow}}$ are
vertically and horizontally linear polarizers respectively. This follows
directly from eq. 18. The two facts stated above explain why both passes of
the double pass filter have the same filter transmission.


# Alternative metrics to select motors for Quasi-Direct Drive actuators

Karthik Urs, Challen Enninful Adu, Elliott J. Rouse, Talia Y. Moore K. Urs and
C. Enninful Adu are with the EMBIR Lab, Robotics, University of Michigan, Ann
Arbor, MI 48109 USA email: ursk, enninful @umich.edu.E. Rouse is with the
Neurobionics Lab, Mechanical Engineering, Robotics, University of Michigan,
Ann Arbor, MI 48109 USA email<EMAIL_ADDRESS>Moore is with the EMBIR
Lab, Mechanical Engineering, Robotics, Ecology and Evolutionary Biology, and
Museum of Zoology, University of Michigan, Ann Arbor, MI 48109 USA. email:
<EMAIL_ADDRESS>
###### Abstract

Robotic systems for legged locomotion—including legged robots, exoskeletons,
and prosthetics—require actuators with low inertia and high output torque.
Traditionally, motors have been selected for these applications by maximizing
the motor gap radius. We present alternative metrics for motor selection that
are invariant to transmission ratio. The proposed metrics reward minimizing
the motor inertia while maximizing the torque and motor constants without
special consideration for gap radius, providing a better balance of properties
for legged locomotion applications. We rigorously characterize the T-Motor
RI50 and demonstrate the use of the metrics by comparing the RI50 to the
widely-used T-Motor U8 as a case study.

##  1 Introduction

Quasi-Direct Drive Actuators (QDDs) have enabled high-speed and robust
locomotion in legged robots by combining a powerful motor with a low-ratio
transmission to minimize output mechanical impedance (OMI) and mass while
meeting torque and power requirements. Designers must carefully balance
competing performance properties when selecting motors to make successful QDDs
for legged locomotion.

Figure 1: Physical characteristics of A) the RI50 and B) the U8 motors.

Legged robotic systems apply large ground reaction forces (GRFs) to execute
demanding actions, such as high-speed galloping, rapid direction changes, and
jumps. These actions impose large torque requirements for actuators. Because
actuators make up a significant portion of total robot mass, they should be as
lightweight as possible to translate actuator torques into high robot
accelerations.

Simultaneously, actuators should have low OMI, which is largely dominated by
the rotational inertia of the actuator. Low OMI, combined with modern
commutation [1], allows the actuators to accurately output torques and act as
torque sensors via measurement of motor current, providing valuable data to
higher-level robot controllers. This capability is analogous to neurological
proprioception, so QDDs are also referred to proprioceptive actuators [2].
With high OMI, every step of a legged robot generates high reaction forces in
the robot structure, potentially resulting in damage. Robot leg actions driven
by the environment must overcome the OMI before acting on the motor, so high
OMI also reduces proprioceptive accuracy (i.e., lower “transparency”).

Thus, motors with a high torque constant $K_{T}$ (torque per current, [Nm/A]),
high motor constant $K_{M}$ (torque per power heat waste,
[Nm/$\sqrt{\text{W}}$]), low inertia $J_{m}$ [kgm2], and low mass $m$ [kg] are
desired. Because these parameters are subject to tradeoffs, designers
typically select a motor with an advantageous balance between key parameters
above some minimum torque or power capability, and select a transmission to
make best use of the motor for the given application (i.e., sufficiently
amplify torque while retaining speed).

Popular design principles suggest that maximizing gap radius ($r_{g}$) of a
motor is the best way to address the challenge of motor selection for QDDs;
this strategy is founded on analysis of ideal motor models, trends shown on a
large range of motors [2, 3, 4, 5], and encouraging performance of modern
robots with high-radius motors [6, 7, 8]. However, for a specific application,
the best motor choice may not have the highest $r_{g}$. We propose a pair of
simple selection metrics independent of $r_{g}$ to identify motors with useful
properties for legged robots (Section 3). We also provide a case study of the
RI50 motor, which defies the larger-scale trends (Section 3.1). Methods for
characterizing the RI50 motor are described in Section 4; results and
comparisons are summarized in Section 5. A physics-based rationale to explain
the observed differences between real motors and scaling-based predictions is
proposed in Section 6.

##  2 Current Design Principles

Figure 2: Commercial motor options and isolines with order-of-magnitude
spacing for $S_{M}$ (10). Motors closer to the bottom-right (such as the RI50)
are considered more suitable for QDDs under the proposed metrics (lower
$S_{M}$). U8 and RI50 empirically characterized; others from [9, 10, 11].

Several metrics and principles have been proposed to guide the motor selection
step of QDD actuator design. Because temperature is frequently the limiting
factor on continuous motor torque and power, motors can be selected by
maximizing thermal specific torque, $K_{\text{ts}}$
[Nm/(kg$\sqrt{\degree\text{C}}$)], proposed in [7]:

$K_{\text{ts}}=\frac{K_{T}}{m}\sqrt{\frac{1}{R_{\text{th}}R_{\phi}}}$ (1)

where $R_{\text{th}}$ is the motor thermal resistance in [$\degree$C/W] and
$R_{\phi}$ is the electrical phase resistance of the motor [$\Omega$].

Others use more general principles derived from physics-based dimensional
scaling analysis of ideal motor models. Under a mass constraint and some given
torque requirement, maximizing motor gap radius $r_{g}$ is often considered
the best way to address the design priorities established above [2, 3, 4, 5].
The following are developed in [2], reproduced with minor nomenclature
adjustments:

$\displaystyle K_{T}$ $\displaystyle\propto r_{g}^{2}l$ (2) $\displaystyle
K_{T}/m$ $\displaystyle\propto r_{g}$ (3) $\displaystyle K_{T}/J_{m}$
$\displaystyle\propto 1/r_{g}$ (4) $\displaystyle K_{M}^{2}$
$\displaystyle\propto r_{g}^{3}l$ (5) $\displaystyle J_{m}$
$\displaystyle\propto r_{g}^{3}l$ (6)

where $l$ is the axial length of the motor. The reference assumes constant
current for thermal similarity, so we have substituted the $\tau$ used in [2]
with $K_{T}$ without a change in meaning. Though (4) appears to suggest a
tradeoff in inertia, actuator inertia $J_{a}$ (9) is invariant when
considering the need for a transmission to meet some given torque requirement
[7].

A scaling analysis in [12] bridges torque production and electrical
responsiveness (i.e., electrical time constant) as another approach, but also
rewards high $r_{g}$ Thermal specific torque (1) increases linearly with
$r_{g}$ in real motors as well [7]. These widely-used principles suggest that
maximizing $r_{g}$ is a favorable design choice: it improves torque density
and motor constant, thereby reducing the necessary transmission ratio and
simplifying mechanical design.

##  3 Proposed Metrics for QDD Motor Selection

We propose alternative metrics for motor selection. Motor properties, when
combined with a transmission selection, yield effective actuator performance
characteristics. The effective torque constant, $K_{Ta}$ in [Nm/A], the
effective motor constant, $K_{Ma}$ in [Nm/W1/2], and actuator inertia,
$J_{a}$, are defined as follows:

$K_{Ta}\doteq N\times K_{T}$ (7) $K_{Ma}\doteq N\times K_{M}$ (8) $J_{a}\doteq
N^{2}J_{m}$ (9)

with the transmission ratio, $N$, defined as a speed reduction. These
equations demonstrate that there is no straightforward motor and transmission
($N$) choice that balances the tradeoff between $K_{Ta}$ and $K_{Ma}$ (to be
maximized) on one hand and $J_{a}$ (to be minimized) on the other.

To understand the relevant design trade-offs of motor selection
quantitatively, we can use the following metrics, which conveniently remove
the transmission selection from consideration (assuming massless transmissions
with no frictional losses). These metrics are one way to understand the
intrinsic advantages of a motor:

$S_{M}\doteq\frac{J_{a}}{K_{Ma}^{2}}=\frac{N^{2}J_{m}}{N^{2}K_{M}^{2}}=\frac{J_{m}}{K_{M}^{2}}$
(10)
$S_{T}\doteq\frac{J_{a}}{K_{Ta}^{2}}=\frac{N^{2}J_{m}}{N^{2}K_{T}^{2}}=\frac{J_{m}}{K_{T}^{2}}$
(11)

Motors that minimize $S_{M}$ and $S_{T}$ have an intrinsically better balance
of properties for the characteristics of interest (maximizing $K_{T}$ and
$K_{M}$ while minimizing $J_{m}$).

Equation (10) also has a physical implication: it approximates the mechanical
time constant of motor speed response to voltage change, $\tau_{\omega}$,
which should be minimized to improve responsiveness:

$\displaystyle T_{m}$
$\displaystyle=J_{m}\dot{\omega}=\frac{K_{T}}{R_{\phi}}(V_{q}-K_{B}\omega)$
(12) $\displaystyle J_{m}\Omega(s)s$
$\displaystyle=\frac{K_{T}}{R_{\phi}}(V_{q}(s)-K_{B}\Omega(s))$ (13)
$\displaystyle\Omega(s)/V_{q}(s)$
$\displaystyle=\frac{\frac{1}{K_{B}}}{\frac{R_{\phi}J_{m}}{K_{T}K_{B}}s+1}=\frac{\frac{1}{K_{B}}}{\frac{J_{m}}{K_{M}^{2}}s+1}$
(14) $\displaystyle\rightarrow\tau_{\omega}$
$\displaystyle=\frac{J_{m}}{K_{M}^{2}}=S_{M}$ (15)

where $T_{m}$ is the motor torque [Nm], $\omega$ is the motor velocity
($\Omega(s)$ in Laplace domain) [rad/s], $R_{\phi}$ is the motor phase
resistance [$\Omega$], $V_{q}$ is the motor quadrature-axis (“q-axis”, [1])
voltage ($V_{q}(s)$ in Laplace domain) [V], $K_{B}$ is the motor back-EMF
constant [Vs/rad], and $s$ is the Laplace variable. By (15), we call $S_{M}$
the Responsiveness Metric. This analysis does not explicitly consider gap
radius. Equations (5) and (6) suggest that responsiveness $S_{M}$ should be
invariant to choices in motor geometry under the assumptions of common ideal
motor models. Therefore, this metric is particularly useful at identifying
motors that gain performance benefits from unmodeled factors (e.g., winding
quality, stator core material, etc.). Metric $S_{T}$, which we call Torque-
Specific Inertia, does not have such an invariance, but is nonetheless
relevant to actuator implementation.

As established in the previous section, robot performance is also sensitive to
motor mass. Therefore, a designer may choose to incorporate a mass weighting
(e.g., using $mS_{M}$ or $mS_{T}$).

###  3.1 Case Study: T-Motor RI50 vs. U8

Considering the many motor selection methodologies that emphasize large gap
radius (Section 2), it is no surprise that one of the most commonly used
motors in legged robots is the large radius, exterior-rotating T-Motor U8 and
“8108” size equivalents (e.g., iFlight EX8, SunnySky M8) [6, 7, 13, 8, 14, 15,
16, 17] (U8 properties in Table LABEL:tab:motordata). For these robots, the U8
style motor is often the largest, commonly available motor that will meet
design size requirements. The U8 is also used as the jumping-off point and
benchmark to propose new motors in [12]. Due to its ubiquity, we use the U8 as
a benchmark.

In contrast, the interior-rotating T-Motor RI50 is a motor that has a better
balance between torque constant, motor constant, and inertia as measured by
the proposed metrics (i.e., low $S_{M}$ and $S_{T}$, see Table
LABEL:tab:motordata) than the dimensional scaling analysis would suggest for
its small gap-radius [2], and is only 193 g.

##  4 Empirical Characterization of RI50 Motor

To facilitate direct comparison, we follow the U8 characterization methods
described in [18] to characterize the RI50 motor. We constructed a dynamometer
that features two actuators (motors) in opposition with outputs connected
through a rotary torque sensor (TRD605-18Nm, TRS605-5Nm, Futek, Irvine, CA).
Both actuators may act as the loading or driving actuator, and can swap roles
mid-test as necessary. The motor drivers (moteus r4.5/r4.8, mjbots, Cambridge,
MA, USA) can be queried for position, velocity, q-axis current, and
diagnostics. A host Linux single-board computer (4B, Raspberry Pi Foundation,
Cambridge, UK) commands and queries all devices and logs data using custom
software written in C++.

###  4.1 Passive Properties

While passing current between two phase terminals (“line-to-line”) to heat the
windings, we observed the motor with a thermal camera (A655, Teledyne FLIR,
Wilsonville, OR, USA) to visually determine the winding style (“Wye” or
“Delta”).

Line-to-line resistance was measured with a multimeter (34461A, Keysight,
Santa Rosa, CA, USA), and line-to-line inductance was measured with a
precision LCR meter (TMPRO 4230, Wayne Kerr, West Sussex, UK).

For both resistance and inductance, the value between all three combinations
of phase terminals were measured and then averaged to yield a final result,
which was then converted to phase-frame (phase resistance $R_{\phi}$ and
effective inductance $L_{e}$) with knowledge of the winding style, per [1].

###  4.2 Motor Coefficients

To empirically determine the torque constant, $K_{T}$, for the RI50 motor, the
motor under test was commanded with q-axis current using the embedded field-
oriented controller (FOC) for 5 s while mechanically stalled. For each $i_{q}$
command 5 s test (-8 to 8 A in 1 A increments), the first and last 0.5 s of
torque data were discarded to remove any transient artifacts.

The back electromotive force (back EMF, or BEMF) constant, $K_{B}$ [Vs/rad],
describes the proportionality between rotational speed and the q-axis voltage
($V_{q}$). To empirically determine $K_{B}$, the motor under test was back-
driven by a cordless drill while the phase line-to-line voltages were measured
with an oscilloscope.

The fundamental motor constant, $K_{M}$ can then simply be calculated as
$K_{M}=\sqrt{K_{T}K_{B}/R_{\phi}}$

###  4.3 Motor Inertia

We determined the motor dynamic properties via random-input testing to observe
the frequency response of the system, as in [18]. The motor under test was
disconnected from the motor driver such that the phase leads were open
circuits. A driving actuator was commanded with a Gaussian random process
(i.e., white noise) low-pass filtered to 40 Hz input and scaled to a current
command in the ranges [-0.5,0.5]/$K_{Ta}$ and [-1.0,1.0]/$K_{Ta}$. Torque data
were collected from the torque transducer and velocity data were collected
from the actuator encoder (sampled at 40kHz and filtered onboard the motor
driver) together at 1100 Hz.

These data were fit with a non-parametric model in the frequency domain via
Welch’s method in the Matlab tfestimate() function [19]. Then, a first-order
parametric model was fit to the non-parametric estimate by optimizing a fit to
the gain frequency response, then finally optimizing for time-domain Variance
Accounted For (VAF) to obtain the final parameter estimate, both with
fmincon(). We chose a time-domain metric (VAF) for the second optimization
stage as fitting in the frequency domain overemphasizes frequencies with low
signal energy contribution.

For this test, the rotor was press-fit and bonded to a plastic arbor. Because
the motor inertia $J_{m}$ was very low compared to the support hardware (shaft
couplers, bearings, etc.), $J_{m}$ was estimated by (1) conducting the test
with the rotor installed behind a lightweight $N=7.5$ planetary transmission
to yield inertia $J_{r}$, then (2) repeating the test without the rotor and
arbor to yield $J_{nr}$. The motor inertia $J_{m}$ was then taken as
$J_{m}=\frac{1}{N^{2}}(J_{r}-J_{nr})$.

The RI50 is frameless, so its rotor mass was independently measured. As an
alternative to the random-input test result, $J_{m}$ was estimated via thin-
ring approximation with size and mass measurements of the rotor.

##  5 Results

The results (summarized in Table LABEL:tab:motordata) indicate that the RI50
motor is more suitable for use in QDDs than the T-Motor U8, according to the
design objectives established in Section 2. For the RI50, a lower $K_{T}$ and
$K_{M}$ can be compensated for by a higher gear ratio (7, 8), while still
maintaining a lower reflected output inertia (9), compared to an actuator with
the U8 or equivalent motor. In other words, for transmission selections that
result in matching actuator inertias ($J_{a}$), the RI50 actuator will produce
more torque per current input and per heat waste than the U8 actuator (higher
$K_{Ta}$ and $K_{Ma}$).

It is particularly important to note that $S_{M}$, which should be invariant
to motor geometry under commonly used models (15), indicates that the RI50
motor is $\sim 3.4\times$ more responsive than the U8. This benefit is visible
in clearer terms and in context with more motors in Fig. 2. While the
predicted trend of $J_{m}\propto K_{M}^{2}$ (6, 5) is visible, there may be
almost an order of magnitude variation of $S_{M}$ (traversing between isolines
in the figure) for some given $K_{M}$ range. These variations offer designers
an opportunity for higher performance. It also appears that TQ RoboDrive
motors are particularly well suited for QDDs over a broad range [11]. Physical
differences between the motors and potential drawbacks of the RI50 are
discussed in Section 6.

Table 1: Case study motors. (T-Motor) RI50 characterized in this paper and
(T-Motor) U8 characterized in [18].

|  |  | RI50 | U8
---|---|---|---|---
Winding Style |  |  | Wye | Delta
Phase Resistance | $R_{\phi}$ | [$\Omega$] | 0.705 | 0.279
Effective Inductance | $L_{e}$ | [µH] | 2.559 | 0.069
Torque Constant | $K_{T}$ | [Nm/A] | 0.105 | 0.14
Motor Constant | $K_{M}$ | [Nm/W0.5] | 0.12 | 0.23
Motor Inertia | $J_{m}$ | [gcm2] | 90.1 | 1200
Mass | $m$ | [g] | 193 | 242*
Gap Radius | $r_{g}$ | [mm] | 14.6 | 40.8
Length | $l$ | [mm] | 16.0 | 8.0
Responsiveness Metric (10) | $S_{M}$ | [ms] | 0.67 | 2.3
Torque-Specific Inertia (11) | $S_{T}$ | [g(A/N)2] | 0.82 | 6.1

*The U8 is a framed motor (assembly with bearings and support structure) while the RI50 is frameless. A hypothetical frameless U8 may be $\sim$25% lighter, but these are not yet commercially available.

###  5.1 Passive Properties

Thermal tests revealed that the RI50 motor is wound in the Wye style by the
pattern of eight hot windings to four cold (Fig. 3). Electrical measurements
yielded a phase resistance of $R_{\phi}=705\pm 3$ m$\Omega$ and an effective
inductance of $L_{e}=2559\pm 2$ µH.

###  5.2 Motor Coefficients

The gradient of a line fit to the torque-current data yielded a torque
constant of $K_{T}=0.105\pm 0.002$ Nm/A. The gradient of a line fit to the
voltage-speed data yielded a BEMF constant of $K_{B}=0.094\pm 0.002$ Vs/rad.
Deviation from ideal PMSM patterns ($K_{T}=K_{B}$ [1]) may be due to
imperfectly sinusoidal back-EMF profiles. With these values for $K_{T}$ and
$K_{B}$, we can then obtain the value of the fundamental motor constant
($K_{M}$) as 0.118$\pm 0.003$ Nm/W0.5.

###  5.3 Motor Inertia

The frequency domain non-parametric estimate had high coherence up to 40 Hz
(Fig. 4), suggesting that the torque and velocity data could be explained by a
first-order linear model:

$\frac{\Omega(s)}{T_{b}(s)}=\frac{1}{J_{e}s+B_{e}}$ (16)

where $\Omega(s)$ is actuator speed, $T_{b}(s)$ is backdriving torque, $J_{e}$
is output inertia [kgm2], $B_{e}$ is viscous damping [Nms/rad], and $s$ is the
Laplace variable, and where the subscript $e\in\\{r,nr\\}$ for rotor installed
and not installed, respectively. The following parameters resulted from
fitting the model: the actuator inertia $J_{r}$ was estimated as $(6.55\pm
0.95)\times 10^{-4}$ kgm2 and $J_{nr}$ was estimated as $(1.48\pm 0.08)\times
10^{-4}$ kgm2. The goodness of fit is evaluated with the Variance Account For
(VAF) metric. The parametric fit showed $92\%-98\%$ VAF, indicating a high-
quality parameter estimation for the two inertias. These two results yielded a
final inertia estimate of $J_{m}=(9.01\pm 1.8)\times 10^{-6}$ kgm2.

The thin-ring inertia estimate yielded $J_{m}$ as $(5.4\pm 0.5)\times 10^{-6}$
kgm2 with a mass of $34\pm 0.5$ g and radius of $12.6\pm 0.5$ mm. Differences
between the random-input result and thin-ring result are likely due to missing
inertial contributions from support bearing inner-races. These contributions
are not captured with the rotor (and arbor) uninstalled.

Figure 3: Averaged thermal image over 50 frames of motor during thermal test.
Eight of the twelve windings appear hotter than the rest, indicating a Wye
winding style.

Figure 4: Mechanical System ID Frequency Response and Coherence for the 7.5:1
with and without the rotor. Legend indicated input amplitude and Variance
Account For goodness of fit. The confidence intervals for the $J$ and $B$ are
2 standard deviations of identified $J$ and $B$ across input magnitudes

##  6 Discussion

We propose two motor selection metrics, $S_{M}$ and $S_{T}$, for use in Quasi-
Direct Drive actuator design. These metrics are simple, invariant to
transmission ratio selection, and readily evaluated on a wide range of motors
via datasheet values (Fig. 2). While maximizing gap radius $r_{g}$ has been a
successful strategy in QDD actuator design (Section 2), we show through a case
study in comparing the T-Motor U8 and RI50 that alternative motors may be more
suitable, as illuminated by the proposed metrics. The mechanical and
electrical properties of the RI50 motor is rigorously characterized (Section
5) for accurate comparison to the U8 motor. Designers may leverage the
proposed metrics to (1) quickly down-select from a range of motors, and (2)
find motors that offer different, potentially useful, design opportunities.
For example, the RI50 has comparable performance properties to the U8, but is
much more compact (Fig. 1), offering alternative packaging strategies than
with the U8.

The RI50’s performance characteristics are indeed surprising, given its very
small gap radius. Here, we propose explanations for the unexpected performance
advantages of the RI50.

###  6.1 Coil Turns Contribute to $K_{T}$

Common motor analyses focus on geometric parameters and use assumptions to
remove number of coil turns, $n_{t}$, from consideration [2, 3]. However,
$n_{t}$ is a useful “lever” for motor designers to use in manipulating motor
properties [5]. The parameter $n_{t}$ is linearly related to the shear force
generated between the stator and rotor in the motor, as it amplifies the
magnetic field of the motor coil. Thus, $K_{T}\propto n_{t}$. In the case of
the U8 motor, $n_{t}\approx 16$ and for the RI50, $n_{t}\approx 36$ by visual
inspection. This greater $n_{t}$ allows the RI50 motor to partially make up
for its radius disadvantage with respect to $K_{T}$, and also explains the
higher $R_{\phi}$ and $L_{e}$ for the RI50. This effect is captured by the
$S_{T}$ metric (11), and is also discussed in [12].

###  6.2 Inner- vs. Outer-rotating Geometry Greatly Affects $K_{M}$

The RI50 is inner-rotating (its rotor is inside its stator) and the U8 is
outer-rotating. With the small gap radius of the RI50, this difference plays
an important role that common geometric analyses do not capture. Specifically,
it is often assumed that the rotor with radial thickness $t_{r}$ and stator
with radial thickness $t_{s}$ (Fig. 1) are thin rings, i.e., $t_{r}\ll r_{g}$,
$t_{s}\ll r_{g}$ [2]. While this assumption is approximately true for the
rotor, it is not for the stator in the case of the RI50. This fact means that
the stator can hold more copper, lending it a higher $K_{M}$ than an outer-
rotating motor of the same gap radius and length [5]. Such an outer-rotating
motor would have $\sim 4\times$ less volume available for the stator. This
effect is captured by $S_{M}$ metric (10).

###  6.3 Additional Design Considerations

The proposed motor selection strategy emphasizes properties useful for QDD
actuator design. However, additional properties can be considered for
specialized applications. For example, although the RI50 has high torque for
its inertia and is lighter than the U8, its torque density, $K_{T}/m$, is
lower than the U8 — designers should determine whether the higher transmission
ratio necessary to meet torque requirements adds untenable mass or complexity.
The RI50 also has a very high inductance, $L_{e}$ thereby limiting current-
control bandwidth, which could potentially limit performance if the bandwidth
is not sufficiently faster than robot mechanical dynamics. Lastly, the RI50
has a high phase resistance, which will naturally limit its peak speed. Higher
voltages can be used to compensate, but may be limited by other system
components (e.g., the motor driver). Future work will consider the relative
importance of these drawbacks in the context of a full robot.

## References

  * [1] J. R. Mevey, “Sensorless field oriented control of brushless permanent magnet synchronous motors,” Master’s thesis, Kansas State University, 2009.
  * [2] P. M. Wensing, A. Wang, S. Seok, D. Otten, J. Lang, and S. Kim, “Proprioceptive actuator design in the MIT Cheetah: Impact mitigation and high-bandwidth physical interaction for dynamic legged robots,” _IEEE Transactions on Robotics_ , vol. 33, no. 3, pp. 509–522, 2017.
  * [3] S. Seok, A. Wang, D. Otten, and S. Kim, “Actuator design for high force proprioceptive control in fast legged locomotion,” in _2012 IEEE/RSJ International Conference on Intelligent Robots and Systems_ , Oct. 2012, pp. 1970–1975, iSSN: 2153-0866.
  * [4] H. Zhu, C. Nesler, N. Divekar, V. Peddinti, and R. Gregg, “Design principles for compact, backdrivable actuation in partial-assist powered knee orthoses,” _IEEE/ASME Transactions on Mechatronics_ , 2021.
  * [5] D. C. Hanselman, _Brushless permanent magnet motor design_. The Writers’ Collective, 2003.
  * [6] B. Katz, J. D. Carlo, and S. Kim, “Mini cheetah: A platform for pushing the limits of dynamic quadruped control,” in _2019 International Conference on Robotics and Automation (ICRA)_ , 2019, pp. 6295–6301.
  * [7] G. Kenneally, A. De, and D. E. Koditschek, “Design principles for a family of direct-drive legged robots,” _IEEE Robotics and Automation Letters_ , vol. 1, no. 2, pp. 900–907, 2016.
  * [8] A. F. Azocar, L. M. Mooney, J.-F. Duval, A. M. Simon, L. J. Hargrove, and E. J. Rouse, “Design and clinical implementation of an open-source bionic leg,” _Nature biomedical engineering_ , vol. 4, no. 10, pp. 941–953, 2020.
  * [9] “maxon’s 2020/2021 catalogue.” [Online]. Available: http://www.maxongroup.co.uk/maxon/view/news/maxon’s-20202021-catalogue
  * [10] “Brushless Motors.” [Online]. Available: https://www.alliedmotion.com/brushless-motors/
  * [11] “RoboDrive | Frameless servo motors.” [Online]. Available: https://www.tq-group.com/en/products/tq-robodrive/servo-kits/
  * [12] A. De, A. Stewart-Height, and D. E. Koditschek, “Task-based control and design of a bldc actuator for robotics,” _IEEE Robotics and Automation Letters_ , vol. 4, no. 3, pp. 2393–2400, 2019.
  * [13] A. De and D. E. Koditschek, “Parallel composition of templates for tail-energized planar hopping,” in _2015 IEEE International Conference on Robotics and Automation (ICRA)_ , 2015, pp. 4562–4569.
  * [14] P. Arm, R. Zenkl, P. Barton, L. Beglinger, A. Dietsche, L. Ferrazzini, E. Hampp, J. Hinder, C. Huber, D. Schaufelberger, F. Schmitt, B. Sun, B. Stolz, H. Kolvenbach, and M. Hutter, “Spacebok: A dynamic legged robot for space exploration,” in _2019 International Conference on Robotics and Automation (ICRA)_ , 2019, pp. 6288–6294.
  * [15] Y. Li, Y. Zhou, J. Huang, Z. Wang, S. Zhu, K. Wu, L. Zheng, J. Luo, R. Cao, Y. Zhang _et al._ , “Design of a flying humanoid robot based on thrust vector control,” _arXiv preprint arXiv:2108.11557_ , 2021.
  * [16] C. Nesler, G. Thomas, N. Divekar, E. Rouse, and R. D. Gregg, “Enhancing voluntary motion with modular, backdrivable, powered hip and knee orthoses,” _IEEE Robotics and Automation Letters_ , pp. 1–1, 2022.
  * [17] J. Duperret, B. Kramer, and D. E. Koditschek, “Core actuation promotes self-manipulability on a direct-drive quadrupedal robot,” in _International Symposium on Experimental Robotics_. Springer, 2016, pp. 147–159.
  * [18] U. H. Lee, C.-W. Pan, and E. J. Rouse, “Empirical characterization of a high-performance exterior-rotor type brushless dc motor and drive,” in _2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , 2019, pp. 8018–8025.
  * [19] P. Welch, “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” _IEEE Transactions on Audio and Electroacoustics_ , vol. 15, no. 2, pp. 70–73, Jun. 1967.