# Modified Martin-Puplett interferometer for magneto-optical Kerr effect
measurements at sub-THz frequencies

A. Glezer Moshe<EMAIL_ADDRESS>National Institute of Chemical Physics
and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia  R. Nagarajan
National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618
Tallinn, Estonia  U. Nagel  National Institute of Chemical Physics and
Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia  T. Rõõm
<EMAIL_ADDRESS>National Institute of Chemical Physics and Biophysics,
Akadeemia tee 23, 12618 Tallinn, Estonia  G. Blumberg
<EMAIL_ADDRESS>National Institute of Chemical Physics and
Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia  Department of Physics
and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA

###### Abstract

We present magneto-optical Kerr effect (MOKE) spectrometer based on a modified
Martin-Puplett interferometer, utilizing continuous wave sub-THz low-power
radiation in broad frequency range. This spectrometer is capable of measuring
the frequency dependence of the MOKE response function, both the Kerr rotation
and ellipticity, simultaneously with sub-milliradian accuracy without the need
for reference measurement. The instrument’s versatility allow it to be coupled
to a cryostat with optical windows enabling studies of a variety of quantum
materials such as unconventional superconductors, two-dimensional electron gas
systems, quantum magnets, and other system showing optical Hall response, at
sub-Kelvin temperatures and in high magnetic fields. We demonstrate the
functionality of the MOKE spectrometer using an undoped InSb wafer as a test
sample.

##  I Introduction

The application of a magnetic field can significantly alter the properties of
materials. It is well-accepted that magneto-resistivity and Hall-effect
measurements provide more comprehensive information compared to standard dc
zero-field transport measurements. Similarly, the magneto-optical response
carries new data about physical systems.[1]

Recent advancements in magneto-optical techniques have allowed for the
acquisition of novel information about electronic systems by extending dc Hall
conductivity measurements into the THz frequency range.[2] This spectroscopic
approach is designed to complement transport experiments, testing the bounds
of standard relaxation time approximation in metals,[3] measuring intrinsic
Berry-phase contributions to the anomalous Hall effect, and probing mechanisms
of spontaneous time-reversal symmetry breaking.[4] Electronic systems where
magneto-optical spectroscopy has recently proven to be particularly
informative are unconventional superconductors,[5, 6] the integer and
fractional quantum Hall systems,[7, 8] quantum anomalous Hall states,[9]
graphene-like systems,[10] magnetic topological insulators,[11, 12] valley
Hall effect 2D crystals,[13] the "strange metals" including cuprates and heavy
fermions near antiferromagnetic quantum critical points,[3] to name a few.

Magnetic fields induce non-reciprocal optical phenomena, such as the Faraday
and Kerr rotations. These magneto-optical rotations play a pivotal role in
characterizing quantum materials. An examples of notable area of recent
interest is chiral superconductors, which possess non-trivial topological
properties and order parameters that spontaneously break time-reversal
symmetry (TRS).[14, 15]

One of the greatest challenges in identifying superconductors that
spontaneously break TRS is the limited availability of suitable probes.[16]
Optical measurements of the polar Kerr effect (PKE), where the rotation angle
of linearly polarized radiation reflected from a sample’s surface is detected,
have emerged as a key method for identifying TRS-broken states in
unconventional materials.[17] The Kerr rotation arises from the ac Hall
conductivity in dimensions perpendicular to the light’s wave vector. For this
Hall conductivity to be finite, the medium must break time-reversal symmetry
and mirror symmetries about all planes parallel to the wave vector.[4] The
detection of non-vanishing PKE polarization rotation at zero external magnetic
field provides unambiguous evidence for finite anomalous ac Hall conductivity,
thus indicates TRS-breaking.[18, 4, 19]

To distinguish the signature of broken TRS from effects due to natural optical
activity, it is critical to measure the Kerr rotation angle rather than the
Faraday rotation. This distinction is necessary because PKE is absent if TRS
is preserved.[4]

Although recent PKE studies using a telecom laser at approximately a
micrometer wavelength have identified spontaneous Kerr responses in several
unconventional superconductors, such as UPt3 ,[20] URu2Si2 ,[21] PrOs4Sb12
,[6] UTe2 ,[22] FeTe0.55Se${}_{0.45},$[23] a significant limitation of these
experiments is their restriction to a single near-visible radiation frequency.
This frequency far exceeds the energy scale relevant to unconventional
superconductivity, typically around 0.1 THz. Consequently, these results have
sparked considerable debates regarding their interpretation. Key questions
include whether the observed effects are intrinsic bulk properties of the
superconductor or influenced by external factors such as pinned vortices,[24,
25] or inhomogeneities of the sample. More importantly, the microscopic origin
and quantitative understanding of the TRS-broken state remain contentious
issues that can only be resolved through ultra-low-frequency spectroscopic
measurements of the Kerr response as a function of frequency, within the range
relevant to superconductivity and magnetism that causes TRS-breaking. This
necessitates a new generation of spectroscopic THz-range magneto-optical Kerr
effect (MOKE) instruments.

There are very few MOKE instruments operating in the THz range. R. Shimano et
al.[26, 27] used a time-domain THz setup to measure the amplitude of reflected
radiation, with an angle of incidence of 45 degrees in a 0.48 T magnetic field
at room temperature, achieving an accuracy of about 10 mrad between 0.5 and
2.5 THz. The setup of D. Drew et al.[28, 29] employs a CO2 laser to generate
THz radiation at selected frequencies of 0.7, 1.3, and 2.5 THz. Using a
rapidly rotating waveplate, the Faraday rotation angle can be measured with a
resolution of about 0.1 mrad in transmission, and the Kerr rotation angle can
be measured with a resolution of about 1 mrad in reflection. In time-domain
methods, it is possible to separate Kerr and Faraday effects by isolating the
Faraday rotation from the first pulse and subtracting it from the reflected
pulse that travels twice through the sample and arrives later in time. The
most recent time-domain setup, which employs dual detectors, reports a Faraday
angle precision of about 0.02 mrad.[30] The challenges of THz-range
ellipsometry are comprehensively discussed elsewhere.[31]

The previously described methods either use high-power radiation or are
limited in the selection and resolution of frequencies. High-power radiation
is highly detrimental for studying samples at sub-Kelvin temperatures,
necessitating a new approach.

We have developed a MOKE spectrometer that uses very low incident power on the
sample, about 0.1-1 $\mu$W, in the range of 0.1-1 THz to simultaneously
measure both the real and imaginary parts of the Kerr response function with
sub-milliradian accuracy. Below, we explain the underlying principle of the
interferometer-based approach and demonstrate the performance of the
instrument using MOKE measurement in an InSb semiconductor as a test sample.

##  II  Polar Kerr effect

###  II.1 Reflection matrix transformation

Here we analyse the state of initially linearly polarized radiation after
reflection under normal incidence from a material where the normal modes of
electromagnetic radiation propagation are circularly polarized. The relation
between linearly polarized radiation, $\mathbf{E}=(E_{x},E_{y})$, and
circularly polarized radiation propagating along the $z$ axis is

$(E_{-},E_{+})=\hat{\mathbf{U}}(E_{x},E_{y}),$ (1)

where the transformation matrix is

$\displaystyle\hat{\mathbf{U}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&-i\\\
-1&-i\end{array}\right).$ (4)

The reflection coefficient from the material-vacuum interface is

$\hat{\mathbf{r}}=\left(\begin{array}[]{cc}r_{-}&0\\\
0&r_{+}\end{array}\right).$ (5)

Using the complex indexes of refraction for the right-handed ($N_{-}$) and
left-handed ($N_{+}$) modes in the material the reflection coefficients for
the vacuum-material interface are

$r_{\pm}=\frac{N_{\pm}-1}{N_{\pm}+1}.$ (6)

The initially linearly polarized radiation $\mathbf{E}^{i}$ after reflection
is transformed into $\mathbf{E}^{f}=(E_{x}^{f},E_{y}^{f})$,

$\displaystyle\mathbf{E}^{f}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)\hat{\mathbf{U}}^{-1}\hat{\mathbf{r}}\hat{\mathbf{U}}\mathbf{E}^{i}=\hat{\mathbf{r}}_{xy}\mathbf{E}^{i},$
(9)

where the matrix

$\displaystyle\left(\begin{array}[]{cc}-1&0\\\ 0&1\end{array}\right)$

inverts the $x$ axis of the coordinate frame. Such transformation of the
coordinate frame also changes the handedness of the reflected radiation what
is defined relative to the propagation direction:

$(E_{+},E_{-})=\hat{\mathbf{U}}\left(\begin{array}[]{cc}-1&0\\\
0&1\end{array}\right)\hat{\mathbf{U}}^{-1}(E_{-},E_{+}).$ (11)

From Eq. 9 we get the reflection matrix for the linearly polarized radiation,

$\displaystyle\hat{\mathbf{r}}_{xy}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\begin{array}[]{cc}r_{+}+r_{-}&i(r_{+}-r_{-})\\\
i(r_{+}-r_{-})&-(r_{+}+r_{-})\end{array}\right).$ (14)

Figure 1:  The length of ellipse’s axes are $E_{a}$ and $E_{b}$ and the major
axis is rotated from the $x$ axis by $\phi$. The electric field in the $xy$
coordinate frame is $(E_{x},E_{y}e^{i\theta})$ where $E_{x}$ and $E_{y}$ are
the amplitudes and $\theta$ is the phase difference between the components.

###  II.2 Kerr rotation and ellipticity

Lets assume the incident linearly polarized radiation is
$\mathbf{E}^{i}=(E_{x}^{i},0)$. Then, the reflected radiation using Eq. 14
becomes,

$(E_{x}^{f},E_{y}^{f}e^{i\theta})=-\frac{E_{x}^{i}}{2}\left(\begin{array}[]{c}r_{+}+r_{-}\\\
i(r_{+}-r_{-}),\end{array}\right)$ (15)

where $\theta$ is the phase difference between $y$ and $x$ components of the
reflected radiation. We define the complex Kerr response function
$\Theta_{\mathrm{K}}$ as

$\Theta_{\mathrm{K}}\equiv\Phi+i\mathrm{X}=\frac{E_{y}^{f}}{E_{x}^{f}}e^{i\theta}=\frac{i(r_{+}-r_{-})}{r_{+}+r_{-}},$
(16)

where the last equality comes from (15). $\Phi$ and $\mathrm{X}$ are the real
and the imaginary parts of $\Theta_{\mathrm{K}}$. With (16) the reflection
matrix (14) for the linearly polarized radiation reads:

$\displaystyle\hat{\mathbf{r}}_{xy}$ $\displaystyle=$
$\displaystyle-\frac{r_{+}+r_{-}}{2}\left(\begin{array}[]{cc}1&\Theta_{\mathrm{K}}\\\
\Theta_{\mathrm{K}}&-1\end{array}\right),$ (19)

The prefactor $(r_{+}+r_{-})/2=-E_{x}^{f}/E_{x}^{i}$ is the complex reflection
coefficient for the linearly polarized radiation.

The Kerr response function $\Theta_{\mathrm{K}}$ is related to the rotation
and ellipticity of radiation as follows. An elliptically polarized radiation,
$(E_{x},E_{y}\exp(i\theta))$, is characterized by two parameters, the phase
difference $\theta$, and the ratio of two orthogonal amplitudes,
$\tan\alpha=E_{y}/E_{x}$. With this notation

$\Theta_{\mathrm{K}}=\tan\alpha(\cos\theta+i\sin\theta).$ (20)

The elliptically polarized state can be transformed into a coordinate frame
where the phase difference between two orthogonal components is $\pi/2$,
$\mathbf{E}=(E_{a},E_{b}\exp(i\pi/2))$, see Fig. 1. The rotation of the
ellipse main axis from the $x$ axis is $\phi$ and the ellipticity is
$\chi=\arctan(E_{b}/E_{a})$. Then, see e.g.,[32, p. 28]

$\displaystyle\tan(2\phi)$ $\displaystyle=$
$\displaystyle\tan(2\alpha)\cos\theta,$ (21) $\displaystyle\sin(2\chi)$
$\displaystyle=$ $\displaystyle\sin(2\alpha)\sin\theta.$

If the ratio of amplitudes is small, $\alpha\ll 1$, we get from (20) and (21)
the relation between the rotation and the ellipticity of radiation and the
real and imaginary parts of the Kerr response function:

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\alpha\cos\theta=\Phi,$
(22) $\displaystyle\chi$ $\displaystyle=$ $\displaystyle\alpha\sin\theta=X.$

Under the normal incidence reflection from the vacuum-material interface, Eq.
(6), the Kerr response function, Eq. 16, depends on the material index of
refraction as:

$\Theta_{\mathrm{K}}=\frac{i(N_{+}-N_{-})}{N_{+}N_{-}-1}.$ (23)

In the small $\alpha$ limit using (23) in (22) we recover Argyres’s result for
the Kerr rotation and ellipticity:[33]

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle-\Im\frac{(N_{+}-N_{-})}{N_{+}N_{-}-1},$ (24)
$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\Re\frac{(N_{+}-N_{-})}{N_{+}N_{-}-1}.$

##  III Working principle of the MOKE spectrometer

###  III.1 The modified Martin-Puplett interferometer

A schematic of the polar MOKE spectrometer is shown in Fig. 2. The central
part of the spectrometer is the modified Martin-Puplett interferometer (MPI)
[34] consisting of a wiregrid beamsplitter WG45 and two roof top mirrors: the
fixed mirror (FM) and the moving mirror (MM). To enable the polar Kerr
spectroscopy under normal incidence a sample is placed into one of the two
ports of MPI, and the source and detector, separated by the wiregrid WG0, are
in the other port 111Interferometer arms are the ones with roof mirrors with
the beamsplitter in between. The modified MP interferometer has two symmetric
ports on the other side of the beamsplitter: input and output..

The beam emitted from the source S after passing through the polarizer
$\mathrm{WG_{0}}$ is vertically polarized, $\mathbf{E}_{0}=E_{0}(1,0)$.

Figure 2:  Schematic of the MOKE interferometer and the radiation path through
the MPI described in Eq. III.1. The coordinate axis in the figure is in frame
of reference of radiation beam emitted from source. H and V refer to
horizontal and vertical direction of polarization of radiation propagating
along $\mathbf{k}$. The optical elements are source (S), detector (D), fixed
roof top mirror (FM) and moving roof top mirror (MM). WGη is the wiregrid
polarizer with wires aligned at angle $\eta$ with respect to the H-axis shown
in the figure. The displacement of the MM is $\Delta x$, where $\Delta x=0$ is
the MM position when it is at the same distance from the beam splitter WG45 as
FM. The direction of wires of WG45 is $\eta=45^{\circ}$ when the beam is
incident on the beamsplitter from the left or from the top in figure. For the
beams incident on the beamsplitter from the right or from the bottom
$\eta=-45^{\circ}$. The roof top mirror flips the polarization of the incident
radiation as described by the Jones matrix in Table 1. If the incident
radiation has polarization at $\pm 45^{\circ}$ relative to the roof line the
reflected radiation will be orthogonal to the incident radiation. (a) The
radiation beam is reflected from MM to sample to MM, adding the path
difference $4\Delta x$, as described by $E_{1}$ in Eq. III.1. (b) The
radiation beam is reflected from FM to sample to FM and path difference for
this beam is zero, as described by $E_{2}$. (c,d) The radiation beam is
reflected from MM (FM) to sample to FM (MM) respectively, adding a path
difference $2\Delta x$, as described by $E_{3}$ ($E_{4}$).

The beamsplitter WG45 splits the beam into two orthogonally polarized beams.
These two beams, $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$, after reflection from
the sample arrive to detector D as shown in Fig. 2 (a) and (b). If the sample
has a non-zero MOKE the beam after the reflection from the sample is split
again by the beamsplitter WG45 and two additional beams, $\mathbf{E}_{3}$ and
$\mathbf{E}_{4}$, reach the detector D, Fig. 2 (c) and (d). The wiregrid WG0
transmits the vertical polarization from the source into the interferometer
and reflects the horizontal polarization from the interferometer output beam
to the detector.

The four beams reaching the detector D are derived by applying the Jones
matrices, Table 1, to the source beam $\mathbf{E}_{0}$:

$\displaystyle\mathbf{E}_{1}$ $\displaystyle=$
$\displaystyle\mathrm{WG}_{0}^{\mathrm{R}}\cdot\mathrm{WG}_{-45}^{\mathrm{T}}\cdot\mathrm{MM}\cdot\mathrm{WG}_{-45}^{\mathrm{R}}\cdot\mathrm{R}_{\Theta}\cdot\mathrm{WG}_{-45}^{\mathrm{R}}\cdot\mathrm{MM}\cdot\mathrm{WG}_{45}^{\mathrm{T}}\cdot\mathrm{WG}_{0}^{\mathrm{T}}\cdot
E_{0}(0,1)=-\frac{1}{2}E_{0}\bar{r}e^{8i\pi\xi}(1,0),$
$\displaystyle\mathbf{E}_{2}$ $\displaystyle=$
$\displaystyle\mathrm{WG}_{0}^{\mathrm{R}}\cdot\mathrm{WG}_{45}^{\mathrm{R}}\cdot\mathrm{FM}\cdot\mathrm{WG}_{-45}^{\mathrm{T}}\cdot\mathrm{R}_{\Theta}\cdot\mathrm{WG}_{45}^{\mathrm{T}}\cdot\mathrm{FM}\cdot\mathrm{WG}_{45}^{\mathrm{R}}\cdot\mathrm{WG}_{0}^{\mathrm{T}}\cdot
E_{0}(0,1)=\frac{1}{2}E_{0}\bar{r}(1,0),$ $\displaystyle\mathbf{E}_{3}$
$\displaystyle=$
$\displaystyle\mathrm{WG}_{0}^{\mathrm{R}}\cdot\mathrm{WG}_{45}^{\mathrm{R}}\cdot\mathrm{FM}\cdot\mathrm{WG}_{-45}^{\mathrm{T}}\cdot\mathrm{R}_{\Theta}\cdot\mathrm{WG}_{-45}^{\mathrm{R}}\cdot\mathrm{MM}\cdot\mathrm{WG}_{45}^{\mathrm{T}}\cdot\mathrm{WG}_{0}^{\mathrm{T}}\cdot
E_{0}(0,1)=-\frac{1}{2}E_{0}\bar{r}e^{4i\pi\xi}\Theta_{\mathrm{K}}(1,0),$
$\displaystyle\mathbf{E}_{4}$ $\displaystyle=$
$\displaystyle\mathrm{WG}_{0}^{\mathrm{R}}\cdot\mathrm{WG}_{-45}^{\mathrm{T}}\cdot\mathrm{MM}\cdot\mathrm{WG}_{-45}^{\mathrm{R}}\cdot\mathrm{R}_{\Theta}\cdot\mathrm{WG}_{45}^{\mathrm{T}}\cdot\mathrm{FM}\cdot\mathrm{WG}_{45}^{\mathrm{R}}\cdot\mathrm{WG}_{0}^{\mathrm{T}}\cdot
E_{0}(0,1)=-\frac{1}{2}E_{0}\bar{r}e^{4i\pi\xi}\Theta_{\mathrm{K}}(1,0),$ (25)

here $\xi=\Delta x/\lambda$ is the reduced path difference where $\Delta x$ is
the position of the moving mirror relative to zero path difference (ZPD)
position, $\lambda$ is the wavelength, and $\bar{r}=(r_{+}+r_{-})/2$. After
adding up the four beams in Eq. III.1 the electric field on the detector is:

$E(\xi)=E_{0}\bar{r}\left[\frac{1}{2}\left(1-e^{8i\pi\xi}\right)-e^{4i\pi\xi}\Theta_{\mathrm{K}}\right].$
(26)

Table 1:  Jones matrices of wiregrid, roof top mirror and back reflection from the sample exhibiting polar Kerr effect. $\eta$ is the angle of wiregrid wires and roof top mirror roof line from the horizontal direction measured towards the vertical direction. Horizontally polarized field has unit vector $\mathbf{e}_{h}=(1,0)$ and vertically polarized has $\mathbf{e}_{v}=(0,1)$ and the beam propagation direction is $\mathbf{e}_{k}=\mathbf{e}_{h}\times\mathbf{e}_{v}$. Sample reflection matrix is Eq. 19 where $\bar{r}=(r_{+}+r_{-})/2$ and the complex Kerr response function $\Theta_{\mathrm{K}}=\Phi+i\mathrm{X}$.  Optical element | Jones matrix | Symbol
---|---|---
WG reflection | $\begin{pmatrix}-\cos^{2}\eta&-\cos\eta\sin\eta\\\ \cos\eta\sin\eta&\sin^{2}\eta\end{pmatrix}$ | $\mathrm{WG}^{R}_{\eta}$
WG transmission | $\begin{pmatrix}\sin^{2}\eta&-\cos\eta\sin\eta\\\ -\cos\eta\sin\eta&\cos^{2}\eta\end{pmatrix}$ | $\mathrm{WG}^{T}_{\eta}$
Roof top mirror | $\begin{pmatrix}-\cos 2\eta&-\sin 2\eta\\\ \sin 2\eta&-\cos 2\eta\end{pmatrix}$ | $\mathrm{RM}_{\eta}$
Sample reflection | $-\bar{r}\begin{pmatrix}1&\Theta_{\mathrm{K}}\\\ \Theta_{\mathrm{K}}&-1\end{pmatrix}$ | $\mathrm{R}_{\Theta}$

To demonstrate how the real and imaginary parts of $\Theta_{\mathrm{K}}$
affect the interferogram we plot using Eq. 26 the normalized amplitude
$|E(\xi)/(E_{0}\bar{r})|$ as a function of mirror displacement in Fig. 3. At
$\xi=0$, the incident radiation at the sample surface is linearly polarized
parallel to the direction transmitted by WG0 and radiation reaches the
detector only if the plane of polarization is rotated upon back reflection
from the sample ($\phi\neq 0$): the real part of $\Theta_{\mathrm{K}}$ lifts
the minima of the normalized amplitude. At $\xi=1/8$ or $\xi=3/8$ we have
right- or left-handed circularly polarized radiation incident on the sample
surface and if an ellipticity $\chi\neq 0$ is introduced by the sample, the
maxima in the detected interferogram have different amplitudes. Finite
ellipticity also shifts the positions of minima away from the position in the
absence of ellipticity, $\xi=n/4$, where $n=0,1,2,\ldots$.

Figure 3:  Theoretical normalized electric field amplitude $|E/E_{0}\bar{r}|$
at the detector calculated with Eq. 26 for different values of
$\Theta_{\mathrm{K}}$, arranged from bottom to top: zero rotation and zero
ellipticity, finite rotation and zero ellipticity, and zero rotation and
finite ellipticity, respectively. The values in the legend are given in units
of radians. The curves are shifted by unity for clarity. It can be seen that
the rotation lifts the minima by $\phi$, while the ellipticity separates the
maxima by $2\chi$. Special points in the interferogram correspond to the
polarization state of the incident radiation on the sample. At the maxima, the
radiation is either left or right circularly polarized, while at the minima,
the radiation is linearly polarized along the horizontal or vertical
direction. The direction of rotation and polarization are shown along with the
local coordinate system. Figure 4:  An example of an interferogram used to
determine the rotation $\phi$ and ellipticity $\chi$ measured from undoped
InSb semiconductor at $T=50$ K and magnetic field 5 mT. From the fit to Eq.
29, we obtained the rotation, $\phi=36\pm 1$ mrad, ellipticity, $\chi=46.8\pm
0.4$ mrad, and the frequency $452.45\,\pm\,0.07$ GHz. The radiation electric
field amplitude, the data points, were fitted to a model (Eq. 29) that
accounts for imbalance between interferometer arms $\delta=-0.0262\pm 0.0006$.
To measure the locations of the minima, the path difference step size $\Delta
x$ was decreased linearly with decreasing amplitude down to 0.10 $\mu$m from
the initial value 3.37 $\mu$m. The 1 s integration time at maxima was
increased as the inverse square root of the step size up to 5.7 s per
interferogram point. This strategy allowed us to limit the total measurement
time of the interferogram with 612 points to 33 minutes (including time for
moving the roof mirror) while still having a relative amplitude error for each
data point equal to about 0.5 %.

###  III.2 Data analysis and accuracy

Following Eq. 26, we show that the normalized electric field intensity for the
interferogram acquires simple form which is the sum of unmodulated rotation
induced intensity $\Phi^{2}$ and ellipticity induced baseline shift of
interference modulation in the second intensity term:

$I(\xi)=\left[\frac{E(\xi)}{E_{0}\bar{r}}\right]^{2}=\Phi^{2}+[\mathrm{X}+\sin{(4\pi\xi)}]^{2}.$
(27)

A difference of intensity interferograms shifted by a quarter of the reduced
path difference enables direct measurement of ellipticity $\mathrm{X}$ from
the normalized intensity modulation amplitude:

$I(\xi)-I(\xi-\frac{1}{4})=4\mathrm{X}\sin{(4\pi\xi)}.$ (28)

Thus, one can obtain both the real and imaginary parts of the complex Kerr
response function $\Theta_{\mathrm{K}}=\Phi+i\mathrm{X}$ from the fit to
measured interferogram amplitudes, Eq. (26), or intensities, Eqs. (27-28).

The figure of merit for MOKE measurements is the accuracy of Kerr rotation
angle $\delta\Phi$ and ellipticity $\delta\mathrm{X}$. Below we first analyze
the theoretical error propagation for the fitting procedure to Eqs. (26-27)
assuming normal distribution function for the radiation amplitude and the path
difference position noise.222The latter can be consider as the interferogram
phase fluctuation noise.

We show below in Sec. IV.1 that the implemented data acquisition procedure
readily achieves measured relative signal amplitude fluctuations $\delta
E_{0}/E_{0}$ better that $5\times 10^{-3}$.333This accuracy can be further
improved by implementing an amplitude modulation measurement schema. For the
wavelength of interest, about a millimeter, commercially available high
precision linear translation stages enable better than $10^{-4}$ accuracy for
the relative path difference control $\delta\xi/\xi$. With these parameters, a
relatively quick interferogram measurement containing $n=200$ points per
wavelength allows to obtain relative ellipticity accuracy
$\frac{\delta\mathrm{X}}{\mathrm{X}}\propto\frac{\delta
E_{0}}{E_{0}\sqrt{n}}\approx 0.35\times 10^{-4}$. Thus, small ellipticity can
be quickly measured with sub-millirad accuracy, which can be further improved
with enhancement of measurement points density.

The situation is less favorable for the Kerr rotation angle accuracy
$\delta\Phi$ as this variable is not modulated by the interferogram scan, see
Eq. (27). It appears that the relative Kerr rotation angle accuracy is
inversely proportional to the square root of number of points measured in the
proximity of interferogram minima $n_{min}$,
$\frac{\delta\Phi}{\Phi}\propto\frac{\delta E_{0}}{E_{0}\sqrt{n_{min}}}$.
Therefore, to obtain the Kerr rotation accuracy at the same order as the
ellipticity accuracy, measurement with about an order of magnitude higher
point density near the interferogram minima has to be implemented, see Sec.
IV.1. Experimentally, the point density is limited by the minimal repeatable
incremental move distance of translational stages.444Stages with air bearing
and interferometric position readout provide better positioning accuracy.

Other contributions to the Kerr response function accuracy include
photocurrent fluctuations in the Toptica detection system, interferences with
intruding spurious back reflection beams from multiple surfaces of
interferometer optical components, imbalance between two interferometer arms,
frequency stability of the radiation, non-orthogonality of the sample surface
to incoming beam, to name a few.

For example, the effect of imbalance between two interferometer arms, which
remains present even after careful alignment of the optical components,
results in a deviation from the ideal interferogram shape shown in Fig. 3,
making the odd and even interferogram minima unequal, see Fig. 4. In practice,
such imbalance between two interferometer arms can be described by a modified
interferometer electric field amplitude function

$\displaystyle E(\xi)$ $\displaystyle=$ $\displaystyle
E_{0}\bar{r}\left[\frac{1}{2}\left(\left(1-\delta\right)^{2}-e^{8i\pi\xi}\left(1+\delta\right)^{2}\right)-\right.$
(29)
$\displaystyle\left.-\,e^{4i\pi\xi}\Theta_{\mathrm{K}}\left(1-\delta^{2}\right)\right],$

where the electric field amplitude in the moving mirror arm is changed from
the average of two by the factor $1+\delta$ and that of the fixed mirror arm
by $1-\delta$. The Eq. 29 introduces an additional, typically small, fitting
parameter $\delta$, see Fig. 4.

The relative signs of $\delta$ and $\Phi$ determine which of the two
interferogram minima is smaller in amplitude. If the signs are opposite then
the even minima, including the one at ZPD, are deeper and with the same signs
of $\delta$ and $\Phi$ odd minima are deeper. Thus, knowing the ZPD position
and following the signs of $\delta$ and $\Phi$ across the spectrum we may
obtain information about the sign of $\Phi$.555Additionally, the sign of
$\Phi$ could be obtained from the behaviour of the phase of the detected
signal. We have not used the phase information so far, because of too large
overall phase drifts that are caused by THz frequency and optical setup
instabilities, including the lasers and fibers that connect the lasers to the
photomixers.

We also note that
$\Theta_{\mathrm{K}}(\omega)=\Phi(\omega)+i\mathrm{X}(\omega)$ is a
spectroscopic response functions for which the real and imaginary parts are
related by the Kramers–Kronig relations. Here $\omega$ is the radiation
frequency. Thus, spectroscopic measurement of frequency dependence of
ellipticity, $\mathrm{X}(\omega)$, which is faster than the direct Kerr angle
$\Phi(\omega)$ measurement, may also give an estimate for the Kerr angle
frequency dependence.

##  IV Spectrometer and performance test

###  IV.1  Details of the MOKE spectrometer

The MOKE spectrometer is comprised of three units: the radiation source, the
modified MP interferometer, and the detector of interfered four beams.

The optics assemblies for source and detector units are similar to each other,
these are designed to (i) collimate the incoming beam from transmitter
$\mathrm{T_{X}}$ and focus the outgoing beam on the receiver $\mathrm{R_{X}}$,
and (ii) isolate spurious back reflections from re-entering into the
interferometer, Fig. 5.

Figure 5:  Coupling of the transmitter/receiver ($\mathrm{T_{X}/R_{X}}$) to
the modified MPI. OAP and I are off-axis parabolic mirrors and iris apertures
of the spacial filters to collimate the beam. The Fresnel rhombs (FR) together
with the output/input linear polarizers ($\mathrm{WG_{0}/WG_{90}}$) act as the
broad band optical isolators of the source/detector units.

The Toptica Terascan 780 frequency domain THz spectroscopy system with GaAs
photomixers coupled to log-spiral antennas is used to generate and detect the
continuous-wave (cw) THz radiation. A voltage of about 10 V is applied in the
$\mathrm{T_{X}}$ to accelerate the charge carriers to generate tunable
circularly polarized 0.1 – 1 THz radiation at the beat frequency of the two
distributed feedback lasers.[40, 41, 42] We detect the photocurrent from
$\mathrm{R_{X}}$ that is resulting from accelerating the charge carriers by
the THz radiation.

An additional phase modulation extension, using two fiber stretchers,
modulates the relative phase of laser radiation reaching the $\mathrm{T_{X}}$
and $\mathrm{R_{X}}$, enabling simultaneous detection of the amplitude and
phase of the THz radiation.[43] During a typical interferogram measurement
time, about 30 minutes for 600 data points, the amplitude drifts less than 0.5
$\%$. The THz radiation frequency may drift over the interferogram measurement
time: at 300 GHz a typical frequency error from the fits is less than 0.08
GHz. The signal to noise ratio of detecting THz radiation amplitude depends on
the integration time of the lockin detection at fiber stretcher frequency,
where we typically reach more than 2 orders of magnitude in detecting the THz
electric field amplitude. Comparable signal to noise ratio is not achievable
with bolometric detection, where intensity (amplitude squared) is being
measured.

Time-domain THz systems also measure the THz radiation amplitude but they have
a high THz radiation power on the sample surface. The advantage of time-domain
system as compared to cw THz system is that it can separate the Fabry-Perot
reflections from the main signal, although at the cost of reduced spectral
resolution.

To suppress back reflections into the interferometer we use broad band optical
isolators in the source/detector units, comprising a linear polarizer and a
Fresnel rhomb, see Fig. 5. The Fresnel rhombs convert the polarization of
radiation from circular/linear to linear/circular at the output/input of
$\mathrm{T_{X}/R_{X}}$ so that the back reflected radiation from
$\mathrm{T_{X}/R_{X}}$ has orthogonal polarization with respect to the
original beam and therefore is reflected away from the interferometer by the
linear polarizer $\mathrm{WG_{0}/WG_{90}}$. We designed and made the rhombs
from TOPAS cyclic olefin copolymer [44] with non-parallel facets to avoid
Fabry-Perot modulation within the rhombs. With this we suppress the back
reflections from $\mathrm{T_{X}/R_{X}}$ by more than an order of magnitude to
less than 1 % level. The parallel THz beam incident on the Fresnel rhomb facet
has 1" diameter, while the beam is expanded to 2" diameter for the modified MP
interferometer.

We use wiregrid (WG) polarization beam-splitters from Specac which are made of
$5~{}\mu$m diameter tungsten wires with the periodicity $12.5~{}\mu$m and have
close to ideal transmittance and reflectance characteristics.[45] At 0.3 THz
we measured better than 1:100 contrast in radiation amplitude between parallel
and perpendicular polarized beams.

The hollow rooftop mirrors comprise of two perpendicular (to 5 arc seconds
accuracy) flat unprotected Au-coated mirrors,[46] with the roof line aligned
parallel to the H axis of the incident radiation beam, Fig. 2. The bare gold
coating of 150 nm is sufficient for the frequency range of interest.

A 12 mm long translation stage [47] with calculated 29 nm resolution of the
actuator and minimal repeatable incremental movement 0.2 $\mu$m was used to
move the rooftop mirror to control the path difference in the modified MPI.

###  IV.2  Performance test

As it was first shown by Feil and Haas,[48] the dispersion of the diagonal
part of the dielectric constant near the plasma frequency resonance in a
metallic system has a strong influence on the MOKE properties. The plasma edge
leads to resonance-like peaks in the Kerr rotation and ellipticity spectra
with strong enhancement of the MOKE magnitude. This prediction was verified
for InAs by Shimano et al.[26] at room temperature using a 0.48 T permanent
magnet.

To test the performance of the MOKE spectrometer we carried out MOKE
measurements from small direct gap undoped InSb semiconductor which upon
cooling to 50 K has low plasma frequency, 0.5 THz, and high mobility,
$\mu=3\times 10^{5}$ $\mathrm{cm}^{2}\mathrm{V}^{-1}\mathrm{s}^{-1}$. The
small effective mass of charge carriers in InSb, $m^{*}=0.014m_{e}$,
guarantees high cyclotron frequency $\omega_{c}=eB/m^{*}$ even for relatively
low magnetic field $B$. Thus, the necessary condition of electron orbit
coherence $\omega_{c}\tau>1$ ($\tau$ is the mean free collision time) can be
reached easily even in a few mT field. For this system Kerr rotation and
ellipticity near the plasma frequency reaches 100 mrad already at 5 mT field.

Figure 6:  Rotation $\phi(\omega)$ and ellipticity $\chi(\omega)$ spectra
measured from undoped InSb at 50 K and in magnetic field of 5 mT. To
demonstrate the reliability of the measurement, we fitted the ellipticity
$\chi$ (full squares) to the Drude model (solid line) and plotted the
theoretical rotation $\phi$ (dashed line) using the fitting parameters
obtained from the ellipticity fit. The calculated rotation aligns well with
the experimental data (full circles). The measurement time for each datapoint
was 30 minutes. Figure 7:  Magnetic field dependence of rotation $\phi$ and
ellipticity $\chi$ at 442.2 GHz and 50 K in undoped InSb. The dashed and solid
lines are obtained by fitting simultaneously the magnetic field dependence of
the rotation and ellipticity to the Drude model, Eq. 33. The measurement time
for each datapoint was 30 minutes.

Under external magnetic field perpendicular to the sample surface,
$\mathbf{B}\parallel\mathbf{z}$, the optical conductivity tensor is [33]

$\displaystyle\hat{\mathbf{\sigma}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\sigma_{xx}&\sigma_{xy}\\\
-\sigma_{xy}&\sigma_{xx}\end{array}\right).$ (32)

Within Drude approximation, the components of $\hat{\mathbf{\sigma}}$ are:

$\displaystyle\sigma_{xx}=\epsilon_{0}\epsilon_{b}\omega_{p}^{2}\frac{\gamma-i\omega}{(\gamma-i\omega)^{2}+\omega_{c}^{2}},$
(33a)
$\displaystyle\sigma_{xy}=-\epsilon_{0}\epsilon_{b}\omega_{p}^{2}\frac{\omega_{c}}{(\gamma-i\omega)^{2}+\omega_{c}^{2}},$
(33b)

where $\omega_{p}=\sqrt{ne^{2}/m^{*}\epsilon_{0}\epsilon_{b}}$ is the plasma
frequency, $n$ the carrier density, $e$ the electron charge,
$\epsilon_{0}=8.85\times 10^{-12}$ F m-1 the vacuum permittivity,
$\epsilon_{b}=15.68$ the background dielectric constant for InSb [49], and
$\gamma=1/\tau$ the scattering rate. In this case, it is convenient to work in
a circularly polarized radiation basis Eq. 1, where the index of refraction
$N$ depends on the handedness of radiation,

$\displaystyle N_{\mp}=\sqrt{\epsilon_{xx}\pm i\epsilon_{xy}},$ (34a)
$\displaystyle\epsilon_{xx}=\epsilon_{b}+i\frac{\sigma_{xx}}{\omega\epsilon_{0}},$
(34b) $\displaystyle\epsilon_{xy}=i\frac{\sigma_{xy}}{\omega\epsilon_{0}},$
(34c)

where we assumed $\mu=1$ for the magnetic permeability. Using Eqs. 23, 20, 21
and 34, we obtain the frequency-dependent rotation $\phi(\omega)$ and
ellipticity $\chi(\omega)$. The three fitting parameters for the Drude model,
Eq. 33, are $\omega_{p}$, $\omega_{c}$ and $\gamma$.

As it is shown in Fig. 6, the frequency dependencies of the Kerr angle
rotation and ellipticity obtained by fits of the individual interferograms
measured from undoped InSb wafer at 50 K and the external magnetic field of 5
mT using the modified interferometer electric field amplitude function, Eq.
29, are in agreement with the Drude model prediction. The result demonstrates
the instrument’s ability to track the sign change. The ideal instrument,
described by Eq. 26 would not yield any information about the sign of $\phi$,
but a small unbalance between the interferometer arms, Eq. 29, gives some
information about the sign. It can be seen that by fitting the ellipticity
alone, the rotation is predicted quite well, thus proving the reliability of
the measurement. The parameters obtained from the Drude fit are $\tau=2.38\pm
0.07$ ps, $\omega_{p}/2\pi=483.2\pm 0.4$ GHz, and $\omega_{c}/2\pi=10.3\pm
0.3$ GHz. Using the relation $\mu=e\tau/m^{*}$, the mobility of the sample is
$\mu=(2.99\pm 0.07)\times 10^{5}$
$\mathrm{cm}^{2}\mathrm{V}^{-1}\mathrm{s}^{-1}$. These values are inline with
time-domain THz spectroscopy (TDS) reflection measurements on undoped
InSb,[49] and with the dc transport measurements performed on a separate
sample cut from the same InSb wafer.

In Fig. 7 we also show the rotation and ellipticity as a function of magnetic
field measured at $T=50$ K and at frequency $\omega=442$ GHz. The magnetic
field dependence of the rotation and ellipticity was simultaneously fitted to
Drude model, Eq. 33. The obtained fitting parameters, $\tau=2.40\pm 0.02$ ps
and $\omega_{p}/2\pi=476.1\pm 0.4$ GHz, are in agreement with those obtained
from fitting the rotation and ellipticity as a function of frequency.

The residuals of $\phi$ and $\chi$ shown in Fig. 7 indicate that the overall
resolution for tracking small changes of the rotation and ellipticity is about
1 mrad at 442 GHz. A typical measurement time of an interferogram (Fig. 4) of
600 data points is 30 minutes and the spacing of points near minima is as
determined by the precision of the moving mirror stage.

##  V Summary

We have developed a MOKE spectrometer designed for simultaneous measurement of
the Kerr rotation angle and ellipticity using continuous wave sub-THz low-
power radiation over a broad frequency range under normal incidence to the
sample, eliminating the need for reference measurements. We demonstrated that
the spectrometer is capable of measuring the frequency dependence of the polar
Kerr effect response function with sub-milliradian accuracy.

The MOKE spectrometer is well-suited for integration with magneto-optical
systems that have direct optical access. The implemented interferometric
technique can be adapted to other radiation sources across a broad frequency
range, including shorter wavelengths, by using polarizing beam splitter cubes.
The instrument’s versatility allows for the studies of magneto-optical effects
in quantum materials such as unconventional superconductors, two-dimensional
electron gas systems, quantum magnets, and systems exhibiting the anomalous
Hall effect, at sub-Kelvin temperatures and in high magnetic fields.

###### Acknowledgements.

The work was supported by the Estonian science agency grants PRG736, MOBJD1103
and by the European Research Council (ERC) under the European Unions Horizon
2020 research and innovation program Grant Agreement No. 885413. The work at
Rutgers was supported by the National Science Foundation (NSF) Grant No.
DMR-2105001.

## Author declarations

### Conflicts of interest

The authors have no conflicts to disclose.

### Author contributions

GB conceived the project. UN, GB and TR invented the measurement method. TR,
GB and UN modelled the response function of the interferometer. UN, GB and AGM
built the interferometer. UN created the LabVIEW code to run the Kerr
interferometer and control the experiments. All authors performed the
experiments and wrote the manuscript.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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