# Band splitting induced Berry flux and intrinsic anomalous Hall conductivity
in NiCoMnGa quaternary Heusler compound

Gaurav K. Shukla  School of Materials Science and Technology, Indian Institute
of Technology (Banaras Hindu University), Varanasi 221005, India  Jyotirmay
Sau  S. N. Bose National Centre for Basic Sciences, Kolkata 700098, West
Bengal, India  Vishal Kumar  School of Materials Science and Technology,
Indian Institute of Technology (Banaras Hindu University), Varanasi 221005,
India  Manoranjan Kumar  S. N. Bose National Centre for Basic Sciences,
Kolkata 700098, West Bengal, India  Sanjay Singh  School of Materials Science
and Technology, Indian Institute of Technology (Banaras Hindu University),
Varanasi 221005, India

###### Abstract

The anomalous transport properties of Heusler compounds become a hotspot of
research in recent years due to their unique band structure and possible
application in spintronics. In this paper, we report the anomalous Hall effect
in polycrystalline NiCoMnGa quaternary Heusler compound by experimental means
and theoretical calculations. The experimental anomalous Hall conductivity
(AHC) was found at about 256 $S/cm$ at 10K with an intrinsic contribution of
$\sim$ 121 $S/cm$. The analysis of Hall data reveals the presence of both
extrinsic and intrinsic contributions in AHE. Our theoretical calculations
show that a pair of spin-orbit coupled band formed by the band splitting due
to spin-orbit interaction (SOI) at the Fermi level produces a finite Berry
flux in the system that provides the intrinsic AHC about 100 $S/cm$, which is
in good agreement with the experiment.

##  I INTRODUCTION

The conventional Hall effect describes the phenomenon of transverse deflection
of moving charges in a current carrying conductor placed in a magnetic field
and this effect leads to a voltage difference perpendicular to both direction
of motion of charges and the magnetic field [1]. However, in ferromagnetic
materials an additional large transverse voltage drop occurs in comparison to
the normal conductors and this phenomena is known as anomalous Hall effect
(AHE) [2, 3, 4, 5]. So far, two kinds of mechanisms have been proposed to
understand the origin of AHE; intrinsic and extrinsic mechanisms [6, 7, 8, 9].

In the intrinsic mechanism, the interband mixing along with spin-orbit
interaction (SOI) results in an anomalous velocity of electrons perpendicular
to the electric field direction [8]. Subsequently, this anomalous velocity was
reformulated in terms of the Berry phase and Berry curvature of Bloch bands
[10, 11, 5]. Berry curvature is a pseudo-magnetic field in momentum space,
which acts just like the magnetic field in electrodynamics [11, 5, 12]. The
SOI has small energy in comparison to the exchange splitting energy of
ferromagnets, however the small energy of SOI can split the band dispersion
near to the Fermi energy and when the Fermi energy lies in the SOI energy gap,
a non zero dissipationless Hall current arises due to non-vanishing total flux
of Berry curvature [13, 14, 1]. The anomalous Hall conductivity (AHC) due to
Berry phase (intrinsic mechanism) does not depend on the longitudinal
conductivity ($\sigma$xx) [15, 16].

The external origin of AHE, which includes the skew scattering and side jump
mechanisms are known as extrinsic mechanism. Skew scattering is an asymmetric
scattering of carriers by impurity potential, which introduces a momentum
perpendicular to both the incident wave vector k and magnetization M. AHC due
to skew scattering is proportional to the $\sigma$xx [6]. Side jump is a
microscopic displacement of wave packet due to SOI coupled Bloch states under
the influence of disorder potential and the AHC due to side jump is
independent of $\sigma$xx [13, 9].

Currently, there is an immense interest in the AHE due to its potential
applications in spintronics such as for magnetic sensors and memory devices
[17, 18, 19]. Several materials have been reported to have large intrinsic AHC
due to the Berry curvature originating from their characteristic spin-orbit
coupled band structure. For example, the intrinsic AHC in Fe (751 $S/cm$) [14]
and MnAs ($\sim$ 103 $S/cm$) [13] have been reported due to the Berry
curvature arising from a pair of spin-orbit coupled band near to the Fermi
level. Co3Sn2S2 [20], Fe3Sn [21] and LiMn6Sn6 [22] show the large intrinsic
AHC arising due to the finite Berry curvature associated with the band
structure. Besides these materials, Heusler alloys [23, 24, 25] show the
exotic anomalous transport properties owing to the distinctive band structure
due to the combined effect of SOI and broken time-reversal symmetry [15, 26,
27, 28]. For example, Co2MnAl [29], Co2MnGa [26], Fe2-based Heusler alloys
[30] show the large intrinsic AHE.

Co2MnGa is a full Heusler compound that crystallizes in the L21 structure with
space group Fm${\bar{3}}$m, consists of three nodal lines near Fermi energy,
derived from the three mirror symmetries present in the system in the absence
of spin-orbit coupling (SOC) [31]. With the consideration of SOC, these nodal
lines gap out according to the magnetization direction and create the large
Berry curvature and intrinsic AHE in the system [28, 31]. In the present
manuscript, we studied the AHE in NiCoMnGa quaternary Heusler compound, which
can be obtained by replacing the one Co atom by its neighbouring Ni atom in
Co2MnGa full Heusler compound keeping the magnetic moments nearly close for
both the compounds. Experimentally, we found the value of AHC around 256
$S/cm$ at 10K with an intrinsic contribution of $\sim$ 121 $S/cm$. The
theoretical calculations give the intrinsic AHC $\sim$ 100 $S/cm$, which is in
good agreement with the experiment. The reduction of the mirror symmetries in
the NiCoMnGa in comparison to the Co2MnGa lead to the absence of nodal line,
nevertheless the band splitting in presence of SOC at Fermi energy leads to
the finite Berry curvature and intrinsic AHC in the system.

##  II METHODS

Polycrystalline NiCoMnGa Heusler compound was synthesized by arc melting
method in the environment of high pure argon atmosphere by taking 99.99 % pure
constituent elements in the water cooled copper hearth. To reduce further
contamination, a Ti piece was used as an oxygen getter. The sample was flipped
several times and re-melted for homogeneous mixing. The energy dispersive
x-ray (EDX) analysis suggests a composition ratio of 1:1:1:1 within the
standard deviation (3%-5%) of the EDX measurement. For the structural analysis
x-ray diffraction (XRD) pattern of powder sample was recorded at room
temperature using Rigaku made x-ray diffractometer with Cu-K$\alpha$
radiation. The magnetization measurements were performed using a vibrating
sample magnetometer (VSM) attached with Physical Properties Measurement System
(PPMS) from Quantum Design. The resistivity and Hall measurement were carried
out on a rectangular piece of sample of dimension 4.34
$\times\,$2.45$\times\,$0.66 mm3 employing four-probe method using Cryogen
Free Measurement System (CFMS). The electronic band structure and magnetic
properties of the NiCoMnGa were calculated by employing density functional
theory (DFT) using the Vienna ab initio simulation package (VASP) [32]. The
generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) type
was used for the exchange-correlation functional [33].

Figure 1: Rietveld refinement of the room temperature powder x-ray diffraction
(XRD) data of NiCoMnGa system. The inset shows an enlarged view of XRD pattern
around (111) superlattice reflection. (b) Simulated XRD pattern of NiCoMnGa
system. Figure 2: Field-dependent magnetic isotherms at 10 K and 300 K. Figure
3: (a) Temperature-dependent longitudinal resistivity $\rho$xx. Inset shows a
schematic drawing of sample device used for Hall voltage VH and longitudinal
voltage Vxx measurements. (b) Field-dependent Hall resistivity $\rho$H at
different temperatures. (c) Temperature variation of anomalous Hall
resistivity $$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$. Inset shows the field-
dependent $\rho$xx at (i) 10 K, (ii) 200 K, and (iii) 300 K. (d) Temperature-
dependent carrier concentration n. (e) Double logarithmic plot between
$$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$ and $\rho$xx (black balls) and the
linear fitting is shown by a red line. (f)
$$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$ and $\rho$xx plot (black balls) and
fitted curve using Eq.(5) is shown in red color . Figure 4: (a) Field-
dependent Hall conductivity $\sigma$H at different temperatures. (b)
Temperature variation of anomalous Hall conductivity (AHC). Inset shows the
variation of AHC with longitudinal conductivity $\sigma$xx.

The kinetic energy cutoff of 520 eV was taken for the plane-wave basis. A 15 ×
15 × 15 k-point mesh was used for the Brillouin zone (BZ) sampling and the
Gaussian smearing method with a width of 0.1 eV was adopted for the Fermi
surface broadening. The cell parameter was relaxed until the forces on all
atoms were smaller than 0.01 eV/Å. The SOC was taken into account in all the
calculations. To explore the non-trivial band topology and the intrinsic AHC,
the tight binding Hamiltonian was constructed with the maximally localized
Wannier functions using the Wannier90 code [34, 35]. Based on the tight-
binding Hamiltonian, the AHC and the Berry curvature were evaluated via the
Kubo-formula approach [36].

##  III RESULTS AND DISCUSSION

###  A Structure and magnetization

The XRD pattern of NiCoMnGa was recorded at room temperature for structural
analysis. The observed XRD pattern (red curve in Fig. 1(a)) fairly shows the
cubic structure of the sample. The quaternary Heusler alloys generally
crystallize in the LiMgPdSn type structure (space group F$\bar{4}$3m)[37]. The
Rietveld refinement of the room temperature XRD pattern of NiCoMnGa was
carried out using FULLPROF software package [38] in the space group of
F$\bar{4}$3m (space group no.216) and the special Wyckoff’s positions; 4d
(0.75,0.75,0.75), 4c (0.25, 0.25,0.25,0.25), 4b (0.5, 0.5, 0.5), and 4a (0, 0,
0) were considered for Nickel (Ni), Cobalt(Co), Manganese (Mn) and Gallium
(Ga) atoms, respectively. The calculated XRD pattern (black curve) depicted in
Fig. 1(a), shows that all the Bragg peaks are well-indexed, which confirms the
single-phase (cubic) of the sample. The refined lattice parameter was found
5.79 Å, which matches well with the value reported in the literature [39]. The
presence of the (111) and (200) superlattice reflections generally mark the
ordered structure of Heusler compounds [15]. For NiCoMnGa the structure
factors for (111), (200) and (220) reflections can be written as [40]

$F_{111}=4[(f_{Ga}-f_{Mn})-i(f_{Ni}-f_{Co})]$ (1)
$F_{200}=4[(f_{Ga}+f_{Mn})-(f_{Ni}+f_{Co})]$ (2)
$F_{220}=4[(f_{Ga}+f_{Mn})+(f_{Ni}+f_{Co})]$ (3)

The negligible intensity of the (111) superlattice reflection survives only
due to the difference of the atomic scattering factors (SFs) of Ga and Mn
atoms as the difference between atomic SFs of Ni and Co atoms is negligible
(both are the consecutive elements in periodic table) [15, 40]. The vanishing
intensity of the (200) reflection can be understood from Eq.2. The intensity
of fundamental reflection is due to the sum of all atomic SFs of constituent
elements (Eq.3). We simulated the XRD pattern of NiCoMnGa using POWDERCELL
software [41] depicted in the Fig. 1(b). The intensity ratio of superlattice
reflection (111) to the fundamental reflection (220) (i.e.
$\frac{I_{111}}{I_{220}}$) was found about 0.01 and 0.012 from the simulated
and observed XRD patterns, which suggests the formation of ordered structure
of NiCoMnGa Heusler compound. Inset of Fig. 1(a) shows the enlarged view of
XRD pattern around (111) superlattice reflection. Figure 2 shows the magnetic
isotherms at the temperature 10 K and 300 K. The magnetic moment was found 4.7
$\mu_{B}$/f.u at 10 K. The magnitude of the observed magnetic moment is close
to the value reported in literature[39].

###  B Transport measurements

Figure 3(a) shows the measured temperature dependence of longitudinal
resistivity ($\rho$xx). The temperature variation of $\rho$xx indicates the
metallic conduction with $\sigma$xx about 1.88$\times$104 $S/cm$ at 300 K. The
residual resistance ratio (RRR=
$\rho\textsubscript{xx}(300K)$/$\rho\textsubscript{xx}(10K)$), which
quantifies the degree of disorder, is found around 2, which suggests a clean
sample of NiCoMnGa [15]. Now, afterward investigating the phase purity,
magnetization and resistivity of the sample, we will discuss the outcomes of
Hall measurement. Inset of Fig. 3(a) shows the schematic diagram of sample
device used for longitudinal voltage (Vxx) and Hall voltage (VH) measurements.
In general the Hall resistivity ($\rho$H) in magnetic materials is sum of two
parts [15, 1];

$$\rho$\textsubscript{\tiny{{H}}}=$\rho$^{0}_{\textsubscript{\tiny{{H}}}}+$\rho$^{A}_{\textsubscript{\tiny{{H}}}}=R_{0}H+R_{s}M$
(4)

$$\rho$^{0}_{\textsubscript{\tiny{{H}}}}$ and
$$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$ are the ordinary and anomalous Hall
resistivity, respectively. R0, Rs, H and M are the ordinary Hall coefficient,
anomalous Hall coefficients, applied external magnetic field and spontaneous
magnetization of material, respectively. R0 which depends on the type of
charge carriers and their density is the inverse of the product of carrier
concentration (n) and electronic charge (e) [42]. The Hall resistivity in
ferromagnetic materials dominates by AHE at the lower field and the role of
ordinary Hall effect usually appears in the higher field region [15]. With the
linear fitting of the high field $\rho$H curve, slope and intercept on the
y-axis give R0 and $$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$, respectively.
$$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$ is alike to Hall response due to
magnetization of material in absence of an external magnetic field. Hall
measurement outcomes are summarized in Fig. 3 (b)-(f) and Fig. 4 (a)-(b).
Figure 3(b) shows the field dependent $\rho$H isotherms up to field of 4 T.
The $\rho$H curves show a sharp jump at low field and change linearly in the
high field regime that signifies an AHE in the present material. Figure 3(c)
displays the variation of extracted $$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$
with temperature. $$\rho$^{A}_{\textsubscript{\tiny{{{H}}}}}$ increases non-
linearly with temperature and achieves the maximum value of about 0.56
$\mu\Omega$-cm at room temperature. We also measured the field-dependent
$\rho$xx at fixed temperatures. The $\rho$xx does not changes significantly
with the magnetic field at a particular temperature as shown in the inset of
Fig. 3(c) for (i) 10 K, (ii) 200 K, and (iii) 300 K. The charge carrier
density (n) was calculated using relation R0=$\frac{1}{ne}$ [42] and
temperature variation of n is depicted in Fig. 3(d). The magnitude of n is a
little scattered with temperature and estimated about
2.5$\times$10${}^{21}\,$cm-3 at 300 K. In order to examine the origin of AHE,
we have plotted the $$\rho$^{A}_{\textsubscript{\tiny{{H}}}}$ versus $\rho$xx
on a double logarithmic scale as shown in Fig. 3(e). A linear fitting was done
to determine the exponent $\alpha$ according to the formula
$$\rho$^{A}_{\textsubscript{\tiny{{{H}}}}}$ $\propto$
$$\rho$^{\alpha}_{\textsubscript{\tiny{{xx}}}}$ [43, 42]. If $\alpha$ = 1, the
origin of AHE is due to skew scattering and if $\alpha$ = 2, the origin of AHE
is assigned to the intrinsic and side jump mechanisms [2, 15, 43]. In this
way, we found the exponent $\alpha=1.50$, which primarily specifies that the
extrinsic and intrinsic mechanisms are involved in the AHE. To disentangle the
intrinsic and extrinsic contributions linked with AHE, we have used the
following equation, which accounts the phonon contribution in the skew
scattering as suggested for alloys system [15, 44, 42, 45]

$$\rho$^{A}_{\textsubscript{\tiny{{H}}}}=a$\rho$\textsubscript{\tiny{{xx}}}+$\sigma$^{int}$\rho${{}^{2}_{\textsubscript{\tiny{{xx}}}}}.$
(5)

Here a is the parameter related to the skew scattering and the notation
$\sigma$int is used for intrinsic AHC. Here we have assumed the ratio of SOI
energy ($\epsilon\textsubscript{SO}$) to the Fermi energy (EF) i.e.
$\epsilon\textsubscript{SO}$/EF is in order of 10-3 to 10-2, which leads the
suppression of side jump contribution in comparison to intrinsic AHC as
observed for other metallic ferromagnets [15, 43]. Fig. 3(f) shows the
$\rho$${}^{A}_{\textsubscript{\tiny{{H}}}}$ versus $\rho$xx plot (black balls)
and the fitting was employed using Eq.(5) as shown by a red curve. From
fitting, the parameter $a$ and $\sigma$int comes out to be 0.0034 and 121
$S/cm$, respectively. The change in
$$\sigma$^{A}_{\textsubscript{\tiny{{H}}}}$ with temperature and/or
longitudinal conductivity ($\sigma$xx) also holds the information about the
involved mechanism in the AHE. Hall conductivity ($\sigma$H) was calculated
using equation [31, 15, 16]

$$\sigma$\textsubscript{\tiny{{H}}}=\frac{$\rho$\textsubscript{\tiny{{H}}}}{($\rho${{}^{2}_{\textsubscript{\tiny{{xx}}}}}+$\rho${{}^{2}_{\textsubscript{\tiny{{H}}}}})}$
(6)

Figure 5: (a) The band structure of NiCoMnGa in the absence of SOC, where the
red and blue colors represent the spin-up and spin-down bands, respectively.
(b) The band structure of NiCoMnGa in the presence of SOC. The inset shows the
splitting of the bands near the Fermi energy EF. (c) Berry curvature
distribution in $\textit{k}_{y}$-$\textit{k}_{z}$ plane at $\textit{k}_{x}=0$.
(d) Berry curvature along the high-symmetry path.

Figure 4 (a) shows the field-dependent Hall conductivity curves at different
temperatures. The value of AHC at a particular temperature is calculated by
zero-field extrapolation of the high field Hall conductivity data with the
y-axis. We found the AHC around 256 $S/cm$ at 10 K. This value is nearly twice
of the magnitude of the intrinsic AHC, which shows the presence of equal
contribution of intrinsic and extrinsic AHC at 10 K. The AHC decreases with
increasing the temperature and reduces to 194 $S/cm$ at 300 K as shown in Fig.
4(b). The variation of AHC with $\sigma$xx is shown in the inset of Fig. 4(b).
The decreasing (increasing) value of AHC with temperature ( $\sigma$xx) is due
to the skew scattering contribution in AHC rather than the temperature
variation of spontaneous magnetization of sample as the temperature has a
little effect on the magnetization of the present system (Fig. 2). It is
worthwhile to mention here that intrinsic AHE is expected to dominate in the
overall behaviour of AHE, when the longitudinal conductivity of sample lies in
the good metallic regime i.e. $\sigma$xx is an order of 104-106 $S/cm$ [15,
16, 43, 1, 46]. Our system shows the deviation from this criterion as the
extrinsic AHE has significant contribution in the AHE despite the $\sigma$xx
is an order of 104 $S/cm$. The AHC due to skew scattering can be given as [47]

$$\sigma$^{skew}_{\textsubscript{\tiny{{H}}}}=$\sigma$\textsubscript{\tiny{{xx}}}S=\frac{2e^{2}}{ha}\frac{E_{F}{\tau}}{\hslash}S$
(7)

where h, a, EF, $\tau$ and S are the Planck constant, lattice parameter, Fermi
energy, mean free path of electron and skewness factor, respectively. The
skewness factor is S $\sim$ $\epsilon_{SO}$vimp/W2, where W is the bandwidth
and vimp is impurity potential [47]. In the superclean limit where the
$\frac{\hslash}{\tau}\textless\textless$ $\epsilon_{SO}$, the skew scattering
dominates, however in the case of $\frac{\hslash}{\tau}$ <$\epsilon_{SO}$, the
skew scattering contribution decreases. From Eq.7, it is clear that the skew
scattering contribution rapidly decays with increasing the
$\frac{\hslash}{\tau}$. Henceforth, the value of AHC due to skew scattering
depends on both the strength of SOI energy and the $\sigma$xx. Our experiment
also suggests that the lineup of the origin of AHE based solely on the
longitudinal conductivity is a rough estimation and need not to be strictly
valid. Since the intrinsic AHE depends on the Berry curvature of the system,
which may have different origin such as presence of gapped nodal line, Weyl
point or interband mixing along a certain high symmetry path. Therefore, to
understand the origin of Berry curvature in present system, we performed the
first principles calculation.

###  C First principles calculation

The relaxed lattice parameter of NiCoMnGa was found 5.782 Å, which is
consistent with the experiment. Our first principles calculation for magnetic
moment suggests that Co and Mn have a large magnetic moment with
$\mu_{Co}$=1.188 $\mu_{B}$/f.u and $\mu_{Mn}$=3.246 $\mu_{B}$/f.u
respectively, whereas Ga has a small vanishing magnetic moment and Ni have a
moment $\mu_{Ni}$=0.563 $\mu_{B}$/f.u. The total magnetic moment per formula
unit is 4.993 $\mu_{B}$, aligned along the (001) direction, which is
consistent with Slater Pauling rule for Heusler alloys [48]. To calculate the
Berry curvature and the intrinsic AHE in the present system, the tight-binding
Hamiltonian was constructed with the maximally localized Wannier functions
[34, 35]. Based on the tight-binding Hamiltonian, the AHC and the Berry
curvature were evaluated via the Kubo-formula [36] approach in the linear
response scheme as follows;

$\sigma_{\alpha\beta}=-{\frac{e^{2}}{\hbar}\sum_{n}\int\frac{d^{3}K}{(2\pi)^{3}}\Omega^{n}_{\alpha\beta}f_{n}}$
(8)

where Berry curvature $\Omega$ can be written as sum over eigen state using
Kubo formula[11]

$\displaystyle\Omega^{n}_{\alpha\beta}=i\sum_{n\neq n^{\prime}}\frac{{\langle
n|\frac{\partial H}{\partial R^{\alpha}}|n^{\prime}\rangle}{\langle
n^{\prime}|\frac{\partial H}{\partial
R^{\beta}}|n\rangle}-(\alpha\xleftrightarrow{}\beta)}{(\epsilon_{n}-\epsilon_{n}^{\prime})^{2}}$
(9)

Here $\ket{n}$ and $\epsilon_{n}$ are the energy eigenstate and eigenvalue of
Hamiltonian H, respectively. fn is the Fermi distribution function. The
intrinsic AHE can be analyzed by exploring the electronic band structure of
the NiCoMnGa. Two major features in electronic band structure make this system
important; first, a twofold degenerate band form a triple point crossing with
a non-degenerate band along high symmetry direction $\Gamma$-L in the absence
of SOC as shown in Fig. 5a by a black dotted circle. This doubly degenerate
band splits in the presence of SOC and also lifts the triple point degeneracy.
This degeneracy may arise due to a symmetry of C3v along $\Gamma$-L high
symmetry direction, whose elements are threefold rotation (C3) and a
$\sigma_{v}$ mirror plane [49]. Second feature is a pair of degenerate
minority spin band at $\Gamma$ point in absence of SOC, however, in the
presence of SOC these bands split into an occupied and an unoccupied bands
separated by a small energy gap in a small k interval near $\Gamma$ point as
shown in Fig. 5b. In Figure 5d, we calculated the Berry curvature
$\Omega^{z}_{xy}$ in the presence of SOC along same symmetry direction and
$\Omega^{z}_{xy}$ has large value near the $\Gamma$ point. This large value of
Berry curvature can be explained in terms of splitting of degenerate band,
which opens a small energy gap and creates intrinsic AHC in the system.

Figure 6: Fermi energy scan of the anomalous Hall conductivity.

Owing to the nearly degenerate conduction and valance band at $\Gamma$ point,
the Berry curvature is large at this point; therefore an intrinsic AHE is
expected in the present compound. In the Fig. 5(c), the color plot of Berry
curvature is shown in $k_{x}$ =0 plane of the first Brillouin zone, which
shows positive and negative distribution of Berry curvature in the plane.
Following the symmetry of NiCoMnGa compound, if the magnetization is taken
along the z-axis, $\Omega^{z}_{xy}$ is the only survival component of the
Berry curvature. We found the intrinsic AHC due to finite $\Omega^{z}_{xy}$
about 100 $S/cm$ at Fermi level, which is consistent with the experiment (121
$S/cm$). We also showed the dependency of AHC as a function of shift in Fermi
energy in Fig. 6, which gives an idea about contribution of various valance
bands by shifting the Fermi energy. The large peak below the Fermi level
(-0.38 eV) is due to large Berry curvature at the triple point, which is
contributing as an occupied valence band to AHC. We observed the decreasing
trend of AHC from -0.38 eV and it again increases at Fermi energy due to a
large positive Berry curvature shown by blue color in Fig. 5(c).

##  IV CONCLUSIONS

We have experimentally measured the AHC in NiCoMnGa quaternary Heusler
compound and theoretically calculated the intrinsic part of AHC due to Berry
curvature of the dispersion bands. The extrinsic and intrinsic mechanisms both
contribute equally in the AHC of the present system. We found a good agreement
between the extracted intrinsic AHC experimentally and the theoretically
calculated AHC. The reduction of the number of mirror symmetries in the
NiCoMnGa in comparison to the Co2MnGa lead to the absence of nodal line,
nevertheless the band splitting in the presence of SOC at Fermi energy leads
to the finite Berry curvature and intrinsic AHC in the system. The presence of
signification contribution of extrinsic mechanism in AHE, despite the
longitudinal conductivity is an order of 104 $S/cm$, suggests that the
relation of the origin of AHE solely with the longitudinal conductivity may
not be strictly valid.

## ACKNOWLEDGMENT

We gratefully acknowledge IIT BHU for experimental support. SS thanks Science
and Engineering Research Board of India for financial support through the
award of Ramanujan Fellowship (grant no: SB/S2/RJN-015/2017) and UGC-DAE CSR,
Indore for financial support through “CRS” Scheme. G.K.S. acknowledges the
DST-INSPIRE scheme for support through a fellowship. M.K thanks DST for
funding through grant no.CRG/2020/000754. J.S. thanks University Grants
Commission (UGC) for Ph.D. fellowship. V.K. thanks CSIR, New Delhi for
financial support.

<EMAIL_ADDRESS>
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figure table

# Physics-informed neural networks for diffraction tomography

Amirhossein Saba  Carlo Gigli  Ahmed B. Ayoub  Demetri Psaltis  Optics
Laboratory, École polytechnique fédérale de Lausanne, CH-1015, Lausanne,
Switzerland

###### Abstract

We propose a physics-informed neural network as the forward model for
tomographic reconstructions of biological samples. We demonstrate that by
training this network with the Helmholtz equation as a physical loss, we can
predict the scattered field accurately. It will be shown that a pretrained
network can be fine-tuned for different samples and used for solving the
scattering problem much faster than other numerical solutions. We evaluate our
methodology with numerical and experimental results. Our physics-informed
neural networks can be generalized for any forward and inverse scattering
problem.

* Corresponding author: Amirhossein Saba<EMAIL_ADDRESS>
† These authors contributed equally to this work.

##  1 Introduction

Optical diffraction tomography (ODT) is an imaging technique for extracting
the three-dimensional (3D) refractive index distribution of a sample, for
example, a biological cell using multiple two-dimensional (2D) images acquired
at different illumination angles. The refractive index of the sample provides
useful morphological information, making ODT an interesting approach for
biological applications [1, 2, 3]. The conventional method to reconstruct the
3D refractive index from multiple projections was proposed by Emil Wolf in
1969 [4]. In Wolf’s method, the 3D Fourier domain of the refractive index is
filled with 2D Fourier transforms of the measured scattered fields. However,
due to the finite number of projections, limited numerical aperture (NA) of
the optical system, and the single scattering approximation, missing
frequencies are causing some distortion, elongation, and underestimation of
the reconstructed refractive index [5].

In the last years, many different iterative methods have been proposed to
reconstruct accurate refractive indices from ill-posed measurements [5, 6, 7,
8, 9]. The main idea behind these iterative approaches is to use a forward
model which predicts the candidate projections for the current estimation of
the refractive index in that iteration, compare this 2D prediction with the
experimental measurements of the projection as a loss function and update the
estimation of the refractive index upon the minimization of this loss function
plus possibly any other prior knowledge, e.g. sparsity conditions.
Importantly, such an iterative scheme requires an analytical/semi-analytical
model in order to backpropagate the computed loss and update the estimation of
the refractive index. This precludes the use of common mesh-based numerical
solvers like finite difference and finite element methods. In Ref. 5 the
authors use a linear (single-scattering) forward model, in the approach
proposed in Refs. 6, 7, 8, referred as learning tomography, the forward model
is beam propagation method (BPM), and in Ref. 9 the authors resort to the
Lippmann–Schwinger equation. The forward models used in these iterative
solutions either have inaccuracies in the cases of multiple-scattering and
high-contrast samples, or they are computationally demanding. As a result,
presenting a fast, accurate, and differentiable forward model is necessary to
be used in iterative ODT. Physics-informed neural networks (PINNs), can be a
good candidate for solving forward scattering problem and being used in
iterative tomographic reconstruction.

PINNs have recently gotten intensive research attention for solving different
complex problems in physics [10, 11]. These networks use physics laws as the
loss function instead of the data-driven loss functions. In conventional
supervised deep learning, a large dataset of labeled examples is used for the
training process: by comparing the known ground truth with the predictions
from a deep multi-layer neural network, one can construct a loss function and
tune the parameters of the network in order to solve complex physical
problems. Different examples of these data-driven neural networks are proposed
for optical applications such as resolution enhancement [12], imaging through
multi-mode fibers [13, 14], phase retrieval [15], and ODT [16]. In these
networks, the knowledge acquired by the network strongly depends on the
statistical information provided in the dataset, and training such a network
requires access to a large dataset. In contrast, PINNs directly minimize the
physical residual from the corresponding partial differential equation (PDE)
that governs the problem instead of extrapolating physical laws after going
through a large amount of examples. In the pioneering approach proposed by
Lagaris et al. [17], the neural network maps independent variables, such as
spatial and time coordinates, to a surrogate solution of a PDE. By applying
the chain rule, for example through auto-differentiation integrated in many
deep-learning packages, one can easily extract the derivatives of the output
fields with respect to the input coordinates and consequently construct a
physics based loss [18]. The correct prediction can be therefore retrieved by
minimizing the loss with respect to the network weights. This approach has
been used to solve nonlinear differential equations [19, 20, 21, 22], to
realize the forward model in the inverse design of optical components [23],
and to extract material parameters in near field microscopy [24].

A different idea was proposed recently in [25] to solve Maxwell’s equations
for a set of examples with different permittivity distributions. The
calculation of physical loss, in this case, is based on the finite difference
scheme, and in contrast to the previous approach which is trained for a
specific example, this model proved to be well-suited for cases in which fast
inference is required. In this paper, we use a similar approach, able to solve
different forward scattering problems, such as light scattering from
biological cells. Then we demonstrate using our PINN as a forward model in an
iterative reconstruction of the 3D distribution of the refractive index.

##  2 Methodology

The main idea of our work, summarized in Fig. 1, consists of two blocks. The
first, MaxwellNet, is a neural network which takes as an input the refractive
index distribution $n(\mathbf{r})$ and predicts the scattered field $U^{s}$.
Its structure is based on the U-Net architecture [26], and the training is
performed on a large dataset of digital phantoms using a physics-defined loss
function. Then, this network is used as a forward model in a second
optimization task which compares the fields predicted by MaxwellNet for a
candidate RI distribution with the ground-truth projections, e.g. computed
numerically or evaluated experimentally, and updates $n(\mathbf{r})$ up to
convergence.

###  2.1 Forward Model: MaxwellNet

In this subsection, we describe the implementation of a PINN which predicts
the scattered field for a known input RI distribution. For the sake of
simplicity, we first describe the method for the 2D case, but we will show the
extension to 3D in the following. In this case, MaxwellNet takes as an input
the RI distribution as a discrete array of shape
${N_{x}{\times}N_{z}}{\times}1$ and we do expect an output with size
${N_{x}{\times}N_{z}}{\times}2$, where the two channels correspond to the real
and imaginary parts of the complex field. Among all the available
architectures, the choice of U-Net appears favorable as we do expect to embed
the latent features of the RI distributions in a lower dimensional space
through consecutive 2D convolutions and then retrieve the complex
electromagnetic field in the same spatial points through the decoding step. A
similar architecture was also proven to provide good accuracy for the
computation of the scattered field from micro lenses [25]. We implement the
present network in TensorFlow 2.6.0. For each step in the encoder, we use two
Conv2D layers, each followed by batch-normalization and elu activation
function. A total number of five layers is adopted to encode the information
and each one is terminated with average pooling to reduce the dimension. The
maximum number of channels that we get in the latent space is 512. On the
decoder side, we used transposed convolutional layers to the output with the
size ${N_{x}{\times}N_{z}}{\times}2$ (or
${N_{x}{\times}N_{y}{\times}N_{z}}{\times}2$ in the 3D case). It should be
noted that we also use residual skip connections from the encoder branch. In
common data-driven training, we would tune the weights of this network by
minimizing the difference between predictions and ground-truth data computed
with numerical solvers, in turn requiring a large database of simulations and
consequently a massive computational cost. Here we do not provide input-output
pairs, instead we train the network by requiring that the Helmholtz equation
is satisfied on the predicted field. In order to speed up the training and
improve performances, we require the network to predict the slowly varying
envelope of the scattered field $U^{s}_{env}$ being the scattered field
obtained after demodulated by the fast-oscillating component along propagation
direction $U^{s}=U^{s}_{env}e^{jk_{0}n_{0}z}$. We define a physics-informed
loss function to be minimized by updating the weights of the network:

---
Figure 1: Schematic description of MaxwellNet, with U-Net architecture, and
its application for tomographic reconstruction. The input is a refractive
index distribution and the output the envelope of the scattered field. The
output is modulated by the fast-oscillating term $e^{jk_{0}n_{0}z}$ to compute
the physics-informed loss for tuning the weights. For tomographic
reconstruction, we minimize a data-driven loss based on the difference between
measured and calculated projections using MaxwellNet. A regularization term
can be added to improve the reconstruction.

$\mathcal{L}_{Ph}=\sum_{r}\frac{1}{N}\left\|\\{{\nabla}^{2}+{k_{0}^{2}}{n^{2}(r)}\\}U^{s}+{k_{0}^{2}}(n^{2}(r)-n_{0}^{2})U^{i}\right\|^{2}$
(1)

where, $k_{0}$ is the wave-number which is $k_{0}=2\pi/\lambda$ and
$\lambda=1.030\mu m$ is the wavelength. $n(r)$ is the RI distribution and
$n_{0}$ is the RI of the background medium. The summation in Eq. (1) is done
over the pixels of the computational domain and $N$ is the number of pixels.
In order to implement the Laplacian in Eq. (1), we follow the Yee grid finite
difference scheme, computing the derivative of variables by 2D convolutions
with a $5\times 5$ kernel [27]. Additionally, light scattering is by
definition an open boundary problem. In order to satisfy Sommerfeld radiation
condition and confine the problem in a finite space we use a stretched-
coordinate perfectly-matched layer (PML) [28] at the edges of the simulation
domain by introducing a complex coordinates transformation ($x\rightarrow
x+if(x)$) when calculating the derivatives inside the PML region. We use the
gradient of the so-computed physical loss function to update the weights of
the neural network, $w$ through Adam optimizer:

$w\rightarrow w-\gamma_{Ph}\frac{\partial{\mathcal{L}_{Ph}}}{\partial w}$ (2)

When we train MaxwellNet for a class of samples, it can accurately calculate
the field for unseen samples from the same class. However, the key point to
mention is that if we want to use MaxwellNet for a different set of RI
distributions, we can fix some of the weights, and adjust only a part of the
network for the new dataset, instead of re-starting the training from scratch.
This process, referred in the following as fine-tuning, is much faster than
the original training of MaxwellNet. We will elaborate and discuss this
interesting feature in section 3.

###  2.2 Optical diffraction tomography using MaxwellNet

Once MaxwellNet has been trained on a class of RI distributions, it can be
used to rapidly backpropagate reconstruction errors with an approach similar
to learning tomography [6]. Let us assume that we measure $L$ projections
$U^{m}_{i}$, with $i=0,...,L$, from an unknown RI distribution $\bar{n}(x,z)$
for different rotational angles. From these data, we can reconstruct a first
inaccurate candidate $n(x,z)$ through the Wolf’s transform using Rytov
approximation. By feeding MaxwellNet with $n(x,z)$ we predict the complex
scattered fields $U^{s}_{i}$ for the same $L$ angles. Consequently we can
construct a data-driven loss function $\mathcal{L}_{D}$ given by the
difference $\|U^{s}_{i}-U^{m}_{i}\|$ plus any additional regularizer, compute
its gradient through auto-differentiation, update $n(x,z)$ and iterate up to
convergence:

$\mathcal{L}_{D}=\sum_{i=1}^{L}\frac{1}{L}{\left\|{U^{s}_{i}(n)-U^{m}_{i}}\right\|}^{2}+\mathcal{R}\textrm{eg}\\{n,U^{s}_{i}(n)\\}$
(3)

$n\rightarrow n-\gamma_{D}\frac{\partial{\mathcal{L}_{D}}}{\partial n}$ (4)

Also in this case, we use Adam optimizer for updating RI values. The
regularizer in Eq. (3) consists of three parts, a total-variation (TV), a non-
negativity and a physics-informed terms,
$\mathcal{R}\textrm{eg}\\{n,U^{s}_{l}(n)\\}=\lambda_{TV}\mathcal{R_{TV}}(n)+\lambda_{NN}\mathcal{R_{NN}}(n)+\lambda_{Ph}\mathcal{L}_{Ph}(n,U^{s})$.
The TV regularizer helps smoothing the RI reconstruction and the non-
negativity regularizer adds the prior information that $n(x,z)$ should be
larger than the background refractive index:

$\mathcal{R_{TV}}(n)=\sum_{r}\sqrt{\left|\nabla_{x}n(r))\right|^{2}+\left|\nabla_{y}n(r))\right|^{2}+\left|\nabla_{z}n(r))\right|^{2}}$
(5a) $\mathcal{R_{NN}}(n)=\sum_{r}{\mathrm{min}(n(r)-n_{0},0)}^{2}$ (5b)

Importantly, we have to remark that MaxwellNet is trained for a specific
dataset and accurately predicts the scattered field for RI distributions that
are not too far from this set. To take into account this effect we add the
physics-informed loss to the regularizer. This further correction term helps
to find RI values in a way that MaxwellNet can predict the scattered field for
them correctly. In contrast to TV and non-negativity constraints that are used
due to the ill-posedness of the ODT problem, the physics-informed regularizer
is necessary in our methodology to ensure that the index distributions remain
within the domain in which MaxwellNet has been trained.

The key advantages of using MaxwellNet with respect to other forward models
are three folds: differently from BPM, it can accurately calculate field
scattering, considering reflection, multiple-scattering, or any other
electromagnetic effects [5, 6, 7, 8]; once trained, the field computations are
performed in milliseconds, much faster than Lipmann-Schwinger model; and
finally, the data-driven error in Eq. (3) can be easily backpropagated
differently from commercially available full-vectorial solvers. We discuss the
reconstruction results and compare them with other methods in subsection 3.2.

##  3 Results and Discussion

###  3.1 MaxwellNet results

In this subsection, we evaluate the performance of MaxwellNet for the
prediction of the scattered field from RI structures such as biological cells.
At first, we check the performance on a 2D sample assuming that the system is
invariant along the y axis. The number of pixels for our model are
$N_{x}=N_{z}=256$ for both the x and z directions, and their size is
$dx=100\mathrm{nm}$. We also use PML with the thickness of $1.6\mathrm{\mu m}$
at the edges of our computational domain. We create a dataset of digital cell
phantoms and divide it into the training and testing sets. MaxwellNet has
$\sim 5.9\mathrm{M}$ parameters to train and we use the Adam optimizer with a
learning rate of $1\times 10^{-4}$ and batch training. The details about the
dataset and training and validation losses are discussed in Appendix B. We
train and test MaxwellNet in Tensorflow 2.6.0 on a desktop computer (Intel
Core i7-9700K CPU, 3.6GHz, 64GB RAM, GPU GeForce RTX 2080Ti).

---
Figure 2: Results of MaxwellNet and its comparison with COMSOL. (a,b) Two test
cases from the digital phantom dataset and the prediction of the real and
imaginary of the envelope of the scattered fields using MaxwellNet, COMSOL and
their difference. (c) Scattered field predictions from the network trained in
(a,b) for the case of an experimentally measured RI of HCT-116 cancer cell and
comparison with COMSOL. The difference between the two is no longer
negligible. (d) Comparison between MaxwellNet and COMSOL after fine-tuning the
former for a set of HCT-116 cells. MaxwellNet predictions reproduces much more
accurately results after fine-tuning.

In Fig. 2(a) and Fig. 2(b), we choose two random examples of the digital
phantoms in the test set (which is not seen by the network during the
training). For each test case, in the second and third rows, we present the
prediction of the envelope of the scattered field by the network, and we
compare it with the result achieved by the finite element method (FEM) using
COMSOL Multiphysics 5.4. We can see a very small difference between the
results of MaxwellNet and COMSOL, which we attribute to discretization error.
There are different schemes of discretization in two methods that can cause
such differences. In order to quantitatively evaluate the performance of
MaxwellNet, we define the relative error of MaxwellNet with respect to COMSOL
as,

$error=\frac{\int{\left\|{{U}_{MaxwellNet}(r)-{U}_{COMSOL}(r)}\right\|^{2}dr}}{\int{\left\|{{U}_{COMSOL}(r)}\right\|^{2}dr}}$
(6)

where ${U}_{MaxwellNet}$ and ${U}_{COMSOL}$ are the total fields calculated
with MaxwellNet and COMSOL. The integration is done excluding the PML regions.
The calculated relative errors for the test case 1 and the test case 2 in Fig.
2 are $4.1\times 10^{-2}$ and $4.6\times 10^{-2}$, respectively.

It should be noted that once MaxwellNet is trained, the scattered field
calculation is much faster than numerical techniques such as FEM. We present a
time comparison in Table 1. For the test phantoms in Fig. 2, it took 17ms for
MaxwellNet in comparison with 13s for COMSOL meaning three orders of magnitude
acceleration.

---
Figure 3: Results of MaxwellNet3D and its comparison with COMSOL. The RI
distribution is shown in (a). The real part of the envelope of the scattered
field calculated by MaxwellNet3D is shown in (b), calculated by COMSOL in (c),
and their difference in (d). The imaginary part of the envelope of the
scattered field calculated by MaxwellNet3D, COMSOL, and their difference are
presented in (e-g), respectively.

Furthermore, performing a physics based instead of direct data-driven training
holds promises for exploiting the advantages of Transfer learning[29]. Maxwell
equations are general but having a neural network which predicts the scattered
field for any class of RI distribution in milliseconds with a negligible
physical loss is usually unfeasible. Most of the previous PINN studies for
solving partial differential equations are trained for one example, and they
will work for that specific example. In our case, U-Net architecture proved to
be expressive enough to predict the field for a class of samples. However, if
we use MaxwellNet for inference on a RI distribution completely uncorrelated
with the training set, the accuracy drops. In order to evaluate MaxwellNet
extrapolation capability, we considered the model trained on phantoms samples
in Fig. 2 and use it for inference on HCT-116 cancer cells. The comparison
between MaxwellNet and COMSOL is shown in Fig. 2(c). The input of the network
is a 2D slice of the experimentally-measured HCT-116 cell in the plane of best
focus. The discrepancy between MaxwellNet and COMSOL is due to the fact that
the former does not see examples of such RI distributions during the training.
As a result, if we require accurate results for a new set of samples with
different features, we have to re-train MaxwellNet for the new dataset, which
would take a long time as reported in Table 1. However, it turns out that
learning a physical law, as Maxwell equations, even though on a finite dataset
is better suited than data-driven training for transfer learning on new
batches. Indeed, we can use the pretrained MaxwellNet on digital phantoms and
fine-tune some parts of the network for HCT cells achieving good convergence
in a few epochs. In this example, we create a dataset of 136 RI distributions
of HCT-116 cancer cells and divide them into the training and validation sets.
Some examples of HCT-116 refractive index dataset are shown in Appendix B. A
wide range of cells with different shapes are included in the dataset. We have
single cancer cells, like Fig. 2(c), examples of cells in the mitosis process,
or examples with multiple cells. In this case, we freeze the weights of the
encoder part and fine-tune the decoder with the new dataset. We can see in
Fig. 2(d) that after this correction step, the calculated field is much more
accurate. As it can be seen in Table 1, the fine-tuning process is two orders
of magnitude faster than a complete training from scratch.

The 2D case is helpful for demonstrating the method and rapidly evaluate the
performances. Nevertheless, full 3D fields are required for many practical
applications. We can straightforwardly recast MaxwellNet in 3D using arrays of
size $N_{x}\times N_{y}\times N_{z}\times 1$ as inputs of the network and
requiring a $N_{x}\times N_{y}\times N_{z}\times 2$ output, with the two
channels corresponding to the real and imaginary of the envelope of the
scattered field. In this case, the network consists of Conv3D, AveragePool3D,
and Conv3DTranspose layers instead of 2D counterparts. As a benchmark test, we
created a dataset of 3D phantoms with 200 examples (180 for training and 20
for testing). The computational domain is defined with $N_{x}=N_{y}=N_{z}=64$,
$dx=100\mathrm{nm}$, and PML thickness of $0.8\mathrm{\mu m}$.

Table 1: Computation time comparison Dataset | 2D Phantoms | 2D HCT-116 | 3D Phantoms
---|---|---|---
MaxwellNet training/fine-tuning | 30.5h | 0.18h | 15.5h
MaxwellNet inference | 17.0ms | 17.0ms | 44.9ms
COMSOL | 13s | 13s | 2472s

The 3D version of MaxwellNet has $\sim 17.2\mathrm{M}$ parameters. We use Adam
optimizer (learning rate $=1\times 10^{-4}$), and a batch size of 10. The
results of the predicted field for an unseen example and its comparison with
COMSOL are shown in Fig. 3. We can see that MaxwellNet performs as good as
COMSOL in field calculation. The quantitative error described in Eq. 6 is
$3.4\times 10^{-3}$ for the 3D example of Fig. 3. There are some marginal
differences due to different discretization schemes. However, we can see in
Table 1 that MaxwellNet is about 50000 faster than COMSOL in predicting fields
(44.9 milliseconds versus 41.2 minutes). This result and the significant
efficiency in the computation time highlight MaxwellNet potential for the
calculation of the field in different applications. In the next subsection, we
demonstrate how this method can be applied for improving ODT reconstruction
fidelity.

---
Figure 4: Tomographic reconstruction of RI using MaxwellNet. (a) The RI
reconstruction was achieved by Rytov, MaxwellNet, and the ground-truth. (b) 1D
RI profile at $z=0$ (plane of best focus), for Rytov (green), MaxwellNet
(blue), and the ground-truth (orange).

###  3.2 Tomographic reconstruction results

To show the ability of MaxwellNet to be used for different imaging
applications, we implement an optimization task with MaxwellNet as the forward
model for ODT as explained in subsection 2.2. In this example, we consider one
of the digital phantoms in the test set of Fig. 2 and we use 2D MaxwellNet as
the forward model to compute the 1D scattered field along the transverse
direction $x$ for $N=81$ different rotation angles $\theta$. We restrict
ourselves to the range $\theta\in[-40^{\circ},40^{\circ}]$ as this is
consistent with the typical conditions in common tomographic setups. As is
shown in Fig. 4(a), the Rytov reconstruction obtained from these field
projections is elongated along z-axis and underestimated due to missing
frequencies. We then minimize the loss function (3) to improve the RI
reconstruction choosing as regularizer parameters $\lambda_{TV}=3.1\times
10^{-7}$, $\lambda_{NN}=1\times 10^{-1}$, $\lambda_{Ph}=5\times 10^{-2}$ and
Adam optimizer with initial learning rate of $3\times 10^{-4}$. We also
scheduled the learning rate, halving it every 1000 epochs to speed-up
convergence. The resulting RI distribution after 3000 epochs is shown in Fig.
4. It can be seen that the reconstructed RI is not anymore underestimated nor
elongated along the z-axis. This is a significant improvement in comparison
with Rytov prediction. The missing details in the reconstructed RI, which can
be better visible in the 1D cutline in Fig. 4(b), can be due to the missing
information in 1D fields that the optimization of RI could not retrieve this
information.

---
Figure 5: Tomographic RI reconstruction of 3D sample using MaxwellNet. The RI
reconstruction is achieved by Rytov, MaxwellNet, and the ground-truth in
different rows at YX, YZ, and XZ planes in the center of the sample. A z-stack
demonstration of the reconstruction is shown in Video 1 (Video 1, MP4, 1.4
MB).

Next, we try a 3D digital phantom from the test set and we use 3D MaxwellNet
as the forward model in our tomographic reconstruction method. Since
generating synthetic data with COMSOL is time-consuming for multiple angles,
we create synthetic scattered fields from the phantom with the
Lippmann–Schwinger equation [9]. We will show later an experimental example,
where we illuminate the sample with a circular illumination pattern with an
angle $\approx 10^{\circ}$. As a result, in this numerical example, we rotate
the sample for 181 angles (including 1 normal incidence), equivalently to an
illumination rotation with a fixed illumination angle of $10^{\circ}$. We keep
the experimental conditions, $\lambda=1.030\mathrm{\mu m}$, and $n_{0}=1.33$.
Then, we use these synthetic measurements for our optimization task along with
TV, non-negativity, and physics-informed regularization. The reconstruction is
achieved after 6000 epochs with $\lambda_{TV}=1.2\times 10^{-8}$,
$\lambda_{NN}=2\times 10^{1}$, and $\lambda_{Ph}$ started with $5\times
10^{-1}$ and divided by two every 500 epochs. The reconstructions are shown in
Fig. 5 in YX, YZ, and XZ planes. The first row shows the Rytov reconstruction
where we can see a significant underestimation and elongation along z-axis
which is due to the small illumination angle ($10^{\circ}$). The details in
the reconstruction achieved using MaxwellNet are slightly blurred in
comparison with the ground-truth as a result of low resolution with
$\lambda=1.030\mathrm{\mu m}$.

---
Figure 6: Tomographic RI reconstruction of a polystyrene micro-sphere immersed
in water. The projections are measured with off-axis holography for different
angles. The RI reconstruction achieved by Rytov, and MaxwellNet are presented
at YX, YZ, and XZ planes in the center of the sample.

We also evaluated our methodology experimentally. We mentioned earlier that
MaxwellNet takes care of reflection as a forward model, and therefore, our
reconstruction technique can be used for samples with high contrast. In our
experimental analysis, we try a polystyrene micro-sphere immersed in water,
where we expect to have a $\sim 0.25$ refractive index contrast. Polystyrene
micro-spheres (Polybead® Polystyrene 2.0 Micron) are immersed in water and
placed between two #1 glass coverslips. We have an off-axis holographic setup
where we use a yttrium-doped fiber laser (Amplitude Laser Satsuma) with
$\lambda=1.030\mathrm{\mu m}$ and we change the illumination angle with two
Galvo mirrors. Using a delay path, the optical length of the reference and
signal arms are matched. We measure holograms for 181 illumination angles and
extract the phase and amplitude of the complex scattered fields using Fourier
holography. More details about the experimental setup are discussed in
Appendix C. Then, we use the extracted scattered fields for different
projections for our optimization task to reconstruct the 3D RI distribution of
the sample. The experimental projections are 2D complex fields that are imaged
in the center of the sample using a microscope objective lens and we can
propagate them in the background medium to calculate the scattered field in
any other plane, perpendicular to $z$-axis, after the sample. This 2D field
can be compared with the output of MaxwellNet in that plane, as described in
Eq. 3. Additionally, the experimental projections are based on illumination
rotation and we interpolate them to achieve the equivalent sample rotation
projections. We iteratively optimize the loss function in Eq. 3 for 2000
epochs where we use the regularization parameters of $\lambda_{TV}=3.8\times
10^{-9}$, $\lambda_{NN}=5\times 10^{1}$, and $\lambda_{Ph}$ started with
$1.5\times 10^{-1}$ and divided by two in every 500 epochs. The reconstruction
is shown in Fig. 6. It can be seen that the underestimation and z-axis
elongation in the Rytov reconstruction are remarkably improved.

##  4 Conclusion

In summary, we proposed a PINN which rapidly calculates the scattered field
from inhomogeneous RI distributions such as biological cells. Our network is
trained by minimizing a loss function based on Maxwell equations. We showed
that the network can be trained for a set of samples and could predict the
scattered field for unseen examples which are in the same class. As our PINN
is not a data-driven neural network, it can be trained for different examples
in different conditions. Even though the network is not efficiently
extrapolating to classes which are statistically very different from the
training data-set, we showed that by freezing the encoder weights and fine-
tuning the decoder branch, one can get a new predictive model in a few
minutes. We believe that this can be further used for changing wavelength,
boundary condition, or other physical parameters.

We used our PINN as a forward model in an optimization loop to retrieve the RI
distribution from the scattered fields achieved by illuminating the sample
from different directions, known as optical diffraction tomography. This
example shows the ability of MaxwellNet to be used as an accurate forward
model in optimization loops for inverse design or inverse scattering problems.

##  Appendix A Calculation of Physics-informed Loss

During the training of MaxwellNet, we calculate at each epoch the loss
function in Eq. 1 for the network output. In order to evaluate the Helmholtz
equation residual, we should numerically compute the term
$\frac{\partial^{2}U^{s}}{\partial x^{2}}+\frac{\partial^{2}U^{s}}{\partial
y^{2}}+\frac{\partial^{2}U^{s}}{\partial z^{2}}$. In the previous PINN papers
for solving PDEs [17, 18, 23, 24, 19, 21, 22], the inputs of the network are
the spatial coordinates $x$, $y$, $z$, and the derivatives with respect to
these variables can be calculated using the chain rule. In this
implementation, the weights of the network can be trained to minimize the loss
function for a single refractive index distribution, $n(r)$ in Eq. 1. In our
approach, the PINN gets the refractive index, $n(r)$, on a uniform grid as the
input and finds the field on the same grid which minimizes the loss function
for that refractive index. The output of the network is the 3D array of the
scattered field envelope, and we use a finite difference scheme to calculate
the derivative of the field with respect to the coordinates:

$\frac{\partial U^{s}}{\partial
x}=\frac{U^{s}((i+1){\Delta}x,j{\Delta}y,k{\Delta}z)-U^{s}((i-1){\Delta}x,j{\Delta}y,k{\Delta}z)}{2{\Delta}x}$
(7)

in which $(i,j,k)$ are the pixel indices and ${\Delta}x$, ${\Delta}y$,
${\Delta}z$, are the pixel sizes along the $x$, $y$, and $z$ axes. This way,
we can calculate $\frac{\partial U^{s}}{\partial x}$ by convolving $U^{s}$
with a kernel of $\left[-1/2,0,1/2\right]$ along the $x$ axis. When computing
electromagnetic fields, since the curl of the electric field gives the
magnetic field and vice versa, a smart technique to improve accuracy is to use
two staggered grids for discretizing fields, commonly referred to as Yee
scheme [27]. In practice, this can be easily implemented through two shifted
convolutional kernels for the two grids, $\left[-1/2,1/2,0\right]$ and
$\left[0,-1/2,1/2\right]$.

In order to minimize the discretization error, one can use a smaller pixel
size, ${\Delta}x$ or higher order approximations. Here, we use the fourth-
order finite difference scheme [30] in which convolutional kernels of
$\left[0,+1/24,-9/8,+9/8,-1/24\right]$ and
$\left[+1/24,-9/8,+9/8,-1/24,0\right]$ are used for the calculation of the
derivatives in Eq. 1.

##  Appendix B Training and Fine-tuning of MaxwellNet

As mentioned in Section 3, we create a dataset of digital cell phantoms to
train and validate MaxwellNet. The dataset for 2D MaxwellNet includes 3000
phantoms with elliptical shapes oriented in different directions. The size of
these phantoms is in the range of $5-10\mathrm{\mu m}$, their refractive index
varies in the range of $(1.38,1.45)$, and the background refractive index is
$n_{0}=1.33$. Two examples of these phantoms are shown in Fig. 2. We divide
this dataset into 2700 phantoms for training and 300 phantoms for testing. We
use batch training with a batch size of 10 for 5000 epochs. This training took
30.5 hours and after 5000 epochs, no significant decrease in the validation
loss could be observed. The training and validation curves of the physical
loss are presented in Fig. 7(a). This figure shows that MaxwellNet performs
very well for out-of-sample cases.

We discussed in Section 3 using MaxwellNet which was trained for cell phantoms
to predict the scattered field for real cells. A dataset of HCT-116 cancer
cells is used for this purpose. The 3D refractive index of these cells is
reconstructed using Rytov approximation with projections achieved with an
experimental setup utilizing spatial light modulator as described in [8].
Then, a 2D slice of the refractive index is chosen in the plane of best focus.
A total number of 8 cells are used and we rotated and shifted these cells to
create a dataset of 136 inhomogeneous cells whose refractive index range is
$(1.33,1.41)$. We use 122 of these images for training and 14 for validation.
Some examples of HCT-116 refractive index dataset are shown in Fig. 7(c). We
freeze the encoder of MaxwellNet and fine-tune its decoder for this new
dataset. The training and validation losses are presented in Fig. 7(b).

For 3D MaxwellNet, a dataset of 200 phantoms is created. These 3D phantoms
have a spherical shape with some details inside them and the range of their
diameter is $1.8-2.4\mathrm{\mu m}$. We randomly choose 180 phantoms for
training and 20 phantoms for testing. We train 3D MaxwellNet with the training
dataset with batch size of 10. The example of Fig. 3 and Fig. 5 is one of the
phantoms in the testing dataset.

---
Figure 7: Training and fine-tuning of MaxwellNet. (a) Training (blue) and
validation (orange) loss of MaxwellNet for Digital cell phantoms dataset. (b)
Fine-tuning the pretrained MaxwellNet for a dataset of HCT-116 cells for 1000
epochs. (c) Examples of the HCT-116 dataset.
---
Figure 8: Experimental setup for multiple illumination angle off-axis
holography. HW: Half-wave plate, P: Polarizer, BS: Beam splitter, L:Lens, Obj:
microscope objective, M: Mirror.

##  Appendix C Experimental Setup for ODT

For ODT, we require complex scattered fields from multiple illumination
angles. The off-axis holographic setup to accomplish that is shown in Fig. 8.
It relies on a ytterbium-doped fiber laser at $\lambda=1.030\mathrm{\mu m}$
whose power is controlled with a half-wave plate and a polarizing beam
splitter. The optical beam is divided into the signal and reference arms using
a beam splitter (BS1). In the signal arm, we use two galvo mirrors, GM-V and
GM-H to control the illumination angle in the vertical and horizontal
directions. Using two 4F systems (L1-L4), we image these galvo mirrors on the
sample plane, so the position of the beam remains fixed while changing the
illumination angle. This way, we can illuminate the sample with a condensed
plane wave. The sample is then imaged on the camera (Andor sCMOS Neo 5.5)
using another 4F system consisting of a 60X water dipping objective (Obj1) and
a tube lens L5. The signal and reference arms are then combined with another
beam splitter, BS2 to create the off-axis hologram on the camera. A motorized
delay line controls the optical path of the reference arm to match the optical
path of the signal arm.

###  C.1 Disclosures

The authors have no competing interests to declare.

###  C.2 Acknowledgments

This project was funded by the Swiss National Science Foundation (SNSF)
funding number 514481. Additionally, A. Saba would like to thank Dr. Babak
Rahmani and Dr. Joowon Lim for their useful comments about the implementation
of MaxwellNet and tomographic optimization in Tensorflow.

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