Acta Polytechnica


https://doi.org/10.14311/AP.2022.62.0023
Acta Polytechnica 62(1):23–29, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence

Published by the Czech Technical University in Prague

PHOTONIC GRAPHENE UNDER STRAIN WITH
POSITION-DEPENDENT GAIN AND LOSS

Miguel Castillo-Celeitaa, ∗, Alonso Contreras-Astorgab,
David J. Fernández C.a

a Cinvestav, Physics Department, P.O. Box. 14-740, 07000 Mexico City, Mexico
b Cinvestav, CONACyT – Physics Department, P.O. Box. 14-740, 07000 Mexico City, Mexico
∗ corresponding author: mfcastillo@fis.cinvestav.mx

Abstract. We work with photonic graphene lattices under strain with gain and loss, modeled by the
Dirac equation with an imaginary mass term. To construct such Hamiltonians and their solutions, we
use the free-particle Dirac equation and then a matrix approach of supersymmetric quantum mechanics
to generate a new Hamiltonian with a magnetic vector potential and an imaginary position-dependent
mass term. Then, we use a gauge transformation that maps our solutions to the final system, photonic
graphene under strain with a position-dependent gain/loss term. We give explicit expressions for the
guided modes.

Keywords: Graphene, Dirac materials, photonic graphene, matrix supersymmetric, quantum mechan-
ics.

1. Introduction
Graphene is the last known carbon allotrope, it was
isolated for the first time by Novoselov, Geim, et al. in
2004 [1]. This material consists of a two-dimensional
hexagonal arrangement of carbon atoms. Graphene
excels for its interesting properties, such as mechanical
resistance, electrical conductivity, and optical opac-
ity [2, 3]. The study of graphene has contributed to the
development of different areas in physics, for example,
in solid-state, graphene has prompted the discovery
of other materials with similar characteristics, such
as borophene and phosphorene. At low energy, the
charge carriers in graphene behave like Dirac massless
particles, and from this approach, graphene has al-
lowed the verification of the Klein tunneling paradox
as well as the quantum Hall effect. These phenomena
have gained a special interest in particle physics and
quantum mechanics [4].

Exploring graphene in an external constant mag-
netic field has allowed identifying the discrete bound
states in the material, the so-called Landau levels.
Moreover, theoretical physicist have analyzed the be-
havior of Dirac electrons in graphene under different
magnetic field profiles as well. Supersymmetric quan-
tum mechanics is a useful tool to find solutions of the
Dirac equation under external magnetic fields [5–9].
Following this approach, a mechanical deformation
in a graphene lattice is equivalent to introducing an
external magnetic field [10, 11].

Graphene has its analog in photonics, called pho-
tonic graphene. It is constructed through a two-
dimensional photonic crystal with weakly coupled
optical fibers in a three-dimensional setting [12–17].
Photonic graphene under strain is modeled through

a deformation in the coupled optical fiber lattice [18–
21].

Compared with the conventional graphene Hamilto-
nian, the photonic graphene Hamiltonian has an extra
term that represents the gain/loss in the fibers. The
literature on this topic always considers a constant
gain/loss in space. With the previous motivation,
we will apply supersymmetric quantum mechanics
in a matrix approach (matrix SUSY-QM) to obtain
solutions of the Dirac equation for strain photonic
graphene with a position-dependent gain/loss.

2. Strain in photonic graphene
The graphene structure consists of carbon atoms in
a hexagonal arrangement similar to a honeycomb lat-
tice. This structure can be described by two triangular
sublattices of atoms, which are denoted as type A and
type B. The base vectors to the unitary cell are given
by

a1 =
a

2
(
√

3, 3), a2 =
a

2
(−

√
3, 3), (1)

where a is the interatomic distance, for graphene a =
1.42 Å (see Figure 1a). The position of the atoms in
the whole lattice can be defined by the set of vectors
Rl = l1a1 + l2a2, with l1, l2 ∈ Z. An alternative
description of graphene is through the first neighbors,
which are connected by the vectors δn

δ1 =
a

2
(
√

3, 1), δ2 =
a

2
(−

√
3, 1), δ3 = a(0, −1). (2)

A reciprocal lattice can be defined in the momentum
space, which is also hexagonal, as shown in Figure 1b.
It is rotated 90◦ with respect to the original carbon
network. A hexagon in the reciprocal lattice is recog-
nized as the first Brillouin zone. In this zone, there
are only two inequivalent points, K± = (± 4π3√3a, 0).

23

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M. Castillo-Celeita, A. Contreras-Astorga, D. J. Fernández C. Acta Polytechnica

A

B
a1a2
δ1δ2

δ3

(a).

b1
b2

K+K-

(b).

Figure 1. (A) Hexagonal graphene lattice. The lat-
tice is constructed by type A and type B atoms, in
this case, a1 and a2 correspond to the lattice unitary
vectors, and δn are the vectors that connect the atoms
A(B) with the nearest neighbors. (B) Reciprocal lat-
tice, which is characterized by the b1,2 vectors and K±
correspond to the two possible inequivalent points in
the lattice.

All subsequent corners are determined from either K+
or K− plus integer multiples of the vectors

b1 =
2π
3a

(
√

3, 1), b2 =
2π
3a

(−
√

3, 1). (3)

Vectors ai and bj fulfill the condition ai · bj = 2πδij .

2.1. Tight-binding model
The tight-binding Hamiltonian describes the hopping
of an electron from an atom A (B) to an atom B (A)

H = −t
∑
Ri

3∑
n=1

(|ARi ⟩ ⟨BRi +δn | + |BRi +δn ⟩ ⟨ARi |), (4)

where t ≈ 3 eV is called the hopping integral, Ri
runs over all sites in the sublattice A, thus |ARi ⟩ is
a state vector in these sites, the same applies to B
and |BRi+δn ⟩, recall that δn connects the atoms of
the sublattice A(B) with its nearest neighbors in the
sublattice B(A). The translational symmetry suggests
the use of Bloch states

|ΨBloch⟩ =
1

√
Nc

∑
Rj

(eik·Rj ψA(k) |ARj ⟩

+ eik·(Rj +δ3)ψB (k) |BRj +δ3 ⟩), (5)

where Nc is the number of the unitary cell [22]. Then
H |Ψ⟩ = E |Ψ⟩ becomes a matrix problem
 0 −t

3∑
n=1

e−ik·δn

−t
3∑

n=1
eik·δn 0



(
ψA
ψB

)
= E

(
ψA
ψB

)
,

(6)
with ψA ≡ ψA(k) and ψB ≡ ψB (k) and the energy
term given by

E± = ±

∣∣∣∣∣t
3∑

n=1
e−ik·δn

∣∣∣∣∣
= ±t

√
3 + 2 cos(

√
3kxa) + 4 cos(

√
3kx

a

2
) cos(3ky

a

2
).

To obtain an effective Hamiltonian at low energy, we
can consider the Taylor series around the Dirac points
H(k = K± +q) ≈ q · ∇kH|K± . Note that E(K±) = 0,
as a consequence, at these points, the valence and
conduction bands are connected. The above calculus
leads to the analog of the Dirac-Weyl equation

HϱΨ = ℏv0(ϱσ1qx + σ2qy )Ψ = EΨ, (7)

where ϱ = ±1 correspond to the K± valleys, v0 is
called the Fermi velocity, in graphene, v0 = 3ta/2ℏ ≈
c/300, with c being the velocity of light, σi are the
Pauli matrices

σ1 =
(

0 1
1 0

)
, σ2 =

(
0 −i
i 0

)
, σ3 =

(
1 0
0 −1

)
, (8)

and Ψ is a bi-spinor. The matrix nature of this equa-
tion is related to the sublattices A and B, this degree
of freedom is called pseudo-spin. Notice that at low
energies, the dispersion relation is linear, given by
E±(q) = ±ℏv0|q|, then, the Dirac cones are connect-
ing at E± = 0, as expected for particles without
mass [23].

2.2. Uniform strain
The photonic analog of a graphene lattice is built with
weakly coupled optical fibers. This kind of photonic
system is described by the same tight-binding Hamil-
tonian in graphene with an additional term γA/B ; that
represents the gain and loss in the optical fibers in the
position A/B, this new term produces an attenuation
or amplification in the optical modes.

If we consider uniform strain in the lattices, which
is represented by a strain tensor

u =
(

u11 0
0 u22

)
, (9)

the Fermi velocity is modified in the following form

vij = v0(1 + (1 − β)uij ). (10)

The hopping integrals are modified with a little per-
turbation t → tn, that, considering the changes in the
orbitals by the modification of the carbon distances

tn ≈ t
(

1 −
β

a2
δn · u · δn

)
, (11)

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vol. 62 no. 1/2022 Photonic graphene under strain with position-dependent . . .

where
β = −

∂ ln t
∂ ln a

(12)

is the Grüneisen parameter that depends on the model;
for graphene, β is between 2 and 3 [10] (see also [24,
25]). In photonic graphene, a is the distance between
adjacent waveguides. The Hamiltonian of a photonic
graphene with a uniform strain reads as

H = γA
∑
Ri

|ARi ⟩ ⟨ARi | + γB
∑
Ri

|BRi +δn ⟩ ⟨BRi +δn |

−
∑
Ri

3∑
n=1

tn(|ARi ⟩ ⟨BRi +δn | + |BRi +δn ⟩ ⟨ARi |). (13)

The deformation of the lattice produces a shift of
the Dirac points KD± ≈ (1 − u) · K± ± A, where
A = (Ax, Ay )

Ax =
β

2a
(u11 − u22), Ay = −

β

2a
(2u12). (14)

Using the Bloch solution, the Hamiltonian under
strain takes the form:

H =


 γA −

3∑
n=1

tne
−ik·(1−u)·δn

−
3∑

n=1
tne

ik·(1−u)·δn γB


 ,
(15)

under the assumption |u·δn| ≪ a. In this work, we will
assume that γA = iγ and γB = γ∗A, then, for positive
γ, the waveguides in the sublattice A (B) present the
energy gain (loss), as in the arrangements proposed
in [14]. Expanding this Hamiltonian around the Dirac
points, through the substitution k = KD± + q, one
arrives at a Dirac Hamiltonian analog with minimal
coupling

H = v0σ · (1 + u − βu)q + iγσ3. (16)

Comparing with (7), the effect of strain is equiva-
lent, to consider magnetic-like field modeled through
a pseudo-magnetic vector potential A. The last term
represents a gain/loss balance in sublattices A/B. In
photonic graphene, strain could be generated by de-
formations in the geometry of the optical-fiber lattice.

2.3. Non-uniform strain
For non-uniform strain, the deformation matrix de-
pends of the position, u → u(r). Thus, the expression
for the Hamiltonian becomes

H = −iσi
√

vij∂j
√

vij + v0σiAi + iγσ3, (17)

considering a strain tensor of the form

u =
(

u11(x) 0
0 u22(y)

)
, (18)

and equations (10) and (14), still apply. We can also
write the strain Hamiltonian as

H(x,y) = − iσ1
√

v11∂x
√

v11 − iσ2
√

v22∂y
√

v22

+ σ1
v0β

2a
(u11(x) − u22(y)) + iγσ3, (19)

where v11 = v11(x), v22 = v22(y).
We can relate the eigenvalue equation of this Hamil-

tonian, HΨ = EΨ, with a strain-free one using the
following transformation. First, we define the coordi-
nates

r =
∫

v0
v11(x)

dx, s =
∫

v0
v22(y)

dy, (20)

and the operator

G(x,y) =
√

v11v22
v0

exp
(
iv0β

2a

∫ x
0

u11(q)
v11(q)

dq

)
, (21)

then, H will be related with a flat Fermi velocity
Hamiltonian H0 as

H(x,y) = G−1(x,y)H0(r(x),s(y))G(x,y), (22)

where

H0Φ =
(

−iv0σ1∂r − iv0σ2∂s −
v0β

2a
u22 + iγσ3

)
Φ, (23)

and u22 = u22(y(r,s)). The solutions are mapped as

Ψ(x,y) = G−1(x,y)Φ(r(x),s(y)). (24)

The energy spectrum is the same for both Hamiltoni-
ans [18, 19, 26].

3. Supersymmetric quantum
mechanics: matrix approach

Supersymmetric quantum mechanics (SUSY-QM) is
a method that relates two Schrödinger Hamiltonians
through an intertwining operator [27, 28]. Another
approach is the matrix SUSY-QM, which intertwines
two Dirac Hamiltonians H0, H1 by a matrix operator
L. In this work, we use the latter to construct an
appropriate Hamiltonian H1 that will be linked via the
operator G introduced in (21) to a photonic graphene
system under strain. For the sake of completeness we
will give a brief review of matrix SUSY-QM (more
details can be found in [29]).

We start by proposing the following intertwining
relation:

L1H0 = H1L1, (25)

where the Dirac Hamiltonians are given by

H0 = −iσ2∂s + V0(s), H1 = −iσ2∂s + V1(s), (26)

and the intertwining operator is

L1 = ∂s − UsU−1, (27)

with U being a matrix function called seed or trans-
formation matrix, the subindex in Us represent the
derivative respect to s, and U must satisfy H0U = UΛ.
Let us write U in a general form and Λ as a diagonal
matrix

U =
(
u11 u12
u21 u22

)
, Λ =

(
λ1 0
0 λ2

)
. (28)

25



M. Castillo-Celeita, A. Contreras-Astorga, D. J. Fernández C. Acta Polytechnica

From the intertwining relation and the given defini-
tions, the potential V1 can be written in terms of the
potential V0 and the transformation matrix as

V1 = V0 + i[UsU−1,σ2]. (29)

Solutions of the Dirac equation H0ξ = Eξ can be
mapped onto solutions of H1Φ = EΦ using the inter-
twining operator as Φ ∝ L1ξ. There are some extra
solutions, usually referred to missing states. They
can be obtained from each column of (UT )−1, named
Φλj , j = 1, 2, which satisfy H1Φλj = λj Φλj . If the
vectors Φλj fulfill the boundary conditions of the prob-
lem, λj must be included in the spectrum of H1. As
a summary, with this technique, we start from H0,
its eigenspinors and spectrum, then we construct H1,
obtain the solutions of the corresponding Dirac equa-
tion and the spectrum. Now, let us mention that it is
possible to iterate this technique. The main advantage
comes from the modification of the spectrum, since
with each iteration, we can add more energy levels.
The second-order matrix SUSY-QM can be reached
through a second intertwining relation

L2H1 = H2L2, (30)

which is similar to (25). The intertwining operator
now takes the form

L2 = ∂s − (U2)sU −12 . (31)
The operator L1 is used to determine the transforma-
tion matrix of the second iteration, U2 = L1U2, where
U2 fulfills the relation H0U2 = U2Λ2. In this case, Λ2
is an Hermitian matrix that we choose diagonal once
again,

Λ2 =
(
λ̃1 0
0 λ̃2

)
, U2 =

(
w11 w12
w21 w22

)
. (32)

The elements of Λ2 are such that (λ̃1, λ̃2) ̸= (λ1,λ2).
Therefore, the second order potential is given by

V2 = V1 + i
[
(U2)sU −12 ,σ2

]
. (33)

The solutions of H2χ = Eχ are obtained from the
eigenspinors of H1 as χ ∝ L2Φ. The second-order
matrix SUSY-QM generates, in principle, two sets of
eigenspinors that correspond to the columns of the
matrices (U T2 )−1 and L2(UT )−1.

4. Photonic graphene under strain
and position-dependent gain and
loss

In this section, we start from the auxiliary Dirac
equation of a free particle with imaginary mass, and
using a matrix SUSY-QM and a gauge transformation
G, we obtain a photonic graphene model with strain
and position dependent gain/loss. We show that we
can iterate the technique to add more propagations
modes.

Figure 2. Graph of the functions v0kr + K(s) (line
blue) and the gain/loss term γ − Γ(s) (dashed red line)
for ϵ = 1.5, kr = π, γ = 1, v0 = 1.0. Notice that
γ − Γ(s) coincides asymptotically with γ.

4.1. Photonic Graphene with a single
mode

Let us start from the free particle Dirac equation
where we included a purely imaginary mass term:

H0Φ =(−iv0σ1∂r − iv0σ2∂s + iγσ3)Φ. (34)

Considering Φ(r,s) = exp(ikrr)(ϕA(s),ϕB (s))T , the
Hamiltonian can be written as

H0(r,s) = −iv0σ2∂s + V0, (35)

where V0 = v0krσ1 + iγσ3. Now, we use the matrix
SUSY-QM to construct a new system. A convenient
selection of the Λ elements is λ1 = ϵ = −λ2. We
build the transformation matrix U with the entries
u21 = u∗22 = cosh(κs) + i sinh(κs), the corresponding
momentum in s is given by κ =

√
k2r − (γ2 + ϵ2)/v20 .

The other two components are found through the
equation:

u1j =
v0

(λj − iγ)
(−u′2j + kru2j ), j = 1, 2. (36)

From (29) we obtain V1 as

V1 = V0 + σ1K(s) − iσ3Γ(s), (37)

where Γ(s), K(s) are given by

Γ = 2γ +
2ϵ (κ(γ sinh(2κs) + ϵ) − γkr cosh(2κs))
κ(γ − ϵ sinh(2κs)) + krϵ cosh(2κs)

,

K =
2v0krϵ (kr cosh(2κs) − κ sinh(2κs))
κ(γ − ϵ sinh(2κs)) + krϵ cosh(2κs)

− 2v0kr.

Figure 2 shows a plot of the functions v0kr + K(s)
and γ − Γ(s). The new Hamiltonian takes the form

H1(r, s) = −iσ2v0∂s + σ1(−iv0∂r + K) + iσ3(γ − Γ). (38)

This system supports two single bound states. They
are the columns of the matrix (UT )−1 = (Φϵ, Φ−ϵ).
The eigenvector associated with ϵ is given by

Φϵ(r,s) =

eikr r

2


 − (γ2+ϵ2)(cosh(κs)+i sinh(κs))v0κ(γ−ϵ sinh(2κs))+v0kr ϵ cosh(2κs)

(γ−iϵ)((κ+ikr ) cosh(κs)−(kr +iκ) sinh(κs))
κ(γ−ϵ sinh(2κs))+kr ϵ cosh(2κs)


 . (39)

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vol. 62 no. 1/2022 Photonic graphene under strain with position-dependent . . .

Our next step is to apply the gauge transformation
defined in (20)-(22). The strain and Fermi velocity
tensors that we consider are

u =
(

0 0
0 − 2aK(y))

β

)
,

v = v0

(
1 1
1 1 − (1 − β) 2aK(y)

β

)
, (40)

see (18). The change of variables in (20) becomes

r = x, s =
∫

1
1 − (1 − β) 2aK(y)

β

dy, (41)

and the operator G(x,y) =
√

1 − (1 − β) 2aK(y)
β

. This
choice leads to the following Hamiltonian

H1(x,y) = − iv0σ1∂x − iσ2
√

v22(y)∂y
√

v22(y)

− σ1
v0β

2a
u22(y) + i(γ − Γ(y))σ3. (42)

Bounded eigenstates of H1 can be found as Ψϵ(x,y) =
G−1(x,y)Φϵ(r(x),s(y)). In this system, there is a sin-
gle mode in the upper Dirac cone and another in the
bottom cone.

The strain generates an analog of a magnetic
field perpendicular to the graphene layer B⃗(y) =
(β/2a)∂y u22ẑ. Since we are working with a photonic
graphene, such a pseudo-magnetic field affects light.
Moreover, the term iΓ(y)σ3 indicates a position de-
pendent gain/loss in the optical fibers of the sublattice
A/B. Figure 3a shows the square modulus of each com-
ponent of Φϵ = (ϕϵA,ϕϵB )T (shadowed curves) and
the intensity |ϕϵA|2 + |ϕϵB |2 (red curve). Figure 3b
shows the same for the mode Ψϵ.

4.2. Photonic Graphene with two modes
In this subsection, we use two iterations of the ma-
trix SUSY-QM, starting again from the free-particle
Hamiltonian. Let us choose an initial system with
zero gain/loss (γ = 0), which is a massless fermion
in graphene. In the first matrix SUSY-QM step we
use the same transformation matrix U as in the ex-
ample above. The first matrix SUSY-QM partner
Hamiltonian has the form

H1 = −iσ2v0∂s + σ1K(s) + iσ3Γ(s), (43)

where K(s) = krv0,

Γ(s) =
2ϵv0κ

κv0 sinh(2κs) − krv0 cosh(2κs)
, (44)

with κ =
√

(krv0)2 − ϵ2/v0. As a result of the first
matrix SUSY-QM step, it is generated a position de-
pendent gain/loss term Γ(s). The iteration of the
method requires to define the second diagonal ma-
trix Λ2, with λ̃1 = −λ̃2 = ϵ1 ≠ ϵ, and the sec-
ond transformation matrix U2. For this example, we

(a).

(b).

Figure 3. (A) Plot of the individual intensities |ϕϵA|2
(gray curve) and |ϕϵB |2 (blue curve) and the total
intensity |ϕϵA|2 + |ϕϵB |2 (red line). (B) Analog of
the (A) plot for the solution Ψϵ of the Hamiltonian
under strain. The parameters in this case are: kr = π,
ϵ = 1.5, a = 1.0, β = 0.8, γ = 1, v0 = 1.0.

Figure 4. Graph of the function v0kr + K2(s) (black
line), and the gain/loss function Γ(s) + Γ2(s) (red
dashed line), for ϵ = 1.5, ϵ1 = 2.0, kr = π, γ = 0,
v0 = 1.0.

choose w21 = cosh(κ2s) and w22 = cosh(κ2s), where
κ2 =

√
(krv0)2 − ϵ21/v0. The other two components

can be found through the equation

w1j =
v0

λ̃j
(−w′2j + krw2j ), j = 1, 2. (45)

The potential V2 can be calculated from (33), V2 =
V1+σ1K2(s)+iσ3Γ2(s) = V0+σ1K2+iσ3(Γ+Γ2). The
functions v0kr + K2(s) and Γ(s) + Γ2(s) are shown
in Figure 4. It is important to highlight that the
gain/loss term remains a pure imaginary quantity.

27



M. Castillo-Celeita, A. Contreras-Astorga, D. J. Fernández C. Acta Polytechnica

(a).

(b).

Figure 5. (A) Intensity of the superposition |Φ̄(s, z)|2
propagating in the z-axis. (B) Intensity of the super-
position |Ψ̄(y, z)|2. The values of the parameters taken
are ϵ = 1.5, ϵ1 = 2.0, kr = π, γ = 0, β = 0.8, a = 1.0,
v0 = 1.0.

The second matrix SUSY-QM step introduces two
new sets of eigenmodes. They can be extracted from
the columns of the matrix (U T2 )−1 = (χϵ1,χ−ϵ1 ). The
eigenmodes added in the first step are mapped as
χ±ϵ = L2Φ±ϵ. Similar to the previous example, it
is possible to perform the gauge transformation (20)-
(22). Then, in the system under strain, the modes
become

Ψ±ϵ1 (x,y) = G
−1(x,y)χ±ϵ1 (r(x),s(y)),

Ψ±ϵ(x,y) = G−1(x,y)χ±ϵ(r(x),s(y)).

Therefore, in this new optical system, two guided
modes are created in the upper Dirac cone and two
more in the bottom Dirac cone. Finally, let us mention
that we can have superpositions of the introduced
modes and let them propagate along the z-axis inside
the photonic graphene. For example, in the flat Fermi
velocity system (before the gauge transformation),

Φ̄(s,z) = A1e−iϵz Φϵ(s) + A2e−iϵ1z Φϵ1 (s), (46)

becomes

Ψ̄(y,z) = A1e−iϵz Ψϵ(y) + A2e−iϵ1z Ψϵ1 (y), (47)

in the photonic graphene system under strain with
the position dependent gain/loss balance. Figure 5a

shows the propagation along z-axis of the intensity
|Φ̄(s,z)|2, while Figure 5b shows |Ψ̄(s,z)|2.

5. Summary
This article shows a natural way to construct Hamil-
tonians associated with a photonic graphene under
strain with a position-dependent gain/loss balance.
The main tools that we use are a matrix approach
to supersymmetric quantum mechanics and a gauge
transformation. With a correct choice of a transforma-
tion matrix U, it is possible to add a bound state to the
free-particle Hamiltonian using the matrix SUSY-QM,
but the Dirac equation will have two new terms in the
potential, V1 = V0 + σ1K(s) − iσ3Γ(s). The function
K could be associated with a magnetic vector poten-
tial, but the function iΓ is related to an imaginary
mass term, which is difficult to interpret or realize in
a solid-state graphene. The gauge transformation G
maps solutions from the flat Fermi velocity system of
the previous step to a graphene system under strain.
At this point, it becomes relevant to work with the
photonic graphene. The magnetic vector potential
translates into deformations of the lattice of optical
fibers, while the iΓ function indicates the gain/loss
of the fibers in the sublattice A/B. We end with the
Hamiltonian of photonic graphene with a single mode.
This mode is confined by the strain and the position-
dependent gain/loss balance. Finally, we show that
the technique can be iterated, to have two or more
modes in the photonic graphene.

Acknowledgements
The authors acknowledge the support of Conacyt, grant
FORDECYT-PRONACES/61533/2020. M. C-C. acknowl-
edges as well the Conacyt fellowship 301117.

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	Acta Polytechnica 62(1):23–29, 2022
	1 Introduction
	2 Strain in photonic graphene
	2.1 Tight-binding model
	2.2 Uniform strain
	2.3 Non-uniform strain

	3 Supersymmetric quantum mechanics: matrix approach
	4 Photonic graphene under strain and position-dependent gain and loss
	4.1 Photonic Graphene with a single mode
	4.2 Photonic Graphene with two modes

	5 Summary
	Acknowledgements
	References