J. Eur. Opt. Society-Rapid Publ. 21, 28( 2025) 299
fermions, which is very promising for making compact nanophotonic devices. Additionally, graphene’ s surface conductivity can be modified by altering the chemical potential, relaxation time, and frequency of its EM waves. The electrical properties of graphene are strongly influenced by these parameters, making it an extremely versatile material with a wide range of applications in a variety of fields such as electronics and optoelectronics [ 17, 18 ]. Furthermore, the carrier concentration in graphene is controlled by electrical gating, and this control is extremely influential over the electronic characteristics of graphene [ 19 ].
Perfect magnetic conductor( PMC) waveguides are becoming increasingly important for the analysis of EM wave propagation. Several researchers have conducted research relevant to PMC boundary conditions [ 20 – 23 ]. It has been reported that PMC surfaces have extremely high surface impedance within certain frequency ranges. In addition, PMC surfaces reflect EM waves without changing the electric field phase unlike perfect electrical conductor surfaces [ 24 ]. The boundary conditions that exist at the surface of a PMC are as follows [ 20, 22, 25 ]:
n H ¼ 0;
n E ¼ 0; ð1Þ ð2Þ
where n represents the unit vector normal to the PMC surface.
Mathematical formulation
Consider the planar waveguide filled with graphene, surrounded by magnetized plasma, and with a PMC as the substrate shown in Figure 1.
For the magnetized plasma, the constitutive relations are given as [ 26 ]: @
e is given below:
D ¼ e 0 e @ E; H ¼ 1 = l 0 B:
|
2 e 1
½ @
6 e
Š¼4je 2
|
�je 2
e 1
|
3
0
0
|
0 |
0 |
e 3 |
ð3Þ ð4Þ
7 5: ð5Þ
e 1, e 2, and e 3 are permittivity tensor values, as reported in [ 27 ]. In terms of H z and E z, the magnetized plasma wave equation can be expressed as follows [ 28 ]:
" # r 2 E
z þ s " #
1 js Ez 2
¼ 0; ð6Þ r 2 H z js 3 s 4 H z where
@ r¼be x
@ x þ be @
y
@ y;
s 1 ¼� b2 e 3
� x 2 l e 0 e 3
1 ð7Þ ð8Þ
Figure 1. Configuration of magnetized plasma – graphene – magnetized plasma – PMC planar waveguide.
s 2 ¼ xl 0be 2 e 1
; ð9Þ s 3 ¼�bxe 3 e 3 e 1
; ð10Þ s 4 ¼ x2 l 0 e 2
1 � x2 l 0 e 2
2 Þ e 1
� b 2; ð11Þ
where the eigenvalues of magnetized plasma are [ 29 ]: |
|
u 2 1 ¼ 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 s 1 þ s 4 þ ðs 1 þ s 4 Þ 2 � 4ðs
1 s 4 � s 2 s 3
Þ
;
|
ð12Þ |
|
u 2 2 ¼ 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 s 1 þ s 4 � ðs 1 þ s 4 Þ 2 � 4ðs
1 s 4 � s 2 s 3
Þ
:
|
ð13Þ |
The EM fields associated with the magnetized plasma media are as follows [ 29 ]:
E z3
H z3
E z1 ¼ c 1 e �u 1x þ c 2 e �u 2x; H z1 ¼ jðc 1 a 1 e �u 1x þ c 2 a 2 e �u 2x Þ;
¼ c 3 cos ðu 1 x Þþc 4 sin ðu 1 x
Þþc 5 cos ðu 2 xÞ ð14Þ ð15Þ
þ c 6 sin ðu 2 xÞ; ð16Þ ¼ jða 1 ðc 3 cos ðu 1 x Þþc 4 sin ðu 1 xÞÞ þ a 2 ðc 5 cos ðu 2 xÞþc 6 sin ðu 2 xÞÞ; ð17Þ
where c 1, c 2, c 3, c 4, c 5; and c 6 are unknown constants. The |
remaining EM field components can be obtained from |
[ 30 ]. The boundary conditions applied at the magnetized |
plasma – graphene – magnetized |
plasma – PMC |
interface |
are given below: |
|
|
|
bx
½ H 1 � H 2
Š ¼ rE;
|
ð18Þ |
bx ½ E 1 � E 2 Š ¼ 0; ð19Þ