2 2
∇ 2 E =
(7)
Taking the curl of Eq.2, we can get
2 2
∇ 2 H =
(8)
The equations 7 and 8 are the standard wave equations.
Electromagnetic light field that propagates along a cylindrical fibre can be represented by a superposition of bound or trapped modes (which are two kinds of modes). Therefore, a cylindrical coordinate system {r, ∅, z} can be established in the optical fibre. It is defined with the z axis lying along the axis of the waveguide. If the electromagnetic waves are to propagate along the z axis, the will have a functional dependence form = 0 (, ∅) ( −𝑧) (9) = 0 (, ∅) ( −𝑧) (10) The electromagnetic waves are harmonic in time t with radian frequency , the parameter is the z component of the propagation vector and is the main parameter in describing wave modes. There are only certain values for which will be determined from the mode fields that satisfy Maxwell’s equations and the electric and magnetic field boundary conditions at the core-cladding interface. (Keiser, 1991)
When Eq.9 and 10 are substituted into Maxwell’s curl equations, we have
Using the determinant curl ∇ × = | 1
1
∅ ∅
𝑧
𝑧
| (Keiser, 1991)
∅
𝑧
)
1
𝑧 ∅
𝑧
1
( ∅
∅
(
)
) ∅
∇ × =
+ ∅
+ (
+
+
(
−
) 𝑧
−
= −𝜇
= − 𝜇
+ 𝜇 ∅
∅
+ − 𝜇 𝑧
Equating the coefficients we get 1 ( 𝑧 ∅
) = − 𝜇
+ ∅
(11)
𝑧
+
= 𝜇 ∅
(12)
)
1
( ∅
∅
(
) = − 𝜇 𝑧
−
(13)
Similarly, from Eq.1
16
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