)
1
𝑧 ∅
𝑧
1
( ∅
∅
(
)
) ∅
∇ × =
+ ∅
+ (
+
+
(
−
) 𝑧
=
− ∅
∅
+ 𝑧
𝑧
Equating the coefficients, we get 1 ( 𝑧 ∅
) =
+ ∅
(14)
𝑧
（ 15 ）
+
= − ∅
)
1
( ∅
∅
(
) = 𝑧
−
(16)
By eliminating variables these equations can be rewritten such that, when 𝑧 and 𝑧 are known, the remaining transverse components , ∅ , , ∅ can be determined. For example, ∅ or ca be eliminated from Eqs 11 and 14 so that the component ∅ or , respectively, can be found in terms of 𝑧 and 𝑧 = − 2 ( 𝑧 + 𝜇 𝑧 ∅ ) (17) = − 2 ( 𝑧 ∅ − 𝜇 𝑧 ) (18) = − 2 ( 𝑧 − 𝑧 ∅ ) (19) ∅ = − 2 ( 𝑧 ∅ + 𝑧 ) (20) Where 2 = 2 𝜇 − 2 = 2 − 2 Substitution of Eq. 19 and 20 into Eq. 16 results in the wave equation in cylindrical coordinates 2 𝑧 2 + 1 𝑧 + 1 2 𝑧 ∅ 2 + 2 𝑧 = 0 (21)
And substitution of Eq.17 and 18 into Eq.13 leads to 2 𝑧 2 + 1 𝑧 + 1 2 2 𝑧 ∅ 2 + 2 𝑧 Then we can use the separation of variables method 𝑧 = 1 () 2 (∅) 3 ( ) 4
= 0 (22)
() (23)
As was already assumed, the time and z-dependent factors are given by 3 ( ) 4 () = ( −𝑧) (24)
Since the wave is sinusoidal in time and propagates in the z direction. In addition, because of the circular symmetry of the waveguide, each field component must not change when the coordinate ∅ is increased by 2𝜋 . We thus assume a periodic function of the form 2 (∅) = 𝑣∅ (25)
17
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