536
E. Kamenetskii
where
J ⊥
n
=
iω
4
ψ n
B
∗
⊥
n
− ψ
∗
n
B ⊥
n
and
J
n
=
iω
4
ψ n
B
∗
n
− ψ
∗
n
B
n
. Along every of the coordinates
r ,
θ, and z, we have the
power flows (currents):
J r
n
= −
iω
4
|C n |
2
|ξ n |
2
μ 0
˜
ϕ n
μ
∂ ˜
ϕ n
∂r
+ iμ a
1
r
∂ ˜
ϕ n
∂θ
∗
− ˜
ϕ
∗
n
μ
∂ ˜
ϕ n
∂r
+ iμ a
1
r
∂ ˜
ϕ n
∂θ
e r ,
(19.27)
J θ
n
= −
iω
4
|C n |
2
|ξ n |
2
μ 0
˜
ϕ n
−iμ a
∂ ˜
ϕ n
∂r
+ μ
1
r
∂ ˜
ϕ n
∂θ
∗
− ˜
ϕ
∗
n
−iμ a
∂ ˜
ϕ n
∂r
+ μ
1
r
∂ ˜
ϕ n
∂θ
e θ ,
(19.28)
J z
n
= −
iω
4
μ 0 |C n |
2
| ˜
ϕ|
2
ξ n
∂ξ n
∂z
∗
− ξ
∗
n
∂ξ n
∂z
e z ,
(19.29)
where
e r ,
e θ , and
e z are the unit vectors. We can see that for membrane function ˜
ϕ,
defined by (19.20), there is a non-zero real azimuth component of the power-flow
density. So, there is a non-zero quantity of the power flow circulation (clockwise or
counterclockwise) around a circle L = 2πr , where 0 < r ≤ R. At the same time,
homogeneous EM boundary conditions imposed on a ferrite disk on the r and z axes
give standing waves without real power flows.
19.3 Near Fields of MDM Oscillations—the ME Near
Fields
The L-mode wave function ψ( r , t) can define a magnetic flux density in a ferrite disk,
as shown above. This scalar wave function is considered as a generating function for
other types of the fields both inside and outside a ferrite disk. It allows analyzing
complex topological properties of vectorial fields, associated with orbital angular
momentum properties of MDM resonances.
When the spectral problem for the MS-potential scalar wave function ψ( r , t),
expressed by (19.18), is solved, distribution of magnetization in a ferrite disk is
found as
m = −
↔
χ ·
∇ψ, where
↔
χ is the susceptibility tensor of a ferrite [18]. Based
on the known magnetization
m, one can find the magnetic field distribution at any
point outside a ferrite disk [1, 65]:
E. Kamenetskii
where
J ⊥
n
=
iω
4
ψ n
B
∗
⊥
n
− ψ
∗
n
B ⊥
n
and
J
n
=
iω
4
ψ n
B
∗
n
− ψ
∗
n
B
n
. Along every of the coordinates
r ,
θ, and z, we have the
power flows (currents):
J r
n
= −
iω
4
|C n |
2
|ξ n |
2
μ 0
˜
ϕ n
μ
∂ ˜
ϕ n
∂r
+ iμ a
1
r
∂ ˜
ϕ n
∂θ
∗
− ˜
ϕ
∗
n
μ
∂ ˜
ϕ n
∂r
+ iμ a
1
r
∂ ˜
ϕ n
∂θ
e r ,
(19.27)
J θ
n
= −
iω
4
|C n |
2
|ξ n |
2
μ 0
˜
ϕ n
−iμ a
∂ ˜
ϕ n
∂r
+ μ
1
r
∂ ˜
ϕ n
∂θ
∗
− ˜
ϕ
∗
n
−iμ a
∂ ˜
ϕ n
∂r
+ μ
1
r
∂ ˜
ϕ n
∂θ
e θ ,
(19.28)
J z
n
= −
iω
4
μ 0 |C n |
2
| ˜
ϕ|
2
ξ n
∂ξ n
∂z
∗
− ξ
∗
n
∂ξ n
∂z
e z ,
(19.29)
where
e r ,
e θ , and
e z are the unit vectors. We can see that for membrane function ˜
ϕ,
defined by (19.20), there is a non-zero real azimuth component of the power-flow
density. So, there is a non-zero quantity of the power flow circulation (clockwise or
counterclockwise) around a circle L = 2πr , where 0 < r ≤ R. At the same time,
homogeneous EM boundary conditions imposed on a ferrite disk on the r and z axes
give standing waves without real power flows.
19.3 Near Fields of MDM Oscillations—the ME Near
Fields
The L-mode wave function ψ( r , t) can define a magnetic flux density in a ferrite disk,
as shown above. This scalar wave function is considered as a generating function for
other types of the fields both inside and outside a ferrite disk. It allows analyzing
complex topological properties of vectorial fields, associated with orbital angular
momentum properties of MDM resonances.
When the spectral problem for the MS-potential scalar wave function ψ( r , t),
expressed by (19.18), is solved, distribution of magnetization in a ferrite disk is
found as
m = −
↔
χ ·
∇ψ, where
↔
χ is the susceptibility tensor of a ferrite [18]. Based
on the known magnetization
m, one can find the magnetic field distribution at any
point outside a ferrite disk [1, 65]: