1 Chiral Coupling to Magnetodipolar Radiation
7
M α (x, ρ, t) = −i
t
−∞
dt
ˆ
M α (x, ρ ρ ρ, t), ˆ
H int (t
)
.
(1.17)
in terms of the retarded spin susceptibility tensor
χ αδ (x, x
; ρ − ρ
; t − t
) = i(t − t
)
ˆ
S α (x, ρ, t), ˆ
S δ (x
, ρ
, t
)
,
(1.18)
where ˆ
S α = − ˆ
M α /(γ )is the spin operator and a sum over repeated indices is
implied. Hence [6–8],
M α (x, k y , ω) = μ 0 (γ )
2
0
−d
dx
χ αβ (x, x
, k y , ω)H β (x
, k y , ω),
(1.19)
where
χ αβ (x, x
, k, ω) = −
2M s
γ
m
(k)
α (x)m
(k)∗
β (x
)
1
ω − ω k + i k
.
(1.20)
Here, k = 2αω k is the reciprocal lifetime in terms of the Gilbert damping constant α.
The excitation efficiency is determined by m
(ky)∗
β
(x
)H β (x
, k y , ω), so the excitation
of circularly polarized spin waves is chiral (or unidirectional) by the polarizationmomentum locking with the stripline magnetic field. Since the amplitudes across
thin films are constant for kd 1, the excited magnetization in time domain and
position space is the real part of the inverse Fourier transform (q ≡ k y ),
M α (x, y, t) =
q
e iqy−iωt M α (x, q)
≈ 2iμ 0 γ d M s m
(q ω )
α m
(q ω )∗
β
1
v q ω
e −iωt
e iq ω y−δ ω y H β (q ω , ω)
e −iq ω y+δ ω y H β (−q ω , ω)
for
y > 0
y < 0
,
(1.21)
where q ω + iδ ω is the positive root of ω q = ω + i q , and v q ω is the modulus of the
group velocity |∂ω q /∂q| q ω . The polarization-momentum locking of the stripline field
generates two different magnetization dynamics. When the excited spin waves are
circularly polarized, they not only propagate in one direction only, but the excitation is
also spatially limited to half of the film, i.e. the chirality is perfect. We can understand
this phenomenon in terms of the interference between the spin waves and the stripline
magnetic field that is constructive and destructive on opposite sides, as illustrated in
Fig. 1.1.
The dominant excitation direction can be switched with the film magnetization.
For a finite angle θ between the saturated magnetization and the stripline, the situation
becomes complicated by the reduced symmetry. It is advantageous to transform (1.21)
following the Supplements of [7, 8]:
m
(k y )
x
→ m
(l)
x , m
(k y )
y
→ cos θ m
(l)
y
7
M α (x, ρ, t) = −i
t
−∞
dt
ˆ
M α (x, ρ ρ ρ, t), ˆ
H int (t
)
.
(1.17)
in terms of the retarded spin susceptibility tensor
χ αδ (x, x
; ρ − ρ
; t − t
) = i(t − t
)
ˆ
S α (x, ρ, t), ˆ
S δ (x
, ρ
, t
)
,
(1.18)
where ˆ
S α = − ˆ
M α /(γ )is the spin operator and a sum over repeated indices is
implied. Hence [6–8],
M α (x, k y , ω) = μ 0 (γ )
2
0
−d
dx
χ αβ (x, x
, k y , ω)H β (x
, k y , ω),
(1.19)
where
χ αβ (x, x
, k, ω) = −
2M s
γ
m
(k)
α (x)m
(k)∗
β (x
)
1
ω − ω k + i k
.
(1.20)
Here, k = 2αω k is the reciprocal lifetime in terms of the Gilbert damping constant α.
The excitation efficiency is determined by m
(ky)∗
β
(x
)H β (x
, k y , ω), so the excitation
of circularly polarized spin waves is chiral (or unidirectional) by the polarizationmomentum locking with the stripline magnetic field. Since the amplitudes across
thin films are constant for kd 1, the excited magnetization in time domain and
position space is the real part of the inverse Fourier transform (q ≡ k y ),
M α (x, y, t) =
q
e iqy−iωt M α (x, q)
≈ 2iμ 0 γ d M s m
(q ω )
α m
(q ω )∗
β
1
v q ω
e −iωt
e iq ω y−δ ω y H β (q ω , ω)
e −iq ω y+δ ω y H β (−q ω , ω)
for
y > 0
y < 0
,
(1.21)
where q ω + iδ ω is the positive root of ω q = ω + i q , and v q ω is the modulus of the
group velocity |∂ω q /∂q| q ω . The polarization-momentum locking of the stripline field
generates two different magnetization dynamics. When the excited spin waves are
circularly polarized, they not only propagate in one direction only, but the excitation is
also spatially limited to half of the film, i.e. the chirality is perfect. We can understand
this phenomenon in terms of the interference between the spin waves and the stripline
magnetic field that is constructive and destructive on opposite sides, as illustrated in
Fig. 1.1.
The dominant excitation direction can be switched with the film magnetization.
For a finite angle θ between the saturated magnetization and the stripline, the situation
becomes complicated by the reduced symmetry. It is advantageous to transform (1.21)
following the Supplements of [7, 8]:
m
(k y )
x
→ m
(l)
x , m
(k y )
y
→ cos θ m
(l)
y