478
E. Hasman and V. Kleiner
is a manifestation of the optical spin-orbit interaction similar to the spin-Hall and
the Stern-Gerlach effects. The additional momentum leads to a geometric phase
accumulation of,
γ g = −
λ ∂dϕ = −λ τ.
(13.3)
Accordingly, the phase of the scattered plasmonic wave will be continuous up to
the factor of 2α everywhere excluding the point r = 0, where the phase dislocation
appears. The phase in Eq. (13.3) is analogous to the phase arising in a AB wavefunction, ψ(r) [48, 49]. The latter effect appears when a beam of particles with charge
of q is scattered from an infinite impenetrable cylinder containing a magnetic flux
δ =
A(r) · dr =
B(r)dS, where A(r) is the vector potential and B(r) is the
magnetic field. The suitable vector potential is given by A(r) = (δ/2αr ) ˆ
ϕ , where
ˆ
ϕ is the unit vector in the azimuthal direction. The corresponding time-independent
Schröedinger equation is then given by,
1
2m (−i∇ − qA(r))
2
ψ(r) =
2 k 2
2m ψ(r),
where m is the particle’s mass and is Plank’s constant. Note that the expression in
parentheses is the generalized momentum term. In the linear approximation in A, the
above equation can be written as,
∇ 2 + k 2 −
2iq
A · ∇
ψ(r) = 0. The resulting
equation resembles the Helmholtz equation in the rotating frame, where the third
term in the parentheses stands for the Coriolis term. In our system, the momentum
correction term along the ϕ coordinate is (Δk) ϕ = λ/r , and is analogous to the qA
term in the AB effect. The main result of the AB experiment is a spiral phase γ = βπ
acquired by the particles scattered from the cylinder, where the topological charge
β = qδ/(2α ) is the magnetic flux parameter. The topological charge of the phase
obtained in our experiment can be found as, l =
1
2α
(Δk) ϕ dϕ = λ . Therefore,
one can conclude that the intrinsic spin in our experiment corresponds to the flux
parameter β in the AB effect. Due to the non-zero topological charge (λ = ±1) a
singularity of the plasmonic field appears in the center of the defect, resulting in a
dark spot (see inset Fig. 13.10b). The vanishing electromagnetic field in the center
corresponds to the impenetrability of the cylinder proposed in the AB experiment.
Accordingly, the geometry of our system affects the resulting plasmonic phase front
in a similar way as a vector potential affects the electrons’ wavefunction. In contrast
to the original AB effect, here the topological charge of the plasmonic spiral phase
is spin-dependent, therefore it can be regarded as the intrinsic AB effect similar to
the Aharonov-Casher effect [54].
A peculiarity of the observed effect lies in its geometric nature. The spiral phase
of the plasmonic waves arises solely due to a rotation of the local anisotropy and is
not the result of an optical path difference. Therefore, the phase dislocation will be
independent of the wavelength or the size of the defect. This is confirmed experimentally on several elements with apertures of different sizes. Figure 13.11a, b presents
the observed fringe patterns for circular slit with a diameter of 1.8 µm and width of
320 nm illuminated with ε = 800 nm. Moreover, a FDTD simulation for the same
defect size, but for ε = 532 nm incident illumination is presented in Fig. 13.11(c, d).
In the measured as well as in the calculated near-field intensity distributions, same
E. Hasman and V. Kleiner
is a manifestation of the optical spin-orbit interaction similar to the spin-Hall and
the Stern-Gerlach effects. The additional momentum leads to a geometric phase
accumulation of,
γ g = −
λ ∂dϕ = −λ τ.
(13.3)
Accordingly, the phase of the scattered plasmonic wave will be continuous up to
the factor of 2α everywhere excluding the point r = 0, where the phase dislocation
appears. The phase in Eq. (13.3) is analogous to the phase arising in a AB wavefunction, ψ(r) [48, 49]. The latter effect appears when a beam of particles with charge
of q is scattered from an infinite impenetrable cylinder containing a magnetic flux
δ =
A(r) · dr =
B(r)dS, where A(r) is the vector potential and B(r) is the
magnetic field. The suitable vector potential is given by A(r) = (δ/2αr ) ˆ
ϕ , where
ˆ
ϕ is the unit vector in the azimuthal direction. The corresponding time-independent
Schröedinger equation is then given by,
1
2m (−i∇ − qA(r))
2
ψ(r) =
2 k 2
2m ψ(r),
where m is the particle’s mass and is Plank’s constant. Note that the expression in
parentheses is the generalized momentum term. In the linear approximation in A, the
above equation can be written as,
∇ 2 + k 2 −
2iq
A · ∇
ψ(r) = 0. The resulting
equation resembles the Helmholtz equation in the rotating frame, where the third
term in the parentheses stands for the Coriolis term. In our system, the momentum
correction term along the ϕ coordinate is (Δk) ϕ = λ/r , and is analogous to the qA
term in the AB effect. The main result of the AB experiment is a spiral phase γ = βπ
acquired by the particles scattered from the cylinder, where the topological charge
β = qδ/(2α ) is the magnetic flux parameter. The topological charge of the phase
obtained in our experiment can be found as, l =
1
2α
(Δk) ϕ dϕ = λ . Therefore,
one can conclude that the intrinsic spin in our experiment corresponds to the flux
parameter β in the AB effect. Due to the non-zero topological charge (λ = ±1) a
singularity of the plasmonic field appears in the center of the defect, resulting in a
dark spot (see inset Fig. 13.10b). The vanishing electromagnetic field in the center
corresponds to the impenetrability of the cylinder proposed in the AB experiment.
Accordingly, the geometry of our system affects the resulting plasmonic phase front
in a similar way as a vector potential affects the electrons’ wavefunction. In contrast
to the original AB effect, here the topological charge of the plasmonic spiral phase
is spin-dependent, therefore it can be regarded as the intrinsic AB effect similar to
the Aharonov-Casher effect [54].
A peculiarity of the observed effect lies in its geometric nature. The spiral phase
of the plasmonic waves arises solely due to a rotation of the local anisotropy and is
not the result of an optical path difference. Therefore, the phase dislocation will be
independent of the wavelength or the size of the defect. This is confirmed experimentally on several elements with apertures of different sizes. Figure 13.11a, b presents
the observed fringe patterns for circular slit with a diameter of 1.8 µm and width of
320 nm illuminated with ε = 800 nm. Moreover, a FDTD simulation for the same
defect size, but for ε = 532 nm incident illumination is presented in Fig. 13.11(c, d).
In the measured as well as in the calculated near-field intensity distributions, same