13 Spinoptics in Plasmonics
473
The above effect can be regarded as a spatial angular Doppler effect (ADE) [38].
In analogy with a temporal ADE, where an observer at a reference frame rotating
with a rate ∂ t registers a spin-dependent temporal frequency shift [39], here a spatial
rotation of the periodic surface corrugation with a rate of ∂ ϕ induces a spin-dependent
spatial frequency shift. Accordingly, the geometric phase arising from this spatial
frequency modulation can be easily calculated to be γ g = −λ in
∂ ϕ dϕ , where the
rotation of the grating is given by ∂ ϕ = dτ/dϕ , the angle τ indicates the local
groove’s orientation, and ϕ stands for a spatial coordinate. In the structure presented
in Fig. 13.5b where ϕ = π, the geometric phase is simply given by the spin-dependent
spiral phase γ g = −λ in π producing the required additional OAM for the surface
wave to satisfy the AM selection rule (Eq. (13.2)). Note that while polarization and
chirality effects in anisotropic inhomogeneous nanoscale structures were investigated
previously [40–43], the mechanism of spin-orbit interaction was not distinguished.
It is very helpful to compare the behavior of the spiral structure discussed above
with properties of the annular aperture fitted with in a circularly symmetric (achiral)
corrugation using incident beam with OAM . The circular symmetry of the coupling
corrugation does not induce a dynamic phase, which means that l S = 0. However,
due to the spin-orbit interaction, the incident spin induces a spiral Berry phase, and,
as before, is converted to the OAM component of the surface mode. The external
spiral phase modifies the complete OAM of the surface mode to be l SM = λ in +
l ext . Therefore, when the external OAM is zero, the resulting surface mode AM is
l SM = ±1, in which case the selection rule is always satisfied and the transmission
of the element is undistinguished for distinct spin states. Moreover, providing an
external spiral phase of l ext = 2, the AM of the surface mode will be either 1 or 3 for
an incident spin |λ − ≈ or |λ + ≈, respectively. For l ext = −2 the surface mode AM will
be correspondingly, −3 for |λ − ≈ and −1 for |λ + ≈. As before, the best overlapping
of the surface mode and the guided mode is obtained for l SM = ±1; therefore, the
transmitted intensity will be strongly dependent upon the incident spin.
Figure 13.7b shows the measured transmissions of a laser light at wavelength
532 nm by use of a spatial light modulator (SLM) generating a beam with OAM
for various combinations of l ext and λ in . The transmission ratio of the two spin
states is shown to be approximately 3 which is close to the ratio measured with the
spiral corrugations. Thus, the suggested AM selection rule is verified for systems of
substantially different symmetry.
Additional understanding of the AM evolution in the enhanced resonant transmission can be obtained by analyzing the AM of the light scattered from the nanoaperture. To do this, one should investigate the scattered light far behind the element.
The transverse electric field component of the guided mode [44] with l G M = 1, can
be described by the Jones vector E G M
r
=
E x E y
T = E 0 (r )
cos π sin π
T e iπ ,
where E 0 (r ) is the radial field dependence. Note that the guided mode in a circular
basis is given by E G M
r
= 2 −1/2 E 0 (r )[|λ − ≈+e i2π |λ + ≈]. This field distribution corresponds to a vectorial vortex (see Fig. 13.8a) with a Pancharatnam topological charge
of 1, resulting in a total AM of j = 1. As was shown [45], such vectorial vortices are
unstable and collapse upon propagation. The guided mode can be verified by propa-
473
The above effect can be regarded as a spatial angular Doppler effect (ADE) [38].
In analogy with a temporal ADE, where an observer at a reference frame rotating
with a rate ∂ t registers a spin-dependent temporal frequency shift [39], here a spatial
rotation of the periodic surface corrugation with a rate of ∂ ϕ induces a spin-dependent
spatial frequency shift. Accordingly, the geometric phase arising from this spatial
frequency modulation can be easily calculated to be γ g = −λ in
∂ ϕ dϕ , where the
rotation of the grating is given by ∂ ϕ = dτ/dϕ , the angle τ indicates the local
groove’s orientation, and ϕ stands for a spatial coordinate. In the structure presented
in Fig. 13.5b where ϕ = π, the geometric phase is simply given by the spin-dependent
spiral phase γ g = −λ in π producing the required additional OAM for the surface
wave to satisfy the AM selection rule (Eq. (13.2)). Note that while polarization and
chirality effects in anisotropic inhomogeneous nanoscale structures were investigated
previously [40–43], the mechanism of spin-orbit interaction was not distinguished.
It is very helpful to compare the behavior of the spiral structure discussed above
with properties of the annular aperture fitted with in a circularly symmetric (achiral)
corrugation using incident beam with OAM . The circular symmetry of the coupling
corrugation does not induce a dynamic phase, which means that l S = 0. However,
due to the spin-orbit interaction, the incident spin induces a spiral Berry phase, and,
as before, is converted to the OAM component of the surface mode. The external
spiral phase modifies the complete OAM of the surface mode to be l SM = λ in +
l ext . Therefore, when the external OAM is zero, the resulting surface mode AM is
l SM = ±1, in which case the selection rule is always satisfied and the transmission
of the element is undistinguished for distinct spin states. Moreover, providing an
external spiral phase of l ext = 2, the AM of the surface mode will be either 1 or 3 for
an incident spin |λ − ≈ or |λ + ≈, respectively. For l ext = −2 the surface mode AM will
be correspondingly, −3 for |λ − ≈ and −1 for |λ + ≈. As before, the best overlapping
of the surface mode and the guided mode is obtained for l SM = ±1; therefore, the
transmitted intensity will be strongly dependent upon the incident spin.
Figure 13.7b shows the measured transmissions of a laser light at wavelength
532 nm by use of a spatial light modulator (SLM) generating a beam with OAM
for various combinations of l ext and λ in . The transmission ratio of the two spin
states is shown to be approximately 3 which is close to the ratio measured with the
spiral corrugations. Thus, the suggested AM selection rule is verified for systems of
substantially different symmetry.
Additional understanding of the AM evolution in the enhanced resonant transmission can be obtained by analyzing the AM of the light scattered from the nanoaperture. To do this, one should investigate the scattered light far behind the element.
The transverse electric field component of the guided mode [44] with l G M = 1, can
be described by the Jones vector E G M
r
=
E x E y
T = E 0 (r )
cos π sin π
T e iπ ,
where E 0 (r ) is the radial field dependence. Note that the guided mode in a circular
basis is given by E G M
r
= 2 −1/2 E 0 (r )[|λ − ≈+e i2π |λ + ≈]. This field distribution corresponds to a vectorial vortex (see Fig. 13.8a) with a Pancharatnam topological charge
of 1, resulting in a total AM of j = 1. As was shown [45], such vectorial vortices are
unstable and collapse upon propagation. The guided mode can be verified by propa-