470
E. Hasman and V. Kleiner
in this intensity distribution is a spin-dependent transverse shift of the focus which
one can easily observe by comparing the cross-sections of the spots (Fig. 13.4c). This
shift can be regarded as a manifestation of the optical Magnus effect [23, 24] and the
optical spin Hall effect [25, 26] which arises in the system due to a spin-orbit coupling
producing a spiral geometric phase. The experimental results are supported by the
FDTD numerical simulation which are also depicted in the Fig. 13.4c. The focal shift
corresponds to the spin-dependent spiral phase modification due to the Berry phase
and can be estimated by Δx ≈ r 0
k
−1
p
∇ (λ ± π) = λ ±
k
−1
p
, (Δx ≈ 160 nm).
The observed shift of the focal intensity distribution is about 200 nm which is in
good agreement with the above estimation as well as with the FDTD results (see
Fig. 13.4c). The slight deviation of the measured results from the simulation can be
attributed to the modified dispersion relation of the SPPs due to impurities caused by
the fabrication process. This geometric effect is exceptionally pronounced when linearly polarized light (the superposition of spin |λ + ≈ and |λ − ≈ beams) is incident upon
the structure. In this case, the focal spot is split in the lateral direction (see Fig. 13.4d)
and clearly reminds us the effect of spin-dependent electron beam splitting in the
classic Stern-Gerlach experiment [27, 28].
13.3 Optical Spin Symmetry Breaking in Nanoapertures
The dynamics of various physical systems, including optical systems, are substantially characterized by their AM. The AM of an optical beam comprises the spin
component, associated with the handedness of the circular polarization, and the
orbital angular momentum (OAM), associated with a spiral phase front [29–32]. In
a paraxial beam with a spiral phase distribution γ ∪ lπ, the total AM per photon, in
units of (normalized AM), was shown to be j = (λ + l), where λ = 1 stands for the
spin state |λ + ≈ and λ = −1 stands for the spin state |λ − ≈ [33]. In accordance with
fundamental physical principles, resonant excitation of an electromagnetic eigenmode requires that the exciting wave match the excited mode both with its linear
and angular momentum. This required matching imposes certain restrictions on the
excitation process—selection rules. Likely the simplest plasmonic nanoscale structure, which demonstrates these common rules is a nanoaperture in a thin metal film.
Let consider a matrix consisted of annular apertures surrounded by a shallow spiral
periodic corrugation. In the experiment [34] a structure with the inner and the outer
radii of the ring slit apertures of 250 and 350 nm, respectively, spiral corrugation
with a period of 500 nm and a pitch of 2 periods, and depth of 70 nm was milled by
a FIB into a 200 nm thick gold film evaporated onto a glass wafer (see Fig. 13.5b).
Transmission of circularly polarized light through the structure is strongly affected
by relationship between helicity of illuminating light and handedness of the spirals.
A picture of the transmitted light for the spin states |λ + ≈ and |λ − ≈, and for the linear
polarization 2 −1/2 (|λ + ≈ + |λ − ≈)is presented in Fig. 13.5a. The experimental results
show that when the incident spin is opposite to the handedness of the spiral grating,
E. Hasman and V. Kleiner
in this intensity distribution is a spin-dependent transverse shift of the focus which
one can easily observe by comparing the cross-sections of the spots (Fig. 13.4c). This
shift can be regarded as a manifestation of the optical Magnus effect [23, 24] and the
optical spin Hall effect [25, 26] which arises in the system due to a spin-orbit coupling
producing a spiral geometric phase. The experimental results are supported by the
FDTD numerical simulation which are also depicted in the Fig. 13.4c. The focal shift
corresponds to the spin-dependent spiral phase modification due to the Berry phase
and can be estimated by Δx ≈ r 0
k
−1
p
∇ (λ ± π) = λ ±
k
−1
p
, (Δx ≈ 160 nm).
The observed shift of the focal intensity distribution is about 200 nm which is in
good agreement with the above estimation as well as with the FDTD results (see
Fig. 13.4c). The slight deviation of the measured results from the simulation can be
attributed to the modified dispersion relation of the SPPs due to impurities caused by
the fabrication process. This geometric effect is exceptionally pronounced when linearly polarized light (the superposition of spin |λ + ≈ and |λ − ≈ beams) is incident upon
the structure. In this case, the focal spot is split in the lateral direction (see Fig. 13.4d)
and clearly reminds us the effect of spin-dependent electron beam splitting in the
classic Stern-Gerlach experiment [27, 28].
13.3 Optical Spin Symmetry Breaking in Nanoapertures
The dynamics of various physical systems, including optical systems, are substantially characterized by their AM. The AM of an optical beam comprises the spin
component, associated with the handedness of the circular polarization, and the
orbital angular momentum (OAM), associated with a spiral phase front [29–32]. In
a paraxial beam with a spiral phase distribution γ ∪ lπ, the total AM per photon, in
units of (normalized AM), was shown to be j = (λ + l), where λ = 1 stands for the
spin state |λ + ≈ and λ = −1 stands for the spin state |λ − ≈ [33]. In accordance with
fundamental physical principles, resonant excitation of an electromagnetic eigenmode requires that the exciting wave match the excited mode both with its linear
and angular momentum. This required matching imposes certain restrictions on the
excitation process—selection rules. Likely the simplest plasmonic nanoscale structure, which demonstrates these common rules is a nanoaperture in a thin metal film.
Let consider a matrix consisted of annular apertures surrounded by a shallow spiral
periodic corrugation. In the experiment [34] a structure with the inner and the outer
radii of the ring slit apertures of 250 and 350 nm, respectively, spiral corrugation
with a period of 500 nm and a pitch of 2 periods, and depth of 70 nm was milled by
a FIB into a 200 nm thick gold film evaporated onto a glass wafer (see Fig. 13.5b).
Transmission of circularly polarized light through the structure is strongly affected
by relationship between helicity of illuminating light and handedness of the spirals.
A picture of the transmitted light for the spin states |λ + ≈ and |λ − ≈, and for the linear
polarization 2 −1/2 (|λ + ≈ + |λ − ≈)is presented in Fig. 13.5a. The experimental results
show that when the incident spin is opposite to the handedness of the spiral grating,