4 Chirality and Antiferromagnetism in Optical Metasurfaces
85
of lè per photon, where è is the reduced Planck constant [72]. It thus connects
the microscopic quantity of OAM with the macroscopic phase profile of e
ilϕ . The
mathematical proof of the OAM carried in optical vortex beam can be found easily
by using a circularly polarized Laguerre-Gaussian beam with its vector potential of
A = u(r,z)e
ilϕ
•e
i(kz-ωt)
• (e x +σ i e y ), where the amplitude part is angle-independent,
r and ϕ are the polar coordinates. The Lorentz gauge yields the similar electric and
magnetic fields in (4.2) and (4.3) after u(x, y, z) is substituted by u(r, z)e
ilϕ . By using
(4.4) under the paraxial approximation, we have the total angular momentum
J z
W
=
l + σ
ω
,
(4.6)
where the spin item σ refers to the SAM of the circular polarized light and l is
related with the OAM of light. For a linearly polarized vortex beam, its SAM is zero,
leaving the OAM of lè per photon. Considering that the OAM of light is a microscopic
variable, optical vortices with the OAM has been intensely investigated in quantum
physics. From the helical phase profiles, one can find that the orthogonality between
two arbitrary OAM states is extremely good, so that it can be used for the purpose of
communication. In quantum entanglement [76] and optical communication [77–79],
the OAM channel could provide the low crosstalk for high-fidelity data transmission.
Meanwhile, because the integer l can take any value, the number of OAM channels
in communication is unlimited, which therefore could increase the capacity of data
transmission.
As denoted in (4.6), the OAM in an optical vortex beam can take the positive or
negative value, which presents its direction of z or –z, respectively. It means that the
energy flow in the vortex beam moves clockwise or anti-clockwise, behaving like
the spin of a photon in a CP beam. When a tightly focused OAM is used to trap the
micro-particles within the ring-shape intensity, the OAM of the photons is transferred
to micro-particles, leaving a clockwise or anti-clockwise moving trajectory that is
determined by the sign of l. All these phenomena suggest the chiral behavior of
optical vortices, which is therefore expected to have the strong interaction with
chiral objects. However, a plethora of theoretical and experimental works show no
obvious interaction with chiral molecules [80]. The rigorous theory proves that the
OAM states can respond to the electric quadrupole excited in molecules [81], but its
corresponding excitation rate is extremely low in real world. With the help of unique
nanoparticle aggregates that could excite the electric quadrupole efficiently, a recent
work has reported the first observation of discriminating two chiral enantiomers
by the optical vortices with opposite topological charges [82]. Note that, since the
electric quadrupoles are excited by the third part, the resulting scattering induced by
two vortices has the little difference, leading to a quite weak helical dichroism (HD)
of 0.8% [82]. From the viewpoint o practical applications, such a tiny HD cannot be
used to detect the chirality of objects in a steady way. More efforts should be made
to enhance the HD for the wide usage. An experimental attempt to solve the problem
[83] is provided under the theoretical consideration in Sect. 4.4.
85
of lè per photon, where è is the reduced Planck constant [72]. It thus connects
the microscopic quantity of OAM with the macroscopic phase profile of e
ilϕ . The
mathematical proof of the OAM carried in optical vortex beam can be found easily
by using a circularly polarized Laguerre-Gaussian beam with its vector potential of
A = u(r,z)e
ilϕ
•e
i(kz-ωt)
• (e x +σ i e y ), where the amplitude part is angle-independent,
r and ϕ are the polar coordinates. The Lorentz gauge yields the similar electric and
magnetic fields in (4.2) and (4.3) after u(x, y, z) is substituted by u(r, z)e
ilϕ . By using
(4.4) under the paraxial approximation, we have the total angular momentum
J z
W
=
l + σ
ω
,
(4.6)
where the spin item σ refers to the SAM of the circular polarized light and l is
related with the OAM of light. For a linearly polarized vortex beam, its SAM is zero,
leaving the OAM of lè per photon. Considering that the OAM of light is a microscopic
variable, optical vortices with the OAM has been intensely investigated in quantum
physics. From the helical phase profiles, one can find that the orthogonality between
two arbitrary OAM states is extremely good, so that it can be used for the purpose of
communication. In quantum entanglement [76] and optical communication [77–79],
the OAM channel could provide the low crosstalk for high-fidelity data transmission.
Meanwhile, because the integer l can take any value, the number of OAM channels
in communication is unlimited, which therefore could increase the capacity of data
transmission.
As denoted in (4.6), the OAM in an optical vortex beam can take the positive or
negative value, which presents its direction of z or –z, respectively. It means that the
energy flow in the vortex beam moves clockwise or anti-clockwise, behaving like
the spin of a photon in a CP beam. When a tightly focused OAM is used to trap the
micro-particles within the ring-shape intensity, the OAM of the photons is transferred
to micro-particles, leaving a clockwise or anti-clockwise moving trajectory that is
determined by the sign of l. All these phenomena suggest the chiral behavior of
optical vortices, which is therefore expected to have the strong interaction with
chiral objects. However, a plethora of theoretical and experimental works show no
obvious interaction with chiral molecules [80]. The rigorous theory proves that the
OAM states can respond to the electric quadrupole excited in molecules [81], but its
corresponding excitation rate is extremely low in real world. With the help of unique
nanoparticle aggregates that could excite the electric quadrupole efficiently, a recent
work has reported the first observation of discriminating two chiral enantiomers
by the optical vortices with opposite topological charges [82]. Note that, since the
electric quadrupoles are excited by the third part, the resulting scattering induced by
two vortices has the little difference, leading to a quite weak helical dichroism (HD)
of 0.8% [82]. From the viewpoint o practical applications, such a tiny HD cannot be
used to detect the chirality of objects in a steady way. More efforts should be made
to enhance the HD for the wide usage. An experimental attempt to solve the problem
[83] is provided under the theoretical consideration in Sect. 4.4.