450
S. V. Boriskina
Madelung transformation E(r) =
3
m=1 ˆ
e m U m (r) exp {i (τ(r) − ωt)} brings the
wave equation to the hydrodynamic form for the steady flow (πε/πt = 0) of ‘photon
fluid’ in the presence of sources and sinks ψ(r) = −k 2
0 λ i (r) [67, 102]:
≈ (ε(r)v(r)) = ψ(r)ε(r)
(v(r) · ≈) v(r) = −≈ (V (r) + Q(r))
.
(12.7)
Similarly to Eq. 12.6, here the intensity plays the role of a ‘photon fluid’ density ε(r) = U(r) · U(r), and the phase gradient–the role of the fluid velocity,
v = ≈τ(r). However, the local pressure term does not appear in the absence of
non-linearity, and the internal quantum potential has the following form: Q(r) =
1/(8ε(r))
3
m=1 (≈ε m (r) · ≈ε m (r)/ε m (r)) − ≈ 2 ε(r)/(4ε(r)), where ε m (r) =
U 2
m (r). Accordingly, the optical flux defined by the Poynting vector transforms into
the analog of a fluid flux (the momentum density): S = 1/(2μ 0 ω) · ε(r)v(r). One
important difference between conventional fluids and the PF is that photon ‘mass’
can be created owing to the linear gain λ i (r) < 0 and dissipated through material
losses λ i (r) > 0. The ‘photon mass’ reduction due to dissipative losses results in the
decrease of the PF density (i.e., field intensity).
In the case of PF steady flow, the field patterns are constant in time, yet local
accelerations/decelerations of the flow can occur between different parts of the plasmonic nanostructure, governed by the convective term (v(r) · ≈) v(r) in the momentum conservation equation. This situation is analogous to e.g. local variations of the
velocity of the fluid flow passing through pipes of varying diameters. In the structures shown in Figs. 12.10–12.11, these local changes of flow rate are driven by
the formation of vortices, with each vortex line inducing a velocity field given by
the Biot-Savart formula [103] (compare to the vortex-driven airplane downwash
velocity profile shown as the inset of Fig. 12.12e). In particular, the tangential velocity of an optical vortex varies inversely with the distance from its center, and the
angular momentum is thus uniform everywhere throughout the vortex-induced circulating flow.
A change in the fluid’s momentum can generate pressure, and this hydrodynamic effect is utilized in hydraulic pumps and motors, which increase the fluid
kinetic energy (angular momentum) and then convert it into usable pressure energy.
Accordingly, the problem of wave scattering by the VNT in Fig. 12.10 can be cast
in new light by invoking the fluid dynamic analogy. Fig. 12.13 follows the evolution
of the PF velocity and density along the z-axis, which passes from the bottom up
through the gap of the VNT central dimer. At the VNT center, the flows generated
by adjacent counter-rotating vortices collide and form a ‘shock wave’ in the form
of a region of high PF density. In the situation depicted in Fig. 12.13, the PF gets
convectively accelerated by the potential forces and impacts onto the narrow interparticle gaps of the VNT dimers. The threading of the PF through the gaps leads to a
sudden change in the flow regime and results in the dramatic increase of the PF local
density driven by the conversion of the PF kinetic energy into pressure energy. In
effect, VNTs shown in Figs. 12.10–12.11 operate as plasmonic analogs of hydraulic
pumps that exploit the convective acceleration of the PF caused by its circulation
S. V. Boriskina
Madelung transformation E(r) =
3
m=1 ˆ
e m U m (r) exp {i (τ(r) − ωt)} brings the
wave equation to the hydrodynamic form for the steady flow (πε/πt = 0) of ‘photon
fluid’ in the presence of sources and sinks ψ(r) = −k 2
0 λ i (r) [67, 102]:
≈ (ε(r)v(r)) = ψ(r)ε(r)
(v(r) · ≈) v(r) = −≈ (V (r) + Q(r))
.
(12.7)
Similarly to Eq. 12.6, here the intensity plays the role of a ‘photon fluid’ density ε(r) = U(r) · U(r), and the phase gradient–the role of the fluid velocity,
v = ≈τ(r). However, the local pressure term does not appear in the absence of
non-linearity, and the internal quantum potential has the following form: Q(r) =
1/(8ε(r))
3
m=1 (≈ε m (r) · ≈ε m (r)/ε m (r)) − ≈ 2 ε(r)/(4ε(r)), where ε m (r) =
U 2
m (r). Accordingly, the optical flux defined by the Poynting vector transforms into
the analog of a fluid flux (the momentum density): S = 1/(2μ 0 ω) · ε(r)v(r). One
important difference between conventional fluids and the PF is that photon ‘mass’
can be created owing to the linear gain λ i (r) < 0 and dissipated through material
losses λ i (r) > 0. The ‘photon mass’ reduction due to dissipative losses results in the
decrease of the PF density (i.e., field intensity).
In the case of PF steady flow, the field patterns are constant in time, yet local
accelerations/decelerations of the flow can occur between different parts of the plasmonic nanostructure, governed by the convective term (v(r) · ≈) v(r) in the momentum conservation equation. This situation is analogous to e.g. local variations of the
velocity of the fluid flow passing through pipes of varying diameters. In the structures shown in Figs. 12.10–12.11, these local changes of flow rate are driven by
the formation of vortices, with each vortex line inducing a velocity field given by
the Biot-Savart formula [103] (compare to the vortex-driven airplane downwash
velocity profile shown as the inset of Fig. 12.12e). In particular, the tangential velocity of an optical vortex varies inversely with the distance from its center, and the
angular momentum is thus uniform everywhere throughout the vortex-induced circulating flow.
A change in the fluid’s momentum can generate pressure, and this hydrodynamic effect is utilized in hydraulic pumps and motors, which increase the fluid
kinetic energy (angular momentum) and then convert it into usable pressure energy.
Accordingly, the problem of wave scattering by the VNT in Fig. 12.10 can be cast
in new light by invoking the fluid dynamic analogy. Fig. 12.13 follows the evolution
of the PF velocity and density along the z-axis, which passes from the bottom up
through the gap of the VNT central dimer. At the VNT center, the flows generated
by adjacent counter-rotating vortices collide and form a ‘shock wave’ in the form
of a region of high PF density. In the situation depicted in Fig. 12.13, the PF gets
convectively accelerated by the potential forces and impacts onto the narrow interparticle gaps of the VNT dimers. The threading of the PF through the gaps leads to a
sudden change in the flow regime and results in the dramatic increase of the PF local
density driven by the conversion of the PF kinetic energy into pressure energy. In
effect, VNTs shown in Figs. 12.10–12.11 operate as plasmonic analogs of hydraulic
pumps that exploit the convective acceleration of the PF caused by its circulation