124
H. Ishihara et al.
P NP (r, ω) = (χ 1 + iχ 2 )E(r, ω) = (χ 1 + iχ 2 )E(x, y)e
i(k 1 +ik 2 )z
.
(5.14)
The real (imaginary) part of NP’s susceptibility is given by χ 1(2) . Substituting (5.13)
and (5.14) in (5.12) and regarding the induced polarization as the point dipole, the
optical force is written as follows:
F(ω) =
1
2
V NP
1
2
χ 1 ∇ x,y |E|
2
− χ 1 k 2 |E|
2 n z + χ 2 k 1 |E|
2 n z
e
−2k 2 z
, (5.15)
where V NP denotes the volume of the NP, ∇ x,y a 2D gradient, and n z a unit vector
along the z-axis.
We classified the terms of (5.15) into dissipative and gradient forces. The first
and third terms in the right hand side denote the conventional gradient force and
dissipative force, respectively. In the presence of a finite k 2 , we should also regard
the second terms as the gradient force. Notably, under the LSP near-field, the gradient
and dissipative forces depend on χ 1 and χ 2 , respectively. This means that one can
control over the trapping/exclusion of the NP by adjusting the balance between χ 1
and χ 2 and by tuning the laser frequencies.
Appendix 2
Dissipative and Gradient Forces Induced by the Detuned Light [25]
When the non-resonant light is radiated, χ 1 and χ 2 are constant for light energy. However, the resonant polarization in the linear optical response results in dispersion-type
χ 1 and Lorentz-type χ 2 . Therefore, their ratio drastically changes near the resonant
energy of NP. Moreover, strong nonlinear optical effects such as absorption saturation and population inversion can result in the sign inversion of χ 1 and χ 2 . Here, we
have regarded the ratio of χ 1 to χ 2 as that of gradient force to dissipative force, and
demonstrated it for the following three cases: weak excitation, strong excitation, and
stimulated emission.
In Fig. 5.9b–d, we depict the real and imaginary parts of the induced polarization
of the NP at the sample position indicated in Fig. 5.9a. The polarization includes the
resonant and non-resonant elements for the 0–1 and 0–2 transitions that are induced
by the manipulation light. The parameter of the NP is the same as that in the main
text. Under the weak excitation, the nonlinear optical effects are negligible, and the
polarization obeys the linear optical response. Notably, the resonant energy (e.g.., the
peak energy of χ 2 ) is slightly red-shifted because of the metal-NP interaction. Under
strong excitation, χ 2 is approximately zero, as the absorption and emission balance
with each other owing to the absorption saturation. With respect to χ 1 , it becomes
large as the manipulation light energy increases. This is because the absorption saturation effect suppresses the polarization with the 0–1 transition, and the polarization
with the 0–2 transition starts to appear. In addition, in the presence of the pump light,
the sign of the susceptibility is inverted, compared with the weakly excited case. The
H. Ishihara et al.
P NP (r, ω) = (χ 1 + iχ 2 )E(r, ω) = (χ 1 + iχ 2 )E(x, y)e
i(k 1 +ik 2 )z
.
(5.14)
The real (imaginary) part of NP’s susceptibility is given by χ 1(2) . Substituting (5.13)
and (5.14) in (5.12) and regarding the induced polarization as the point dipole, the
optical force is written as follows:
F(ω) =
1
2
V NP
1
2
χ 1 ∇ x,y |E|
2
− χ 1 k 2 |E|
2 n z + χ 2 k 1 |E|
2 n z
e
−2k 2 z
, (5.15)
where V NP denotes the volume of the NP, ∇ x,y a 2D gradient, and n z a unit vector
along the z-axis.
We classified the terms of (5.15) into dissipative and gradient forces. The first
and third terms in the right hand side denote the conventional gradient force and
dissipative force, respectively. In the presence of a finite k 2 , we should also regard
the second terms as the gradient force. Notably, under the LSP near-field, the gradient
and dissipative forces depend on χ 1 and χ 2 , respectively. This means that one can
control over the trapping/exclusion of the NP by adjusting the balance between χ 1
and χ 2 and by tuning the laser frequencies.
Appendix 2
Dissipative and Gradient Forces Induced by the Detuned Light [25]
When the non-resonant light is radiated, χ 1 and χ 2 are constant for light energy. However, the resonant polarization in the linear optical response results in dispersion-type
χ 1 and Lorentz-type χ 2 . Therefore, their ratio drastically changes near the resonant
energy of NP. Moreover, strong nonlinear optical effects such as absorption saturation and population inversion can result in the sign inversion of χ 1 and χ 2 . Here, we
have regarded the ratio of χ 1 to χ 2 as that of gradient force to dissipative force, and
demonstrated it for the following three cases: weak excitation, strong excitation, and
stimulated emission.
In Fig. 5.9b–d, we depict the real and imaginary parts of the induced polarization
of the NP at the sample position indicated in Fig. 5.9a. The polarization includes the
resonant and non-resonant elements for the 0–1 and 0–2 transitions that are induced
by the manipulation light. The parameter of the NP is the same as that in the main
text. Under the weak excitation, the nonlinear optical effects are negligible, and the
polarization obeys the linear optical response. Notably, the resonant energy (e.g.., the
peak energy of χ 2 ) is slightly red-shifted because of the metal-NP interaction. Under
strong excitation, χ 2 is approximately zero, as the absorption and emission balance
with each other owing to the absorption saturation. With respect to χ 1 , it becomes
large as the manipulation light energy increases. This is because the absorption saturation effect suppresses the polarization with the 0–1 transition, and the polarization
with the 0–2 transition starts to appear. In addition, in the presence of the pump light,
the sign of the susceptibility is inverted, compared with the weakly excited case. The