Comprehensive surface-wave description for the nano-scale
energy concentration with resonant dipole antennas
(Supplementary Material)
Hongwei Jia,† Philippe Lalanne‡ and Haitao Liu†,*
†
Key Laboratory of Optical Information Science and Technology, Ministry of Education,
Institute of Modern Optics, Nankai University, Tianjin 300071, China
‡
Laboratoire Photonique, Numérique et Nanosciences, Université Bordeaux 1, CNRS,
Institut d’Optique, 33405 Talence cedex, France
* Email：liuht@nankai.edu.cn
1. Techniques for solving complex resonance wavelength
For solving the nonlinear equation 1/F(λ)=0 (defined in Section 4 of the main text), where
F(λ) denotes the complex-valued component of the excited vectorial field that depends on the
wavelength λ of the incident plane wave. Note that some field components at the symmetry plane
of the dipole antenna, for instance, (Ey, Ez, Hx) at x=0, are always zero and cannot be selected as
F. With the notation f(λ)=1/F(λ), an iteration process with the use of linear interpolation is
employed to solve the nonlinear equation f(λ)=0. In the process, two initial values λ0 and λ1 near
resonance wavelength should be set at first. For the iteration, we assume that λn2 and λn1 are
known. Then f(λ) can be approximated with a linear form f(λ)=p+qλ, where the complex
coefficients p and q can be determined by λn2 and λn1, p=[f(λn1)λn2f(λn2)λn1(λn2λn1),
q=(λn2λn1)/[f(λn2)f(λn1)]. Then we get λn=p/q by solving f(λ)=p+qλ=0. Repeating the
iteration process, we obtain a sequence of λn, which approaches λc as n increases. In the iteration,
the complex gold permittivity for complex wavelengths is obtained through an analytical
continuation. The analytical expression for the gold dispersion is obtained through a polynomial
fitting of experimental data of the gold permittivity for real wavelengths [1]. The iteration
process is illustrated in Figs. S1(a1), S1(b1) and S1(c1) for antenna lengths L=0.21, 0.88,
1.56μm [corresponding to resonances m=0, 2, 4 in Fig. 2(a)], respectively, where we set initial
values λ0=0.9μm and λ1=1μm. It is seen that λn converges rapidly to λc with two or three steps of
iteration, and we obtain λc=0.99+0.10i, 1.00+0.026i, and 1.00+0.025i for the three antenna
lengths, corresponding to Q=4.9, 19.3 and 20.2.
For solving λc from Eq. (8) of the SPP model, since all the quantities u=exp(ik0neffL), neff, 
and  are dependent on k0, Eq. (8) forms a transcendental equation with the complex k0=2/λc as
the unknown. To seek the solution, Eq. (8) is rewritten as
k0  
ln(| r ||    |)  i[arg(r )  arg(    )  2 N ]
,
i 2neff L
(S1)
where N is an integer corresponding to different orders of resonant modes. The form of Eq. (S1)
benefits from the fact that the expression on the right side [denoted with g(k0) hereafter] is a
slowly varying function of k0, since the scattering coefficients r, ρ and τ [see Figs. S1(d1)-S1(d3)]
and the complex effective index neff all vary slowly with the wavelength. In the context of
numerical algorithms for solving nonlinear equations [2], g(k0) forms a compressed map of k0, so
that Eq. (S1) can be solved with the iterative formula k0,n=g(k0,n1). The obtained sequence of
k0,n=2/λn will converge to k0,c=2/λc with the increase of the iteration number n. For the iteration,
the initial value is selected as λ0=0.9μm, and the iteration process is shown in Figs. S1(a2)-S1(c2)
for antenna lengths L=0.21, 0.88, 1.56 μm, respectively [setting N=1, 3, 5 in Eq. (S1),
respectively]. It is seen that λn converges rapidly to λc with few iteration steps. The obtained
resonance wavelengths are λc=1.01+0.064i, 1.00+0.029i and 1.00+0.025i, corresponding to Q
factors 7.9, 17.2 and 20.2, respectively.
Thanks to the compressed-map form of Eq. (S1), the iteration process for solving Eq. (S1) is
insensitive to the selection of the initial value λ0. The obtained sequence of λn can be controlled
to converge to λc of different resonant modes by simply setting different values of N. This can be
understood by considering the extreme case that g(k0) is a constant function of k0, for which Eq.
(S1) simply provides an analytical solution of resonance wavelengths that correspond to different
resonant modes for different orders N. In comparison, for solving the nonlinear equation f(λ)=0
with the fully-vectorial a-FMM, the iteration process is sensitive to the selection of the two initial
values λ0 and λ1. The obtained sequence of λn will converge to λc that is closest to the selected
initial values λ0 and λ1.
Next we make a cross check for the solved Q factors. The Q factor of a resonant mode can
be estimated with Q=λres/∆λ, with λres being the real wavelength at the resonance peak of the
spectrum of EF, and ∆λ being the full width at half maximum of the resonance peak [3]. The
spectra of EF for L=0.21, 0.88, 1.56 μm are calculated with the a-FMM and are plotted in Figs.
S1(a3)-S1(c3), respectively. The estimated Q factors are 4.3, 18.8 and 20.0, respectively, well
agreeing with the earlier results obtained with the a-FMM.
Fig. S1 (a1)-(c1) Iteration process for solving the complex resonance wavelength λc with the fully-vectorial
a-FMM. The results are obtained for dipole antennas with arm lengths L=0.21, 0.88 and 1.56μm
[corresponding to resonance orders m=0, 2, 4 in Fig. 2(a)], respectively. (a2)-(c2) Iteration process for solving
λc with Eq. (S1) of the SPP model. (a3)-(c3) Enhancement factor (EF) as a function of wavelength λ obtained
with the a-FMM. (d1)-(d3) |ρ|, |τ| and |r| plotted as functions of wavelength.
2. Calculation of the residual field
For the calculation of the SPP field and the residual field shown in Fig. 5, we firstly
calculate the SPP coefficients c+ and c (defined in Section 5 of the main text) with the a-FMM.
c+ and care calculated with the mode orthogonality theorem [Eq. (2)], which is similar to the
calculation of β and α. Then the residual field is obtained by removing the SPP field from the
field scattered by the antenna. Note that the c+ and care not exactly the SPP coefficients a and b
predicted by the SPP model [see Eqs. (3)], but result from two contributions: the first
contribution (a and b) predicted with the SPP model, and the second contribution excited by the
residual field that is excluded in the model. So we conclude that if the residual field is weak (for
instance, for dipole antennas with long arms that support high-order resonances), there should
have c+≈a and c≈b. This is numerically confirmed in Fig. S2, which shows that at high-order
resonances (m≥1), there are c+≈a and c≈b, while deviation can be observed at resonance m=0.
Fig. S2 SPP amplitude coefficients on the right antenna arm plotted as functions of the antenna length L.
(a1)-(a2) Moduli and arguments of c+ (blue circles, obtained with fully-vectorial a-FMM) and a (black-solid
curves, obtained with the SPP model) for forward-going SPP. (b1)-(b2) Moduli and arguments of c (a-FMM)
and b (SPP model) for backward-going SPP. The antenna parameters are the same as those in Fig. 2(a).
3. Reproducing scattered far field with the model
The field of the dipole antenna in free space (denoted by ) can be expressed as,
Ψ  Ψ PW  Ψ G,s  Ψ R,s  Ψ L,s ,
(S2a)
G,s
G,s
ΨG,s  ΨG,s
PW  b1uΨSPP,+  b2uΨSPP,  ,
(S2b)
R,s
ΨR,s  ΨR,s
PW  a2uΨSPP,+ ,
(S2c)
L,s
ΨL,s  ΨL,s
PW  a1uΨSPP,  .
(S2d)
In Eq. (S2a), Ψ PW is the field for an infinitely-long nano-wire illuminated by a plane wave,
Ψ G,s is the field scattered by the nano-gap, Ψ R,s is the field scattered at the right termination of
the antenna, and Ψ L,s is the field scattered at the left termination. Ψ G,s results from three
contributions given by Eq. (S2b): the first contribution Ψ G,s
PW from the scattering of the incident
plane wave [Fig. 1(c)], and the other two contributions from the scattering of the right-going SPP
G,s
G,s
( ΨSPP,+
with coefficient b1 and damping u) and the left-going SPP ( ΨSPP,
with coefficient b2 and
damping u), see Fig. 1(e). Ψ R,s results from two contributions given by Eq. (S2c): the scattering
R,s
of the incident plane wave Ψ R,s
PW [Fig. 1(d)], and the scattering of right-going SPP ( Ψ SPP+ with
coefficient a2 and damping u), see Fig. 1(f). Ψ L,s is expressed as Eq. (S2d) in a similar way. All
the scattered fields can be obtained with fully-vectorial calculations, from which the incident
field for an infinitely-long nano-wire has been removed.
Figure S3 shows the scattered far-field diagram of the dipole antenna in free space. The
scattered field of the dipole antenna is expressed as Ψsca=ΨΨinc, where Ψ is the total
electromagnetic field of the dipole antenna, and Ψinc is the field of the incident plane wave
without the antenna. Figures S3(a)-S3(c) show the results for the first three orders of resonance
[m=0, 1, 2 in Fig. 2(a)], respectively. The far-field angular distributions of the scattered power
are shown in the left column in Fig. S3, where the blue circles show the results obtained with the
fully-vectorial a-FMM. They are obtained by calculating the modulus of the Poynting vector (i.e.
energy flux density) of Ψsca on a circle with a radius 5μm surrounding the antenna (in the plane
y=0). The corresponding spatial distributions of energy flux density (in y=0 plane) obtained with
the a-FMM are shown in the middle column.
The scattered far-field diagram can be reproduced with Eqs. (S2) of the SPP model. The
predicted angular distribution of scattered power is shown with red-solid curves in the left
column of Fig. S3. The corresponding spatial distribution of energy flux density obtained with
the SPP model is shown in the right column. It is seen that for high-order resonances (m≥1), the
predictions of the model well agree with the a-FMM results. For resonance m=0, the model can
well predict the profile of the far-field diagram, but shows deviation in predicting the absolute
value, which should result from the impact of surface waves other than SPPs.
Fig. S3 Scattered far-field diagram of the dipole antenna. The results for (a)-(c) are obtained for dipole
antennas with arm lengths L=0.21, 0.54 and 0.88 μm [corresponding to resonance orders m=0, 1, 2 in Fig. 2(a)],
respectively. The left column shows the angular distribution of the scattered power, which are obtained with
the fully-vectorial a-FMM (blue circles) and the SPP model (red-solid curves). The middle and right columns
show the modulus of the Poynting vector for the scattered field obtained with the fully-vectorial a-FMM and
the SPP model, respectively. The results are obtained in y=0 plane and the incident plane wave is normalized
with unitary amplitude (with wavelength =1m). The wave impedance in vacuum is set to be 1.
References
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