Download Chapter III. Scattering and Extinction of Evanescent waves by small

Chapter III. Scattering and Extinction of Evanescent waves by small particles
III
Scattering
and
Evanescent
of
observed for size parameters of order unity
Small
or larger. This critical size parameter
Extinction
Waves
by
depends on the real part n of the index of
Particles
The theory laid out in chapter II.2 showed,
refraction. It decreases, if n increases. In
that, due to the strong transverse intensity
order to obtain strong MDR’s, it is
gradient of evanescent waves, high-order
necessary to have a low absorption, i.e. a
multipoles
low imaginary part of
are
strongly
enhanced
in
the index of
scattering and extinction of evanescent
refraction. The MDR’s of large dielectric
waves, compared to plane waves. In many
spheres are also called whispering gallery
cases this leads to strongly enhanced
modes (WGM’s).
resonances in the corresponding spectra.
If the radius of the particles increases,
of
the number of MDR’s and their amplitudes
homogeneous spheres and for coated
increase, whereas their half widths and the
spheres are discussed in this chapter. In
distances between them decrease in a given
addition, the range of refractive indices,
spectral
particle sizes, and thicknesses of the
applications
coating layer is investigated, for which
resonances can be important, e.g. in order
well-defined multipolar resonances are
to allow Rabi-splittings, other nonlinear
obtained in the visible spectral range. A
effects or altered spontaneous emission
plane
generally
characteristics to be observed at low pump
produced by total internal reflection from
power. E.g. for a silica sphere with radius
an interface between two media. We
65 µm and refractive index n = 1.4518 at a
initially want to assume, that the distance
wavelength of 900 nm the half-width
of the particles from the interface is large
amounts to 270 kHz, corresponding to a
enough to justify the neglect of multiple
quality factor Q = 1.4.109 )1. This value is
scattering effects. The latter will be
still limited by residual absorption and
discussed later in this chapter.
does not represent the intrinsic Q value.
Specific
examples
evanescent
for
wave
the
is
case
range.
very
In
quantum
sharp
and
optical
strong
An example for the significant changes
III.1
Isolated Particles in Free Space
Homogeneous dielectric spheres
For dielectric spheres, the only possible
resonances are the so-called morphologydependent resonances (MDR’s), which are
due to constructive interference and are
in the scattering spectra due to evanescentwave excitation is shown in Fig. III.1,
which displays the scattering cross sections
of a spherical diamond particle of diameter
2a = 600 nm. Similar results would be
obtained for particles of lower refractive
40
Chapter III. Scattering and Extinction of Evanescent waves by small particles
2
TE 6
TM TM 5
TE7 6 TE
TE3
5
TM 4 TE
4
TM 3
TM 2
The quantum numbers shown in Fig. III.1
correspond to the multipolar order of the
2
Scattering Cross Section [ µm ]
1
Fig. III.1 and is not explicitly given there.
plan e wa ve
resonance and, hence, describe the angular
0
50
dependence of the scattered field. From the
TM 7
structure of the evanescent waves it
TM 6
p-p ol
TM 5
TM 4
0
100
TM 3
follows, that TM modes can be excited
more efficiently by p-polarized (TM)
evanescent waves and TE modes by s-
TE 7
s-pol
TE6
polarized (TE) evanescent waves.
50
TE5
0
0.5
0.6
TE 4
0.7
In total, Fig. III.1 demonstrates the
TE3
0.8
0 .9
W avelength [ µm]
Fig. III.1: Scattering spectra of a spherical
diamond particle of diameter 2a=600 nm for an
incident plane wave (a) and for p- and spolarized evanescent waves (b, c), respectively.
For the evanescent waves an angle of incidence
of 60° of the totally reflected wave and a
constant index of refraction of the prism np = 1.5
is assumed. Note the different scales in b, c.
extremely efficient excitation of MDR’s by
evanescent waves compared to the case of
plane-wave excitation2. For comparison,
the different scales of the ordinate axes
should
be
taken
into
account.
The
difference is easily understood by looking
at Fig. III.2, which shows the instantaneous
intensity distribution of the total field for
index (e.g. polystyrene spheres), if a
somewhat larger diameter was considered.
Obviously, evanescent-wave excitations
do not modify the peak positions and the
half widths, but merely the amplitudes of
the resonances. Due to the oscillatory
nature of the Ricatti-Bessel functions in the
denominator of the wavelength-dependent
Mie-coefficients multipole resonances of
the same orders n exist in the UV regime
but are not considered here. For this
reason, the modes are characterized by a
second quantum number marking the radial
dependence. This radial quantum number
is equal to 1 for the resonances visible in
the TM14 resonance at λ = 386 nm of a
polystyrene sphere, r = 700 nm, excited by
a p-polarised evanescent wave. The figure
suggests
a
mechanical
analogue
for
evanescent-wave excitation of WGM’s
consisting of a chain (representing the
evanescent wave) driving a gear (representing the WGM). Fig. III.3 displays the
time-averaged intensity of the scattered
field alone in the same case on linear and
logarithmic intensity scales. It may be
easily recognised, that the intensity is
confined to a narrow ring in the plane of
incidence close to the inner surface of the
sphere. If, contrary to the procedure laid
41
Chapter III. Scattering and Extinction of Evanescent waves by small particles
wave excitation of WGM’s demonstrated
in Fig. III.1. Correspondingly, the field
distribution shown in Fig. III.4 contains
significant
contributions
from
other
multipoles.
Another remark should be made here
concerning the angular momentum of the
electromagnetic field: In the case of
evanescent-wave excitation of the MDR of
a homogeneous particle (Fig. III.2, Fig.
III.3) the ring-shaped pattern of intensity
Fig. III.2: Total instantaneous intensity for the
case of the TM14-resonance (λ=386nm) of a
polystyrene sphere (radius r=700 nm) excited by
a p-polarized evanescent wave which propagates
from top to bottom of the image. Multiple
scattering at the interface, where the evanescent
wave is generated by total internal reflection, is
neglected here.
packets is rotating in the plane of incidence
out in chapter II, the axis of quantization
is chosen perpendicular to this ring, then,
for evanescent-wave excitation, the WGM
contains predominantly the m = n component of the vectorial multipole. For
comparison, Fig. III.4 shows the same
quantity for the case of plane-wave
excitation. Here, the plane wave propagates from left to right. The intensity
Fig. III.3: Same model as in Fig. III.2, but here the
time-averaged intensity of the scattered field is
shown. Left: Intensity shown in the same plane as
in Fig. III.2. Right: Intensity perpendicular to it.
Top: linear scale, Bottom: logarithmic scale
distribution is now very different from the
one discussed above, and, in particular, is
not confined to a narrow ring. Whereas in
Fig. III.3 the WGM strongly dominates the
intensity, the sphere acts mainly as a ball
lens, when excited by a plane wave, and,
correspondingly, the WGM is hardly
recognisable in Fig. III.4. This is in
Fig. III.4: Same as Fig. III.3, but for plane-wave
excitation. Left: Time-averaged intensity, Right:
Instantaneous intensity
agreement with the inefficiency of plane42
Chapter III. Scattering and Extinction of Evanescent waves by small particles
corresponding to a nonvanishing angular
excitation, where two counterpropagating
waves are generated, which travel close to
the inner surface of the sphere and produce
a
nonrotating
intensity
pattern
with
vanishing angular momentum and much
smaller intensity.
TE 2/ TM 1
2
TE 3 TM
2
TM 3 TE
TM 2
TM 3
0.4
Scattering Cross Section [ µm ]
This is contrary to the case of plane-wave
TE 2
0.6
momentum of the electromagnetic field.
a
1
0.2
0.0
TM 2
2
b
TM 3
1
TM 3
0
TE3
4
c
2
Homogeneous semiconductor spheres
TE3 TE 2
0
As a further example, Fig. III.5 depicts the
0.4
sections of a compact crystalline silicon
0.6
sphere with 2a = 200 nm excited by a plane
respectively.
Compared to diamond (n=2.4), silicon
has a much larger refractive index (n=3.56) in the visible range. In addition, because
the excitation energy is above the bandgap
TM 1 /TE 2 TM 3 TE2
whereas diamond is transparent in this
range.
The
scattering
spectra
are
0.8
0.9
TM 3 TE3
TM 2
a
TE1
2
0.2
TM 2
0.0
TM 3
TM 2
b
2
TE3
TM 3 TM 2
1
0
TE3
6
c
4
for Si, there is considerable absorption,
0.7
0.4
Extinction Cross Section [ µm ]
polarized evanescent waves (Fig. III.5b,c),
0.6
W avelength [ µm]
optical scattering and extinction cross
wave (Fig. III.5a) and by p- and s-
0.5
TE4
2
TE 2 TE 3
TE
2
0
0.4
0.5
0.6
0.7
0.8
0.9
W avelength [ µm]
qualitatively similar to those of the
diamond
particle.
Contributions
from
higher-order multipoles to the extinction
cross section are, however, stronger than
for the scattering cross section, as the
Fig. III.5: Spectra of the scattering and extinction
cross sections for a homogeneous silicon particle
in vacuum with 2a=200 nm for plane-wave
excitation (a), and p-polarized (b), s-polarized (c)
evanescent-wave excitation. Note the different
scale in Fig. III.5b,c. For the calculations, indices
of refraction were taken from ref. 3.
squared absolute values of the Mie
coefficients, which appear in (II.60) of
wave excitation, the absorption (extinction
chapter II.2.2.2 decrease faster with order n
minus scattering) is marginal in the case of
than the real parts (II.59). E.g. Fig. III.5b
the TM2-mode, whereas the absorption
shows, that for p-polarized evanescent-
cross section case is of the same order of
43
Chapter III. Scattering and Extinction of Evanescent waves by small particles
magnitude as the scattering cross section
polarised evanescent-wave excitation, re-
for the TM3-mode. Comparison of the TE2
spectively (compare eqs. II.59, II.60 and
and TE3 modes for the case of excitation
Fig. II.3 of chapter II.2.2.2). The compari-
by a s-polarised evanescent wave yields
son between the cross sections for plane-
qualitatively the same results (Fig. III.5c).
wave and evanescent-wave excitation de-
It should be added, that the optical
pends on the definition of the normalizati-
absorption cross section generally exhibits
on factor of the cross sections, but the
a stronger contrast between off-resonant
comparison between both polarizations in
and
the case of the evanescent wave excitation
on-resonant
conditions
than
the
scattering cross section.
does not.
For larger silver particles, like those of
Homogeneous metallic spheres
Fig. III.7 (2a=200nm), the extinction cross
Very small particles exhibit the well-
section is strongly enhanced for shorter
known dipolar surface plasmon resonance
wavelengths in the case of evanescent-
at around 365 nm due to collective electron
wave excitation. This effect is particularly
oscillation. As a first example, we consider
distinct for the case of excitation by a p-
the extinction cross sections of a very
polarized evanescent wave. The ratio of the
small silver particle (Fig. III.6). Because in
excitation-dependent cross sections in Fig.
this case, the Mie coefficients an of the
III.7 now depends on the wavelength, in
electric multipoles are much larger than the
the spectra. Furthermore, because the
particle is so small, only the dipolar
coefficient a1 contributes to the spectra to
any significant extent. The shape of the
extinction spectrum for evanescent waves
is therefore unchanged in this case
compared to plane-wave excitation, and
the polarisation-dependent cross sections
for extinction as well as for scattering are
for all wavelengths very nearly in the ratio
N −0 1Τ1 / 1 / N 0−1Π 1 ( Π 1 = 1 ) for p-pola-
rised evanescent-wave, plane-wave, and s-
1.5 x10
-4
1.0 x10
-4
2
multipoles, the TE modes do not appear in
Exti n ctio n Cross Se ction [ µm ]
Mie coefficients bn of the magnetic
p -p ol
plan e
wave
5.0 x10
-5
s-p ol
0 .0
0.3
0.4
0 .5
0 .6
W avele ngth [ µm ]
Fig. III.6: Wavelength dependence of the cross
section for extinction of plane waves and
evanescent waves by a spherical particle with
2a=10 nm. The solid lines are calculated using
(eqs. II.59 and II.60) for evanescent-wave
excitation and the formulae of standard Mie
theory for plane-wave excitation, respectively.
The squares indicate numerical results obtained
by means of the multiple multipole (MMP)
method. Multiple scattering at the prism surface
is neglected here.
44
Chapter III. Scattering and Extinction of Evanescent waves by small particles
2
Extinction Cross Section [ µm ]
excitation
0 .4
contains
stronger
contributions from higher multipoles due
0 .3
to the large transverse field gradient of the
p-pol
0 .2
evanescent wave. The difference in the
0 .1
cross sections for s-polarized and p-
plane wave
s-pol
0 .0
itself
0 .4
0 .6
0 .8
1 .0
W avele ngth [ µm]
Fig. III.7: Wavelength dependence of the cross
sections for extinction of plane waves and
evanescent waves by a spherical particle with
2a=200nm. As in Fig. III.6a the solid lines are
the analytical results from (II.59 and II.60) and
the squares indicate the numerical results
obtained by means of the MMP method.
polarized
evanescent-wave
excitation,
respectively, for a spherical particle is
caused by the different internal structure of
the exciting field in both cases. In the case
of s-polarization, the electric field of the
wave oscillates perpendicular to the plane
of incidence, whereas in the p-polarized
contrast to the spectra in Fig. III.6. In order
to understand the spectra shown in Fig.
III.7, Fig. III.8 displays the decomposition
case, the electric field rotates in the plane
of incidence due to the complex phase of
the totally reflected wave. For this reason,
of the extinction spectra into the different
Multipole deco mposition
multipolar contributions for p-polarized
0.4
plane waves (Fig. III.8b), the numbers
0.3
indicating the multipolar order n of the
TMn-resonances.
0.2
1
3
4
maxima in the extinction spectra are
0.15
2
σex t [µm ]
0.0
multipoles in the case of evanescent-wave
2
0.1
Fig. III.8 clearly reveals, that the
caused by a strong enhancement of higher
a
2
σex t [µm ]
evanescent waves (Fig. III.8a) and for
b
0.10
0.05
1
2
excitation. The electric quadrupole and
particularly the electric octupole give rise
to increased extinction in the range of
shorter wavelengths and a double-peak
structure, if the excitation is a p-polarized
evanescent wave. The enhancement of the
higher multipoles is not caused by an
excitation-dependent susceptibility of the
particles, but merely by the fact, that the
3
0.00
0.3 0.4 0.5 0 .6 0.7 0.8 0.9 1.0
W avelength [µm]
Fig. III.8: Decomposition of the extinction cross
section into individual multipole contributions for
a silver particle with 2a=200 nm. Excitation ppolarized evanescent wave (a), plane wave (b).
The numbers indicate the order of the multipole,
whose contribution to the total extinction cross
section is shown. Results for s-polarized
evanescent waves are not shown here, but are
similar to those for p-polarization.
45
Chapter III. Scattering and Extinction of Evanescent waves by small particles
TM 4
TM 5
*
ε''
0
-5
Extinctio n Efficie ncy
D i ele ctric function of silve r
5
R e so na nc e Po sition s
H om o g en eo u s S ph ere :
n=1
n=2
n=3
Co a te d S ph ere :
n=1
n=2
n=3
ε'
-10
-15
TM 3
TM 2
TM 1
10 0
50
0
0.4
-20
0.5
0.6
0.7
0.8
0.9
Wa velen gt h [ µm]
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
W a velen gth [ µm]
Fig. III.9: Resonance positions of Mie coefficients
an, bn with n<=3 for silver spheres of diameter
2a=22 nm and for silver-coated spheres with core
diameter 2a=20nm and shell thickness d=1 nm.
The surrounding medium is assumed to be
vacuum.
for a spherical particle in front of a
metallic mirror a polarization-dependent
Fig. III.10: Extinction spectra of silica particles
coated with a silver shell of thickness d=10 nm
for p-polarized evanescent-wave excitation and
core radius a=20nm to 150 nm in 10-mm steps
(arrow marks increasing core size). To account
for the variation in the cross section for the
different particle sizes the cross sections have
been normalized to the geometric particle cross
sections to obtain the extinction efficiency. Ppolarized evanescent-wave excitation. The line
marked with a star arises from all multipoles
appearing in the spectrum (compare Fig. III.9).
Again εM =1.
cross section is also expected even for the
case of plane-wave excitation. In order to
demonstrate, that the numerical calculations yield the same results as the analytical
formulae, the symbols representing the
numerical results have been added in Fig.
III.6 and Fig. III.7. The numerical method
will be used below to take multiple
coated spheres can be calculated by
formula (II.84 and II.85). The imaginary
part of the index of refraction can be
neglected
for
silver-coated
particles,
because it is sufficiently small for silver
and does not vary drastically with the
wavelength. The index of refraction for
silver, as well as the resonance positions
scattering at the interface into account.
for a compact silver sphere with diameter
2a=22 nm and for a coated sphere with a
Coated spheres
Whereas
for
homogeneous
dielectric
spheres either a large sphere radius or a
sufficiently
large
refractive
index
is
required to obtain morphology-dependent
resonances, comparatively small particles
can exhibit resonances in the visible
spectral region, if they are coated by a
metal. The resonance position for the
silica core of diameter 2a= 20 nm and a
silver shell of thickness d=1nm are shown
in Fig. III.9 for a broad spectral range.
Fig. III.10 displays the dependence of
the extinction spectra of silica particles
coated with a silver shell of thickness d =
10 nm for the case of p-polarised
evanescent-wave excitation and core radii
46
Chapter III. Scattering and Extinction of Evanescent waves by small particles
1.0
1.0
n=1
0.9
n=2
0.8
0.8
n=3
0.7
0.7
d = 1 nm
0.6
d = 2 nm
0.6
0.5
0.4
0.5
0
20
40
60
80
100
120
140
0
20
Partic le Radius a [nm ]
40
60
80
100
120
140
0.4
Partic le Radius a [nm ]
1.0
1.0
n=2
d = 5 nm
0.9
d = 10 nm
0.9
0.8
n=1
0.7
0.7
n=2
n=4
0.6
0.8
n=1
n=3
n=3
n =5
0.6
0.5
0.5
W avel ength [µ m ]
Wavelength [ µm]
W avelength [µm ]
Wavelength [µ m]
n=1
n=2
0.9
n=5
0.4
0
20
40
60
80
100
120
140
Partic le Radius a [nm ]
0
20
40
60
80
100
120
140
0.4
Partic le Radius a [nm ]
Fig. III.11: Resonance positions of the electric multipoles TMn for silver-coated silica particles as a function of
core radius and thickness of the shell. The straight lines are guidelines for the eye.
a = 20 to 150 nm in 10 nm steps. For
orders broaden and shift to the red. The
clarity the spectra are shifted vertically by
resonance at λ ≈ 340 nm corresponding to
a
individual
ε1(Ag) ≈ 0 remains at approximately the
multipolar contributions are identified. The
same position for all core sizes. It should
weak line marked with a star arises from
be mentioned, that the calculated spectra
all multipoles appearing in the spectrum
shown in Fig. III.10 do not take into
and corresponds to the resonances close to
account size-dependent corrections of the
ε1(Ag) = 0, compare Fig. III.9. For the
dielectric function, which are known4 to be
smallest core radius of a = 20 nm only the
necessary for small metallic structures.
dipole contribution is observed. With
These corrections essentially lead to a
increasing core size higher multipoles
broadening of the calculated resonances.
constant
amount.
The
appear and the splitting between the
Fig. III.11 shows the dependence of the
resonance positions increases. At the same
observable resonance positions on core
time resonances due to lower multipole
size for thicknesses d = 1,2,5,10 nm. The
47
Chapter III. Scattering and Extinction of Evanescent waves by small particles
lines through the calculated positions are
discussed. The following discussion is also
only guidelines to the eye, but to a good
relevant for apertureless near-field optical
approximation a linear relationship is
microscopy (chapter VI).
observed.
In an attempt to verify the calculations
Homogeneous silver spheres:
for metal-coated spherical particles shown
As an example, the cross sections for a
above homogeneously silver-coated latex
spherical silver particle in contact with a
spheres have been prepared in our group in
glass prism were caclulated via the MMP
the meantime5. Smooth silver coatings of
technique and are shown in Fig. III.12: a
thickness d=11.5 nm were obtained.
redshift of the spectra due to the presence
Confocal wavelength-resolved scattering
of the interface is universallyly recognised.
microscopy of individual particles of this
kind did not reproduce, however, the
discussed
above,
and
a
particle. This is ascribed to the strong
influence
of
the
unavoidable
Extinction
spectra was obtained from partricle to
2
considerable variation of the scattering
Cross Section [µm ]
resonances
grain
0,2
plane wave
Scattering
Effects of Multiple Scattering at
p-pol
0,2
plane wave
0,0
p-pol
In evanescent-wave scattering experiments
the particle is almost always in contact
with the interface, where the evanescentwave is generated by total internal
Absorption
2
the Surface of a Substrate
Cross section [µm ]
2
Cross Section [µm ]
scattering spectra (compare here also
III.2
p-pol
0,0
structure of the silver shell on the
chapter V).
0,4
0,2
0,0
plane wave
0,4
0,6
W avelength [nm]
reflection. Modifications of the scattering
spectra due to backscattering from the
interface must then be taken into account.
These effects will now be discussed for a
Fig. III.12. Comparison of the cross sections of a
silver sphere with diameter 2a=200 nm in free
space (solid lines) and on the glass prism (squares)
for excitation with a plane (lower two curves) and
a p-polarized evanescent wave (upper two curves).
number of examples. Applications to total
internal
reflection
microscopy6
are
48
Chapter III. Scattering and Extinction of Evanescent waves by small particles
Moreover, the scattering cross section is
1.5
enhanced in the wavelength range λ > 0.4
absorption is strongly reduced in ppolarisation for λ < 0.4 µm.
Homogeneous dielectric spheres
In
this
case,
again
considerable
modifications of the scattering spectra
occur. As an example, in Fig. III.13 the
2
S ca tte rin g Cros s S ec tion [µm ]
µm in both polarisations, whereas the
1.0
TM
TE 5
TM 6 6 TE
5
TM 4 TE4
TM 3
a
0.5
0.0
TM 6
TM 7 TM 5
5
b
TM 4
TM 3
TM 2
0
10
T E7
TE 6 TE5
c
TE 4
TE 3
5
cross sections for a spherical diamond
particle in contact with a glass prism are
shown: Compared to the isolated particle,
the resonance wavelengths are again
slightly shifted to longer wavelengths. The
resonances are broadened considerably,
and, correspondingly, their amplitudes are
reduced substantially.
0
0.5
0.6
0.7
0.8
0.9
W avelength [µm]
Fig. III.13: Scattering spectra for the spherical
diamond particle of Fig. III.1 (diameter 2a = 600
nm), when it is in contact with the prism surface.
Multiple scattering is included in the calculation.
Refractive index of the prism nP = 1.5. Evanescent
waves as in Fig. III.1. In the plane-wave case a
plane wave is assumed to be incident normally on
the prism surface from the prism side.
An equivalent model consisting of a
spherical
scattering
object,
which
is
surface, which means that the impedances
embedded in a medium with a higher index
are parallel. Simplifying the surface as an
of refraction approximately describes the
equivalent electric circuit consisting of two
spectra. If the index of refraction of the
serial or parallel capacitances, respectively,
environment increases, the resonances shift
a larger effective dielectric constant for the
to longer wavelengths and in this way this
TM modes is obtained. Comparison of Fig.
simple model can describe the observed
III.1 and Fig. III.13 makes it clear, that
red shift.. The red shift is larger for TM
large resonance enhancements are possible,
modes than for TE modes. This may be
when the particle is held at some distance
explained in a heuristic fashion by the fact,
from the prism surface. On the other hand,
that for TE modes the electric field is
even when the sphere is in contact with the
parallel to the surface, which corresponds
surface, the spectra for evanescent-wave
to a serial arrangement of the impedances
excitation differ strongly from those for
of both halfspaces, whereas for TM modes
plane-wave excitation, and resonances with
there is a field component normal to the
large peak amplitudes are observed for s49
Chapter III. Scattering and Extinction of Evanescent waves by small particles
as well as p-polarised evanescent waves.
changes of the scattered field due to
The broadening and the reduction of the
multiple scattering processes, if the particle
peak heights may be explained by the
is in contact with the interface.
disturbance of the total internal reflection
Multiple scattering effects like those
due to multiple scattering processes. The
shown in Fig. III.14 are obviously highly
latter result in a smaller transverse field
relevant for Total Internal Reflection
gradient, equivalent to reduction of the
Microscopy (TIRM)6. In this experimental
multipolar enhancement factors, and lead
technique light scattering by particles,
to a strong energy flux into the prism,
which move in an evanescent wave, is used
equivalent to a reduction of the quality
to study the motion of the particle
factor. This energy flux is described by the
perpendicular to the interface with high
Poynting vector. Fig. III.14 shows the
sensitivity to small changes in the distance
time-averaged Poynting vector for the TM6
to the interface, and to characterise in this
resonance of the particles of Fig. III.1 and
way particle-interface potentials.
Fig. III.13 and demonstrates the large
It is thereby generally assumed, that the
Fig. III.14: Time-averaged Poynting vector of the scattered field for a diamond particle of radius a=300nm
and for p-polarized evanescent-wave excitation at the TM6 resonance at 456 nm in free space (a), and at 458
nm, when the particle is in contact with the prism surface (b). The latter is indicated by the straight line, and
the prism is to the right of the line. The evanescent wave propagates from the top of the figure to the bottom
and decays from the right to the left.
50
Chapter III. Scattering and Extinction of Evanescent waves by small particles
scattered power is proportional to that of
size. Scattering cross sections for a p-
the exciting evanescent wave at the center
polarized evanescent wave at two different
of the sphere, and, hence, decreases expo-
wavelengths, λ = 458 nm, corresponding to
nentially with the distance d to the inter-
the resonance ( TM 16 ) of the isolated
face. Under this assumption, the particle-
sphere, and λ = 488 nm, corresponding to
wall interaction potential can be calculated
from the probability distribution of z and
the use of the Boltzmann probability
distribution: w(d) ~ exp(φ(d)/kT). This was
first applied by Prieve et al.6 in order to
study colloidal forces in the proximity of a
surface.
In
recent
experiments7
the
Coulomb interaction between negatively
charged particles (assumed to be infinitely
hard
in
first
approximation)
and
a
negatively charged surface was determined
in this way. Gravity appears as an
additional potential in this experiment.
The method is very sensitive, because
off-resonant excitation, were determined as
a function of d. The differential scattering
cross section was integrated I) over a solid
angle of 4π in order to obtain the total
power scattered by the particle and II) over
a finite solid angle (aperture angle 53°) in
the vacuum half space above the particle in
order to calculate the signal obtained
experimentally with a N = 0.8 microscope
lens.
Fig. III.15 displays the results for the
case of resonant excitation of the whispering-gallery resonance TM 16 at λ = 458 nm,
the forces acting on the particles can be
as well as for off-resonant excitation λ =
measured in the range of 10 fN. The
488 nm. As in Fig. III.14b strong scattering
calculated potential, however, will be
into the prism is easily recognised, when
affected by any deviation from the
the particle is in contact with the substrate
exponential dependence due to multiple
(Fig. III.15a,b). Moving the particle away
scattering effects in the vicinity of the
from the surface by a few hundred
surface. The method can be used for the
nanometers results in a drastic reduction of
examination of surfaces, e.g. to distinguish
this directional scttering into the prism
between surfaces of different polarity.
(Fig. III.15c,d). Furthermore, in the case of
We have calculated the power scattered
resonant excitation of a WGM, the diffe-
from a diamond sphere (r=300 nm, same as
rential cross section for scattering scatter-
above) as a function of the distance d from
ing into the vacuum half space exhibits
the interface, where the evanescent wave is
much stronger modulations as a funtion of
generated. Similar results are expected for
the scattering angle than in the case of off-
polymeric particles of somewhat larger
resonant excitation. This holds independent
of the particle-interface distance. The
51
Chapter III. Scattering and Extinction of Evanescent waves by small particles
Fig. III.15: Absolute value of time-averaged Poynting vector for a diamond particle of radius a=300nm,
which is excited by a p-polarized evanescent-wave of wavelength of λ=458 nm (TM16 resonance: part (a,c))
and 488 nm (off-resonant excitation, b,d), respectively. The images a,b show the results for the particle in
contact with the interface, where the evanescent wave is generated, and b,d those for the particle 550 nm
above it. The boundary is indicated by the straight line, and the optically denser medium is to the right of this
line. The evanescent wave propagates from the top of the figure to the bottom and decays from the right to
the left. The scattered fields are shown (left and middle part) in the plane of incidence. The modification of
the corresponding potentials (explanation in the text) due to multiple scattering processes are shown in the
right part for the resonant (e) and the non-resonant (f) case. The inserts in (e,f) show the integrated powers
on a logarithmic scale. The main graphs show the integrated powers on a linear scale after normalisation to
the expected exponential dependence. The black line represents the corrections necessary in TIRM.
inserts in the graphs (Fig. III.15e,f) show
exponential is clearly observed. This is
the integrated powers on a logarithmic
related to the strong scattering into the
scale. In the main graphs these results are
prism, when the particle is close to contact
shown on a linear scale after normalisation
with the interface. The power scattered into
to the expected exponential dependence.
the lens (black lines), on the other hand
Due to multiple scattering effects
decreases by about 25% in the case of
involving the substrate and particle boun-
resonant excitation and by about 10% for
daries, an increase of the total scattered
off- resonant excitation, relative to the
power
expected exponential dependence. This can
(red
lines)
over
the
simple
52
Chapter III. Scattering and Extinction of Evanescent waves by small particles
be explained by the strong directional
almost tangential to the interface. This
emission into the medium with the higher
applies to the scattered intensity in both
index of refraction discussed above, which
leads to reduced scattering into the vacuum
half space. The inserts in both graphs
show, that the intensities have to be
determined
with
high
accuracy
over
several orders of magnitude, which is
experimentally quite difficult.
Our calculations show, that it is
important at which wavelength the TIRM
experiment described above, is carried out
and that sizable deviations from the
exponential distance dependence indeed
exist, especially at wavelengths, where
whispering gallery modes (WGM’s) of the
sphere are resonantly excited. These
corrections lead, of course, to altered
particle-wall potentials.
We have investigated not only diamond
particles in air, but also polystyrene
spheres in water on a substrate surface,
because real experiments often employ this
set-up. The corresponding results at a
wavelength of λ = 500 nm are presented in
Fig. III.16. If the particle is in contact with
the surface, a highly directional emission
into the optically denser medium is again
observed. Because of the relatively small
difference of the index of refraction of
water and polystyrene and the relatively
small particle diameter, the light is
scattered mainly in the direction of
propagation of the evanescent wave,
Fig. III.16: Absolute value of the time-averaged
Poynting vector for a polystyrene particle of radius
a = 300 nm in water, which is excited by a ppolarized evanescent wave at a wavelength of λ =
500 nm. The images (a,c) show the results for the
particle in contact with the interface, where the
evanescent wave is generated, and the images (b,d)
those for the particle 550 nm above it. The
boundary is indicated by the straight line, and the
optically denser medium is to the right of this line.
The evanescent wave propagates from the top of the
figure to the bottom and decays from the right to the
left. The scattered fields are shown in the plane of
incidence (a,b) and perpendicular to it (c,d). The
main graphs in part (e) show the integrated powers
on a logarithmic scale. In the insert, these results are
shown on a linear scale after normalisation to the
expected exponential dependence.
53
that most of the scattered power does not
enter the lens in this case.
The total scattered power depends on
the surface-to-surface distance in a way
quite different from the case considered
above. Here, not only the power scattered
2
media (glass, H2O). It is easy to recognize,
scattering cross section [ µm ]
Chapter III. Scattering and Extinction of Evanescent waves by small particles
2x10
monomer, p-pol
monomer, s-pol
dimer, p-pol
dimer, s-pol
2
T E14
T M 14
1x10
2
T E13
T E15
TM 13
T M 12
0
380
400
420
440
wav elength [n m]
into the solid angle in water, corresponding
to the numerical aperture of the lens, but
also the total scattered power decreases
with
decreasing
surface-to-surface
Fig. III.17: Scattering cross sections for two
contacting polystyrene spheres (r=700nm) excited
by an evanescent wave. The spectra were
calculated using the MMP technique. Multiple
scattering processes due to the presence of the
substrate surface have been neglected here.
distance. Besides, the power scattered into
the lens decreases already for a distance of
about 500 nm. This must be ascribed to the
change in the field distribution with
increasing distance, which is different in
the present case, compared to the case
considered in Fig. III.15.
III.3
Interacting Dielectric Spheres in
free Space
interact due to the evanescent fields of
these modes outside of the particles. The
coupling is, however, relatively weak, and
the condition for constructive interference
of the WGM is approximately unchanged.
Therefore, the resonant modes of two
interacting spheres (‘photonic molecule’8)
can still be classified according to the TE
and TM modes of the individual spheres.
In many cases scattering and extinction of
The main effect of the interaction is then a
more than one particle is of interest. If the
splitting of the whispering-gallery resonan-
particles are close to each other the most
ces. This is demonstrated in Fig. III.17,
important change in the spectra is a
which displays the scattering spectra of
splitting of the resonances due to the
two interacting polystyrene spheres, r =
interaction of the resonant modes. For
700 nm, in a narrow spectral range. The
metallic particles the situation becomes
spectra have been calculated using the
actually quite involved the delocalised
MMP technique, and multiple scattering at
nature of the electronic wavefunction. This
the substrate surface has been neglected.
is discussed in detail in chapter V. In the
Fig. III.19 displays the absolute value of
present paragraph only dielectric particles
the time-averaged (a,c) and instantaneous
are considered. In this case the whispering-
(b,d) Poynting vector for the components
gallery modes of the individual spheres
of the split TM14 resonance at 384 nm (a,b)
54
Chapter III. Scattering and Extinction of Evanescent waves by small particles
Fig. III.18: Dimer
(same configuration
as in Fig. III.17 and
Fig. III.19) on the
substrate of a surface. (a), (b): instantaneous intensity
(c):
timeaveraged intensity
of the scattered
field. (a) linear
scale (ratio of the
maximum and the
minimum intensity:
10), (b,c) logarithmic scale (30dB).
The bright colors
feature the regions
of
high
field
intensity
(see
colorscale at the
bottom)
Fig. III.19: Absolute value of the time-averaged
(a,c) and instantantaneous (b,d) Poynting vector
of the scattered field (total field minus field in the
absence of the particles) for the split components
of the TM14 mode of the same configuration as in
Fig. III.17. λ=384nm (a,b) and λ=388nm (c,d),
respectively.
and 388 nm (c,d). The two resonances
III.4
Interacting Dielectric Spheres
on the Surface of a Substrate
differ, because the phase difference of the
two WGM’s of both spheres is different at
If the particle dimer is now placed on a
the two wavelengths. This phase difference
substrate surface, multiple scattering pro-
is determined by the wavelength of the
cesses lead again to a substantial modifi-
evanescent wave and the distance of the
cation of the intensity distribution (Fig.
two particles. It controls the coupling of
III.18). As in the case of an isolated
the two WGM’s at the contact point. Fig.
particle on a substrate (Fig. III.14, Fig.
III.19 shows, that the almost isotropic
III.15) strong emission into the substrate is
intensity distribution for a single particle in
observed. The emission pattern, however,
the plane of incidence (compare the left
is much more complicated than in the case
column of images in Fig. III.3 becomes
of isolated spheres. This applies also to the
highly anisotropic for a dimer of such
emission pattern into the vacuum half
particles. Light is emitted predominantly in
space.
two directions corresponding to the upper
right and lower left corners of the images
in Fig. III.19.
It is immediately evident, that the field
distribution
will
become
even
more
complicated, if particle arrays containing
more than two particles are considered.
55
Chapter III. Scattering and Extinction of Evanescent waves by small particles
Such arrays may be interesting, e.g. for
rapid intensity variations as a function of
optical computing purposes, or in order to
excitation wavelength can hence be related
create
to multiple splitting of the single-particle
photonic
crystals
with
band
structures in two or three dimensions.
resonances. For large chains or two-
Using the MMP method, we have
dimensional structures, which may be
modeled the scattering of evanescent
considered as ‘photonic crystals’, these
waves by a chain of six latex spheres
splittings correspond, of course, to the
(r=0.7µm) on the surface of a substrate
band structure of the crystal. The k vector
(n=1.5) for different wavelengths of the
of the mode excited by the evanescent
exciting light. Multiple scattering effects at
wave is then determined by the angle of
the boundary of the surface are taken into
incidence of the wave inside the substrate.
account in this calculation. We have
A scan of the latex chain with a SNOM
chosen a range of wavelengths, where
tip at a constant height above the chain
resonances of the single particle occur. The
would scan images depending strongly on
scattering spectrum of such a particle was
the scan height. The images displayed in
shown in Fig. III.17. The images in Fig.
Fig. III.20 show highly directed radiation
III.20 show the absolute value of the time-
into the optically denser medium and, less
averaged Poynting-vector in the plane of
intense, into the vacuum. It is possible to
incidence. Similar models were recently
address different spheres by choosing the
examined by other groups9. Within the
appropriate wavelength. For example, the
spheres
immediate
first image in the top row of Fig. III.20
intensity
(λ=408nm) shows a high intensity at the
distribution is obtained. Noticeable is the
top sphere, and the first image in the
high intensity contrast in the images, and
bottom row (λ=414nm) a high intensity at
the sensitive dependence of the local
the second latex sphere from the bottom.
intensity on the excitation wavelength. In
This ‘addressing’ of individual spheres
order to understand the basic features of
concerns both, the radiation into the
these results, it is useful to reconsider the
substrate and into the vacuum domain.
spectra for two such spheres in contact
This may be of interest for the construction
with each other (Fig. III.17). These spectra
of optical switches in the field of optical
show a splitting of the whispering-gallery
computing. In order to address individual
resonances into two modes. It is expected,
spheres in a 2-dimensional lattice of latex
that in the case of a chain of n spheres, the
spheres, it is necessary to generate two
resonances split into n components. The
exciting evanescent waves, which pro-
and
neighborhood
in
a
the
complicated
56
Chapter III. Scattering and Extinction of Evanescent waves by small particles
pagate in two different directions on the
substrate surface. If the latex array
becomes
very
large,
however,
the
addressing of the inner spheres will
become more difficult, because the inner
spheres in the array become more and
more equivalent. As may be recognized
from Fig. III.20, the radiation is emitted
mainly from the spheres at the end of the
chain and only weakly from the spheres in
the middle part.
Whispering-gallery
modes
can
be
efficiently excited not only by evanescent
waves, but also by light-emitting dipoles
within the spheres. Experimentally, such
active spheres can be realized by doping
the spheres with fluorescent molecules,
semiconductor quantum dots or other
fluorescent particles10. The latter may be
excited by a plane wave, whose frequency
fits to an absorption of the dopant. Lasing
in micro-cavities is currently studied in
many
groups,
and
in
this
context,
stimulated emission in photonic crystals
made of latex spheres is of considerable
interest.
57
Chapter III. Scattering and Extinction of Evanescent waves by small particles
Fig. III.20: Linear chain of six latex spheres (radius r=700 nm) in vacuum on a glass substrate (n=1.5), where an evanescent wave is generated by total internal reflection
of a plane wave (white arrows) incident under an angle of 60° on the glass-vacuum interface. In the glass medium a standing wave is formed. The evanescent wave
propagating in the vacuum domain is scattered by the latex spheres. The intensity (absolute value of the time-averaged Poynting vector) was calculated by the MMPmethod for 8 different wavelengths (values given on top of the individual images) and plotted using a logarithmic color-coded scale.
58
Chapter III. Scattering and Extinction of Evanescent waves by small particles
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L. Colet, V. Levèvre-Seguin, M. Brune, J. M. Raimond, S. Haroche, Europhys. Lett. 23, 327 (1993)
Any sizable absorption, for example due to impurities in the material, would of course broaden the resonances
and strongly reduce the resonant cross sections. In fact the line widths found experimentally in ref. 1 were
limited by residual absorption.
3
E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, New York, 1996)
4
H. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, (Springer-Verlag, Berlin, 1995)
5
A. B. R. Mayer, W. Grebner, R. Wannemacher, J. Phys. Chem. B, 7278 (2000)
6
D. C. Prieve, J. Y. Walz, Appl. Opt. 32, 1629 (1993)
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C. Bechinger, D. Rudhardt, P. Leiderer, Phys. Rev. Lett. 83, 3960 (1999)
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T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, M. Kuwata-Gonokami, Phys. Rev. Lett. 23, 4623 (1999)
9
M. Haraguchi, T. Nakai, A. Shinya, T. Okamoto, T. Fukui, Optical modes of two-dimensionally ordered
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10
H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A 13, 396 (1976)
2
59