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Transmission of plasmons through a nanowire Peter Geisler, Enno Krauss, Gary Razinskas, and Bert Hecht ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 13 Jun 2017 Downloaded from http://pubs.acs.org on June 14, 2017
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Transmission of plasmons through a nanowire Peter Geisler,
†, ¶
Enno Krauss,
†, ¶
Gary Razinskas,
†, ¶
and Bert Hecht
∗,†,‡
NanoOptics & Biophotonics Group, Experimentelle Physik 5, Physikalisches Institut, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany ‡ Röntgen Research Center for Complex Material Systems (RCCM), Am Hubland, 97074 Würzburg, Germany ¶equally contributing †
E-mail:
[email protected] Abstract Exact quantitative understanding of plasmon propagation along nanowires is mandatory for designing and creating functional devices. Here we investigate plasmon transmission through top-down fabricated monocrystalline gold nanowires on a glass substrate. We show that the transmission through nite-length nanowires can be described by Fabry-Pérot oscillations that beat with free-space propagating light launched at the incoupling end. Using an extended Fabry-Pérot model, experimental and simulated length dependent transmission signals agree quantitatively with a fully analytical model.
Keywords surface plasmons, nanowires, plasmon propagation, decay length, Fabry-Pérot resonator
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At optical frequencies gold and silver nanowires can be used as subwavelength waveguides due to the considerably shortened eective wavelength of surface plasmon polaritons compared to free-space light. for sensing
57
16,17
Their high eld con nement makes them an ideal candidate
and sub-diraction information processing,
couple to quantum emitters networks.
14
14,15
813
and their ability to strongly
quali es them as a building block for nano quantum optical
High-precision experiments of simple physical systems often reveal subtle but
important eects or can be used to test theoretical descriptions of experimental results. Validated theoretical descriptions can then be used with con dence to model more complex systems. Yet, systematic high-precision experiments of light transmission through nanowires to date hardly exist. The characterization of such waveguides often relies on leakage radiation or mode imaging with uorophores,
19,20
18
which both lead to increased damping during ongo-
ing propagation and may be aected by photobleaching. Coherent white-light transmission spectra of single wires can also be used to analyze their transmission.
21
However this method
requires knowledge of the waveguide's wavelength-dependent optical functions
22
in a broad
wavelength range as well as the analysis of wavelength dependent incoupling and outcoupling e ciencies.
23
As a result, the transmission e ciency of light through such nanowires in
many experiments signi cantly deviates from theoretical expectations based on bulk dielectric constants.
24
To date the origin of such deviations remains unknown since the structural
uncertainties of bottom up and top-down fabricated nanowires are not small enough to allow for conclusive analyses. Here we present a systematic study of the monochromatic light transmission through more than 300 monocrystalline gold nanowires of equal cross section but variable length ranging between 1940 to 8040 nm in length with a 20 nm increment. By varying the length and keeping the operation wavelength xed complexity is avoided since each unknown, like the in- and outcoupling e ciency, can be described by a single (complex) number.
We
demonstrate by experiments and simulations that a quantitatively correct description of the length-dependent nanowire transmission can be obtained by also taking into account free-
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space propagating modes launched by scattering of the excitation spot at the wire input in addition to Fabry-Pérot-type internal plasmon resonances. These free-space propagating modes interact with the outcoupling end of the wire and beat with the regularly emitted photons originating from the wire plasmon's radiative decay. This leads to signi cant amplitude modulations of the Fabry-Pérot transmission resonances. The quantitative agreement between our model, numerical simulations, and measurements validates our model and yields values of propagation parameters that are compatible with bulk dielectric constants and for which remaining sources of uncertainties are clearly identi ed.
Setup To experimentally characterize the transmission properties of nanowires of dierent lengths we use a home-build inverted microscope setup.
25
Nanofabricated single crystal gold nanowires
supported by a cover glass are mounted above an oil immersion microscope objective (Leica, 1.30 NA,
∞,
PL Fluotar 100x) which is used to focus a laser beam ( λ = 800 nm, 12 nm
FWHM spectral linewidth, 80 MHz repetition rate, 50 nW average power measured in front of the objective, NKT Photonics, SuperK Power with SpectraK AOTF, masterseed puls duration 5 ps, after AOTF about 300 ps) via a
λ/2-plate
(Foctec, AWP210H NIR) to a
diraction-limited (390 nm diameter) spot at the airglass interface that is linearly polarized along the wire axis. The same objective is used to image the emitted and re ected light onto a CCD camera (Andor, DV887AC-FI EMCCD) via a 50/50 non-polarizing beamsplitter (Thorlabs, CM1-BS013). In order to avoid saturation of the CCD, the strong re ection of the excitation spot is suppressed by a small beam block (OD 2) introduced in an intermediate image plane. The exact position of the excitation spot with respect to the wire end can be adjusted with nm-precision by moving the sample using a piezo translation stage (Physik Instrumente, P-527) and was optimized to obtain maximum signal intensity at the wire end.
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Model t0 t1
2 1 7
6
5
3
8 9
4
Figure 1: (color online) Sketch of the free-space and guided elds involved in the experiment in a plane through the nanowire long axis perpendicular to the substrate (wire height stretched for better visibility).
The eld intensity distributions are represented in a loga-
rithmic scale and were obtained by FDTD simulations. To illustrate progress in time the intensities of two short pulses with a short time delay
t1 − t0
are combined.
(1) focused
Gaussian source illumination from the glass half space; (2) scattered and transmitted light above the glass surface leading to (3) refracted waves at the airglass interface; (4) scattered and re ected light below the airglass interface; (5) launched plasmon pulse at time
t0 ;
(6) plasmon pulse at time
t1 ;
light emission from the wire end (7) above and (8) below
the airglass interface; (9) scattered and re ected light below the airglass interface at time
t1
leading to additional scattered elds at the wire end. The (green) lines below the glass
surface symbolize the limited accepting angle of the objective with NA = 1.3.
The principle of the experiment and all light propagation channels are sketched in Fig. 1 showing free-space and guided elds as obtained from numerical simulations.
The tightly
focused laser at the incoupling end of the wire (Fig. 1(1)) launches a plasmon with about 30% e ciency (Fig. 1(5)) that propagates towards the outcoupling end (Fig. 1(6)) where it is partly radiated into the surrounding media (Fig. 1(7) and (8)) while about 40% of the plasmon eld is re ected and propagates back along the wire leading to Fabry-Pérot-type standing waves.
26,27
Additional propagating elds - in the following referred to as air-wave
and glass-wave (Fig. 1(2) and (4)) - are launched by the partial scattering of the excitation
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source at the incoupling end.
While about 30% of the laser energy is coupled into the
nanowire, about 50% of the energy is scattered into the glass half space leading to a spherical wave originating from the incoupling end of the wire. The remaining 20% are transmitted into the air half space where they also evolve as spherical waves. Another contribution to propagating waves in the glass half space arises from light refracted at the airglass interface (Fig. 1(3)) which propagates into the glass as a plane wave under an angle of about 43
◦
(critical angle for total internal re ection) and thus within the
acceptance angle of the objective. This wave leads to a distinct pattern in the wire far- eld images that vanishes for low-NA imaging. The air-wave and the glass-wave originating from the excitation position are not directly detected by the camera for dierent reasons. While the air-wave propagates away from the collecting objective lens above the interface the glasswave is strongly suppressed by the beam block at the intermediate image plane. However, as we detail below, the interaction of these propagating waves of dierent eective wavelengths with the wire end leads to interference and beating eects in the light intensity emitted by the wire ends which is detected by the camera. A video showing the time evolution of a focused laser pulse coupled to the end of a wire and all resulting free-space and guided elds involved in the experiment can be found in the supporting information (SI, see section "Propagation Video" and Figure S1). Using an eigenmode solver (Lumerical Solutions, MODE Solutions) it can be con rmed that the propgation along the present nanowire is single mode at the chosen vacuum wavelength (see SI, section "Mode dispersion" and Figure S2). Based on Fabry-Pérot theory the eld amplitude
ψT
that is transmitted by a waveguide of length
and emitted into the detection path can be expressed as
ψT =
Here,
ψ0
L
via one single eigenmode
27
ψ0 η t e−(α+iβ)L
2.
(1)
1 − (re−(α+iβ)L )
is the amplitude of the Gaussian excitation beam,
η
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is a (complex valued) e ciency
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factor comprising the combined eects of incoupling into the waveguide and detecting the emitted signal, and
r = Reiφ
and
t
are the complex plasmon re ection and transmission
coe cients, respectively. These coe cients are assumed to be identical for both wire terminations. Furthermore,
α = 1/2ldecay
and
β = 2π/λe
ldecay
attenuation and wave number, respectively, with length and
λe
denote the propagating wire mode's being the mode's intensity decay
its eective wavelength. Both air-wave and glass-wave can be approximated
by propagating spherical waves. The respective amplitudes scattered from the wire far end at a distance
L
away from the incoupling end towards the detector can be expressed as
ψmedium = ψ0 ηmedium
where
βmedium = 2π/λmedium
e−iβmedium L , L
is the wave vector and
respective medium, i.e. air and glass.
λmedium
(2)
is the wavelength of light in the
The complex quantity
ηmedium
denotes a combined
e ciency factor accounting for the e ciency of scattering of the excitation eld
ψ0
at the
incoupling wire end, thus generating the spherical wave in the respective medium, as well as for the e ciency for scattering of this wave at the wire's far end into the detection path. All elds originating from the waveguide end interfere at the detector according to
Itotal = |ψT + ψair + ψglass |2 .
(3)
Experiment and Simulation To perform a high-precision experiment that can reveal the eects of the additional freespace waves on the overall apparent wire transmission we prepared a sample consisting of 306 monocrystalline
28,29
gold nanowires ranging from 1940 nm to 8040 nm in nominal length
with a length increment of 20 nm (Fig. 2). All nanowires were fabricated by focused-ion beam (FIB) milling of a single monocrystalline gold platelet to ensure uniform milling conditions. For such high-precision structuring, dierences between the structuring parameters and the
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(b)
(a)
(c) 290 nm
20 µm
290 nm
70 nm
(d) height
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lateral position Figure 2: (color online) (a)-(c) Scanning electron microscopy (SEM) images of the sample showing (a) the full platelet including the focused-ion beam milled area of the platelet with the array of single wires of random wire lengths as well as (b), (c) two closeups at dierent zoom levels. (d) Atomic force microscopy line pro le along one wire's cross section.
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exact nal results depend on the ne adjustments of the whole setup.
To ensure these
dierences to be identical for all structures, we use single patterning step to create the whole array.
The wire lengths were distributed randomly over the array to avoid artifacts due
to changes of fabrication or measurement conditions. Simulations show that the remaining gold frames (Fig. 2 (b)) around the wires do not aect the length dependent behaviour (see SI, section "Sample" and Figure S3).
All resulting wires are of uniform quality showing
no observable dierences in SEM images apart from the wire length.
To provide optimal
conditions for focused-ion beam milling the sample was fabricated on a conductive substrate (silicon) and then transferred to a clean and at glass substrate (no adhesion layers). Transfer to the glass substrate avoids the presence of a glass ridge below the nanowires and excludes fabrication-induced surface roughness as well as Ga area.
30
+
-Ion implantation within the milling
The simulation results are obtained by modeling the structure using a nite-dierence
time-domain (FDTD) solver (Lumerical Solutions, FDTD Solutions).
The geometry was
chosen to match high resolution SEM and AFM images and includes e.g. the soft edges (Fig. 2 (c,d)). Optical properties of the gold are taken from literature, index of the glass was set to
n
λe
= 505 nm and
ldecay
while the refractive
= 1.46. Using the eigenmode solver the fundamental mode
properties for the present geometry at vacuum wavelength be
22,31
= 4960 nm.
λ
= 800 nm are calculated to
In order to simulate far- eld images of the wire
near- eld intensities are recorded 10 nm below the structure and projected into the far eld.
32,33
Simulated and experimentally obtained far- eld images for a wire of 8
µm
length
are displayed in Fig. 3 (a) and (b), respectively. The eect of the beam block used for spatially blocking the high-intensity re ection spot resulting from the focused laser excitation of the nanowire input terminal is visible by the nearly circular areas of reduced intensity around the excitation spot. Apart from a somewhat increased scattering in the experiment simulated and experimental images agree exceptionally well.
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1
(a)
Intensity
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(b)
4 µm Figure 3:
(color online) (a) Far- eld intensity from FDTD simulation of an 8
0 µm
long
wire and (b) the corresponding experimental CCD image. The (yellow) circle visualizes the position of the beam block and the (green) corners mark the borders between gold platelet on glass and bare glass (see Fig. 2 and SI, section "Sample" and Figure S3). The red (blue) square marks the area that was integrated to obtain the intensity in simulation (experiment). Experimental data was scaled to match simulated peak intensity within the integration area.
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Results and discussion For a detailed analysis of plasmon transmission through the nanowires we extract for each nanowire the simulated far- eld intensity as well as experimental CCD image counts by integrating 1
µm x 1 µm squares centered at the wire end (red and blue squares in Fig. 3).
From
all 306 measured nanowires, only about 1% showed unexpected signals that we attribute to structural defects (see SI, section "Measurement" and Figure S4) and therefore excluded them from the further analysis. The resulting values for the experimentally determined wire transmission are plotted as a function of the wire length (Fig. 4 (blue +)). A corresponding plot resulting from simulated far- eld images can be found in the SI in the section "Simulation" and Figure S5. The data is normalized such that the resulting decay curve (interleaved dark green line) passes an intensity value of 1/e at a wire length matching ldecay . The detected intensity as a function of the wire length exhibits an exponential decay modulated by an oscillatory behavior. A short-wavelength and a superimposed longer-wavelength oscillation are distinguishable. We rst t the model to the intensity obtained by the simulated data. In order to reduce the number of free parameters the mode's propagation properties ( λe = 505 nm,
ldecay
φ = 1.39)
= 4960 nm) and its re ection coe cient at the wire termination ( R = 0.42,
are obtained from FDFD and FDTD simulation, respectively. The initial phase
osets of both air-wave and glass-wave with respect to the propagating plasmon are set to a xed value of
π.
With these constraints the amplitudes of the launched wire plasmon, the
air-wave, and the glass-wave at the incoupling position remain the only free parameters of the model. The resulting t to the simulation data (see SI, section "Simulation" and Figure S5) shows perfect agreement.
The amplitude ratio of the three contributions is adopted
from this t, so that the nal t to the experimental data (Fig. 4 (blue +)) has only one remaining amplitude parameter. A fabrication-induced length oset of 85 nm has to be introduced, which accounts for a systematic dierence between the nominal and the actual wire length. Its value is precisely determined by tting the position of the Fabry-Pérot oscillations (see SI, section "Uncertainties" and Figure S6). While high-resolution SEM measurements
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252 nm
1.0
1372 nm
6520 nm
1.0
Detected intensity [a.u.]
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0.8
0.5 0.0
102
103
Frequency−1 [nm]
0.6
0.4
0.2
Experiment Model Decay Decay+Beating
0.0 2000
4000
6000
Wire length [nm]
8000
Figure 4: (color online) Detected signal (blue +, thin blue line to guide the eye) within the integration box at the wire end. Superimposed is the tted model (solid red line) together with the intensity decay curve (interleaved dark green line) and a curve of the model with Fabry-Pérot re ectivity
R = 0
visualizing the oscillations from the beating between the
Fabry-Pérot model and the scattered free-space waves in air and glass at the wire end (dashed light green line); (inlay) Fourier transformed experimental data. The expected peak positions from the Fabry-Pérot oscillation (252 nm), the beating between the transmitted light and the air-wave (1372 nm), and the beating between the transmitted light and the glass-wave (6520 nm) are indicated.
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support this eect, contrast and charging-induced uncertainties in the measured length are larger than the uncertainty in the tted length-oset. in Fig. 4 together with the experimental data.
The model is plotted as a red line
The Fourier transformation of the length-
dependent data (inset of Fig. 4) shows two distinct peaks. The highest frequency component corresponds to a periodicity of 252 nm and therefore matches
λe /2
as anticipated by the
Fabry-Pérot model. The slower oscillation corresponding to a wavelength of about 1300 nm corresponds to the calculated beating wavelength of the excitation's vacuum wavelength and the Fabry-Pérot modulated plasmon emission at
λairfpt beating
= 1372 nm. We thus attribute it
to the interference of the spherical air-wave scattered at the wire end and the emitted plasmons.
This origin is further supported by numerical simulations using a mode source to
directly excite the plasmons without launching spherical waves and the resulting absence of the beating (see SI, section "Mode source" and Figure S7). In addition, the model predicts a beating between the plasmon emission and the glass-wave showing a periodicity of about
λglass-fpt beating
= 6520 nm. This beating wavelength is about the same as the length-dierence be-
tween longest and shortest measured wire ( ∆Lmax = 6100 nm) and close to the length scale of the plasmonic intensity decay ( ldecay = 4960 nm). While it is not clearly resolved in the Fourier transformation data it is important to note that - within the observation window - it will appear as an additional slope of the exponential decay curve. Neglecting this additional component will - depending on it's relative phase - lead to either an under- or overestimation of the decay length. In our case a t to the experimental data without taking the glass-wave into account leads to a 20% underestimation of the decay length as compared to the simulated plasmon decay. This deviation can also be inferred from Fig. 4 by observing that the dashed light green curve oscillates (visible beating
λairfpt beating
= 1372 nm) above the interleaved
dark green decay curve due to the additional intensity caused by the beating between the spherical wave in glass and the Fabry-Pérot modulated plasmon emission. To observe and distinguish these dierent contributions in the experiment the requirements on the sample's geometrical precision are demanding. By validating our experimental data against a model
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that includes arti cial errors, i.e. by comparing the residuals, we determine the upper limits for the uncertainties (standard deviation) of our structures geometrical parameters and experimental conditions, i.e. wire length, width as well as random intensity uctuations to be 32 nm, 8 nm and 7%, respectively (see SI, section "Uncertainties" and Figure S6). For the length of the wires this corresponds to a relative error of below 0.4%, which is, taking into account the non-conductive substrate, smaller than the experimentally accessible resolution limit of current state of the art SEM techniques.
Conclusion We conclude that the high precision and reproducibility of the fabricated nanowires allowed us to reveal the nonnegligible in uence of air and substrate waves on their apparent lengthdependent transmission.
Our experiments also show that by inclusion of these additional
waves simulated and experimentally data agree quantitatively within the remaining small experimental uncertainties. The eect of the air wave reveals itself as a beating superimposed to the Fabry-Pérot standing wave pattern. The role of the substrate wave is less obvious. We show that the very long beating wavelength of the substrate wave with the wire plasmon causes unavoidable uncertainty for the tting of the decay length because the corresponding oscillatory behaviour cannot be captured experimentally even for the longest wires since the overall damping of the plasmon becomes too strong. The resulting uncertainty regarding the starting phase of the substrate wave is likely responsible for measurements of decay lengths that reported too long or too short decay lengths.
19,24
The presence of additional waves due
to the dierent group velocities could in uence the temporal structure of plasmon pulses transmitted through plasmonic nanowires.
18,34
The interference eect inherent to plasmon
propagation in a single wire could also lead to ultracompact realizations of interferometric sensing schemes which may exploit changes of the launching phases of the involved excitations.
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Supporting Information Available The following les are available free of charge.
•
si.pdf: A le containing additional information as referenced to in the main text.
•
NanowireExcitation.avi: A video of a simulation showing the time evolution of a focused laser pulse coupled to the end of a wire and all free-space and guided elds involved in the experiment.
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