Download Bandgap-assisted surface-plasmon sensing Arnaud J. Benahmed* and Chih-Ming Ho

Bandgap-assisted surface-plasmon sensing
Arnaud J. Benahmed* and Chih-Ming Ho
Department of Mechanical Engineering, University of California, Los Angeles, 420 Westwood Plaza,
Los Angeles, California 90095
*Corresponding author: [email protected]
Received 1 September 2006; revised 30 January 2007; accepted 30 January 2007;
posted 30 January 2007 (Doc. ID 74665); published 15 May 2007
Surface-plasmon resonance (SPR) is a sensing technique widely used for its label-free feature. However,
its sensitivity is contingent on the divergence angle of the excitation beam. The problem becomes
pronounced for compact systems when a low-cost LED is used as the light source. When the Kretschmann
configuration with a periodically modulated surface is used, a bandgap appears in the surface plasmon
dispersion relation. We recognize that the high density of modes on the edge of the surface-plasmon
bandgap permits the coupling of a wider range of incidence angles of excitation photons to surfaceplasmon polaritons than what is possible in the traditional Kretschmann configuration. Here, the
numerical simulation illustrates that the sensitivity, detection limit, and reflectivity minimum of an
amplitude-based SPR bandgap-assisted surface-plasmon sensor are almost independent of the divergence angle. Two different bandgap structures are compared with the Kretschmann configuration using
the rigorous coupled-wave analysis technique. The results indicate that the bandgap-assisted sensing
outperforms traditional SPR sensing when the angular standard deviation of the excitation beam is above
1°. © 2007 Optical Society of America
OCIS codes: 000.3110, 120.4640, 120.5700, 130.6010, 240.6680, 260.6970.
1. Introduction
A.
Surface-Plasmon Sensing
Surface-plasmon waves (SPWs) are electromagnetic
waves propagating on the surface of a plasma and
created by fluctuations of the surface charge density
[1]. At optical frequencies, SPWs can be observed on
the surface of metals such as gold and silver. SPWs are
extremely sensitive to any changes of optical index on
the interface and are therefore used in a wide range of
surface-sensing applications [2,3], the most common
being called surface-plasmon resonance (SPR). On a
flat surface, the SPW dispersion relation has the following analytical form, with kSP being the complex
SPW wavenumber, k0 being the wavenumber of light
propagating at the same frequency in vacuum, and ⑀2
and ⑀3 being the complex relative permittivities of the
metal and the dielectric, respectively [1]:
0003-6935/07/163369-07$15.00/0
© 2007 Optical Society of America
冑
kSP ⫽ k0
⑀2⑀3
.
⑀2 ⫹ ⑀3
(1)
SP sensing is the monitoring of ⑀3 by measuring kSP.
In the SPR technique, the coupling between TM polarized light and SPW is used to measure kSP using
the momentum-matching condition at the resonance.
A convenient technique is to use the Kretschmann
configuration [4], which consists of a high-index
prism with a thin layer of gold or silver deposited on
one face. TM-polarized light is sent inside the prism
where it internally reflects at the metal– dielectric
interface. When the transverse momentum of the incident beam kx is equal to kSP, the energy of the incident light is transferred to SPW and the specular
reflectivity becomes zero, if the metal thickness is
well chosen.
This phenomenon can be used for sensing in several ways, using the measure of the resonance angle
[3,5], the resonance wavelength [6], or the phase [7].
The easiest technique to implement in a compact
format is the amplitude technique [8], which can be
described as follows: When 冑⑀3 ⫽ n3 changes while
1 June 2007 兾 Vol. 46, No. 16 兾 APPLIED OPTICS
3369
Fig. 1. Reflectivity on each structure for a Gaussian beam of
divergence 0° and 2° at 633 nm and with a central angle of incidence of 51°.
the incident angle, ␪i, and the wavelength, ␭, are
kept fixed at some well-chosen values, the reflectivity, R共n3兲, presents a dip whose minimum corresponds to the momentum-matching condition
kSP共n3兲 ⫽ kx. In the range of n3 where the slope of R
is maximal, it is possible to measure n3 by monitoring the reflectivity.
The two important characteristics of the reflectivity dip are its slope [9], which defines the sensitivity
⭸R兾⭸n, and the value of the reflectivity minimum,
which has an effect on the detection limit. Villatoro
and Garcia-Valenzuela [10] pointed out that these two
parameters are strongly dependent on the quality of
the excitation beam. Particularly, for a Gaussian beam
with a small waist, ␻0, the slope becomes lower and the
minimum of reflectivity is higher. For example, when
the beam waist is 32 ␮m at 633 nm, which corresponds to a 0.2° divergence, the reflectivity minimum
is 30 times larger and the sensitivity is 65% less than
a perfect beam [10]. The reflectivity plots for zero divergence 共␴ ⫽ 0兲 SPR in Fig. 1 illustrate this effect.
Such a situation occurs when the light source has a
small aperture or the optical system cannot achieve
perfect collimation. Also, the beam can be decollimated
for focused measurements in a very precise location
such as a spot in DNA or protein microarray. To remedy this problem, we propose to take advantage of the
increase of the SP density of modes on the edge of the
surface-plasmon bandgaps (SPBG) to decrease the influence of the beam divergence on the signal.
B.
Surface-Plasmon Bandgap
When a SPW propagates through a periodic structure, its dispersion relation is modified [11,12]. Notably, when the periodicity of the structure, ␭g (See
Fig. 2), is equal to half of the periodicity of the SPW,
the SPW is Bragg reflected, and the propagation of
the wave through the structure is blocked [13,14].
This phenomenon, which is similar to a photon in a
photonic crystal [15], creates energy gaps in the dispersion relation of the surface plasmon. This type of
structure is of particular use in the field of plasmonics,
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APPLIED OPTICS 兾 Vol. 46, No. 16 兾 1 June 2007
Fig. 2. Schematic of the different structures studied.
where they are used as SPW mirrors and waveguides
for integrated plasmonic circuits [16,17]. This phenomenon has been analytically explained and described by Barnes et al. [14] and by Kitson et al. [18]
in the case of a corrugated surface on a conical mount.
There are different methods for defining the
surface-plasmon density of modes in one dimension.
A simple definition for the density of modes ␳共␻兲 can
be written as
␳共␻兲 ⫽
dkSP
.
d␻
(2)
The SPW dispersion is flat on the edge of the
bandgap. Therefore, according to Eq. (2), the density
of modes will be increased on the band edge. This is
a recurrent phenomenon in most bandgap situations.
In this paper, we propose to take advantage of this
phenomenon to reduce the SPR signal decay when
the beam divergence increases. The principle is as
follows: the system is first prepared so that an excitation beam excites SPWs on the edge of the bandgap
(see Fig. 3). Since the density of modes is large at this
point, the coupling between a larger range of incident
Fig. 3. Illustration of the concept of the sensor. On the band edge,
a wider range of excitation angles can be coupled to SPWs because
the density of modes is higher.
transverse momenta and SPW modes is possible.
Even if the divergence of the incoming beam is large,
the reflectivity will be low at this point. When the
optical index of the medium changes, so does the gap
position causing the excitation beam to no longer be
located on the band edge. The corresponding reflectivity increase and change can be used for sensing. In
the next sections, we describe how we simulated such
a system and derive the performances of this SPBG
sensor.
2. Numerical Simulations
We numerically investigated the SPBG sensor by
simulating three different types of structure. The
simulation technique consisted of two steps. First, a
map of the reflectivity of the structures for plane
waves was calculated. Then, the reflectivity for a
Gaussian beam was computed by integrating the
plane-wave reflectivity over a range of angles of incidence.
A.
Simulated Structures
In two dimensions, periodic structures modify the SP
dispersion relation by creating new branches that
correspond to ⫾kSP共␻兲 shifted m times by the reciprocal periodicity of the grating, kg, where m is an
integer. Each branch is indexed by its corresponding
m and the sign of the direction of propagation in the
superscript. For example, the branch that corresponds to the SPW propagation on a flat surface in
the forward direction is indexed 0⫹.
Bandgaps appear at branch crossings. In the past,
SP bandgaps have been observed at normal incidence
at the crossing of the ⫹1⫺ and ⫺1⫹ branches. In this
situation, the first order of the grating is used for
momentum matching between the photon and the
SPW, while the second order is used for creation of
the surface plasmon bandgap at kx ⫽ 0 [14]. The main
advantage of this technique is that no prism is necessary since momentum matching is ensured by
the grating. Since it is difficult to precisely control the
second order of corrugation of the structure at optical
scale, we instead use in this paper the gap appearing
at the 0⫹ and 1⫺ branch intersections, which is controlled by the the first order of the grating. However,
in order to probe this gap by the SPR technique, a
prism needs to be used to increase the excitation light
transverse momentum.
As illustrated in Fig. 2, we simulated three different types of structure. The first, further referred to
as (a), corresponds to the classic Kretschmann configuration. It is a thin 共39 nm兲 layer of silver on top of
a glass prism 共n1 ⫽ 冑⑀1 ⫽ 1.5兲. The values for the
permittivity 共⑀2兲 of silver have been linearly extrapolated at each wavelength from tables [19]. We chose
the silver thickness in order to minimize the reflectivity minimum at 600 nm. The second structure (b)
is a direct modification of the Kretschmann configuration, where a square-shaped dielectric layer is patterned on the metallic layer. We chose nd ⫽ 冑⑀d ⫽
1.5 as the optical index of the dielectric layer and a
layer thickness of 200 nm, so that the full extent of
the surface-plasmon field is perturbed by the grating.
We also chose a filling ratio of 0.5 to maximize the
size of the bandgap. The last structure (c) that we
simulated was a uniform thin layer of silver 共39 nm兲
on top of a sinusoidally corrugated prism. The corrugation height was 30 nm, so that the bandgap was
large enough to be observed. It is also a convenient
structure to fabricate in an optical lithography setup.
In the last two devices, the pitch of the grating was
chosen so that the bandgap was observable at
⬃600 nm. The position of the bandgap can be approximated by the relationship:
kg
⫽ ᑬ共k̃SP兲.
2
(3)
In this equation, k̃SP is the wavenumber of the SPW
that would propagate through the grating without
the Bragg reflection, and ᑬ is the real part operator.
In the assumption that k̃SP ⫽ kSP at 600 nm and for
⑀3 ⫽ 1, we obtain
冉冑 冊
kg
2␲
ᑬ
⫽ ᑬ共kSP兲 ⫽
2
␭0
2␲
⑀2⑀3
⬇ 1.0378
.
⑀2 ⫹ ⑀3
␭0
(4)
This leads to a pitch, ␭g, of 280 nm. The assumption
is that k̃SP ⫽ kSP is reasonable in structure (c) but
excessive in structure (b). In this case, an effective
optical index should be used instead of ⑀3. We observed that a simple weighted average between ⑀3
and ⑀d leads to reasonable results.
For these three structures, the reflectivity maps,
R共␭, ␪i兲, were calculated in the range 关400–800 nm兴
and 关40°–70°兴 and by varying the optical index n3 in
the range 关1–1.2兴.
B.
Reflectivity Calculations
In structure (a), which corresponds to the classic
Kretschmann configuration, we calculated the reflec1 June 2007 兾 Vol. 46, No. 16 兾 APPLIED OPTICS
3371
tivity of the interface using the Fresnel coefficients
[10]. To calculate the reflectivity of structures (b) and
(c), we implemented a code based on rigorous coupled
wave analysis (RCWA). This technique is well known
[20 –22] and will not be described in further detail
here. RCWA simulates the transmission and the reflectivity for a periodic stack of square-shaped dielectric layers. Structure (b) can be exactly simulated
with a four-layer system. To simulate structure (c),
we had to discretize the sinusoidal shape by approximating it with a large number of flat dielectric slabs
[see Fig. 2(c)]. RCWA is notoriously known to have
convergence problems when simulating sinusoidal
cases [23,24]. In our particular simulation, we did
obtain convergence for a 30-layer approximation and
61-wave decomposition.
C. Calculation of the Reflectivity of a Gaussian Beam
Once we calculated the reflectivity, R共␭, ␪兲, for each
wave, we used the technique described by Villatoro
and Garcia-Valenzuela [10] to calculate the intensity
of the reflected beam. The Gaussian beam can be
expressed as a continuum of plane waves traveling in
different directions. The angular intensity follows a
normal distribution around the average direction, ␪i.
In the approximation of low angular divergence and
by assuming that the changes of reflectivity in the
polar angle of incidence are neglected, the reflectivity
of a Gaussian beam with a central angle of incidence
␪i and angular divergence ␴ can be calculated by
RGauss共␭, ␪i兲 ⫽
1
冑2␲␴
冕
⫹⬁
⫺⬁
冋
R共␭, ␪兲exp ⫺
共␪ ⫺ ␪i兲2
2␴2
册
d␪.
(5)
This integral was computed using a simple Riemann
approximation technique in the range of 关␪i ⫺ 3␴,
␪i ⫹ 3␴兴. Divergence ␴ can be related to the diameter
of beam waist w0 by the relationship ␴ ⫽ 1兾w0k1.
3. Results
A.
Reflectivity Maps
Figure 4 represents the results obtained for the calculation of the reflectivity for each structure for n3
⫽ 1 and n3 ⫽ 1.2. In these maps, the reflectivity
minima (dark areas) represent conditions for which
the incident beam excites SPW and is absorbed. The
center of these minima corresponds to the SP dispersion relation. As expected, the dip of reflectivity shifts
to a higher angle of incidence when the optical index
of the medium, n3, increases, which corresponds to
the increase in k̃SP. The value of the resonant angle of
incidence (corresponding to kx ⫽ k̃SP) is higher for
structure (b) than for structures (a) or (c). This is to be
expected since half of the SPW is in a region of high
optical index, nd. However, the resonant angle for
structure (c) is very close to that for normal SPR
[structure (a)]. This means that the corrugation only
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APPLIED OPTICS 兾 Vol. 46, No. 16 兾 1 June 2007
Fig. 4. Reflectivity maps for each of the structures. The left column corresponds to n3 ⫽ 1 and the right n3 ⫽ 1.2. On all the
figures, the arrow represents the position where the signal was
monitored in the further sensing results. The black curves in (b)
and (c) correspond to kx ⫽ kg兾2. The white dashed curve in (b1)
corresponds to k̃SP (see text). The white dashed curve in (c1) corresponds to the SP excitation on the glass–metal interface.
weakly perturbs the SP propagation outside the gap,
and our earlier assumption k̃SP ⫽ kSP is valid in this
case. For structures (b) and (c), the locus corresponding to kx ⫽ k̃SP can be visualized as the prolongation
of the 0⫹ SP resonance branch through the gap
[dashed curve in Fig. 4(b1)].
In Figs. 4(b1), 4(b2) and 4(c1), 4(c2), the SP
bandgap appearing at ⬃600 nm for (b) and 550 nm
for (c) can clearly be seen. As explained in the previous section, the expected position of the gap is at
k̃SP ⫽ kg兾2. The black curve in Figs. 4(b1), 4(b2) and
4(c1), 4(c2) represents the set of 共␭, ␪i兲 for which kx is
equal to kg兾2. As predicted, the bandgap appears at
the intersection of the black and dashed curves, and
the slope of the dispersion relation is zero on each
side of the gap. This confirms the expected increase in
the SPW density of modes on the edge of the gap and
justifies our approach for SP sensing tolerant to beam
divergence. By comparing Figs. 4(b1) and 4(b2), it can
be seen that for structure B, the size of the bandgap
decreases substantially when the optical index in-
creases. In this type of structure, the size of the gap
is intimately related to the contrast between the optical index of the medium n3 and the dielectric that
forms the grating nd [13]. The reflectivity maps corresponding to structure (c) present two distinctive
features. First, the average reflectivity, even outside
the SPR, is lower than what is obtained for structures
(a) and (b). The main reason for this is due to the fact
that the ⫺1 reflected diffractive mode is propagative.
Indeed, around the bandgap position, the transverse
momentum of the ⫺1 diffracted mode is ⫺kx and
therefore necessarily propagative. Note that this corresponds to the Littrow mount configuration: the ⫺1
reflected beam propagates in the direction opposite
the excitation beam. The diffraction efficiency of this
order depends primarily on the shape of the metal–
glass interface. In structure B, this interface is flat;
the diffraction efficiency for this mode is low. Alternatively, the metal– glass interface in structure (c) is
corrugated, and the diffraction efficiency is much
higher while the specular reflectivity is lower.
The second notable feature on the reflectivity map
corresponding to structure (c) is the transverse minimum of reflectivity for longer wavelengths [white
dashed curve in Fig. 4(c1)]. This feature corresponds
to the excitation of SPW on the metal– glass interface
in the negative direction. These modes are usually
not accessible since their wavenumber kSPglass,metal is
greater than n1k0. However, in this case, it is possible
to excite the glass–metal SPW propagating in the
opposite direction when ⫺kSPglass,metal ⫽ kx ⫺ kg.
B.
Bandgap-Assisted Surface Plasmon Sensing
Once the reflectivity maps were computed, we calculated the reflected intensity for a Gaussian beam of
angular deviation ␴ using Eq. (5). The arrows in Fig.
4 point to the value of the mean angle of incidence, ␪i,
and wavelength, ␭, for which we chose to monitor the
reflectivity, while n3 was varied from 1 to 1.2. Figure
1 is the plot of the reflectivity obtained for each structure for beam divergence of 0° and 2°. The choice of
the side of the bandgap where the resonance was
monitored was dictated by the angular width of the
resonance on the edge of the bandgap. We observed
that the resonance below (low energy) the bandgap
was wider than the one above it, which is advantageous to diminish the effect of the divergence upon
the sensitivity but detrimental for the overall sensitivity. However, in the case of structure (b), the size of
the bandgap is dictated by the contrast of refractive
indices between the ridges of the grating and the
medium and diminishes when n3 increases. We took
advantage of this effect by monitoring the resonance
above the band for structure (b) in Fig. 1 and below
the gap for structure (c).
While the degradation of the traditional SPR signal [structure (a)] when the divergence increases is
clearly visible, the signal for structures (b) and (c)
stays almost the same because the dip of reflectivity
corresponds to a position where the density of SP
modes is high. This is the effect that we were seeking
in order to maintain the quality of the sensing when
the beam divergence increases.
However, at ␴ ⫽ 0 (plane wave) the width of the
reflectivity is much larger, and the slope is lower in
the two SPBG cases when compared with the SPR
technique. This corresponds to a sensitivity as measured by dR兾dn3 that is approximately 10 times
lower. If we assume that the effect of changing n3 is
only to shift the reflectivity map, it is possible to
express the sensitivity as
SBG ⫽
dR ⭸R ⭸␪BG ⭸R ⭸␭BG
⫽
⫹
,
dn3 ⭸␪ ⭸n3
⭸␭ ⭸n3
(6)
with ␪BG and ␭BG the angle and wavelength (in vacuum) corresponding to the center of the SPBG. Since
we probe the signal on the band edge, we can make
the approximation ⭸R兾⭸␪ ⬇ 0. Also, since the bandgap is situated at k̃SP ⫽ kg兾2, we have
n1
␲
2␲
sin ␪BG ⫽ .
␭BG
␭g
(7)
By derivating this expression and introducing it in
Eq. (6):
dR ⭸R ⭸␭BG
⫽
dn3 ⭸␭ ⭸n3
⭸R ⭸␭BG ⭸␪BG
⫽
⭸␭ ⭸␪BG ⭸n3
SBG ⫽
⫽ ⫺2n1␭g cos ␪BG
⭸R ⭸␪BG
,
⭸␭ ⭸n3
(8)
making the approximation k̃SP ⫽ kSP, valid in structure (c) results in ␪BG ⫽ ␪SP, with ␪SP the angle of SP
resonance without corrugation. This allows us to
compare SBG with the sensitivity of traditional SPR,
SSPR:
SSPR ⫽
⭸R ⭸␪SPR
,
⭸␪ ⭸n3
SBG
2␭g⭸R兾⭸␭
⫽ ⫺n1 cos ␪BG
.
SSPR
⭸R兾⭸␪
(9)
(10)
The relative sensitivity of the SPBG sensor compared with the traditional SPR sensor depends primarily on the comparison between the slopes of R in
the wavelength direction versus the angle direction.
By measuring slopes on the reflectivity map, and by
using ␪BG ⫽ 53°, we obtain SBG兾SSPR ⬇ 12%, which is
in good agreement with our calculations for ␴ ⫽ 0.
When the optical index of the prism is increased, ␪BG
decreases, which increases the comparative sensitivity of the sensor. Alternatively, if n3 increases, the
comparative sensitivity of the bandgap sensor will
reach zero in the limit of high angles. This is a problem in the case of fluid sensing where the angle of
1 June 2007 兾 Vol. 46, No. 16 兾 APPLIED OPTICS
3373
Fig. 5. Plots of the sensitivity, the detection limit, and the reflectivity minimum versus the angular deviation of the excitation
beam for each structure.
incidence is high and the use of a high index ratio
prism is required to keep cos ␪BG relatively high. Note
that in the limit of a high angle of incidence, the term
⭸␪SP兾⭸n3 diverges and this model is no longer valid.
Figure 5 represents the values of the sensitivity,
reflectivity minimum, and detection limit against ␴
for the three structures. The sensitivities were calculated at the point of maximum slope. The maximum
theoretical detection limit, which is the minimum
amount of change in n3 that will produce a change of
signal equal to the shot noise is proportional to [8]
␦n3,min ⬀
冑R
⭸R兾⭸n3
.
(11)
We plotted the minimum of this quantity for the
three structures in Fig. 5(b).
The SPBG sensor is notably unaffected by increases in the divergence of the beam in sensitivity,
reflectivity minimum, and detection limit, which
proves that the increase in SP density of state can be
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APPLIED OPTICS 兾 Vol. 46, No. 16 兾 1 June 2007
used for sensing. The SPBG sensor outperforms the
traditional SPR in sensitivity for ␴ ⬎ 1.1° and in the
detection limit for ␴ ⬎ 0.7°. The threshold is lower for
the detection limit because the minimum reflectivity
stays low in the case of SPBG sensing and this reduces the factor 冑R (shot noise). An angular beam
divergence of 1° corresponds to a waist radius of approximately 6 ␮m at 600 nm. This is much smaller
than the typical diameter of an optical fiber, and it is
therefore unlikely that the SPBG technique would be
useful in this case. However, this number corresponds to a perfectly Gaussian beam, which is difficult to realize in a compact format. Therefore, the
main application of a SPBG sensing technique would
be in situations where the divergence is nonzero because the beam is focused on a small area or when the
optical system cannot be practically made perfectly
collimated such as when the source is not a laser
(diode or arc lamp). Another technique for mitigating
the effect of the divergence on the sensitivity is to
widen the resonance peak by changing the thickness
of the metallic layer [8] in the Kretschmann configuration. However, changing the thickness of the metallic layer increases the reflectivity minimum, which
has negative effects on the shot noise. Conversely,
SPBG sensors have a sensitivity independent of divergence and a low reflectivity minimum.
The differences in signal between the two BG structures (b) and (c) are minimal. The inherent lower
sensitivity of structure (b), due to the fact that only
part of the metal is in contact with the sensing medium, is compensated for by a higher reflectivity average than that of structure (c). Again, the lower
reflectivity average for structure (c) is due to the corrugation on the metal– glass interface, which yields
to a high diffraction efficiency in the ⫺1 reflected
order. Since structure (c) is easier to fabricate, it is a
good candidate for experimental validation. An ideal
structure would be a metallic film that is corrugated
on the metal– dielectric interface and flat on the
glass–metal interface. In such a structure, the SPW
would be totally perturbed by the changes in optical
index, and the diffraction efficiency in the ⫺1 reflected order would be low.
4. Conclusion
We have analyzed the use of a SP bandgap for sensing applications in amplitude-based SP sensors. Specifically, we showed that the increase in the density of
modes on the gap edge can be used to remedy the
degradation of the sensor signal when a divergent
Gaussian beam is used to excite SPWs. Using RCWA,
we analyzed two types of SPBG structure and compared their performances to traditional SPR sensing.
For a plane wave, the sensitivities of SPBG sensors
are 1 order of magnitude below the sensitivity of
SPR, which is due to the differences in slopes in the
momentum and energy axis. However, SPBG sensors
outperform SPR sensors in sensitivity and in the detection limit as soon as the angular deviation is over
a degree. We envision that this technique will be used
for the design and fabrication of compact and low-cost
SP sensors.
This work was made possible by the National Science Foundation Center for Embedded Networked
Sensing (CENS) under contract CCR-0120778.
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