index matching of surface plasmons

Index matching of surface plasmons

M.J.A. de Dooda, E.F.C. Driessena, D. Stolwijka, M.P. van Extera, M.A. Verschuurenb andG.W. ’t Hoofta,b

aHuygens Laboratory, P.O. Box 9504, 2300 RA Leiden, The Netherlands;bPhilips Research, High Tech Campus 4 (mailstop 12), 5656 AE Eindhoven, The Netherlands

ABSTRACT

We have measured the angle and wavelength dependent transmission of index matched metal hole arrays, andof arrays with a dielectric pillar in each hole. Index matching enhances the transmission, but also broadens theresonances due to an enhanced coupling between plasmon and radiation modes. Hole arrays that are coveredwith glass or have a glass pillar in each hole are created using an imprinting technique. We observe additionalwaveguide modes in the transmission spectra of these arrays and discuss the avoided crossing that we observefor the hybrid structure with dielectric pillars in the holes.

Keywords: Surface plasmons, diffraction, Fano resonance, index matching

1. INTRODUCTION

It is well known that an optically thick metal film perforated by a regular array of sub-wavelength holes showsextraordinary transmission.1 The enhanced transmission over a set of independent holes2 is due to excitation ofsurface plasmons via diffraction from the regular lattice of holes. These surface plasmons are electromagneticwaves bound to a metal-to-dielectric interface. Interaction between modes on the same interface via Braggscattering has been observed for 1-D arrays of slits or wires.3–5 Coupling between plasmons on different interfacesis more difficult to observe, because most systems involve an asymmetric geometry of a metal film on a dielectricsubstrate. This detunes the frequencies of plasmon modes on different sides of the metal film. The asymmetrycan be lifted by fabricating symmetric samples or by using an index matching liquid.6–9 The measured spectraresemble the calculated spectra,7, 10 but the finite size of the array and the large numerical aperture of theincoming beam in the experiments make it impossible to resolve the coupling between the surface modes.

Here, we present details on an index matching experiment11 using liquids and experiments on metal holearrays with either an array of pillars12 or a continuous dielectric layer on top. In these experiments we makeuse of large 1×1 mm2 or 0.5×0.5 mm2 arrays and we use a white light beam with a small enough numericalaperture (<0.01) to resolve the coupling between modes. The article is organized as follows: we first introducethe measured angle and wavelength-dependent transmission spectra for an index matched metal hole array insection 3.1 and discuss the origin of the different resonances. In section 3.2, we introduce a small differencein refractive index and use a phenomenological Fano model to describe the resonances. The resonance aredue to surface plasmons on the two metal-to-dielectric interfaces. A coupled mode theory that includes bothconservative and dissipative coupling is used to describe the observations. In the last two sections we describeresults on an array made via an imprinting technique. These include the observation of coupling between surfaceplasmon modes on the same interface for a metal hole array with pillars in each hole in section 3.3, and the effectof waveguide modes in a thin index-matched dielectric layer on top of a metal hole array in section 3.4.

2. EXPERIMENTAL

Two kinds of metal hole arrays were prepared. For liquid index matching experiments we used a metal holearray fabricated by e-beam lithography followed by a lift-off procedure. The 1×1 mm2 square array of holeswith diamter d = 200 nm in an optically thick (200 nm) gold layer has a lattice constant a = 700 nm. A 2nm Ti bonding layer between the gold and the glass substrate ensures proper adhesion of the gold layer. The

Send correspondence to M.J.A. de Dood – E-mail: [email protected]

Metamaterials III, edited by Nigel P. Johnson, Ekmel Özbay, Nikolay I. Zheludev, Richard W. ZiolkowskiProc. of SPIE Vol. 6987, 698713, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.780524

Proc. of SPIE Vol. 6987 698713-12008 SPIE Digital Library -- Subscriber Archive Copy

hole array was placed in a closed glass cuvette with a 2.5 mm optical path length and immersed in solutionswith different refractive index n in the range 1.36–1.66. The glass substrate (Schott-BK7) has a nearly constantrefractive index n = 1.51 in the wavelength range of interest.

A second set of metal hole arrays was prepared using an imprinting technique.12 An array of pillars wasdefined by e-beam lithography and was used to create a rubber stamp. The stamp was then used to create areplica of the original array by pressing it into a layer of liquid sol-gel glass (n = 1.41) coated on an AF45 glasssubstrate (n = 1.52). Subsequently, the substrate was coated with a layer of gold and the gold was selectivelyremoved from the pillars by making use of the fact that the gold layer deposited on the side of the pillars isthinner than the layer on the bottom. The resulting 0.5 mm × 0.5 mm metal hole array has a lattice constanta = 760 nm and holes with a diameter d = 135 nm in a 200 nm thick gold layer. In each hole there is a∼650 nm long glass pillar. After measuring the transmission of the array, the pillars were selectively removedusing hydrofluoric acid. Finally, the samples were coated with a top layer of sol-gel glass that serves as anindex-matching coating.

The optical transmission of the samples was measured as a function of wavelength and angle of incidence. Afiber-coupled incandescent lamp was used to illuminate a ∼300 µm diameter spot on the sample. The transmittedlight was collected in a fiber and sent to a fiber-coupled grating spectrometer with a Si CCD array (resolution1.2 nm, 550-1000 nm) or an InGaAs linear array (resolution 3.0 nm, 900-1700 nm). Apertures in the incidentand transmitted light beam were used to limit the numerical aperture to less than 0.01. Polarizers were placedin parallel parts of the incident and transmitted beams to measure the p-polarized component of the light. Thesubstrate was mounted on a computer controlled rotation stage with the rotation axis aligned with the (0, 1)lattice direction of the hole array. The wave vector of the incident wave was perpendicular to the rotation axis.

3. RESULTS AND DISCUSSION

3.1 Liquid index matching

Typical transmission spectra of the hole arrays show a number of resonant peaks for which the transmission isenhanced. Figure 1 shows a grayscale plot of the transmission of p-polarized through a metal hole array immersedin an index matching liquid (methoxybenzene, n = 1.518). Data are shown as function of wavelength (verticalaxis) and angle of incidence (horizontal axis). The Fano-type lineshape of the resonances is strongly asymmetricand each maximum is accompanied by a deep minimum on the high frequency side.5, 13–16 These minima areclearly visible in Fig. 1 as the black bands.

The resonances in the transmission spectra are the result of coupling of the incident radiation to a surfaceplasmon mode of the metal hole array. The condition for this coupling is given by the relation17

k‖ = kSP + G, (1)

where k‖ is the component of the incident wave vector along the interface, kSP is the wave vector of the surfaceplasmon and G is a reciprocal lattice vector. As a result, each resonance can be labeled by a specific choiceof the reciprocal lattice vector, which can be conveniently expressed for a square lattice as (Nx, Ny), with Nx

and Ny integer. To calculate the position of the resonances, the wave vector of the surface plasmon on a metalhole array can be well approximated by that of a surface plasmon propagating on a smooth metal-to-dielectricinterface, given by

k2SP =

(ω2

c2

)εdεm(ω)

εd + εm(ω)(2)

Here ω is the angular frequency, and c is the speed of light. The symbols εd and εm(ω) refer to the dielectricconstant of the dielectric and the metal respectively. To correctly describe the observed dispersion of the modesthe frequency dependence of the dielectric constant of the metal should be taken into account. The resonancesas calculated from Eqs. (1) and (2) using literature values for the dielectric constant of gold,18 are shown by thedashed lines in Fig. 1 and closely follow the position of the minima.

At normal incidence, several resonances are visible, each corresponding to a different set of reciprocal latticevectors G. The lowest frequency resonance is due to coupling to the (1, 0) modes, the next lowest resonance is

Proc. of SPIE Vol. 6987 698713-2

0

1

2

3

4

Tra

nsm

issi

on

(%)

-60 -45 -30 -15 0 15

Angle (degrees)

0.6

0.8

1.0

1.2

1.4

1.6

Wav

ele

ngth

(µm

) (-1,0)

(+1,0)

(0,±1)

Figure 1. Grayscale plot of the transmission of a metal hole array immersed in an index matching liquid with n = 1.518as function of angle of incidence and wavelength. The hole array has square lattice symmetry with a lattice constanta = 700 nm and was rotated along the (0, 1) lattice direction to change the angle of incidence. The dashed lines in thefigure indicate the calculated position of the resonances. In particular, the (+1, 0), (−1, 0) and (0,±1) resonances areindicated in the figure.

kSP

H//MetalAir

kSP

G

k//

(+1,0)

kSP-G

k//

(-1,0) (0,±1)

G

k// kSP

-G

Figure 2. Field distribution of the H-field of a surface plasmon on a metal-to-air interface (left). The direction ofpropagation of the surface plasmon is indicated by the vector kSP . The images on the right give the parallel componentof the incoming light wave k‖, the reciprocal lattice vector G and the corresponding vector kSP for the (±1, 0) and (0,±1)modes of a metal hole array with square lattice symmetry

due to coupling to the (1, 1) resonances, followed by the (2, 0) and (1, 2) modes. At normal incidence each ofthese modes is degenerate. When the angle of incidence is changed, the degeneracy is lifted and the resonancesplits into several resonances. For instance, at non-zero angles of incidence, the (1, 0) resonance splits into a(+1, 0), a (−1, 0) and a two-fold degenerate (0,±1) mode as indicated by the labels. The nature of the different(1, 0) modes is shown in Fig. 2 that shows the field distribution of the H-field of a surface plasmon on a metal-to-dielectric interface (left). Also shown are vectorial drawings (right) of the parallel component of the incominglight wave k‖, the reciprocal lattice vector G and the corresponding vector kSP for the (+1, 0), (−1, 0) anddegenerate (0,±1) modes of a metal hole array with square lattice symmetry.

The measurement in Fig. 1 shows that the (0,±1) mode is not efficiently excited with p-polarized light forsmall angles of incidence. This mode becomes clearly visible for angles of incidence larger than ∼20◦ only. Thisbecomes clear from the drawing in Fig. 2 that shows the propagation direction of the different modes. Theexcitation efficiency of the (0,±1) modes is proportional to the component of kSP parallel to the incoming wavevector, which vanishes at normal incidence.


0.7 0.8 0.9 1.0

Frequency (µm -1)

0.0

4.0

8.0

Tra

nsm

issi

on(%

)

1.01.21.4

Wavelength (µm)

nL=1.36

0.7 0.8 0.9 1.0

Frequency (µm -1)

1.01.21.4

Wavelength (µm)

nL=1.50

0.7 0.8 0.9 1.0

Frequency (µm -1)

1.01.21.4

Wavelength (µm)

nL=1.66

Figure 3. Measured transmission (dots) as function of frequency at normal incidence for a metal hole array immersed ina liquid with a refractive index of 1.36 (left), 1.50 (middle) and 1.66 (right). The solid line is a fit to a Fano model (seetext).

3.2 Tuning of the liquid refractive index

When the refractive index of the liquid is no longer matched to the index of the substrate the structure isasymmetric and it becomes possible to distinguish surface plasmon modes on the liquid-to-metal from modes onthe glass-to-metal interface. This is demonstrated in Fig. 3, which shows the measured transmission at normalincidence as function of frequency (dots) for different values of the liquid refractive index. Each of the situationsshow a central peak around a frequency of 0.9 µm−1. For the nearly index-matched situation (center), only asingle peak is visible. When the refractive index of the liquid is lowered (left), an additional peak appears onthe high-frequency side. If the refractive index of the liquid is higher than that of the glass (right) the extrapeak appears on the low-frequency side. This extra peak corresponds to the resonance from the surface plasmonmode on the liquid-to-metal interface.

We adopt a Fano description to describe the transmission spectra. The transmission through a metal holearray is then described by a combination of a non-resonant direct transmission through the holes, and a reso-nant component that couples to surface plasmons.5, 10, 14–16 Interference between these two channels results inan asymmetric line-shape T (ω) = |t(ω)|2, where t(ω) is given by a sum over a finite number of (uncoupled)resonances:

t(ω) = anrω2 +

∑j

bjΓj exp(iϕj)(ω − ωj) + i(Γj + γj)

. (3)

Here each of the modes j has a radiative loss rate Γj , an intrinisic Ohmic loss γj , a frequency of the resonance ωj

with an amplitude bj and a phase ϕj . The slowly varying non-resonant contribution has an amplitude anr that wechose to be proportional to ω2 to reflect the fact that the transmitted intensity through a single sub-wavelengthhole is proportional to ω4.2

The solid lines in Fig. 3 show fits of the data to Eq. (3) using up to three resonances and setting the phases ϕj

to π. Two resonant terms are needed to fit the two resonances in the non-index matched case. The background inthe spectra that is visible in the high frequency part is a combination of the tails of higher frequency resonancesand the non-resonant contribution anr. Unfortunately, it is difficult to unambiguously separate the non-resonantcontribution from a contribution due to resonances at higher frequencies.

The fitting procedure was repeated for the measured transmission spectra of metal hole arrays in contact withliquids with a refractive index between 1.33 and 1.66. This analysis yields the resonance frequencies, linewidthsand amplitudes of the lowest two resonances. These resonances correspond to a coupling to the (1, 0) surfacemodes on the glass and the liquid side of the metal film. The frequency and linewidth thus obtained are shownin Fig. 4 for the two lowest energy modes (symbols) and are compared to different coupled mode theories (lines,see below). The solid symbols refer to the lowest energy mode, while the open triangles refer to the higher energymode. When the refractive index of the liquid is tuned towards the index matched situation, the linewidth of


0.80

0.90

1.00

Fre

quen

cyω

j(µ

m-1

)

a)

Conservative

b)

Dissipative

c)

Conservative & Dissipative

1.3 1.4 1.5 1.6 1.7

Refractive index

0.00

0.01

Line

wid

thΓ(

µm-1

)

1.3 1.4 1.5 1.6 1.7

Refractive index

0.00

0.01

Line

wid

thΓ(

µm-1

)

1.3 1.4 1.5 1.6 1.7

Refractive index1.3 1.4 1.5 1.6 1.7



Refractive index

Figure 4. Frequencies and linewidths of the (1, 0) resonance as function of the refractive index of the liquid. The valuesare obtained from a Fano fit of the measured transmission spectra at normal incidence and compared to coupled modetheory with only conservative coupling (a), only dissipative coupling (b) or both conservative and dissipative coupling(c). The closed symbols (data) and dashed lines (theory) refer to the low-energy mode, while the open triangles and solidlines refer to the high-energy mode.

the low-energy mode grows at the expense of the linewidth of the high-energy mode. At the same time, theamplitude of the low-energy mode increases, while that of the high-energy mode decreases (data not shown).11

This behavior shows that the two modes on different sides of the metal film are coupled.

To describe this coupling we employ a coupled-mode theory with only 2 modes. These modes correspond tothe two surface plasmons that propagate on either the substrate or the liquid side of the metal film, treatingthe plasmon modes on the same interface that are coupled via Bragg reflection3–5 as a single mode. The timeevolution of the amplitudes a and b of the two modes in our model is given by the equation of motion

iddt

(ab

)= H

(ab

). (4)

The Hamiltonian H that describes the coupled system has the following form:

H =(

ωa + V − iΓa W + iΓC

W + iΓC ωb + V − iΓb

). (5)

The diagonal elements of the matrix contain the frequencies ωa,b and linewidths Γa,b of the uncoupled modes,and a frequency shift V . The off-diagonal elements contain parameters W and ΓC that describe conservativecoupling, and dissipative coupling, respectively.19, 20 Conservative coupling corresponds to a lossless couplingprocess and an avoided crossing is observed. For the case of dissipative coupling, the coupling itself involves lossand mode pulling or frequency locking is observed.

The complex eigenvalues of the Hamiltonian H give the frequencies ω1,2 and linewidths Γ1,2 of the coupledmodes. The lines in Fig. 4 correspond to a fit of the coupled mode theory to both the frequency and linewidthdata. The dashed and solid lines refer to the low-energy and high-energy mode, respectively. The different figurescorrespond to a different nature of the coupling, i.e. either conservative (a), dissipative (b) or both (c). Whenonly conservative coupling is included, the coupled mode theory describes the frequencies very well, but failsto predict the observed narrowing of the modes. Similarly, to describe the fact that the damping of one mode


increases while the damping of the other mode decreases, a dissipative component in the coupling is needed.However, the mode pulling in the fit for the dissipative coupling (Fig. 4b, top) is not observed in the experiment.

For the best fit in Fig. 4 we find a conservative coupling rate W of 0.0046 ± 0.0015 µm−1 and a dissipativecoupling rate ΓC of 0.0068 ± 0.0011 µm−1. Furthermore, we find a slightly negative frequency shift V of−0.013 ± 0.028µm−1 compared to the resonance condition calculated by Eqs. (1) and (2), consistent with thetheory in Ref. 10. The fitted values of Γa = 0.0067±0.0004 µm−1 and Γb = 0.0100±0.0005 µm−1 for the dampingrate of the two surface plasmon modes indicate that the mode on the liquid side is somewhat more lossy thanthe mode on the glass side. This is consistent with atomic force microscopy measurements of the roughness ofthe glass substrate compared to the top of the metal layer which yields a root mean square values of 4.4 nm forthe gold surface and 0.8 nm for the substrate. As a refinement of our model, we have also used damping ratesΓa,b that depend on the refractive index of the liquid via the optical density of states.5, 21, 22 Introducing theseextra fit parameters indeed results in a better fit. However, this does not significantly alter the values of the fitparameters V , W and ΓC , and thus does not change the interpretation of the measurements.

The modes in our coupled-mode theory are consistent with the states calculated using a scattering formal-ism.7, 10 On resonance, the eigenmodes correspond to a situation where the plasmons on the two interfacesoscillate in phase (low-frequency mode) or out of phase (high-frequency mode). Close to resonance, the calcu-lated spectra show a split resonance (avoided crossing) and a linewidth of the low-energy mode that becomeslarger, while the linewidth of the high-energy mode becomes smaller.

3.3 Metal hole arrays with pillars

Adding a dielectric pillar to each of the holes drastically changes the transmission spectra.12 The measuredtransmission for p-polarized light as function of angle of incidence for an array with pillars is compared to thatof the same hole array with the pillars removed in Fig. 5. Again, the dark bands in the figure correspond tothe minima in the transmission spectra and are accompanied by a maximum in transmission at slightly longerwavelength. The white lines correspond to the calculated resonances using Eqs. 1 and 2. Because the array isnot index matched, the dispersion of surface modes is different for the two sides of the metal film. The solid linesin the figure refer to modes that exist on the glass-to-metal interface, while the dashed lines refer to modes onthe air-to-metal interface. In order to correctly describe the dispersion of the surface mode on the glass-to-metalinterface we used an effective index neff = 1.46 for the glass substrate. This value is lower than the index of theglass (≈1.52) due to the ∼100 nm thick layer of low index sol-gel material (n = 1.42) between the gold film andthe substrate. This is explained in more detail in section 3.4.

When comparing the transmission of the array with pillars (Fig. 5b) to that of the same array without pillars(Fig. 5a) several important differences can be observed:

I. The average transmission of the array with pillars is larger than that of the array without pillars. Thiscan be explained by the fact that the dielectric pillars in the hole effectively enlarge the hole size. Thisis consistent with the observation that also the linewidth increases when pillars are placed in the holes,12

while the position of the minima does not change.13, 23

II. For the conventional array (Fig. 5a) the (0,±1) resonance is barely visible, consistent with earlier work,24

while this resonance is strikingly visible in the transmission of the array with pillars (Fig. 5b). At normalincidence, the (±1, 0) and (0,±1) modes on the air side are degenerate. When the angle of incidence ischanged, the degeneracy is partially lifted and the resonance splits into a (1, 0), a (−1, 0), and a degenerate(0,±1) resonance. The (1, 0) and (−1, 0) modes have a strong dispersion, because the parallel componentof wavevector of the incoming light �k‖ and the reciprocal lattice �G are parallel. For the (0,±1) modesthe dispersion is limited, corresponding to the fact that �k‖ and �G are perpendicular. The measurementsin Fig. 1 for an index matched array without pillars show that the (0,±1) mode can be excited with p-polarized light, albeit with small efficiency for near-normal angles of incidence. Therefore, the role of thepillars is to enhance the excitation of the (0,±1) mode.

III. In addition to the enhanced excitation of the modes, the pillars also enhance the coupling between the (1, 0)surface plasmon modes on the same interface. This results in an avoided crossing of these modes on the


0.0

0.5

1.0

1.5

2.0

2.5

3.0

Tra

nsm

issi

on

(%)

-15 -10 -5 0 5 10 15

Angle (degrees)

b)

(1,1)Air

(2,0)Glass

(1,0)Air

(1,1)Glass

-15 -10 -5 0 5 10 15

Angle (degrees)

0.6

0.7

0.8

0.9

Wav

ele

ngth

(µm

)

a)

(1,1)Air

(2,0)Glass

(1,0)Air

(1,1)Glass

Figure 5. Grayscale plot of the measured transmission of p-polarized light as function of angle of incidence for a metalhole array without (a) and with dielectric pillars (b). The dashed lines show the calculated resonance for a hole arraywith a square lattice symmetry and a lattice constant a = 760 nm. When pillars are added, the (0,±1) resonance on theair side becomes clear visible even at small angles of incidence and a splitting between the (1, 0) modes occurs at normalincidence.

air interface (the side with the pillars) for normal incidence. For the array with pillars, we observe minimain the spectra at 775 and 825 nm, of which the latter coincides with the minimum of the (1, 1) mode onthe glass-to-metal interface. The two resonances occur at different energies depending on the position ofthe nodes and antinodes of the standing waves.3, 5, 25 The energy of the mode that has anti-nodes at theposition of the pillars is lowered. For our array, with a fill fraction of pillars of only 2.3%, the observedsplitting between the two minima is 6% of the center frequency∗. We find that this splitting is muchlarger that the splitting calculated either from a standard two-band model or from a model that uses thepolarizability per unit volume.21, 27

The effect of the pillars may be increased by using a larger fill fraction of the pillars or by using pillars with ahigher dielectric constant. At the same time, this will also increase the coupling to radiation modes and increasethe damping of the surface waves,5 thereby increasing the linewidth of the observed Fano resonances. Similareffects can occur in metal hole arrays covered by a dielectric hole array, although in general the interaction withp-polarized light is expected to be weaker for a hole when compared to a pillar. Both kind of structures, usingeither dielectric pillars or holes, can be used to locally change the dispersion of the surface modes by selectivelyremoving or changing the pillars or holes. Such devices would allow control over the excitation and propagationof plasmon modes on length scales comparable to the wavelength of light.

3.4 Solid-state index matching

After removing the pillars from the metal hole array from section 3.3 and measuring the transmission, the samplewas covered with a thin (∼400 nm) layer of sol-gel glass. This layer serves as an index matching layer and shouldmake the surface plasmons that propagate on different sides of the metal film indistinguishable. Figure 6 showsthe measured transmission for p-polarized light as function of wavelength and angle of incidence. Several strong,relatively broad, minima (black lines in Fig. 6) are clearly visible. These minima in transmission are associatedwith the Fano-type lineshape caused by the coupling to a surface plasmon mode. Two such modes can beobserved, which correspond to a surface plasmon mode propagating on the substrate side and a surface plasmon

∗We use the separation between the minima as a measure of the splitting, because the minima do not shift whenradiation losses are increased23, 26


0.0

0.5

1.0

1.5

2.0

2.5

3.0

Tra

nsm

issi

on

(%)

-15 -10 -5 0 5 10 15

Angle (degrees)

0.6

0.7

0.8

0.9

Wav

ele

ngth

(µm

)

(1,0):TE0

(1,0):TM0

(1,1):SP

(1,1):TE0

(1,1):TM0

(2,0):SP

(2,1):SP

Figure 6. Grayscale plot of the measured transmission of a metal hole array covered with a dielectric layer as functionof angle of incidence. The lines in the figure correspond to the calculated position of the different resonances. Each ofthe resonances corresponds to a specific reciprocal lattice vector and has an additional label to indicate the nature of themode; i.e. surface plasmon (SP), fundamental TE waveguide mode (TE0) or fundamental TM waveguide mode (TM0).

mode propagating on the side covered with the dielectric layer. These modes can be distinguished because theAF45 substrate has a higher refractive index (n ≈ 1.52) than to the sol-gel material (n ≈ 1.40), making thestructure slightly asymmetric.

In addition to the surface plasmon modes that are clearly visible, several much weaker resonances are alsovisible. Most notably, at normal incidence a resonance appears at a wavelength of 700 nm that cannot beexplained by a surface plasmon mode. The extra resonances are characterized by a sharp symmetric peakinstead of the asymmetric lineshape of the surface plasmon modes, and are caused by coupling to a waveguidemode supported by the structure.28 Similar to the case of surface plasmons, the resonance condition to excite amode is given by

k‖ = βm + G, (6)

where βm is the propagation constant of the mode, and k‖ and G are defined as before. For the layered structure,βm has to be found numerically. The modes can be classified as either transverse electric (TE) or transversemagnetic (TM) depending on whether the E-field or the H-field vector is perpendicular to the surface normal ofthe structure. Following the notation of Ref. 29, we define the x-direction as perpendicular to the interface andthe z-direction as the propagation direction of the guided mode. The dielectric constant ε(x) of our structureis stepwise continuous and equal to ε1 for 0 ≤ x, ε2 for −d ≤ x ≤ 0 and equal to 1 for x ≤ −d. Here d is thethickness of the dielectric layer, ε1, ε2 and ε3 are the dielectric constants of the metal, the dielectric layer andthe dielectric above the structure. The metal-to-dielectric interface is positioned at x = 0. The H-field of them-th order TM mode, that propagates in the positive z-direction can be written as:

Hm(r, t) = Hm(x) ei(ωt−βmz), (7)

where βm is a propagation constant which can be interpreted as the component of the wavevector parallel tothe interface. For modes that are guided, the H field is confined either to the metal-to-dielectric interface(surface plasmon) or to the dielectric layer. Outside the region of the dielectric layer, the field Hm(x) decaysexponentially. The field is given by:

Hm(x) =

⎧⎨⎩

(ε1S2)/(ε2S1) exp(−S1x) 0 ≤ x(ε1S2)/(ε2S1) cosh(S2x) − sinh(S2x) −d ≤ x ≤ 0[(ε1S2)/(ε2S1) cosh(S2d) + sinh(S2d)] exp[S3(x + d)] x ≤ −d

(8)


Here, the transverse momentum in a layer with dielectric constant εi is given by S2i = β2

m − (ω2/c2)εi. A modeof this form can exist whenever the boundary conditions from Maxwell’s equations are fulfilled at x = 0 andx = −d, leading to the condition:

tanh(S2d) = −S2

ε2

(S1

ε1+

S3

ε3

)/((S2

ε2

)2

+S1

ε1

S3

ε3

)(9)

This equation can be used to numerically find the propagation constant βm as function of frequency for all TMmodes, including the surface plasmon mode. A similar approach can be applied to find all TE waveguide modes.

The lines in Fig. 6 are the numerically calculated resonance conditions for the surface plasmon on the sol-gelside (solid lines) and the waveguide modes in the structure (dashed lines). The calculated resonances due thesurface plasmon on the substrate side are slightly red shifted and have been omitted from the figure for reasonsof clarity. Each of the calculated resonances in the figure is labeled by a combination of reciprocal lattice vectorand an identifier to indicate the nature of the mode. The modes are classified as a surface plasmon (SP), afundamental TE waveguide mode (TE0) or a fundamental TM waveguide mode (TM0). In our calculation, weadjusted the thickness d of the sol-gel layer to get good agreement with the measured data and find a valueof 480±10 nm. For this layer thickness, the cut-off frequency of higher order TE and TM waveguide modes isbeyond the frequency range probed in the experiment.

For applications (e.g. as sensors), it may be interesting to design structures in which the plasmons on differentsides are index matched. The same formalism can be used to design such structures, which use a dielectric layerinstead of a liquid to achieve index-matching. In this case, the numerical algorithm is used to find the propagationconstant β of the surface plasmon at a given wavelength as function the layer thickness d of the dielectric layer.This calculation is done for the surface plasmon mode on the “air” (n = 1.0) side of the structure as well as onthe substrate (n = 1.52) side of the structure. In both cases the dielectric layer has a refractive index of 1.4. Thiscalculation allows to find the minimum required thickness of the dielectric layer, on each side, that is needed tocreate a structure in which the surface plasmon modes on the two interfaces are indistinguishable.

Figure 7 shows the real part of the calculated effective index, defined as (βm(ω)c/ω), for gold covered with adielectric film as function of the layer thickness d for a wavelength of 0.8 µm (solid line). The dotted lines showsimilar calculations for slightly different wavelengths of 0.7 µm and 0.9 µm. We have included results for bothsides of the metal film, with insets next to the curves to show the configuration that corresponds to the curves.For a thin dielectric layer inserted between the metal and air, the surface plasmon propagates mostly in the airand the effective index is close to that of a surface plasmon propagating on a bare air-to-gold interface. When thelayer thickness is increased, the effective index approaches that of a surface plasmon on the dielectric-to-metalinterface. From the figure it is clear that this situation is reached for a layer thickness ∼0.5µm. As expected,the required thickness becomes smaller for shorter wavelengths.

The effective index for the configuration that corresponds to the substrate side of the metal film behavesdifferently. As can be seen in the figure, the curves start at an effective index of a surface plasmon on a glass-to-gold interface. As long as a very thin layer of dielectric is used, the effective index of the surface plasmon isalways larger than that of the dielectric with the highest ε for frequencies below the plasmon frequency.30 Ifthe layer thickness is increased this is no longer the case and bound modes cease to exist for thicknesses largerthan 0.2 µm in Fig. 7. At this point, the effective index of the surface plasmon mode becomes equal to therefractive index of the substrate and the surface mode transforms into a leaky mode. However, for a sufficientlythick dielectric layer the coupling to radiation modes will be weak and the mode resembles the plasmon modeof a single glass-to-gold interface with n = 1.4. Based on the calculation on the air side, we estimate that thetypical thickness required to be in this regime is also around 0.5 µm for a wavelength of 0.8 µm. It should thusbe possible to create a metal hole array in which the surface plasmon modes on the two glass-to-gold interfacesare indistinguishable by inserting a layer of sol-gel material of at least 500 nm on both sides.

4. SUMMARY AND CONCLUSIONS

Index matching of metal hole arrays makes the surface plasmon modes on different sides of the metal layerindistinguishable. Immersing a metal hole array in a liquid has the distinct advantage that the refractive index


0.0 0.5 1.0 1.5

Layer Thickness (µm)

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Effe

ctiv

eIn

dex

n = 1.40n = 1.00

n = 1.40n = 1.52

Figure 7. Effective index for a gold interface covered with a dielectric film as function of the thickness of the dielectriclayer. The dielectric layer has a refractive index n = 1.4. Results are shown for an ambient with a lower refractive index(n = 1.00) and with higher refractive index (n = 1.52). The solid curves are calculated for 0.8 µm wavelength. Thedotted lines indicate the change when the wavelength is 0.7 µm and 0.9 µm.

of the liquid can be tuned to match the refractive index of the substrate. By using large arrays (1×1 mm2) andan optical setup with a small numerical aperture (<0.01) it becomes possible to resolve an avoided crossing oftwo modes that propagate on different sides of the metal film. The two modes are coupled and the linewidth andamplitude of the high-energy mode vanish, while the linewidth and amplitude of the low energy mode increasewhen the sample is index matched. This behavior can be described by a coupled mode theory in which bothconservative and dissipative coupling is used.11

The use of solids to alter the dispersion of surface modes has the distinct advantage that the resulting holearrays are easier to handle. To realize such arrays we implemented a new fabrication method that involves andimprinting technique. After the first step, this creates a metal hole array with a dielectric pillar in each of theholes. A conventional hole was created by selectively removing the pillars after the transmission measurements.The angle dependent transmission measurements for p-polarized light of a metal hole array with pillars showsthat the (0,±1) mode is strongly enhanced for small angles of incidence when compared to the transmissionthrough a conventional hole array.12 The dielectric pillars effectively act as antennas that enhance the excitationefficiency of this particular mode.

In the last stage, the conventional hole array was covered by a thin layer of sol-gel glass with a refractiveindex n = 1.4. This cover layer partly index matches the surface plasmon modes on different sides of the metallayer. We find that this index matching is not perfect, because the surface plasmon on the substrate side has alarger effective index than the surface plasmon on the side with the cover layer. This is predominantly due tothe higher refractive index of the AF45 substrate (n = 1.52). In addition to the two different surface plasmonmodes, we identified another type of resonance in the measured transmission spectra. These resonances are dueto waveguide modes confined to the thin dielectric film. We compared the observed resonances as function ofangle of incidence to numerical calculations of the resonances for different thickness d of the film and find goodagreement for a film thickness d = 480±10 nm. Based on the numerical results, we estimate that a layer ofsol-gel material of ∼500 nm between the substrate and the metal layer and on top of the metal layer is sufficientto index-match the surface plasmon modes on both sides of the metal layer.

ACKNOWLEDGMENTS

We thank Arjen van Zuuk (DIMES, Delft) for fabrication of the hole array and Federica Galli for help withAFM measurements. The refractive index measurements on the liquids were done by Paul Junger of the “LeidseInstrumentmakers School (LIS)”. This research was funded by the Dutch Association for Scientific Research(NWO) and the Foundation for Fundamental Research of Matter (FOM).

Proc. of SPIE Vol. 6987 698713-10

REFERENCES[1] Ebbesen, T. W., Lezec, H. J., Ghaemi, H. F., Thio, T., and Wolff, P. A., “Extraordinary optical transmission

through sub-wavelength hole arrays,” Nature 391, 667 (1998).[2] Bethe, H. A., “Theory of diffraction by small holes,” Phys. Rev. 66, 163 (1944).[3] Chen, Y. J., Koteles, E. S., Seymour, R. J., Sonek, G. J., and Ballantyne, J. M., “Surface plasmons on

gratings: coupling in the minigap regions,” Sol. State Comm. 46, 95 (1983).[4] Lochbihler, H., “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795 (1994).[5] Ropers, C., Park, D. J., Stibenz, G., Steinmeyer, G., Kim, J., Kim, D. S., and Lienau, C., “Femtosecond

light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94, 113901 (2005).[6] Krishnan, A., Thio, T., Kim, T. J., Lezec, H. J., Ebbesen, T. W., Wolff, P. A., Pendry, J., Martın-Moreno,

L., and Garcıa-Vidal, F. J., “Evanescently coupled resonance in surface plasmon enhanced transmission,”Opt. Comm. 200, 1 (2001).

[7] Martın-Moreno, L., Garcıa-Vidal, F. J., Lezec, H., Pellerin, K. M., Thio, T., Pendry, J. B., and Ebbesen,T. W., “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev.Lett. 86, 1114 (2001).

[8] Pang, L., Tetz, K. A., and Fainman, Y., “Observation of the splitting of degenerate surface plasmon polaritonmodes in a two-dimensional metallic nanohole array,” Appl. Phys. Lett. 90, 111103 (2007).

[9] Henzie, J., Lee, M. H., and Odom, T. W., “Multiscale patterning of plasmonic metamaterials,” NatureNanotechnology 2, 549 (2007).

[10] Garcia de Abajo, F. J., “Light scattering by praticle and hole arrays,” Rev. Mod. Phys. 79, 1267 (2007).[11] de Dood, M. J. A., Driessen, E. F. C., Stolwijk, D., and van Exter, M. P., “Observation of coupling between

surface plasmons in index-matched hole arrays,” Phys. Rev. B 77, (in press) (2008).[12] Stolwijk, D., Driessen, E. F. C., Verschuuren, M. A., ’t Hooft, G. W., van Exter, M. P., and de Dood, M.

J. A., “Enhanced coupling of plasmons in hole arrays with periodic dielectric antennas,” Opt. Lett. 33, 363(2008).

[13] Wood, R. W., “On the remarkable case of uneven distribution of a light in a diffractived grating spectrum,”Philos. Mag. 4, 396 (1902).

[14] Fano, U., “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces(sommerfelds waves),” J. Opt. Soc. Am. 31, 213 (1941).

[15] Genet, C., van Exter, M. P., and Woerdman, J. P., “Fano-type interpretation of red shifts and red tails inhole array transmission spectra,” Opt. Comm. 225, 331 (2003).

[16] Sarrazin, M., Vigneron, J., and Vigoreux, J., “Role of wood anomalies in optical properties of thin metallicfilms with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003).

[17] Ghaemi, H. F., Thio, T., Grupp, D. E., Ebbesen, T. W., and Lezec, H. J., “Surface plasmons enhanceoptical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).

[18] Johnson, P. B. and Christy, R. W., “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972).[19] Spreeuw, R. J. C., Centeno Neelen, R., van Druten, N. J., Eliel, E. R., and Woerdman, J. P., “Mode

coupling in a he-ne ring laser with backscattering,” Phys. Rev. A 42, 4315 (1990).[20] Spreeuw, R. J. C., van Druten, N. J., Beijersbergen, M. W., Eliel, E. R., and Woerdman, J. P., “Classical

realization of a strongly driven two-level system,” Phys. Rev. Lett. 65, 2642 (1990).[21] Johnson, S. G., Ibanescu, M., Skorobogatiy, M. A., Weisberg, O., Joannopoulos, J. D., and Fink, Y.,

“Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611(2002).

[22] Kim, D. S., Hongh, S. C., Malyarchuk, V., Yoon, Y. C., Ahn, Y. H., Yee, K. J., Park, J. W., Kim,J., Park, Q. H., and Lienau, C., “Microscopic origin of surface-plasmon radiation in plasmonic band-gapnanostructures,” Phys. Rev. Lett. 91, 143901 (2003).

[23] van der Molen, K. L., Klein Koerkamp, K. J., Enoch, S., Segerink, F. B., van Hulst, N. F., and Kuipers, L.,“Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwave-length holes: Experiment and theory,” Phys. Rev. B 72, 045421 (2005).

Proc. of SPIE Vol. 6987 698713-11

[24] Barnes, W. L., Murray, W. A., Dintinger, J., Devaux, E., and Ebbesen, T. W., “Surface plasmon polaritonsand their role in the enhanced transmission of light through periodic arrays of subwavelength holes in ametal film,” Phys. Rev. Lett. 92, 107401 (2004).

[25] Barnes, W. L., Preist, T. W., Kitson, S. C., and Sambles, J. R., “Physical origin of photonic energy gapsin the propagation of surface plasmons on gratings,” Phys. Rev. B 54, 6227 (1996).

[26] Wood, R. W., “Anamolous diffraction gratings,” Phys. Rev. 48, 928 (1935).[27] Vos, W. L., Megens, M., van Kats, C. M., and Bosecke, P., “Transmission and diffraction by photonic

colloidal crystals,” J. Phys. Cond. Matter 8, 9503 (1996).[28] Adams, A., Hansma, J. M. P. K., and Schlesinger, Z., “Light emission from surface-plasmon and waveguide

modes excited by N atoms near a silver grating,” Phys. Rev. B 25, 3457 (1982).[29] Yariv, A. and Yeh, P., [Optical waves in crystals ], Wiley, New York (1984).[30] Karalis, A., Lidorikis, E., Ibanescu, M., Joannopoulos, J. D., and Soljacic, M., “Surface-plasmon-assisted

guiding of broadband slow and subwavelength light in air,” Phys. Rev. Lett. 95, 063901 (2005).

Proc. of SPIE Vol. 6987 698713-12

index matching of surface plasmons

Documents