The Excitation of Surface Plasmons by Light is Denoted as a Surface Plasmon Resonance

7/29/2019 The Excitation of Surface Plasmons by Light is Denoted as a Surface Plasmon Resonance

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The excitation ofsurface plasmons by light is denoted as a surface plasmon resonance (SPR)

for planar surfaces orlocalized surface plasmon resonance (LSPR) for nanometer-sized

metallic structures.

This phenomenon is the basis of many standard tools for measuringadsorptionof material onto

planar metal (typically gold and silver) surfaces or onto the surface of metalnanoparticles. It isthe fundamentals behind many color basedbiosensorapplications and different lab-on-a-chip

sensors.

Surface plasmons, also known as surface plasmonpolaritons, are surface electromagnetic waves

that propagate in a direction parallel to the metal/dielectric(or metal/vacuum) interface. Since

the wave is on the boundary of the metal and the external medium (air or water for example),these oscillations are very sensitive to any change of this boundary, such as the adsorption of

molecules to the metal surface.

To describe the existence and properties of surface plasmons, one can choose from various

models (quantum theory, Drude model, etc.). The simplest way to approach the problem is totreat each material as a homogeneous continuum, described by a frequency-dependentrelative

permittivity between the external medium and the surface. This quantity, hereafter referred to as

the materials' "dielectric constant," is complex-valued. In order for the terms which describe the

electronic surface plasmons to exist, the real part of the dielectric constant of the metal must benegative and its magnitude must be greater than that of the dielectric. This condition is met in the

IR-visible wavelength region for air/metal and water/metal interfaces (where the real dielectric

constant of a metal is negative and that of air or water is positive).

Localized surface plasmon polaritons (LSPRs) are collective electron charge oscillations in

metallic nanoparticles that are excited by light. They exhibit enhanced near-field amplitude at the

resonance wavelength. This field is highly localized at the nanoparticle and decays rapidly awayfrom the nanoparticle/dieletric interface into the dielectric background, though far-fieldscattering by the particle is also enhanced by the resonance. Light intensity enhancement is a

very important aspect of LSPRs and localization means the LSPR has very high spatial

resolution (subwavelength), limited only by the size of nanoparticles. Because of the enhancedfield amplitude, effects that depend on the amplitude such as magneto-optical effect are also

enhanced by LSPRs.[1][2]

In order to excite surface plasmons in a resonant manner, one can use an electron orlight beam

(visible and infrared are typical). The incoming beam has to match itsimpulse to that of theplasmon. In the case ofp-polarized light (polarization occurs parallel to the plane of incidence),

this is possible by passing the light through a block of glass to increase the wavenumber(and theimpulse), and achieve the resonance at a given wavelength and angle. S-polarized(polarizationoccurs perpendicular to the plane of incidence) light cannot excite electronic surface plasmons.

Electronic and magnetic surface plasmons obey the following dispersion relation:
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where stands for the dielectric constant, and for the magnetic permeability of the materials (1:

the glass block, 2: the metal film).

Typical metals that support surface plasmons are silver and gold, but metals such as copper,titanium or chromium are also known to be applicable.

Using light to excite SP waves, there are two constructions which are well known. In the Otto

setup, the light is shone on the wall of a glass block, typically a prism, and totally reflected. A

thin metal (for example gold) film is positioned close enough, that the evanescent waves caninteract with the plasma waves on the surface and excite the plasmons.

In the Kretschmann configuration, the metal film is evaporated onto the glass block. The light is

again illuminating from the glass, and an evanescent wave penetrates through the metal film. The

plasmons are excited at the outer side of the film. This configuration is used in most practicalapplications.

[edit] SPR emission

When the surface plasmon wave hits a local particle or irregularitylike on a rough surface,

part of the energy can be re-emitted as light. This emitted light can be detected behindthe metal

film in various directions.

[edit] Applications

Surface plasmons have been used to enhance the surface sensitivity of several spectroscopic

measurements including fluorescence,Raman scattering, andsecond harmonic generation.

However, in their simplest form, SPR reflectivity measurements can be used to detect molecular

adsorption, such as polymers, DNA or proteins, etc. Technically, it is common, that the angle ofthe reflection minimum (absorption maximum) is measured. This angle changes in the order of

0.1 during thin (about nm thickness) film adsorption. (See also the Examples.) In other cases thechanges in the absorption wavelength is followed. [3] The mechanism of detection is based on that

the adsorbing molecules cause changes in the local index of refraction, changing the resonance

conditions of the surface plasmon waves.

If the surface is patterned with different biopolymers, using adequate optics and imaging sensors(i.e. a camera), the technique can be extended to surface plasmon resonance imaging (SPRI).

This method provides a high contrast of the images based on the adsorbed amount of molecules,

somewhat similar to Brewster angle microscopy (this latter is most commonly used together with

aLangmuir-Blodgett trough).

For nanoparticles, localized surface plasmon oscillations can give rise to the intense colors of

suspensions orsols containing the nanoparticles. Nanoparticles or nanowires of noble metals

exhibit strongabsorption bands in the ultraviolet-visiblelight regime that are not present in thebulk metal. This extraordinary absorption increase has been exploited to increase light absorption

in photovoltaic cells by depositing metal nanoparticles on the cell surface.[4] The energy (color)

of this absorption differs when the light is polarized along or perpendicular to the nanowire.[5]
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Shifts in this resonance due to changes in the local index of refraction upon adsorption to the

nanoparticles can also be used to detect biopolymers such as DNA or proteins. Related

complementary techniques include plasmon waveguide resonance, QCM,extraordinary opticaltransmission, and Dual Polarisation Interferometry

[edit] Data interpretation

The most common data interpretation is based on theFresnel formulas, which treat the formed

thin films as infinite, continuous dielectric layers. This interpretation may result multiple

possiblerefractive index and thickness values. However, usually only one solution is within thereasonable data range.

Metal particle plasmons are usually modeled using theMie scattering theory.

In many cases no detailed models are applied, but the sensors are calibrated for the specific

application, and used with interpolationwithin the calibration curve.

Layer-by-layer self-assembly

SPR curves measured during the adsorption of apolyelectrolyte and then a clay mineralself-

assembled film onto a thin (ca. 38 nanometers) gold sensor.

One of the first common applications of surface plasmon resonance spectroscopy was themeasurement of the thickness (and refractive index) of adsorbed self-assembled nanofilms on

gold substrates. The resonance curves shift to higher angles as the thickness of the adsorbed film

increases. This example is a 'static SPR' measurement.

When higher speed observation is desired, one can select an angle right below the resonance

point (the angle of minimum reflectance), and measure the reflectivity changes at that point. Thisis the so called 'dynamic SPR' measurement. The interpretation of the data assumes, that the

structure of the film does not change significantly during the measurement.
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[edit] Binding constant determination

Association and dissociation signal

Example of output from Biacore

When the affinity of two ligandshas to be determined, thebinding constantmust be determined.

It is the equilibrium value for the product quotient. This value can also be found using thedynamical SPR parameters and, as in any chemical reaction, it is the association rate divided by

the dissociation rate.

For this, a so-called bait ligand is immobilized on the dextran surface of the SPR crystal.

Through a microflow system, a solution with the prey analyte is injected over the bait layer. As

the prey analyte binds the bait ligand, an increase in SPR signal (expressed in response units,RU) is observed. After desired association time, a solution without the prey analyte (usually the

buffer) is injected on the microfluidics that dissociates the bound complex between bait ligandand prey analyte. Now as the prey analyte dissociates from the bait ligand, a decrease in SPR

signal (expressed in response units, RU) is observed. From these association ('on rate', ka) and

dissociation rates ('off rate', kd), the equilibrium dissociation constant ('binding constant',KD) can

be calculated.

The actual SPR signal can be explained by the electromagnetic 'coupling' of the incident light

with the surface plasmon of the gold layer. This plasmon can be influenced by the layer just a

few nanometer across the gold-solution interface i.e. the bait protein and possibly the prey

protein. Binding makes the reflection angle change;
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[edit] Magnetic plasmon resonance

Recently, there has been an interest in magnetic surface plasmons. These require materials with

large negative magnetic permeability, a property that has only recently been made available withthe construction ofmetamaterials.

When light moves from a medium of a given refractive indexn1 into a second medium with

refractive index n2, bothreflection and refraction of the light may occur.

Variables used in the Fresnel equations.

In the diagram on the right, an incident light rayPO strikes at point O the interface between twomedia of refractive indices n1 and n2. Part of the ray is reflected as ray OQ and part refracted as

ray OS. The angles that the incident, reflected and refracted rays make to the normal of theinterface are given as i, r and t, respectively. The relationship between these angles is given bythe law of reflection: i = r ; and Snell's law: sin(i) / sin(t) = n2 / n1 .

The fraction of the incidentpowerthat is reflected from the interface is given by thereflectance

R and the fraction that is refracted is given by thetransmittanceT.[1] The media are assumed to be

non-magnetic.

The calculations ofR and Tdepend onpolarisation of the incident ray. If the light is polarised

with the electric fieldof the light perpendicular to the plane of the diagram above (s-polarised),

the reflection coefficient is given by:

where t can be derived from i by Snell's law and is simplified using trigonometric identities.
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If the incident light is polarised in the plane of the diagram (p-polarised), theR is given by:

As a consequence of the conservation of energy, the transmission coefficientin each case is

given by Ts = 1 Rs and Tp = 1 Rp.[2]

If the incident light is unpolarised (containing an equal mix ofs- andp-polarisations), the

reflection coefficient isR = (Rs +Rp)/2.

Equations for coefficients corresponding to ratios of the electric fieldamplitudesof the waves

can also be derived, and these are also called "Fresnel equations". These take several different

forms, depending on the choice of formalism andsign convention used. The amplitudecoefficients are usually represented by lower case rand t. In some formalisms they satisfy:

[3]

At one particular angle for a given n1 and n2, the value ofRp goes to zero and ap-polarisedincident ray is purely refracted. This angle is known as Brewster's angle, and is around 56 for a

glass medium in air or vacuum. Note that this statement is only true when the refractive indices

of both materials are real numbers, as is the case for materials like air and glass. For materialsthat absorb light, like metalsand semiconductors, n is complex, andRp does not generally go to

zero.

When moving from a denser medium into a less dense one (i.e., n1 > n2), above an incidence

angle known as the critical angle, all light is reflected andRs =Rp = 1. This phenomenon isknown as total internal reflection. The critical angle is approximately 41 for glass in air.
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When the light is at near-normal incidence to the interface ( i t 0), the reflection and

transmission coefficient are given by:

For common glass, the reflection coefficient is about 4%. Note that reflection by a window is

from the front side as well as the back side, and that some of the light bounces back and forth a

number of times between the two sides. The combined reflection coefficient for this case is 2R/(1 +R), when interference can be neglected (seebelow).

The discussion given here assumes that thepermeability is equal to the vacuum permeability 0in both media. This is approximately true for mostdielectricmaterials, but not for some other

types of material. The completely general Fresnel equations are more complicated.

[edit] Effect from multiple surfaces

When light makes multiple reflections between two or more parallel surfaces, the multiple beams

of light generally interferewith one another, resulting in net transmission and reflection

amplitudes that depend on the light's wavelength. The interference, however, is seen only whenthe surfaces are at distances comparable to or smaller than the light's coherence length, which for

ordinary white light is few micrometers; it can be much larger for light from alaser. An example

of this effect is the iridescent colours seen in a soap bubble or in thin oil films on water.
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Applications include FabryProt interferometers,antireflection coatings, andoptical filters. A

quantitative analysis of these effects is based on the Fresnel equations, but with additional

calculations to account for interference.

The transfer-matrix method, or the recursive Rouard method[4] can be used to solve multiple-

surface problems.

The refractive index orindex of refraction of a substance is a measure of the speed of lightinthat substance. It is expressed as a ratio of the speed of light in vacuum relative to that in the

considered medium.[note 1] The velocity at which light travels in vacuum is aphysical constant,

and the fastest speed at which energy orinformationcan be transferred. However, light travels

slower through any given material, or medium, that is not vacuum. (See: light in a medium).[1][2][3]

[4]

A simple mathematical description of the refractive index is as follows:

n = velocity of light in a vacuum / velocity of light in medium

For example, the refractive index of water is 1.33, meaning that light travels 1.33 times faster in

a vacuum as it does in water.

As light exits a medium, such as air, water, or glass, it may also change itspropagationdirection

in proportion to the refractive index (see Snell's law). By measuring the angle of incidence and

angle of refraction of the light beam, the refractive index n can be determined. Refractive indexof materials varies with the frequency ofradiated light. This results in a slightly different

refractive index for eachcolor. The cited values of refractive indices, such as 1.33 for water, are

taken foryellow light of a sodium source which has the wavelength of 589.29 nanometers.[1][2][5]

The refractive index, n, of a medium is defined as the ratio of the speed, c, of a wavephenomenon such as light orsound in a reference medium to thephase speed, vp, of the wave in

the medium in question:

It is most commonly used in the context oflightwith vacuum as a reference medium, although

historically other reference media (e.g. airat a standardizedpressureandtemperature) have been

common. It is usually given the symbol n. In the case of light, it equals
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where is the material's relativepermittivity, andr is its relativepermeability. For most

naturally occurring materials,r is very close to 1 at optical frequencies,[6] therefore n is

approximately . Contrary to a widespread misconception, the real part of a complex n maybe less than one, depending upon the material and wavelength (see dispersion (optics)). This has

practical technical applications, such as effective mirrors for X-rays based on total external

reflection. Another example is that the n of electromagnetic waves in plasmas is less than 1.

Thephase speedis defined as the rate at which the crests of thewaveformpropagate; that is, therate at which thephase of the waveform is moving. Thegroup speedis the rate at which the

envelope of the waveform is propagating; that is, the rate of variation of theamplitude of the

waveform. Provided the waveform is not distorted significantly during propagation, it is thegroup speed that represents the rate at which information (and energy) may be transmitted by the

wave (for example, the speed at which a pulse of light travels down an optical fiber). For the

analytic properties constraining the unequal phase and group speeds in dispersive media, refer to

dispersion (optics).

A closely related quantity is refractivity, which in atmospheric applications is denotedNanddefined asN= 106

(n - 1). The 106 factor is used because for air, n deviates from unity at most a few parts per

thousand.

Speed of light

Refraction of light at the interface between two media of different refractive indices, with n2 > n1.Since thephase speed is lower in the second medium (v2 < v1), the angle of refraction 2 is less

than the angle of incidence 1; that is, the ray in the higher-index medium is closer to the normal.

The speed of all electromagnetic radiation in vacuum is the same: approximately 310 8

meters/second, and is denoted by c. Therefore, ifv is thephase speed of radiation of a specificfrequency in a specific material, the refractive index is given by

or inversely
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The numbern is typically greater than one. However, at certain frequencies (e.g. nearabsorption

resonances, and forX-rays), n will actually be smaller than one[7] (see alsoCherenkov radiation).

This does not contradict the theory of relativity, which holds that noinformation-carrying signalcan ever propagate faster than c, because thephase speed is not the same as the group speedor

the signal speed.

Sometimes, a "group speed refractive index", usually called the group index is defined:

where vg is the group speed. This value should not be confused with n, which is always defined

with respect to the phase speed. When the dispersion is small, the group velocity can be linked to

the phase velocity by the relation [8]

In this case the group index can thus be written in terms of the wavelength dependence of the

refractive index as

where is the wavelength in the medium. When the refractive index of a medium is known as a

function of the vacuum wavelength (instead of the wavelength in the medium), the

corresponding expression for the group index is[9]

where 0 is the wavelength in vacuum.

At the microscale, an electromagnetic wave's phase speed is slowed in a material because the

electric fieldcreates a disturbance in the charges of each atom (primarily the electrons)

proportional to thepermittivity of the medium. The charges will, in general, oscillate slightly outofphasewith respect to the driving electric field. The charges thus radiate their own

electromagnetic wave that is at the same frequency but with a phase delay. The macroscopic sumof all such contributions in the material is a wave with the same frequency but shorter

wavelength than the original, leading to a slowing of the wave's phase speed. Most of theradiation from oscillating material charges will modify the incoming wave, changing its velocity.

However, some net energy will be radiated in other directions (see scattering).
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If in a given region the values of refractive indices n orng are found to differ from unity (whether

homogeneously, or isotropically, or not), then this region is distinct from vacuum in the above

sense for lacking Poincar symmetry.

Interaction of light and the medium

Whereas light can be considered as particles or waves, it is treated as waves in optics.[10] Waves

travel at different speeds in different media. An example would be the difference in the speed of

sound through water as opposed to its speed through air. As light crosses from one medium intoanother, the same wave will travel at different speeds. This process can be understood as follows:[citation needed] A light wave enters a transparent material and excites an electron of an atom that

makes up that material. The excited electron now emits a light wave of its own, which in turnexcites another electron of another atom. Between atoms, light is traveling at light speed (c =

3108 m/s). However, the time it takes to excite and emit contributes to the average time it takes

for a light wave to "travel" through the medium. In glass, the apparent light speed is about 2108

m/s.

The index of refraction characterizes not only the light propagation speed, but also the bending

angle and the amount of radiation transmitted and reflected by a material. It also defines the

Brewster angle, the slant angle at which onepolarizationis totally absorbed.[11]

The amount of light reflected from the material under normal incidence is proportional to thesquare of the index change at the face:

R = [(n1-n2)/(n1+n2)]2

For common glass in air, n1 = 1 and n2 = 1.5; thus about = 4% of light is reflected.[11]

Negative refractive index

See also:Negative index metamaterials

Recent research has also demonstrated the existence of the negative refractive index, which can

occur ifpermittivity andpermeability have simultaneous negative values. This can be achieved

with periodically constructed negative index metamaterials. The resulting negative refractive

index (i.e., a reversal ofSnell's law) offers the possibility of the superlens and other exoticphenomena.[12][13][14][15]
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Dispersion and absorption

See also: Dispersion (optics)

The variation of refractive index with wavelength for various glasses.

In real materials, thepolarization does not respond instantaneously to an applied field. This

causes dielectric loss, which can be expressed by apermittivitythat is bothcomplex andfrequency dependent. Real materials are not perfectinsulatorseither, i.e., they have non-zero

direct currentconductivity. Taking both aspects into consideration, a complex index of refraction

can be defined:

Here, the real part of the refractive index n indicates the phase speed, while the imaginary part

indicates the amount ofabsorption loss when the electromagnetic wave propagates through the

material. (See Mathematical descriptions of opacity.) is often called the extinction coefficientin physics. ("Extinction coefficient" has a different definition within chemistry.) Both n and are

dependent on the frequency (wavelength). In most circumstances > 0 (light is absorbed) or =

0 (light travels forever without loss). In special situations, especially lasers, it is also possible that < 0 (light is amplified).

An alternative convention uses instead of , but where > 0 still

corresponds to loss. Therefore these two conventions are inconsistent and should not be

confused. The difference is related to defining sinusoidal time dependence as Re(e it) versusRe(e+ it). See Mathematical descriptions of opacity.

The effect that n varies with frequency (except in vacuum, where all frequencies travel at the

same speed, c) is known as dispersion, and it is what causes aprism to divide white light into its

constituent spectral colors, explains rainbows, and is the cause ofchromatic aberration in lenses.In regions of the spectrum where the material does not absorb, the real part of the refractive
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index tends to increase with frequency. Near absorption peaks, the curve of the refractive index

is a complex form given by the KramersKronig relations, and can decrease with frequency.

Since the refractive index of a material varies with the frequency (and thus wavelength) of light,it is usual to specify the corresponding vacuum wavelength at which the refractive index is

measured. Typically, this is done at various well-defined spectralemission lines; for example, nDis the refractive index at theFraunhofer"D" line, the centre of the yellow sodium double

emission at 589.29 nm wavelength.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and

Sellmeier coefficients are often quoted instead of the refractive index in tables. For some

representative refractive indices at different wavelengths, seelist of indices of refraction.

As shown above, dielectric loss and non-zero DC conductivity in materials cause absorption.Good dielectric materials such as glass have extremely low DC conductivity, and at low

frequencies the dielectric loss is also negligible, resulting in almost no absorption ( 0).

However, at higher frequencies (such as visible light), dielectric loss may increase absorptionsignificantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through use of the

KramersKronig relations. For example, one can determine a material's full complex refractive

index as a function of wavelength from an absorption spectrum of the material.

Relation to dielectric constant

The dielectric constant(which is often dependent on wavelength) is simply the square of the

(complex) refractive index in a non-magnetic medium (one with a relativepermeability of unity).

The refractive index is used for optics in Fresnel equations and Snell's law; while the dielectricconstant is used in Maxwell's equations and electronics.

Where , , , n, and are functions of wavelength:

Conversion between refractive index and dielectric constant is done by:

[16]
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Documents

The Excitation of Surface Plasmons by Light is Denoted as a Surface Plasmon Resonance