COMSOL 6.4 - Surface Plasmon Polariton Excitation via Otto and Kretschmann Configurations

Surface Plasmon Polariton Excitation via Otto and Kretschmann Configurations

Total internal reflection refers to the complete reflection of light, when it travels from a dense medium with a high refractive index to a medium with a lower refractive index. This phenomenon occurs at the interface between the two media, when the incident angle of light is larger than a limiting value, namely the critical angle. Even though the reflection is total, the tangential components of the electromagnetic field must be continuous across the interface and nonzero. However, the spatial distribution of the transmitted field should be limited in the sense that a continuous power flow within the medium of lower refractive index is not permitted. This demonstrates that the total internal reflection mechanism is accompanied by a nonradiative electromagnetic surface wave propagating along the interface with an evanescent wavefront in the normal direction within the transmitted domain.

In the case that another dense medium is brought close to the evanescent field, the total internal reflection mechanism may get disrupted, thereby leading to some power transmission into the new medium. This phenomenon, referred to as evanescent-wave coupling, is commonly used in nanophotonic devices such as waveguide sensors, couplers, and fiber tapping amongst many others.

The evanescent-wave coupling is the fundamental physics that enables excitation of surface plasmon polaritons (SPPs) via the Otto and Kretschmann configurations. For instance, when a metallic structure is brought into the vicinity of the evanescent field generated by the total internal reflection, it excites the free electrons in the metal and introduces SPPs propagating along the dielectric–metal interface. The optimal excitation of SPPs is achieved through the precise angle phase matching, which refers to the process of matching the tangential wave vector of the incident light to the propagation constant of the SPPs.

This model discusses the simulation setup for exciting SPPs using the Otto and Kretschmann configurations. It also analyzes the reflectance, transmittance, and SPP coupling of the systems. Both configurations involve the total internal reflection and evanescent-wave coupling phenomena. Angle phase matching is also performed to calculate the requirements to achieve optimal SPP excitation.

Model Definition

Figure 1 shows the two simulation setups: (a) Otto and (b) Kretschmann configurations. In both cases, SPPs are excited on a metal structure thanks to the evanescent-wave coupling mechanism. The evanescent wave generated due to the total internal reflection at the prism layer, interacts with the plasma waves in metal, and excites SPPs.

Figure 1: Excitation of surface plasmon polaritons using the (a) Otto and (b) Kretschmann configurations due to the total internal reflection mechanism. In both setups, the surface plasmon propagates in the x direction, along the metal–air interface, and decays in the y direction.

Imagine an electromagnetic wave propagating from a high index (refractive index ni) to a low index medium (refractive index nt), with the plane of incidence being the xy-plane at z = 0. Assuming that the interface between the two media is at y = 0, the transmitted wave vector can be expressed as

(1)

Here, kx = kt sin θt and ky = kt cos θt, where kt is the wavenumber of the transmitted wave band and θt is the angle of refraction.

Following Snell’s law, ni sin θi = nt sin θt, where θi is the incident angle, the y-component of the transmitted wave vector can be written as

(2)

When the condition of total internal reflection, sin θi > nt/ni, is satisfied, Equation 2 can be written as

(3)

Assume a transverse magnetic (TM) incident wave with magnetic field polarized in the z direction (perpendicular to the plane of incidence). The magnetic field component of such waves is gives by

(4)

Here, H0 is the magnetic field amplitude. From Equation 4, the transmitted magnetic field component leads to

(5)

Here, Ht0 is the transmitted magnetic field amplitude, α is the attenuation constant, and the ‘+’ sign of ky in Equation 2 is ignored as it corresponds to wave amplification along the y direction. Equation 5 shows that the transmitted wave is evanescent along the y direction, as the total internal reflection condition is satisfied.

In the Otto configuration, as is shown in panel (a) of Figure 1, the incident light is totally internally reflected back from the prism–air interface. A metal structure, that supports SPPs, is positioned near the interface within a few wavelengths of distance so that the evanescent wave can interact with the metal plasma wave and excites SPPs. By adequately tuning the angle of incidence, maximum SPP coupling can be achieved. In this situation, the reflectivity of the system almost vanishes as most of the waves are refracted into metal. Such a resonance condition appears over a very narrow range of incident angles.

The incident angle at which the optimum SPP excitation is achieved can be calculated using the angle phase matching. At this angle, the tangential wavenumber of the incident wave

is equal to the propagation constant

of the SPPs. The angle at which the optimum SPP excitation occurs is expressed as

(6)

In the Kretschmann configuration, as is shown in panel (b) of Figure 1, a metal thin film is deposited on a prism block. Here, the evanescent wave generated due to the total internal reflection penetrates through the metal layer, and SPPs are excited on the outer (metal–air) boundary. When the metal layer is thick, SPPs cannot be excited as the evanescent wave will be completely absorbed. The Kretschmann configuration is more popular in practical applications as it resolves the challenging control of the precise prism–metal distance in the Otto configuration.

This model simulates excitation of SPPs using Otto and Kretschmann configurations, and perform angle phase matching analysis to calculate the required incident angle for the optimum SPP excitation. Silver is considered as the metal with material properties available from the built-in Optical Material Library. Periodic ports are used on the top and bottom boundaries of the model. The top port, with excitation turned on, launches the incident TM-light, and the bottom port, with excitation turned off, absorbs the transmitted wave. A Frequency Domain study step is used to solve for the domain field. An Auxiliary sweep of the incident angle is performed in the study step to evaluate the surface plasmon resonance condition by analyzing reflectance, transmittance, and SPP coupling of the systems.

Results and Discussion

Figure 2 shows the y-component of the electric field using the Otto configuration. The incident light is TM-polarized with an angle of incidence of 49°. The evanescent wave created due to the total internal reflection is strongly coupled to the metal in the form of SPPs propagating along the x direction. The decay rate in the metal is much faster than in air thanks to the strong absorption.

Figure 2: The y-component of the electric field for the Otto configuration. The wavelength is set to 500 nm and the angle of incidence is 49°. Red arrows indicate the incident light wave vector and black arrows indicate the electric field polarization.

Figure 3 shows the reflectance, transmittance, and SPP coupling curve versus the incident angle. The surface plasmon resonance condition is fulfilled at an angle of 49° where maximum SPP coupling is observed.

Figure 3: Reflectance, transmittance, and SPP coupling versus the incident angle.

Figure 4 shows a similar study, but with Kretschmann configuration. Here, SPPs are excited on the outer boundary of the thin metal layer (that is, the metal–air interface).

Figure 4: The y-component of the electric field for the Kretschmann configuration. The wavelength is set to 500 nm and the angle of incidence is 49°. Red arrows indicate the incident light wave vector and black arrows indicate the electric field polarization.

Figure 5 shows that the surface plasmon resonance condition is fulfilled at a 49° angle of incidence, similar to the case of the Otto configuration.

Figure 5: Reflectance, transmittance, and absorptance versus incident angle.

Figure 6 shows the tangential wavenumber of the incident wave and the propagation constant of the SPPs versus the angle of incidence. It shows that the angle phase matching is obtained when the incident angle is approximately 49° and optimum SPP is excited at this angle.

Figure 6: The tangential wavenumber of the incident wave and the propagation constant of the surface plasmon polariton wave versus the incident angle. The wavelength is set to 500 nm in free space.

For a modeling setup to use for calculating the dispersion relation and additional SPP simulation examples, see the blog post in Ref. 1.

Reference

1. X. Chen, “Modeling Surface Plasmon Polaritons in COMSOL®,” COMSOL Blog, 12 Oct. 2022; www.comsol.com/blogs/modeling-surface-plasmon-polaritons-in-comsol/.

Wave_Optics_Module/Waveguides/spp_otto_kretschmann

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, Use two physics interfaces to simulate the Otto and Kretschmann configurations independently, and another physics interface to calculate the angle phase matching.

click

In the tree, select > > .

Click .

Click , to add the second physics interface.

Click , to add the third physics interface.

Click

In the tree, select > .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
L	1[um]	1E-6 m	Simulation domain length
H	3[um]	3E-6 m	Simulation domain height
lda0	500[nm]	5E-7 m	Wavelength
angle	0[deg]	0 rad	Incident angle
n_prism	1.4	1.4	Refractive index, prism