OAM & Surface Plasmon Resonance

Background and Goal

The ultimate goal of the Surface Plasmon Resonance (SPR) experiment is to understand angular momentum states of surface plasmons (SPs). This wiki article assumes that the reader has general knowledge of photonics and electrodynamics

Surface Plasmon Resonance

A surface plasmon is very similar conceptually to a photon confined to a 2-dimensional conductor. The coupling of light to a conductor generates charge density waves that propagate based on the (complex) dielectric constant and thickness of the conductor. These two conditions generally dictate the momentum of a surface plasmon (and therefore the associated wave number, k).

When coherent light undergoes total internal reflection in at an interface in a dialectric, an evanescent field propagating parallel to the component of the incident light in the plane of the dielectric interface is generated. When a conducting film as described above, whose associated plasmon wave number matches the wave number of the evanescent field, a surface plasmon resonance is is generated. In order to tune the resonance, the incident angle of the light can be adjusted, thereby adjusting the wave number of the evanescent field. The wave number of the evanescent field can also be changed by using light of a different wavelength.

In practice, an approximately 43nm thick gold film or 40nm thick silver film applied to the back of a glass prism of index n=1.5 makes for an excellent plasmon setup. While silver is more conductive than gold, thereby allowing for a deeper resonance, its corrosion in atmosphere makes it less practical.

When a surface plasmon encounters a defect such as a bump or a hole in the conducting surface along which it is propagating, the defect acts as a scatterer for the SP (as it is locally off-resonance for that wavelength of SP) and the SP either deflects off the defect or is re-radiated from the surface, converting back to a photon. In order to conserve linear momentum, a re-radiated SP leaves the dielectric film at the angle of incidence matching its momentum in the plane of propagation. In other words, if a plasmon generated by light of a given wavelength whose resonant angle is θ is ejected from the conductive surface, it will be ejected at the same wavelength as the generated light at the angle θ.

Since any azimuthal angle in the plane of the conductor can satisfy the above constraint on the exit of a re-radiated plasmon, a hollow cone of radiation is observed when a surface plasmon resonance is generated in a film with defects sufficiently large to scatter and eject SPs.

A great deal of information can be inferred by measurements of this radiation cone. Because it is a cone whose vertex is at the dielectric/conductor interface, it is convenient to use a prism with cylindrical or spherical symmetry in the axis normal to the dielectric/conductor interface.

Angular Momentum

Diagram of wavefront helicity and associated topological charge

Light can carry both orbital and spin angular momentum. Orbital angular momentum (OAM) of light is a consequence of its spacial distribution, such as helicity, while spin angular momentum (SAM) corresponds to circular polarization states. OAM states are quantized by topological charge m which carriers any integer value, while SAM states are quantized by 1, 0, or -1 as photons are bosons.

Techniques

This section contains information pertaining to the generation of optical OAM and the detection therein.

Generation of Optical OAM

The most intuitive method for generating optical OAM is by use of a spiral phase plate. Such a phase plate retards the input wave by an amount depending on the azimuthal angle with respect to the optical axis (pictured below) giving rise to a helical wavefront characteristic of an $LG_{0,m}$ beam, where m represents the topological charge. The shelf distance d of a phase plate made of a material with refractive index n must equal $d=m\lambda /n$. In practice, these are expensive optical elements that may produce questionable mode purity for m>3

Holographic generation of optical OAM is a far more common practice. A forked diffraction grating with an m-pronged fork dislocation can be easily made using a film camera to downscale a printed rendering, and will produce $LG_{0,m}$ beams with exceptional mode purity in the first order. This is pictured below.

A spiral phase plate

A holographic grating for generation of an $LG_{0,2}$ beam in the first order

Detection of Optical OAM

Detecting OAM in the form of an $LG_{0,m}$ beam is fairly simple in principle and can be done easily one of two ways.

Firstly, if m is a known or suspected quantity, a holographic grating of charge -m can be used to convert the beam to an Failed to parse (syntax error): {\displaystyle LG_}0,0}} beam. When focused through a pinhole, only LG modes with m=0 will be transmitted as LG beams with nonzero m present with a central vortex.

Secondly, if m is unknown, a modified Mach-Zehnder interferometer can be used to infer m. A dove prism is used to flip the beam in one arm of the interferometer. Reinterference of the flipped beam with the original will result in an interference pattern with 2m spokes.

Both of the above methods are discussed in the 2002 paper by J Leach, M J Padgett, et al. found in Phys. Rev. Lett 88, 257901. The purpose of this paper is to outline a method for single-photon detection of OAM, which is necessary for the detection of OAM in an optical field of arbitrary distribution and polarization.

The method outlined in the paper is a modification of the interferometer proposed above, making use of two dove prisms in order to give the two beams in the interferometer an arbitrary rotation relative to eachother. For example, if this rotation is 180 degrees, m=0 light will be unaffected, while m=1 light will interfere destructively in the output port. In fact, for 180 degrees of rotation, this interferometer acts as an OAM mode parity sorters, with even modes passing through one port and odd modes passing through the other. Other rotations can be used to further sort these modes. For example, to decouple m=1 modes from m=3 modes, a rotation of 60 degrees will work.

Angular Momentum Conservation in the Generation of Surface Plasmons

This experiment seeks to understand the conversion of light carrying angular momentum into surface plasmons. That is, when an SPR is generated by light with angular momentum, the angular momentum must be conserved one way or another. It could be that surface plasmon resonance supports modes with angular momentum, or that the angular momentum is simply converted to torque at some point in the conversion of light to SPs.

Currently, this project consists of a laser system designed to generate OAM light with linear polarization and topological charge ±1, and a nearly hemispherical prism with a fairly smooth silver film in which to generate an SPR. No detection of angular momentum of plasmons has been attempted yet, but possible methods for detection are outlined below:

1. Assume that colinear and spatially identical OAM/non-OAM light can be generated. Then observation of angular momentum in radiation from an SPR generated by OAM light, and subsequent observation of no angular momentum in radiation from an SPR generated by non-OAM light is a likely indicator that angular momentum is conserved in the electromagnetic fields through the plasmon conversion processes. This may not be conclusive, but may be rigorous.
2. It is plausible that milling defects into the conductive film that break circular symmetry, such as skewed "escape arms," could provide insight into the local optical field.