Higher Order Gaussian BeamsJennifer L. Nielsen

B.S. In progress – University of Missouri-KC

Modern Optics and Optical Materials REUDepartment of Physics University of Arkansas

Summer 2008

Faculty Mentor: Dr. Reeta Vyas

Transverse Modes of a laser Cross sectional

intensity distribution Intriguing Properties

Angular momentum Polarization properties Applications in optical

tweezing

Different shapes described in different coordinate systems (rectangular, cylindrical, parabolic cylindrical, elliptical, etc...)

Analytical WorkTo derive the higher order Gaussian beam modes, we start out with the paraxial (beam-like) approximation of the wave equation. We then plug in a suitable trial function (ansatz) and work to obtain a solution.

Coordinate systems used in derivationsCartesian coordinates – standard rectangular x, y, z axes

Cylindrical coordinates – basically the polar coordinate system with a z axis.

Parabolic Cylindrical Coordinates - (fancy!)

Special Functions Used Hermite generating

function:

Laguerre generating function:

Parabolic cylindrical functions:

Same functions used in quantum mechanics, as we shall see....

For Cartesian modes, start with this ansatz:

Plug into paraxial—after simplifying and plugging in terms, get this:

Hermite-Gaussian ModesPlotted in Mathematica using “DensityPlot”

Note TEMmn label. TEM stands for transverse electromagneticmode.

The m index – number of intensity minima in the the direction of the electric field oscillation

The n index - number of minima in direction of magnetic field fieldoscillation

For LG modes (cylindrical coordinates): Ansatz

Plug into paraxial in cylindrical coordinates

Laguerre-Gaussian Modes Plotted in Mathematica as Density Plots

TEMpl

p = radial

l = Φ dependence

plotted fromCosine based function

Reference:Opticsby Karl Dieter Moller

TEM11 – A Close Up

“ContourPlot” “Plot3D”

HG modes plotted in Mathematica using our code.

TEM11 – 3D rotation

Rendered in Mathematica 6 and screen captured Left-Rectangular/Hermite; Right-Cylindrical/Laguerre

Orbital Angular Momentum Properties

Azimuthal component Means beam posseses

orbital angular momentum Can convey torque to particles Effect results from the helical phase--

rotation of the field about the beam axis Optical Vortex -field corkscrew

with dark center OAM/photon = ħl

Angular Momentum Properties

A beam that carries spin angular momentum, but no orbital angular momentum, will cause a particle to spin about its own center of mass. (Spin angular momentum is related to the polarization.)

On the other hand, a beam carrying orbital angular momentum (from helical phase)and no spin angular momentum induces a particle to orbit about the center of the beam.

Image Credit: Quantum Imaging, Mikhail Kolobov, Springer 2006

Correlations with Quantum Harmonic Oscillator

(Above: QHO ; Below: LG Modes)

Comparisons with 3D Quantum Harmonic Oscillator

The harmonic oscillator is not z dependent

The equations are analogous but not identical.

Parallels with quantum probability densities obvious.

Hydrogen atom probability densities shown. Plottedin Mathematica.

n = 4 , l = 1, m = 1

n=3, l = 1, m = 1 n=4, l = 0, m = 0 n = 4, l = 2 , m = 1

Solve via separation of variables...

Parabolic Beams

Parabolic, cont'd

Convert to parabolic cylinder equation

Further research on parabolic beams necessary....

We are working to plot the beams and plan to study their angular momentum properties.

Special thanks to....

Dr. Reeta Vyas Dr. Lin Oliver Ken Vickers The National Science Foundation The University of Arkansas And everyone who makes this REU possible!

And on a slightly different note....Human beings aren't the only ones fascinated with the properties of lasers....

Any Questions? Ask now or write Jenny at JLNielsen@umkc.edu