Transcript Higher Order Gaussian Beams

Higher Order Gaussian Beams
Jennifer L. Nielsen
B.S. In progress – University of Missouri-KC
Modern Optics and Optical Materials REU
Department 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 Work
To 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 derivations
Cartesian 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
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Hermite generating
function:
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Parabolic cylindrical
functions:
Laguerre generating
function:
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 Modes
Plotted in Mathematica using “DensityPlot”
Note TEMmn
label. TEM
stands for
transverse
electromagnetic
mode.

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

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Plug into paraxial in cylindrical coordinates
Laguerre-Gaussian Modes

Plotted in Mathematica as Density Plots
TEMpl
p = radial
l=Φ
dependence
plotted from
Cosine based function
Reference:
Optics
by 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. Plotted
in 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
Parabolic Beams
Solve via separation of variables...
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 [email protected]