Research Problems & Potential Student Projects
(Last updated May 2000)
This is a random collection of research problems and calculations, mostly
connected with optical beams and resonators, that I would throw out for
students to attack—or that I might get around to tackling myself some day.
Some of them could be the basis for publishable journal articles; others
are more in the nature of instructive homework exercises or teaching problems.
I'll be glad to receive answers or solutions to any of these, or to
correspond (siegman@stanford.edu)
with anyone who is interested in any of them.
[Back to AES home page]
Beam Quality
& M-Squared Calculations & Analyses Back
to top
Tables of
M-Squared Values For Various Cases
It would be useful to develop a comprehensive collection of the theoretical
M^2 values for a wide variety of simple coherent and incoherent beam profiles,
including VRM output beams, supergaussians, beams with near-field diffraction
effects, pedestal beams, spherically aberrated gaussians, partially coherent
beams (mixtures of coherent & incoherent), cosine and cosine-squared
beams with N cycles, and so on.
Relationships
Between M-Squared and Other Beam Quality Measures
It would also be useful to calculate (analytically or numerically) both
the M^2 values and a wide variety of other beam measures to power-in-bucket,
modified Strehl ratio, other beam spread or beam brightness measures, for
all the above beam profiles. Evaluate relationships between MTF, PSF, and
beam propagation factor.
Beam Properties for Nonorthogonal Beams
Review and summarize ten-moments analysis results for nonorthogonal or
``tridimensional'' beams; and develop a clean and simple formalisms for
the generalized Q properties and M-squared values of arbitrary beams.
More Extensive Binary Phase Plate and ``Bolt Hole'' Beam Results
Do analysis of add'l cases of binary phase plates Ñpower in bucket
analyses for more half cycles, multiple gaussians, etc.; also do similar
analysis of bolt hole beams, with and without center beam.
Array Beam Quality Limits?
Analyse potential beam propagation factor improvement from letting a uniphase
array or a bolt hole beam propagate a fractional Talbot distance, until
phase modulation develops, then eliminating this phase variation. (Opt.
Lett., 15 Oct 93).
M-Squared and Spot-Size Definitions for Discontinuous Beams?
Understand existing papers and develop general spot size and M-squared
definitions for beams with discontinuous steps in the beam profile.
Fiber Coupling vs M-Squared Value
There is some experimental evidence that the coupling efficiency of a coherent
optical beam into a single-mode fiber is directly proportional to 1/M^2.
Attempt to verify this—or more likely see how good an approximation it
may be, at least for not too large values of M^2— by calculating and making
scatter plots of single-mode fiber coupling efficiency vs M^2 values for
random coherent superpositions of HG or LG modes, single-cycle cosine or
cosine-squared beams, triangular beams, and other elementary beam profiles.
Stable Resonator
Analyses & Calculations Back
to top
Modal Properties of Coherent Hermite-Gaussian or Laguerre-Gaussian Mixtures
It would be useful to carry out the straightforward though somewhat messy
exercise of developing analytical expressions for all the significant properties
of optical beams consisting of coherent superpositions of Hermite-gaussian
or Laguerre-gaussian modes as functions of the complex-valued expansion
coefficients. This could include M^2 value, kertosis, equivalent
radius of curvature, coupling to single-mode fiber
Perturbation Analysis of Losses in Apertured Stable Resonators
Do HG mode analysis of apertured stable resonators with small losses: How
many modes are needed? [Ref : Appl. Opt. 30, 1899 (20 May 1991)]
Spherical Aberration and Edge Effects in Apertured Stable Resonators
What effect does spherical or quartic aberration have on the mode profile
and mode losses of the stable mode in a stable gaussian aperture? (Use
PARAXIA to find out). As an extension of previous topic, do careful Fox-
and-Li calcs of stable resonators and look at fine details of mode patterns
at outer edges. Fringes? Edge wave effects? How many HGs needed to expand
these?
Analysis of Mode Competition in Apertured Stable Resonators
Follow up on the Aachen work, and on Jean-luc's calculations, using FRESNEL
with multiple waves propagated around and saturable gain medium at one
end.
Explore ``M'' and ``W'' modes in hemifocal resonators
Check out higher-order modes and mode discrimination in these cavities
Kerr Lens Effects: Mode Shapes and Properties in Kerr Lens Cavities
Explore the steady-state modes in stable cavities containing optical Kerr
lenses. (Cancel out the ``dc'' part of the lensing resulting from the Kerr
effect, explore losses versus aperture size and Gouy phase shift of the
underlying cav ity.)
VRM Mirrors versus VRM Apertures
Suppose you want a good mode-controlled efficient type of laser using the
geometrically unstable variable-reflectivity mirror approach. What are
the comparative virtures of using a VRM output mirror (which can be rather
difficult and expensive to fabricate and adjust) as compared to an intracavity
``VRM aperture'' (which may steal some intracavity energy and therefore
lower the laser effiociency). (See my VRM Refs file for references.)
Resonator Axis Stability in Nonplanar Ring Resonators
Explore axis stability in nonorthogonal ABCDEF systems, following Irl Smith's
SPIE manuscript and the ``ray axis'' concept in LASERS, misaligned ABCD
section.
Optical Resonators with Non-Wavefront-Matched Intracavity Elements
Optical resonators often contain intracavity elements such as tuning etalons,
laser rods, Q-switches, harmonic generation crystals, dye cell windows,
or diode laser end faces or substrates. The surfaces of these elements,
assuming they are not tilted at Brewsters angle, are usually are antireflection
coated to eliminate or minimize surface reflections. In some cases
these elements can also be tilted so that any residual back reflections
are directed out of the cavity rather than feeding back into the cavity
mode.
In some cases, however, the surfaces cannot be tilted, may not be perfectly
AR-coated, and in addition may not be wavefront-matched to the cavity mode
itself, as for example in the case of a flat interface located at a transverse
plane where the cavity mode has a curved wavefront.
Unstable Resonator
Analyses & Experiments Back
to top
Unstable Resonator Mode Experiments Using Pound-Drever Stabilization
Could one build a simple Pound-Drever system locked to an unstable resonator
cavity, then tune either the incident laser or the unstable cavity and
identify modes and their losses from the Pound-Drever signal?
Unstable Resonator Mode Experiments Using ``Energy Ringdown''
Still no careful detailed measurements of unstable resonator mode losses
and transverse modes, even at this late date. Possible experiments using
transient digitizer?
Finish Virtual Source Manuscript!
Analyse One-Sided Stable-Unstable Resonator Designs
Effect of adjustable knife-edge on small side of positive-branch case
Especially consider and understand the one-sided negative-branch case Koji
has good start on this Ñ a bit more could be done
Analyse Unstable Resonator with On-Axis Delta-Function Gain
Mode properties and spiking behavior. Use H-G expansion method.
Explore Mode Build-Up Behavior in Unstable Resonators
Review Anan'ev ideas on mode build-up time in unstable resonators, and
the ideas of a ``diffraction core'' in unstable resonators, and relate
these to mode discrimination in unstable resonators.
Explore Relation Between Mode Crossings and ENF in Unstable Resonators
Why does the ENF become so excessively large just at the mode crossing
points?
Unstable Resonators with Small Output Mirrors and Large Magnification
Analyse Anan'ev's variant on SFUR unstable resonator
Effects of spherical aberration on unstable resonator beam quality
Do some simple numerical simulations to evalue the effects of internal
spherical aberration on beam quality or beam output profiles for unstable
resonators, either hard-edged or gaussian-mirror.
Other
Beam Propagation & Resonator Analysis Problems Back
to top
Generalized ABCDEF Matrix Formulation
Develop most general formulas for Huygens integral and gaussian beam propagation
with misaligned complex elements, including beam center and beam slope
concepts. (Need to take a gaussian quadratic system, consider displaced
and tilted input/output planes, evaluate beam propagation, compare with
linear wedges, ``loss wedges'', and related elements.)
Variable-Element Coupled-Array Mode Analyses
Explore effects of tapered coupling and tapered element parameters
Fresnel Zone Lensguides or Cavities?
Would it make any sense to think of a ``Fresnel zone waveguide'' (i.e.,
a guide defined by index stripes spaced as in a Fresnel zone plate? What
about a ``Fresnel zone plane cavity'' defined by two Fresnel zone plates
set up as confocal (or concentric) mirrors? (Could this be a way to get
higher interaction factor in diode lasers, by having multiple layers on
each side of the active region.
Talbot Imaging and Radial Talbot Mirrors
What is the radial (i.e., circularly symmetric) analog of Talbot imaging?
Is there a simple general analysis for Talbot mirror cavities?
Real-Beam Radius of Curvature for Coherent Superposition of Gaussian
Modes
Develop an analysis of the effective radius of curvature for a beam made
up of a superposition of Hermite-gaussian or Laguerre-gaussian modes, and
explore the behavior of this radius for various random amplitudes and phases
of the HG or LG modes.
Real-Beam Radius of Curvature for Fresnel Plate Output
What is the effective radius of curvature for a collimated beam passed
through a Fresnel zone plate or strip? (Consider both ``black and white''
Fresnel zone plate and 1 phase shift zone plate cases.) Also, what's the
radius of curvature for the spherical aberration case.
Diode Lasers with
Unstable Resonators Back
to top
External-Cavity Unstable-Resonator Diode Laser Experiments
Set up well-designed external-cavity wide-stripe diode experiment Study
modes and diode optical properties, evaluate problem of coherence collapse
GRIN Slab Resonator Designs for Wide-Stripe Unstable Diode Lasers
Carry out ABCD analysis, and trial design analysis
Brewster Angle Diode Laser Output Coupling
Analyse what happens to a very wide stripe ``gaussianized'' diode laser
beam (assumed gaussian in both transverse dimensions) as it comes out through
a wide Brewster angle interface. If stripe is wide enough, is there mode
conversion in the Brewster outputcoupling process?
``Levenson'' Type Experiments on External-Resonator Diode Lasers
Evaluate wide-stripe diode optical characteristics from mode-beat behavior
Dente's Rings
We still don't understand the Dente ring observations.
Nonnormal
Mode Expansions & Excess Noise Analyses Back
to top
Explain the Critical Theta Value for the CHG Expansion
Examine the ratio of cn+1/cn for the CHG case as n->infinity, and see if
this leads to a critical value for theta.
Carry Out Additional Complex Hermite-Gaussian Mode Expansions
Do CHG expansions for delta function at origin, top hat, or other? input
modes.
Explore Nonorthogonal Eigenmode Expansion Mathematics
Do some library research on the basic mathematics underlying this.
Analyse Coupled-Mode Seeding of Pulsed Unstable Resonators
Suppose a seed beam is sent into a pulsed or Q-switched unstable resonator
with a given kind of coupling, either ``adjoint'' or ``matched''. Evaluate
the initial excitation of modes, both by the seeding and by excess noise
(including cross-coupling between modes). What degree of initial excitation
is needed to get the initial seeding above noise, both in the unstable-resonator
case and the gain-guided case. Use the method of analysis from the AES
Phys. Rev. papers on ENF, and consider cases like the Kuo, Smithery and
Raymer or Mike Duncan papers mentioned in the ENF refs list or the ENF
Refs file. (See also Geoff New's most recent paper.)
Distributed-Output-MOPA-Amplifier Phase Noise Analysis
Consider effects of spontaneous emission in an amplifier with distributed
gain and outcoupling (a la SDL's MAG-MOPA concept). Analyse phase noise
due to distributed spontaneous emission, and its effect on far-field Strehl
ratio or M-squared values for the output from this system. [Ref: Gordon
and Mollenauer, OL 15, 1351 (1 Dec 90).]
Accuracy concerns and criteria for Existing PARAXIAª programs
Exact Fresnel number for digitized Fourier and Hankel calculations Criteria
for number of points in digitized Fourier and Hankel calculations.
Compare with GLAD theory.
Add an ABCD Layout routine?
Add an M2 routine?
Be able to extract curvature, compute spot size and M2 for wavefront.
Add a PRONY routine?
Accept output from Fresnel
Extensions to ABCD ¥ Collins chart within ABCD? ¥ Add field
expansion and decomposition capabilities to ABCD?
Be able to take a field and decompose into multiple HG modes, then propagate
and reexpand in HG modes?
Extensions to FRESNEL ¥ Add effective radius of curvature R(z) calculation
to Fresnel? ¥ Add an adjoint mode capability to FRESNEL (or VSOURCE)?
¥ Add a two-transverse-D capability? ¥ Add a beam rotation capability?
UR-90's and nonorthogonal resonators
Extensions to VSOURCE ¥ One-sided negative-branch unstable resonator
still a problem
Miscellaneous Laser Problems
Back to top
Higher-Order Modes in
FM Lasers
The concepts of active AM and FM mode locking and FM laser operation in
homogeneous laser transitions were analyzed by Kuizenga and Siegman and
by Harris in the late 1960s, with extensions by many others, notably Haus,
in later years. At some point ??? and ??? pointed out that an AM
mode-locked laser could have not only a gaussian lowest-order pulse solution
but higher-order Hermite-gaussian solutions as well. Longhi and Laporta
(Phys Rev, 1999) have also pointed out that an FM laser can have higher-order
modes, and noted that these are not orthogonal.
It seems obvious to me that there must in fact be a strong similarity
or even duality between the temporal (pulse) modes of an actively AM or
FM mode-locked laser and the transverse spatial modes of a complex index-
and/or gain-guided duct, i.e., both have Hermite-gaussian solutions trapped
in a complex-valued potential well. It also seems to me that a pure
Harris-type FM laser (active FM modulation with detuning) must have higher-order
modes as well, which may be analyzable as higher-order Bessel solutions
or solved for numerically. It would be particularly interesting to
analyse or calculate these and see what they look like physically, as solutions
in time, as well as understanding their optical beam propagation analogs.
Superluminal (?) Light
Propagation
A recently
[1] L. J. Wang, A. Kuzmich, and A. Dogarlu, "Gain-assisted superluminal
light propagation," Nature, vol. 406, pp. 277--279, 20 July 2000.
Student Laboratory
Design Projects Back
to top
Diode Laser Traffic Counter/Speed Recorder (Dave Bloom? John Fox?)
Electrical Power Outage Recorder
Diode Laser Class Demos & Lab Experiments
Set up, test, and write notes for a whole bunch of diode laser teaching
demos, as per the notes in my ``Diode Laser Demons'' file.
Computer Simulations
for Classroom Teaching Back
to top
Oscillating Atom Movies
Use Mathematica and simple expressions for H atom wave functions to make
3D QuickTime movies of oscillating electric-dipole charge distributions
in atoms.
Other Demos
Many, many other demos and simulations, e.g. beam propagation, inhomogeneous
hole burning, multimode oscillation, mode competition, rate equations,
etc., etc.
[Top of this page]
[AES home page] [Stanford University
home page]