Dispersion Compensation in Mode-locked
Thulium/Holmium Doped Fiber Lasers

Kelley Daenzer

Supervisor: Brian R. Washburn

Kansas
State University Physics Department REU Program

This
work is partially funded by the National Science Foundation (NSF) and the Air
Force Office of Scientific Research (AFOSR) through NSF grant number PHYS-1461251.
Any opinions, findings, and conclusions or recommendations expressed in
this material are those of the author(s) and do not necessarily reflect the
views of the NSF or AFOSR.

**We are using an
all-fiber laser with thulium/holmium doped gain fiber. When mode-locked, the
light inside this laser travels in pulses. Short pulses of light are useful for
measuring things that happen on short timescales – chemical reactions,
biological functions, etc. Our goal is to get these pulses as short in duration
as possible, so that they can be used to measure over shorter and shorter
timescales. However, due to dispersive effects caused by the fiber, the light
experiences group velocity dispersion and third order dispersion, which
distorts and lengthens the pulses. Both types of dispersion can be managed by
using several varieties of fiber whose dispersive components cancel each other.
We will try a type of ultra-high numerical aperture fiber called UHNA-7 to
compensate for both.**

__Group Velocity
and Third Order Dispersion__

Group velocity dispersion is a change in the velocity of spectral
components of a wave pulse with respect to the carrier frequency. Specifically,
different spectral components of the pulse reach a given point at different
times, causing the pulse to distort in what is called “chirping”. A simplified
example of chirping can be seen below:

Because of this separation of spectral components, the pulse becomes
longer in duration than it was initially. The goal is to create shorter and
shorter pulses of light, which means we need to find a way to compensate for
this dispersion.

The parameter which determines group velocity dispersion is β_{2}, which is related to the second derivative of the Sellmeier equation for index of refraction. The graph of β_{2} for bulk fused silica appears as

although the graph shifts to the right inside optical fiber.
As is seen here, β_{2} can be
either positive or negative.

Third-order dispersion is the rate of change of group velocity dispersion
with respect to frequency; i.e. β_{3} is the
derivative of β_{2}.

__Net Cavity
Dispersion__

The thulium/holmium doped fiber that we are using in our laser cavity has
a positive β_{2}, while
the standard single-mode fiber that makes up the rest of the cavity has a
negative β_{2}. Net
dispersion in the laser cavity is found by adding the products of the β_{2} value of a fiber and that fiber’s length. The net
cavity dispersion for our current setup, with only these two types of fiber, is

Finding a third fiber with a positive β_{2} value that will compensate for this dispersion is
difficult. At 2µm light, where our laser operates, most β_{2} values are highly negative. Some fibers with an
ultra-high numerical aperture, however, have a positive β_{2}. We have chosen a fiber of this type, called UHNA-7
and having a β_{2} value
of 0.13ps^{2}/m, to add to our laser.

When group velocity dispersion is near zero, then third-order dispersion dominates,
so we need to compensate for that as well. Luckily, UHNA-7 fiber has a β_{3} value which is also opposite to that of SMF, so it
can compensate both group velocity dispersion and third-order dispersion
simultaneously. The equation for net third-order dispersion, with UHNA-7 fiber
included, is:

__Calculating Fiber
Lengths__

The length of existing Tm/Ho fiber in the laser is known: it was measured
at the same time it was inserted into the laser, and that length has not been
altered. The length of SMF, however, has been altered as time goes on to aid in
mode-locking. The existing length of SMF fiber can be found two ways. The first
is by direct measurement with a tape measure. The second is by measuring the
repetition rate of the laser cavity – that is, how many times the laser pulse
travels around the laser cavity per second. The length can be found from this
value by solving for L in the relationship

where 1/T is the repetition rate. The length of SMF fiber
can be found by subtracting the length of Tm/Ho fiber from the length given by
the above equation.

Using the formula for net cavity dispersion, we can solve for the length
of UHNA-7 fiber needed to compensate the dispersion caused by the other two:

We have selected 0.03ps^{2} as our desired net dispersion value –
this is the value at which we have taken successful data in the past. Then, L_{UHNA7}=3.05m.
We can use even less of this fiber if we alter the amount of single-mode fiber
in the cavity.

__Spectral Data__

Once the laser is mode-locked, we can take measurements to find out what
wavelength(s) it is lasing at. We were able to successfully mode-lock the laser
with only Tm/Ho and SMF fibers in the cavity. Since the gain fiber is doped
with both thulium and holmium atoms, it can lase at 2 wavelengths, as seen by
the two distinct spectra below:

The two wavelengths are found by mode-locking the laser in different
polarization states.

The spectral width (FWHM) is approximately 10nm. If the UHNA-7 fiber
successfully compensates for dispersion, that should get wider by about a
factor of three.

We added 2.889m of UHNA-7 fiber to the laser and successfully mode-locked
it – something that has not been done before with this type of compensation
fiber.

To check our numbers, we did a length comparison. Using the repetition
rate of the cavity (26.56MHz), we found that the total fiber length in the
laser was 7.85m, which corresponds to the measured length of 7.76m.

We took some spectral measurements:

The spectrum is wider, which is what we predicted, but it is an odd shape.
We are still trying to understand why the spectrum looks the way it does.

__Next Steps__

**Our next step
will be to take an interferometric autocorrelation trace in order to find the
temporal pulse duration. We will also cut back the amount of UHNA-7 fiber in
the laser cavity to see how that affects the pulse. Both of these tasks will be
done in order to find out whether the UHNA-7 is adequately compensating for
group velocity dispersion and third-order dispersion. **

__Acknowledgements__

I would like to thank Drs. Kristan Corwin and
Bret Flanders, Ms. Kim Coy, and Kansas State University for preparing and
hosting the physics REU program during the summer of 2016.

__References__

[1] R. Kadel and B. R. Washburn, “All-fiber
passively mode-locked thulium/holmium laser with two center wavelengths,” *Applied
Optics,* vol. 51, no. 27, pp. 6465-6470, Sept. 2012.

[2] R. Kadel and B. R. Washburn,
“Stretched-pulse and solitonic operation of an
all-fiber thulium/holmium-doped fiber laser,” *Applied Optics,* vol. 51,
no. 4, pp. 746-750, Feb. 2015.

__About Me__

I am working towards a physics degree at University of Colorado, Colorado
Springs. At the conclusion of this program, I will have three semesters left,
after which I intend to pursue a graduate degree.

Physics has always made the most sense to me out of any field of science,
and the more I learn, the more I am convinced that this is the closest thing we
have to magic. There is so much mystery in the universe, and it’s exciting! I
first became interested in physics because of my high school physics teacher –
he was excellent at finding new ways to explain concepts, and tried to help
everyone understand in their own way, even if he had to find ten different ways
to explain it before it clicked. I believe that anyone can understand physics
at least on a basic level, if they have it explained to them in a way they can
wrap their head around. I would love to teach undergraduate physics for this
reason; so many people are so intimidated by physics and I want to help make it
more accessible to people with different learning styles.

Outside of physics, I love reading, playing and listening to music,
writing poetry, knitting, going to church, and practicing parkour.

You can download my
poster here.

You can download my
final presentation here.

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