Download Homewo ork Set #4

Homewo
ork Set #4
Due: 2-5-14
1) The Etalon.
E
In thiss problem you
u’ll derive thee basic properrties of a Fabbry-Perot etaloon. If you hadd a good E&M
M or
opticss course durin
ng undergrad, you will hav
ve seen this beefore. There aare different ttypes of etalonn, but
this problem captu
ures the essenttials. Etalons are most com
mmonly used for spectral m
measurement, but
they are
a also used for spectral sh
haping in laseer systems, paarticularly narrrow line laseers, but also short
pulse lasers. A laseer cavity is itsself an etalon,, so this probllem will help aid your undderstanding off laser
operaation and laserr modes. The main goal heere, however, is simply to uunderstand w
what an etalon is
and how it acts as a spectral filteer by using wave
w
interferennce. This is a classic probllem widely
us texts and you
y are welcom
me to refer too them, althouugh everythinng you need iss
descriibed in variou
here. There will bee follow-up qu
uestions in latter homeworkk sets. Finallyy, note there iis an alternatee
oach to that sp
pecified below
w that involvees matching bboundary condditions betweeen fields inside
appro
and ou
utside the etaalon. Feel freee to take that approach
a
if yoou like.
In thee figure, light of wavelengtth  is inciden
nt on a
rectan
ngular medium
m of index n with
w flat, paraallel
sides and dimensio
ons as shown. A medium in
n this
here is no abssorption,
shapee is often calleed a “flat”. Th
but bo
oth sides of th
he flat are coaated so that th
he electric
field reflectivity
r
off each face is r and r’. Thuss, if A is
the co
omplex electriic field magn
nitude of an in
ncident
plane wave, Areflecteed = r * Aincideent, for the leftt face.
Likew
wise, the electtric field transsmissions are t and t’.
For th
his problem, assume
a
that r = r’, t = t’, th
here is no
2 2
absorp
ption (so r +tt =1) and therre is no phase shift due
to refl
flection at an interface.
i
Thiis last is clearlly wrong,
but th
hese assumptions do not thrrow away any
y
intereesting physicss and simplify
y the math.

(i)
We can writee A1 = t2 Ai. Likewise
L
A2 can
c be writtenn as A2 = C ei Ai. The facttor ei accountts for
the phase shiift between th
he outgoing fiields A1 and A 2 because off the path lenggth differencees
between them
m. Find C and
d in terms of
o t,r,n,l,,. ((Consider a reeference plane perpendicullar to
the reflected rays.) Explaiin why I am allowed
a
to negglect the phasse shift in myy expression ffor
A1 .
(ii)
d field, At, as an infinite seeries of terms: At = A1 + A2 + …
Write the tottal transmitted
This series converges to a finite result, so find a sim
mple analytic eexpression foor At.
he fractional trransmitted po
ower, T = |At|2 / |Ai|2 is giveen by:
(iii) Show that th
T
1
4R
andd R  r 2 .
with F 
2
2
1  F sin ( / 2)
(1  R )
R is the refleection coefficient for the in
ntensity.
(iv)
The transmission is periodic in frequency. In other words, light with a frequency of:
 n  m
will be completely transmitted with no reflection losses, even if R = 99.999%.  is called the
free spectral range (FSR). Find the FSR. Sketch T versus frequency.
(v)
Sketch the transmission function over several free spectral ranges for two values of F: F1 and F2
where F1 < F2. Convince yourself that F determines the bandpass of the etalon.
2) Etalons and short pulses
Suppose you send a single, short, transform limited Gaussian pulse through a “high finesse” etalon (R
very close to 1) whose free spectral range is much smaller than the pulse bandwidth. Assume normal
incidence.
(a) Show that this means the travel time through the etalon is large compared to the pulse width.
(b) Here are two arguments predicting the spectrum of the output from the etalon as measured, say,
by a spectrometer and energy meter. Critique both arguments. Which argument, if either, is
correct? If neither argument is correct, describe the output spectrum qualitatively. (An essay is
not required, but clarity is.)
#1) The etalon is a linear device. Therefore, I can analyze its operation by imagining that
I send “each” spectral component of the pulse through individually and then
superpose the results to get the output waveform. Clearly, only the spectral
components in resonance will get through, so the output spectrum will be a set of
equally spaced spikes in a Gaussian envelope. The spacing between spikes will be
the same as the free spectral range, completely independent of the pulse
characteristics.
#2) Etalons depend on interference between light that has traveled different path lengths.
Consider the fraction of the pulse that is directly transmitted and the fraction that is
transmitted after one round trip through the etalon. These pulses are separated in
time by an interval that is much larger than their pulse widths (note part (a)), so they
cannot interfere. This is true for the next pulse that has traveled two-round trips and
so on. Note that the spectrum of each pulse is unchanged after a reflection. Since
interference is not possible, the output spectrum will be unchanged. A pulsed laser
can “beat” an etalon and achieve full transmission of the entire bandwidth so long as
the pulses are sufficiently short.
3) Adding to your laboratory tool-chest. The peak intensity and peak power of a short laser pulse are
convenient figures of merit. In the laboratory, however, the total pulse energy is the most easily
accessible experimental parameter so let’s relate these for a Gaussian. Suppose you have a Gaussian
pulse in space and time:
I(r,t) = Io exp[-4ln2 r2/w2] exp[-4ln2 t2/
where r is the radial distance from the beam axis. (Note, in this parameterization of the intensity, w
and  are intensity FWHM quantities.) Find convenient expressions relating the peak intensity and
peak power to the total energy U, w and t.
4) Two p = 20 fs Gaussian pulses, one at 1064 nm and the other at 532 nm (1st and 2nd harmonics from a
Nd:YAG laser), are traveling in the same beam path and are overlapped in time, meaning they both
reach peak intensity at the same time. They pass through a 3 mm thick window at normal incidence
into a vacuum chamber. The window is made of SF2 glass (see the Schott optical catalog from the
public web site).
For each pulse determine:
(a) The propagation time of a phase front through the window,
(b) The propagation time of the pulse,
(c) The pulse width after exiting the window (use our approximation formula for this).
This provides a good figure of merit for how well they are overlapped after the window.
5) A transform limited 620 nm, 5 fs Gaussian pulse propagates through 15 cm of BK7.
(a) Using our approximate formula derived in class, what is the pulse width?
(b) The 5 fs pulse considered here has 20 times the bandwidth of a 100 fs and is still (even in
these advanced times) not commonly used. Is the Taylor-series expansion approach we
use to understand linear propagation valid for a 5 fs pulse? Evaluate what the true effect
of the BK7 would be using a numerical analysis (see below for discussion). Describe
your approach in detail, plot the result as you might for a publication and report the pulse
width.
One of the things you should plot to explain your calculation is the spectral phase of your
pulse after propagation through the BK7. However, if you simply plot the phase of the
pulse after the window it will be dominated by uninteresting phase contributions, so
remove them. Include plots of the phase with and without these contributions. Also,
include plots of the pulse intensity envelope in the time domain with and without these
contributions. Why the difference?
Discussion: solving this problem numerically in the frequency domain.
(i) Define a pulse in time that has the correct characteristics; (ii) Fourier transform it into
the frequency domain; (iii) apply a frequency domain transformation that accounts for the
effect of the BK7; and (iv) reverse Fourier transform to get the resulting field.
Use the numerical techniques you developed in the last homework assignment. Note that
the output pulse will be longer than the input pulse so the initial representation of the
field should be over a time scale long enough to accommodate the final output pulse. In
other words, if your initial array (or column or list or …) representing the electric field
extends from -200 fs to 200 fs, you’ll get in trouble trying to use that array to represent a
field that has significant energy at t = +250 fs after going through the BK7. Although it is
possible to determine good values in advance, you may have to experiment with different
choices of sampling time and number of samples.
Your representation of the pulse in time can either be the full electric field including the
carrier (real or complex representations: E(t) or ‫ܧ‬෨ ା ሺ‫ݐ‬ሻ) or just the complex envelope. I
recommend the latter. If you try to represent the electric field including the carrier, the
sampling time will need to be smaller than half the optical period (as discussed in
lecture). That will result in a lot of samples. If you stick with working with the envelope,
the sample time need only be small compared to the pulse width. This is an interesting
result. You can use a large sampling time but still get the correct answer.
(c) Would an 800 nm pulse broaden less, the same or more? This can be answered without
calculation. In any case, as always, explain your reasoning.