Download Homework Set #7 Due: 4-4-14

Homework Set #7
Due: 4-4-14
This homework set covers the same material as hw#6, but explores different configurations. It
includes a tutorial on a vital technique used in many short pulse laser labs: OPG/OPA.
(1) A beam enters a negative uniaxial crystal as shown. The crystal
was cut so that the optic axis is perpendicular to the interface. If
the input beam is polarized to be pure ordinary or extraordinary, it
will have a well defined direction inside the crystal, shown as
making an angle  with respect to the optic axis. This is the angle
used in the equation that determines the extraordinary index of
refraction, ne(q).
(a) Suppose the beam is polarized perpendicular to the page,
making it an ordinary wave. Find .
45
o

optic-axis
no = 1.50
n = 1 45
polarization for (b)
polarization for (a)
(b) Suppose, instead, that the beam’s polarization is in the plane of the page making it an
extraordinary wave. Find . (Rather than try and solve a complicated equation, I like to use an
iterative approach here. You have the correct angle when Snell’s Law and the equation for the
extraordinary index of refraction are self-consistent.)
(c) Just for practice, identify if the polarization is s or p for both problems (a) and (b).
Introduction to problems 2 and 3: Optical Parametric Generation, Optical Parametric Amplification
The calculation I’m asking for next is fairly simple. However, this problem gives me a chance to discuss
an important technique without using class time!
Optical parametric generation (OPG) is a (2) process in which an input photon is split into two output
photons, conserving energy. The input field is called the pump field and there are two output fields. For
historical reasons, the output field with the higher frequency is labeled the “signal”, and the other one is
labeled the “idler”. Since energy is conserved, we have:
hp = hs + hi
where p, s and i stand for pump, signal and idler respectively. As an example, a 500 nm pump field
(green) could give rise to a 700 nm signal field (near IR) and a 1750 nm idler field (IR).
The problem is more complicated than this, of course. Classically, an input field incident on a nonlinear
crystal with a nonzero (2) can only give rise to amplification of pre-existing signal or idler fields, not
actual generation. Thus, a 500 nm pump field alone would not be expected classically to generate either
the 700 nm or 1750 nm fields given above. However, if a strong 500 nm pump field was accompanied by
a weak 700 nm field, the 700 nm field could grow at the expense of the pump, and a 1750 idler would be
generated as well. This process is called optical parametric amplification (OPA), and we have already
seen this, only we called it difference frequency generation. However, a full quantum treatment reveals
that OPG is possible. Even if only the 500 nm pump field were present, vacuum fluctuations would
provide the initial signal or idler field. In this regard, OPA and OPG are analogous to the Raman gain and
Raman generation discussed in class.
n generation, the pump en
nergy and thee energy spaacing betweenn the states iin the
In the casse of Raman
nonlinear medium deteermine the neew wavelengtths generatedd. What determ
rmines whether a 500 nm pump
m and 1750 nm
m fields, or 800 nm and 13333 nm fieldss, or any otheer combinatioon that
field geneerates 700 nm
is energettically alloweed? The answ
wer is that iniitially all eneergetically alllowed combinnations will ooccur,
but usuallly some mod
des will have higher gain than
t
others an
and quickly coome to dominate. In partiicular,
phase maatching must still be satissfied for efficcient generattion, and cann be used to select the deesired
signal/idleer wavelength
hs. However,, in general, OPG based oon the ampliffication of vaacuum fluctuaations
followed by
b competitio
on between co
ompeting outtput modes is unstable andd great care m
must be taken to get
useful outtput.
One effecctive approach
h is to place the
t nonlinear crystal insidee an optical cavity, such ass is used for llasers.
A cavity can readily be
b used to prreferentially select
s
a givenn mode of thhe electromaggnetic field. W
When
OPG is done
d
in this fashion,
f
it is called opticaal parametricc oscillation ((OPO). Laserrs are often ccalled
oscillatorss because of the
t close anallogy between lasers and ellectrical circuuit based oscilllators. An OP
PO is,
in fact, a laser with the
t nonlinearr crystal actin
ng as the gaiin medium. OPO’s workk well, but arre not
particularly relevant fo
or ultrafast wo
ork, at least no
ot currently.
However,, there is anotther approach to achieving stable outputt that is of greeat importancce. Rather thaan rely
on vacuum
m fluctuations to start the process,
p
a wh
hite light conttinuum can be generated uusing (3) proccesses
and this can act as the seed (initial signal
s
field). The
T nonlinearr crystal then,, simply perfoorms OPA. Since a
white ligh
ht continuum provides ligh
ht over a widee range of waavelengths (4000 nm to 10000 nm is comm
mon),
a wide ran
nge of outputt fields are av
vailable by ph
hysically rotaating the crysttal to select w
which energettically
allowed fields
fi
will be phase
p
matcheed. Note that the continuum
m generation need only prrovide the signal or
the idler, it does not have
h
to proviide both. This is importannt because thee idler is ofteen too far intto the
infrared to
o be seeded by
b continuum generation.
beam spliitter
(low refleectivity)
mirrorr
800 nm
n input
B
BBO crystal for SHG
residual 800
nm light no
longer needed
dichroic
beamsplitter
sapphire
window
continuum
c
generation
400 nnm light
(pum
mp)
BBO for OPA
residual pump
plus signal and
idler beams
mirror
beam combinerr:
400 nm +
continuum
Suppose we
w have a Ti:Sapphire laaser system producing
p
1000 fs, 1 mJ puulses at 800 nm (commerrcially
available)). While this is all well an
nd good, whaat do we do iff our experim
ment requires ultrafast pullses at
1200 nm or 550 nm? A possible solution
s
using
g continuum generation sseeded OPA is indicated iin the
figure abo
ove. This is provided
p
just so
s we have a simple, conccrete examplee; there are m
many configuraations
and this one
o is not, in general,
g
going
g to produce much
m
energy.. Better system
ms have an addditional stagge.
Here, ~300 J of 800 nm light is frequency doubled using BBO to generate ~100 J of 400 nm light.
BBO has a large nonlinearity and only modest beam walk-off, so it is a good choice for this application.
The 400 nm light will serve as our pump, thus allowing us to generate light throughout most of the optical
range as well as in the infrared. The 400 nm light is separated from the residual 800 nm light using a
dichroic beamsplitter. A small portion of the 800 nm light, of order 10 J, is also split off for continuum
generation in a thin fused silica or similar window. We’ve discussed continuum generation and you know
that this is primarily a (3) effect and, thus, can be excited in any medium. The 400 nm light and the white
light continuum are combined in a 2nd BBO crystal where OPA takes place. Naturally, the system must be
constructed so that the continuum pulses and the 400 nm pulses arrive at the same time. (The delay line to
adjust the timing is not shown.) The BBO crystal converts some of the 400 nm light into signal and idler
fields. The wavelengths of the signal and idler are determined by rotating the BBO crystal until an angle
is found so that the desired output wavelengths are phase matched when traveling, in this case, collinearly
with the pump. The pump, signal and idler can be separated using dichroic beamsplitters or other
apparatus if desired. Reports of output energy for a simple system like this vary greatly, but at least
several J is easily obtained. Often, a second BBO crystal is employed as an amplifier. Conversion
efficiency in an optimized system can be very good, giving rise to signal pulses with ~30 J.
Another way to adjust the output wavelengths is to vary the relative delay between the continuum and
pump pulses. As you recall, continuum generation is accompanied with moderate chirp, thus it is possible
to arrange somewhat for the pump pulse to overlap with different spectral portions of the continuum
pulse. In any case, continuum generation plays a vital role stabilizing the OPG output as does a second
amplifier stage.
A few additional comments.

Both BBO crystals must be sufficiently thin that phase matching does not constrain the output
wavelengths too much. We still want the output pulses to be short, so a sufficient bandwidth must be
phase matched. For BBO, crystals with a length of a few mm’s provides ample gain and yet allows
the generation of short pulses. In fact, pulses shorter than the pump pulse are sometimes reported.

Additional nonlinear steps involving sum- and difference-frequency generation between the residual
800 nm, pump, signal and idler fields can greatly extend the spectral range of this system.

The system described here used a collinear geometry with the pump, signal and idler beams following
the same path inside the crystal. Non-collinear OPA can be better.
The problems.
(2) What is the phase matching angle for Type I, degenerate, collinear OPG in KD*P using a 532 nm
pump? “Type I” means, in this case, that the pump is extraordinary and the signal and idler are
ordinary. “Degenerate” means that both the signal and idler have the same wavelength.
(3) What is the phase matching angle for the OPG of 900 nm light from a 500 nm pump in KD*P? You
will have to decide which beams to make ordinary and extraordinary. Some index data is provided
below.
ne
no
500 nm
1.4697125
1.5089041
900 nm
1.4598833
1.4960746
1125 nm
1.4575388
1.4919834
Wavelength