Saturated
Absorption in Rubidium
When
a beam of photons is passed through a vapor, photons whose energy matches
transition energies from the ground state to excited states in the atoms of the
vapor will be resonantly scattered. This means they are removed from the beam,
but appear at other angles. By measuring the apparent absorption in the beam as
a function of photon energy one can thus map out excited level structure of the
absorbing vapor atoms. In this
experiment you will use the absorption of light from a variable frequency, and
tunable, laser by Rb to explore the level structure of rubidium. You should read Melissinos and Napolitano,
pp243-250.
The
level scheme for Rb is shown below:
Fig. 1. Rubidium level
scheme.
As
you scan the laser in the vicinity of 780.23 nm, you will see four absorption
peaks, a,b,c,d:
Fig. 2. Absorption signal in
detector 2.
These
correspond to the transitions indicated on the (schematic) diagram. The
absorption lines are broad and overlap. Why? Because the atoms are moving due
to room temperature, and the Doppler effect makes
different atoms absorb at different frequencies.
Q:
At room temperature, how big do you expect the Doppler spread of a line at 720
nm in rubidium to be, in MHz?
To
overcome this, we use “Saturated Absorption Spectroscopy”. We send the laser
beam on two different (probe) paths through the Rb cell, each onto its own
photodiode detector. One of these paths is also illuminated by a strong (pump)
beam which propagates in the opposite direction. This situation produces sharp
features on the absorption lines which will show the high resolution hyperfine
structure within each spectrum. These are the “saturated absorption” features.
How
does is work? Consider line a, for example. The pump beam is strong enough to
excite all the atoms which are in resonance with the laser for transition a,
“saturating” the transition by removing all the atoms from the F=2 state of
87 Rb. Consider
the special case of the atoms at rest. For them there is no Doppler shift, so
this resonant depletion occurs when the laser frequency exactly matches the
frequency for transition a. For these atoms, the probe beam coming the opposite
direction at this frequency would normally be absorbed by this resonant transition,
but because F=2 is depleted, this absorption is removed, and the signal goes
up. For any moving atoms the pump will still deplete F=2 for
those atoms which are Doppler shifted to be in resonance, but since the probe
beam is going the other way the Doppler shift for them is in the opposite
direction, so these atoms are not the ones interrogated by the probe beam. Thus
the depletion has no effect on the absorption of the probe beam except for
those atoms which are at rest. By taking the difference between the absorption
of beams 1 and 2 (see below), one of which samples the “pumped” rubidium and
one of which does not, only the “saturated absorption” features remain in the
spectrum, with high resolution since the effect has now pre-selected the atoms
at rest and eliminated the Doppler broadening.
In
the saturated absorption spectrum the resolution is high enough to resolve the
hyperfine splitting of the upper 5p3/2 states. So each of the four transitions (a,b,c,d,) should be split into three lines. For line “a” for
example, the transitions are F=2 to F’=1,2 or 3 (F=2
to F’=0 is not allowed) , three transitions separated by frequencies 157 and 267 MHz . But, oops, there is one
more complication: there are actually six lines. In addition to these three
lines, there is a saturated absorption signal at frequencies corresponding to
the average of any combination of two of the above frequencies. This occurs
because, at this frequency, those atoms which are moving toward the pump so as
to be in resonance with the lower frequency are also moving away from the probe
and Doppler shifted to the higher frequency. So the probed atoms are again
depleted in the initial state. These are called “crossover” lines. See MN for a
lengthier discussion. In order to interpret your data, you will have to be able
to identify which lines are direct and which are crossover lines in your
spectrum
Q:
For the transition “a”, calculate, from frequencies of Fig.1, what the
(relative) locations of the three direct and the three crossover lines are, and
make a diagram showing these as vertical lines with frequency on the x axis.
Calculate the separation in MHz of all five pairs of adjacent peaks in the
spectrum and make a table of these values.
Experiment:
The
setup is shown below:
Fig. 3. Schematic of
apparatus.
Turn
on the laser (key) and let if warm up for about 20 minutes while you are
tracing out the optics. Then push “power on”, and set the current to about 60
mA, voltage to around 40 V. You will be able to see the laser spot on the IR
card provided. Using the card, trace the laser beams through the schematic
shown above. If the beams do not follow this schematic, adjust the mirror so
that they do. The output of each detector goes through one channel of the
amplifier box and thence to the oscilloscope. Set the scope in YT mode and a
slow sweep speed. Set the function generator to scan at about 3 Hz, triangle
wave, and check that the output is going to the “laser frequency” input of the
laser controller, through an attenuator box. The voltage from this generator,
added to the voltage you set on the controller box, determine the angle of the
grating inside the laser which in turn determines the frequency at which the
laser lases. Scanning this voltage scans the frequency. Set the function
generator to scan the maximum range (0-10V), adjust the gains and offsets of
the amplifiers so that you have a signal from the photo detectors, and adjust
the DC offset and/or the voltage set on the laser control box until you find
absorption features, as show in the figure above. Position the TV camera so
that it views the absorption cell, and watch what happens as the laser passes
through the resonance lines. Be sure you understand this.
(1) Calibrating
the scale:
To
calibrate the scale, you will use a Michelson Interferometer setup, as seen on
the left hand side of the schematic diagram. Pass the signal from detector 3
through one channel of the amplifier and adjust the mirrors of the Michelson so
that both beams hit the center hole on detector 3. Seek a signal from each arm
of the interferometer. When both signals are present, you should see
interference oscillations as the wavelength is swept. The frequency separation
of the peaks is given by c/2(L2-L1), where L1
and L2 are the different distances from the half-silvered mirror to
the reflecting mirrors on the two arms of the interferometer.
Q:
Why? Explain and derive this equation in your report.
Put
the scope in XY mode with the function generator into X (Channel 1) and
1V/division. Calibrate the X scale in terms of frequency. As long as you stay
in XY mode, this relationship between voltage and frequency will now be a
constant for your whole experiment, regardless of what X scale you choose on
the scope. Print out your fringe spectrum for your report and show how you got
the calibration constant. You will record the spectra from the Tektronix 210
Oscilloscope using the PC and software provided. This software allows you to
either capture the screen or record the data points. When you are using XY
display, getting the data points for use in, for example, Excel, is a little
tricky. You first should get the picture you want in XY mode on the scope, then
switch to YT mode and do a single sweep. Channel 1 is now the X vs t, and
Channel 2 is Y
vs t, digitized over 2500 points. You
can now transfer the data to the PC, selecting both channel 1 and channel 2,
and read the file into excel, where you will do a scatter plot of channel 1
versus channel 2. You do not really use the time at all except to keep the data
points correlated.
(2) Measuring
the ground state splittings in the two isotopes of rubidium:
Set
the function generator voltages to obtain a spectrum in detector 2, similar to
that seen in fig. 2. (Actually, you probably cannot do this in one sweep, so
you will have to take two or more separate pictures and connect them in Excel.)
You can use a 3-4 Hz sweep to set it up, but when you finally take the data,
make a single sweep through the spectrum only, at the lowest sweep speed
possible. Read out this spectrum (these spectra) and use it to determine the
ground state hyperfine splittings in rubidium.
(3) Measuring
hyperfine splittings for the 5p3/2 state of 87 Rb :
Using
XY scan on the scope, put the outputs from detectors 1
and 2 into channels 1 and 2 of the amplifier and adjust the amplifiers so as to
obtain about the same signals on both channels. The saturated absorption lines
will appear as fine structure in the detector 2 signal. Play around with it:
block each probe, block the pump, etc., and see if it does what you expect.
Make fine adjustments in the mirrors. Etc. For quantitative measurements, put
the difference output from the amplifier into the Y side of the scope . You will have to adjust the function generator and
gains to sweep over line
“a”, absorption in 87Rb starting on F=2. Try to set
the scan so it just covers the necessary region to show the six fine absorption
features plus a little on each side. Some of these features but some appear as
very weak bumps or shoulders. Using the same procedure as in (1), record the
spectrum, take it to Excel or some other plotting program, and analyze it. You
should produce a plot of absorption versus frequency, identify all the features
on the figure and determine the hyperfine splittings in the5p3/2
state of 87 Rb.
Q:
What is “fine” structure splitting? What is “hyperfine” splitting? What do the
quantum numbers L,S.J and F mean ?
Q:
What is the resolution (width in frequency or energy) of this spectrum? Convert
this to a lifetime. What is this the lifetime of? (Note: in real life this
frequency width may be
power broadened beyond the natural line width.)
If
things are going swimmingly, you might want to see if you can measure the
hyperfine splitting for the 5p3/2 state of 85 Rb.