^M
# Sodium Doublet

##
1 Introduction

A sodium lamp contains vapor of the volatile metal sodium. A
current excites the sodium atoms, and they emit a surprisingly
simple spectrum, almost monochromatic. You can be sure of this
because if you look at this light with a Michelson interferometer,
you see nice yellow interference fringes. You can measure the
wavelength of this light in the same way you measured the
wavelength of the HeNe laser light, by counting fringes as the
moveable mirror traverses a measurably long distance. Part of the
experiment below is to measure the wavelength of this yellow
light.
As you do this, you might notice a more interesting and subtle
feature of the fringes. Sometimes they are clear and easy to see,
alternately yellow and black, like tiger stripes. Other times
they look pale and washed out. This is not a problem with your
eyes or with the alignment of the interferometer, but a real
effect. There are actually TWO wavelengths present here with
comparable intensity. Each of them forms a pattern of fringes.
What you explore with the interferometer is two fringe patterns
overlapping. Where they happen to be aligned, with dark fringes
coinciding and bright areas coinciding, you see clear fringes. But
where the bright parts of one fringe pattern overlap the dark
parts of the other fringe pattern, and vice versa, you don't see
much contrast. The applet below shows the intensity pattern
obtained by adding two fringe intensity patterns with slightly
different periods.
(Start the applet).

The distance you have to move along the
pattern from one washout to the next is a measure of the
difference in the two spatial periods. The spectrum of a sodium
lamp is often called "the sodium doublet" because in a
spectrometer capable of resolving these nearby wavelengths, you
see two bright yellow lines next to each other, a doublet. The
Michelson interferometer sees the doublet rather indirectly, by
the overlapping of the two fringe patterns.

Here is a more formal treatment of what is going on. Suppose we
think about just one wavelength for the moment, and consider
adding the two contributions from the two arms of the
interferometer, imagining that the light travels a distance L in
each arm, except that the moveable mirror has been moved an extra
distance x. Then the two amplitudes together at some fixed
observation point are

E_{1}=cos(k_{1}L-w_{1}t)+cos(k_{1}(L+2x)-w_{1}t) = 2cos(k_{1}(L+x)-w_{1}t)cos(k_{1}x) |
| (1) |

Squaring and averaging over time gives the intensity at this
point,

I_{1}=2cos^{2}(k_{1}x)=1+cos(2k_{1}x) |
| (2) |

We check that if x, the moveable mirror position, changes by
l_{1}/2, the intensity oscillates through one full period,
i.e., we see one fringe go by. Now suppose we are actually
looking at both contributions to the intensity,

I_{1}+I_{2}=2+cos(2k_{1}x)+cos(2k_{2}x)=2+2cos((k_{1}+k_{2})x)cos((k_{1}-k_{2})x) |
| (3) |

The rapid modulation of the intensity with x (i.e., the fringe
pattern) is at spatial frequency k_{1}+k_{2}, but an envelope
modulates the contrast with spatial frequency k_{1}-k_{2}. This is
what we see in the figure above. The distance d you must move
the mirror to go from one minimal contrast to the next satisfies
(k_{1}-k_{2})d=p. Therefore p/d=Dk=D(2p/l) » -2pDl/l^{2}, so in
magnitude Dl/l » l/2d.

##
2 Experiment

Use the Michelson interferometer to measure l and
Dl for sodium light. Be sure to think about turning
the screw which controls the mirror position always in the same
direction when you are doing the measurement.
To find the distance d between "washouts", you don't need to
count fringes, only to count "washouts." You will get better
accuracy if you find the distance to go through a large number of
them than if you only try to measure the distance between one and
the next, since they are not easy to locate precisely.

File translated from
T_{E}X
by
T_{T}H,
version 3.00.

On 23 Aug 2001, 15:23.