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Bryn Mawr College

Department of Physics

Undergraduate Teaching Laboratories

Longitudinal and Transverse Modes of a Laser


Introduction



Simplified descriptions of lasers emphasize that they can emit electromagnetic
radiation of a single frequency (and,
therefore, a single wavelength) in a highly
directional beam. The light is said to be "temporally coherent" because the
electromagnetic waves are all in phase, and the light is said to be "spatially coherent"
because the emissions in different transverse
positions in the beam have fixed phase
relationships. However, these are ideal properties that are rarely true without
exceptional technological effort. In this laboratory experiment you will explore what
characteristics of a laser determine the frequenc
ies (wavelengths) and intensity patterns
emitted by a laser. You will investigate a variety of the transverse patterns that can be
found for a laser beam, and what changes in the characteristics of the design of the
laser cause changes in those transverse

patterns. You will also get to investigate how a
laser can be made from discrete pieces: mirrors and a gas of atoms (by a current of
electrons in a discharge), which can spontaneously emit and amplify a range of
wavelengths of light.


A Few Words of Caut
ion



One of the technological advantages, and practical dangers, of laser radiation
arises from the spatial and temporal coherence of the beam. This permits all of the
energy of the beam to be focused by a lens to an extremely small spot, with the
conseq
uent delivery of all of the power of the beam to a very small area. The HeNe
lasers (


= 0.6328

m) you will use in the teaching laboratories range in output powers
from 1
-
5 mW (milliwatts). This is just at the level where discomfort might occur if all o
f
the power is focused to a small spot on your retina. It is below the levels that have been
found to cause retinal damage. Despite the "safety" of these weak beams, it is wise
practice never to look directly at a laser beam. It is safest to remember th
at you should
not put your head at the level of the laser beam or in line with the reflection of a laser
beam from a mirror. The diffuse reflection of a laser beam from the wall, or a piece of
paper, loses both intensity and some of the spatial coherence
because of the roughness
of the surface, and so it is safe to view beams in this way.
Have a look at the laser
safety Wikipedia entry for a discussion of how lasers are classified and safely used:
http://en.wikipedia.org/wiki/Laser_safety
.


Theory: Introduction



A string fixed at both ends can vibrate in many different spatial modes: the first
mode (longest wavelength and lowest frequency) has nodes only at the fixed ends, the
second mode has a t
hird node in the middle and so on. (See the picture on page 484 of
Modes of a Laser .
. . . . . . .

2


Serway, "Physics for Scientists and Engineers," Third Edition.) These are one
-
dimensional modes. These modes are analogous to the longitudinal modes we will
measure for a laser. Two
-
dim
ensional modes are a little more complicated than one
-
dimensional modes. Serway has nice pictures of two
-
dimensional modes in a drum
where the circumference of the vibrating material is a node. For a laser, there is usually
no sharp boundary at the sides
, only at the mirrors at the ends. However, the curvature
of the mirrors and narrow cross
-
sectional region where there is an amplifying medium
limit the laser in the transverse direction. To satisfy Maxwell's equations for the electric
and magnetic field
s that make up a laser beam inside a laser there must be a balance
between diffraction (spreading of a spatially limited beam) and focusing along with
amplification to keep the intense part of the beam at the center.


Theory: Longitudinal Modes



When th
ere are no boundary limitations for electromagnetic waves, they can
exist at any frequency
f

(or wavelength


since
c = f


where
c

is the speed of light).
Electromagnetic waves can be created from the oscillating dipole moments in atoms
when the electrons

of the atoms are in a mixture to two appropriate atomic states. The
frequency of oscillation of the dipole moment is related by Planck's formula to the
difference in the energies of the two atomic states:

E = hf
, where h is Planck's
constant,
f

is the f
requency of the oscillation, and

E

is the energy difference. For many
reasons most observations of atomic dipole transitions correspond to emission or
absorption over a narrow range of energies, typically of order 10
-
2000 Megahertz (10
7
-
10
9

Hz) wide. Th
is is to be compared to the transition energy itself

E
, typically on the
order of 10
15

Hz, which is the frequency for visible light. Laser emission involves a
process of stimulated emission, in which atoms are stimulated by some of the light that
is pres
ent to emit their energy in phase with the initial radiation. This process amplifies
an incident beam of light.


One way to confine the light is with two plane and mutually parallel mirrors. Light
that travels back and forth between the mirrors forms a

standing wave that must satisfy
the boundary conditions. If we take the
z
-
axis perpendicular to the mirrors and place
the mirrors at
z=0
and
z=L
, then the spatially varying part of the electric field of the
standing electromagnetic wave can be expressed
as
E
(
z,t
)

= E
0
sin(
kz
) cos(

t
), where
k

is the wave number (
k

= 2

/

) and


is the angular frequency (


= 2

f
). If there is to be
a node at both ends then an integer number of half wavelengths must fit into the
distance
L
. Therefore at
z = L
,
L=q



where
q

is an integer. Since
c = f

,
the values of
f

are also restricted to
f
q

= qc/2L
where the subscript
q

is a convenient way to keep track
of the possible values of
f
. Hence the boundary conditions give us a discrete set of
allowed frequencies separa
ted by

f = c/2L
. The frequencies
f
q

and their differences
are shown in Figure 1. Figure 1 also shows a schematic curve of the emission of the
lasing transition. If the wavelength is 0.6328 µm and the laser cavity length is 30 cm,
what is the approximat
e value of “
q
”? Most commercial HeNe lasers emit light at
several of these "cavity frequencies" or "longitudinal modes" because the range of
Modes of a Laser .
. . . . . . .

3


colors emitted on the red transition of neon is broadened to about 1.5 GHz by Doppler
shifts of the neon atoms in
the discharge.






Theory: Transverse Modes



The solutions for the transverse modes are relatively complicated. Two families
of these modes are the most common ones: the Gauss
-
Hermite modes and the Gauss
-
Laguerre modes. These modes are called "TEM" modes since the electric and
magnetic fields are b
oth transverse to the direction of propagation, even though this is
not necessarily the case for confined EM fields. Generally three indices,
m
,
n

and
q

are
used to designate a particular modal pattern using the notation TEM
mnq
. The "
q
"
specifies the long
itudinal number of half wavelengths, as discussed above, and the
m

and
n

indices are the integer number of transverse nodal lines in the
x
-

and
y
-

directions
respectively, across the beam. Because the spatial variation differs for the different
modes, the

frequencies of these modes are also different. Generally for each of the
longitudinal modes
q
, of the plane wave resonator, there is an infinite family of
transverse modes (
m, n
), with the frequencies increasing as the complexity of the
modes increases.

Figure 2 shows a schematic picture of the frequencies of longitudinal
and transverse modes in the vicinity of the lasing frequency.





Modes of a Laser .
. . . . . . .

4






For the Gauss
-
Hermite modes, the transverse intensity patterns are shown in
Fig. 13.9 (pp. 592) of Hecht's
Optics

and these patterns are given by


I
mn
(
x
,
y
,
z
)

0
I
2
0
w
w
z


m
H
2
x
w
z




n
H
2
y
w
z
















exp

2
2
x

2
y
2
w
z


















,




(1)


where
x

and
y

are the transverse coordinates and
H
m

and

H
n

are Hermite polynomials.
The first few Hermite polynomials are:
H
0
(x) =
1
, H
1
(x) =
2
x, H
2
(x) =
4
x
2
-
2
.

These are
the same functions as those used to describe the quantum mechanical harmonic
oscillator. The position
z
=0 corresponds to the where in the re
sonator the phase fronts
are a plane. This would be at the plane mirror if one of the mirrors is plane, and if the
mirrors have equal radii of curvature, it would be halfway between the mirrors. At
z
=0,
w
(
z
) has its minimum value
w
0
, which is called the
"beam waist".


We see that the intensity patterns involve the product of the Hermite polynomials
and a Gaussian function. The Gaussian falls to 1/e
2

of its value when
x
2
+
y
2

=
w
2
(
z
).
So roughly speaking, the transverse "width" of the beam intensity patter
n at longitudinal
position
z

is given by 2
w
(
z
) with




w
(
z
)

0
w
1/
2
1

2

z

2
0
w




.








(2)


The radius of curvature of the wavefront, as a function of the distance
z
, is given by:


Modes of a Laser .
. . . . . . .

5



R
(
z
)

z
1

2

4
0
w
2

2
z












.









(3)


Returning to the equation for the
lowest order mode, we see that



00
I

0
I
2
0
w
w
(
z
)


exp

2
2
r
2
w
(
z
)


,







(4)


where
r
2

=
x
2

+
y
2
. This is a circularly symmetric pattern (a spot), and since the
transverse profile is simply a Gaussian function of the distance from the center of the
beam, this mode
is often called the "Gaussian mode", or fundamental mode. See
Figure 13.11 in Hecht, pp. 593.


The second family of modes involves replacing the product of the Hermite
polynomials for
x

and
y

with the Laguerre polynomials for
r

and

, the polar coordinate
s
in the transverse plane. When the circular (azimuthal) symmetry is nearly perfect, the
Gauss
-
Laguerre modes are more commonly found. When the circular symmetry is
broken because of the tilted Brewster Angle window or a tilted mirror, then the Gauss
-
Her
mite modes are more commonly found.


Equipment


There are two lasers on the bench. One is the white Metrologic Neon Laser,
which we’ll refer to as the “commercial laser.” The other is a glass tube mounted on the
rail at the back on the bench, which we’ll refer to as the “open cavity laser.” The power

supply for the open cavity laser is on the shelf above the laser.


There are also two Fabry
-
Perot interferometers on the table. One is very old and
brass. You should look carefully at this device and identify the function of each of the
adjustment knobs
. The other interferometer is sealed inside a silver tube and mounted
in a black disk with two knobs that adjust the orientation of the cylinder.

This
interferometer is powered by a supply on the shelf and its output can be viewed on the
neighboring osci
lloscope.


The final instrument that you will use is a ccd camera. This camera is connected
to the computer, where you can view its output.


Experiment One: Longitudinal Modes of a Laser



Using the old
-
style Fabry
-
Perot interferometer consisting of two c
losely spaced
parallel mirrors, observe the ring pattern that is created in transmission when the mirrors
are illuminated by a commercial laser beam that is focused by a lens to illuminate a
range of angles.
To align this interferometer it is best to begin

by sending the laser
beam through the two mirrors without the lens. Tape a piece of paper to the wall so that
you can see the light that is transmitted through the pair of mirrors. Adjust the
orientation of the cavity so that the laser beam is normal to

the mirrors. Adjust the
Modes of a Laser .
. . . . . . .

6


adjustable mirror so that the dots you see transmitted through the pair of mirrors all fall
on top of each other. Now insert the diverging lens into the beam between the laser
and the cavity. Fine
-
tune the adjustable mirror of t
he Fabry
-
Perot interferometer to get
circular rings. Note that every time the distance between the Fabry
-
Perot mirrors
increases or decreases by half of the laser wavelength, the rings decrease or increase
in size by one ring spacing.


Next, use the scanni
ng Fabry
-
Perot Spectrum Analyzer made by Coherent to
display the optical spectrum of the commercial laser. This analyzer is a commercial
version of the same kind of Fabry
-
Perot interferometer you have just investigated,
except that it has a detector place
d at the center of the transmitted pattern and has
curved mirrors so that the light is almost always limited to the fundamental transverse
spatial mode. A linearly ramped voltage supplied by the controller changes the voltage
on a piezoelectric crystal to

which one of the Fabry
-
Perot Analyzer mirrors is mounted.
The crystal changes its length in proportion to the voltage (about one micrometer for a
few hundred volts), thereby scanning the separation between the two mirrors of the
interferometer.

Steer the

output of the commercial laser beam
(
by using mirrors

if necessary)

so
that it enters perpendicularly into the aperture of the Coherent Fabry
-
Perot Spectrum
Analyzer. Look at the output of the analyzer on an oscilloscope. Trigger the
oscilloscope with th
e “trig out” signal from the back of the controller. Set the offset
voltage to zero and the scanning voltage to ~200 volts. Set the oscilloscope to 2 mV/div
and the time scale to 1 msec per/div. Adjust the tilt of the analyzer until you see sharp
peaks
corresponding to the longitudinal modes of the laser. The Fabry
-
Perot has a
"free spectral range" (FSR) of about 7.5 GHz, so that the transmission pattern is
repeated with a spacing equivalent to 7.5 GHz. Using the 7.5 GHz spacing as a
calibration, measu
re the spacing between the longitudinal modes of the commercial
laser. From your measured values, estimate the actual distance between the mirrors of
the HeNe laser.


Insert a polarizer between the laser and the Fabry
-
Perot Spectrum Analyzer. By
rotati
ng the polarizer determine the polarization of the different longitudinal modes of the
lasers. What polarization pattern do you observe in the modes?

Finally, use the scanning Fabry
-
Perot Spectrum Analyzer to display the optical
spectrum of the open cavit
y HeNe laser.

Alignment is more difficult with this laser since
the beam is weaker that the commercial laser.

Examine and identify the discharge tube
and the two mirrors making up the open cavity laser. Note that one mirror can be
moved along the rail t
o give different cavity lengths. Steer the output of the laser beam
by using

a mirror or

two to direct it perpendicularly into the aperture of the Fabry
-
Perot
Spectrum Analyzer. Again, look at the output of the Analyzer on an oscilloscope.
Adjust the til
t of the analyzer until you see sharp peaks corresponding to the longitudinal
modes of the laser. Using the 7.5 GHz FSR of the Fabry
-
Perot Spectrum Analyzer as a
calibration, measure the spacing between the longitudinal modes of this laser for a
given cav
ity length L. From your measured values, estimate the actual distance
between the mirrors of the HeNe laser. How does it compare to the length of the cavity
measured with a ruler? Now assume you do not know the FSR of the Analyzer. Can
Modes of a Laser .
. . . . . . .

7


you devise a way

to use the open cavity laser to measure it? Do this and determine the
uncertainty in your result.


Experiment Two: Transverse Modes of a Laser



For each longitudinal position of the movable mirror of the open cavity HeNe
laser, study the different tran
sverse modes you can obtain by adjusting the variable
aperture in the cavity and by small adjustments in the tilt of the external mirror. Repeat
this for several different longitudinal positions of the external mirror. Shine the different
mode patterns o
nto the CCD camera and record the intensity profiles with the
associated computer using the beam analysis software.


Using the camera, make measurements of the separation of bright or dark points
in a transverse mode as a function of the distance from the
output end mirror of the
laser. The TEM
10

or TEM
20

modes work well. Use mirrors to fold the beam back and
forth across the room to get the longer distances. Plot the separation of the mode
structure as a function of distance. Use this information to de
termine the divergence of
the laser beam and to estimate the location of the beam waist inside the laser. From
this result, estimate the curvature of the fixed end mirror, assuming the output mirror is
flat.


Modes of a Laser .
. . . . . . .

8


Aligning the Open Cavity Laser



If the open
cavity laser does not emit light when the discharge is turned on,
probably the external output mirror is misaligned. To align this mirror, we use a working
commercial laser as a "straight line" reference. First the external output mirror is
removed. Sec
ond the optical discharge tube is aligned horizontally parallel to the rails
of the optical bench at the desired height and at the desired transverse position. Third,
by use of two or more steering mirrors, the beam from the commercial laser is directed
a
long the optical bench, through the Brewster Angle window of the discharge tube,
through the inside of the discharge tube (with minimal touching of the sidewalls) and
onto the sealed end mirror. A small amount of light is transmitted by this mirror and it

can be viewed on a card. It will consist of a compact circular spot when the light is
aligned properly through the discharge tube. The reflected light from this end mirror
travels back through the tube and can be viewed with a card placed around the out
put
of the commercial laser. By fine tuning the alignment of the steering mirrors to make
the commercial laser beam exactly perpendicular to the sealed end mirror, both the
transmitted and reflected beams form overlapping circular spots. Then the externa
l end
mirror can be inserted and centered on the commercial laser beam path. The tilt of this
mirror is adjusted to make the reflection from its surface a nice round spot on the card
around the exit aperture of the commercial laser. At this point, if the

open cavity laser
discharge tube has a voltage applied across it, and the commercial laser is blocked,
some spontaneously emitted red laser light should be visible from the open cavity laser.
Then the alignment can be improved by minor tweaking of the ti
lt adjustment screws of
the external output mirror.