Middleton W.M. (ed.) Reference Data for Engineers: Radio, Electronics, Computer and Communications
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C 384-90a, Standard Test Method for Impedance and
Absorption of Acoustical Materials by the Imped-
ance Tube Method
C 423-90a, Standard Test Method for Sound Absorp-
tion and Sound Absorption Coefficients by the Rever-
beration Room Method
C 522-87, Standard Test Method for Airflow Resist-
ance of Acoustical Materials
C 634-89, Standard Terminology Relating to Environ-
mental Acoustics
C 635-91, Standard Specification for the Manufacture,
Performance, and Testing of Metal Suspension Sys-
tems for Acoustical Tile and Lay-In Panel Ceilings.
C 636-91, Standard Practice for Installation of Metal
Ceiling Suspension Systems for Acoustical Tile and
Lay-In Panels
C 800-89, Standard Specification for Glass Fiber Blan-
ket Insulation (Aircraft Type)
C 919-84 (Reapproved
88),
Standard Practice for Use
of Sealants in Acoustical Applications
C 971-82, Standard Guide for Selection and Applica-
tion of Insulation Systems for Heating, Ventilating,
and Air Conditioning Duct Work
C 1014-88, Standard Specification for Spray-Applied
Mineral Fiber Thermal of Acoustical Insulation
C
1071-86, Standard Specification for Thermal and
Acoustical Insulation (Mineral Fiber, Duct Lining
Material)
C 1149-90, Standard Specification for Self-Supported
Spray Applied Cellulosic Thermal/Acoustical Insula-
tion
D 1779-65 (Reapproved
83),
Standard Specification
for Adhesive for Acoustical Materials
E 90-90, Standard Test Method for Laboratory Mea-
surement of Airborne Sound Transmission Loss
of
Building Partitions
E 477-90, Standard Test Method for Measuring Acous-
tical and Airflow Performance of Duct Liner
Materials and Prefabricated Silencers
E 492-90, Standard Test Method for Laboratory Mea-
surement of Impact Sound Transmission Through
Floor-Ceiling Assemblies Using the Tapping Ma-
chine
E 497-89, Standard Practice for Installing Sound-
Isolating Lightweight Partitions
E
557-77 (Reapproved 90), Standard Practice for Ar-
chitectural Application and Installation of Operable
Partitions
E
580-91, Standard Practice for Application of Ceiling
Suspension Systems for Acoustical Tile and Lay-In
Panels in Areas Requiring Seismic Restraint
E 596-90, Standard Test Method for Laboratory Mea-
surement of the Noise Reduction of Sound-Isolating
Enclosures
E 597-91 (Reapproved
87),
Standard Practice for De-
termining a Single-Number Rating of Airborne
Sound Isolation for Use in Multiunit Building Speci-
fications
E
717-84 (Reapproved 89), Standard Guide for Prepa-
ration of the Accreditation Annex of Acoustical Test
Standards
E
989-89, Standard Classification for Determination of
Impact Insulation Class (IlC)
E 1041-85 (Reapproved 90), Standard Guide for Mea-
surement of Masking Sound in Open Offices
E
1042-85, Standard Classification for Acoustically
Absorptive Materials Applied by Trowel or Spray
E
1050-90, Standard Test Method for Impedance and
Absorption of Acoustical Materials Using a Tube,
Two Microphones, and a Digital Frequency Analysis
System
E
11
10-86 (Reapproved 90), Standard Classification
for Determination of Articulation Class
E
1111-88, Standard Test Method for Measuring the
Interzone Attenuation of Ceiling Systems
E
1123-86 (Reapproved 90), Standard Practice for
Mounting Test Specimens for Sound Transmission
Loss Testing of Naval and Marine Ship Bulkhead
Treatment Materials
E
1124-86, Standard Test Method for Field Measure-
ment of Sound Power Level by the Two-Surface
Method
E
1130-90, Standard Test Method for Objective Mea-
surement of Speech Privacy in Open Offices Using
Articulation Index
E
1179-87, Standard Specification for Sound Sources
Used for Testing Open Office Components and Sys-
tems
E
1224-88, Standard Guide for Categorizing Fields of
Testing for Laboratory Accreditation Purposes
E
1264-90, Standard Classification for Acoustical
Ceiling Products
E
1265-90, Standard Test Method for Measuring Inser-
tion
Loss
of Pneumatic Exhaust Silencers
E
1289-91, Standard Specification for Reference Spec-
imen for Sound Transmission Loss
E
1374-90, Standard Guide for Open Office Acoustics
and Applicable ASTM Standards
E 1375-90, Standard Test Method for Measuring the
Interzone Attenuation of Furniture Panels Used as
Acoustical Barriers
E
1376-90, Standard Test Method for Measuring the
Interzone Attenuation
of
Furniture Panels Used
as
Acoustical Barriers
E
1408-91, Standard Test Method for Laboratory Mea-
surement
of
the Sound Transmission Loss of Door
Panels and Door Systems
E 1414-91a, Standard Test Method for Airborne Sound
Attenuation Between Rooms Sharing a Common
Ceiling Plenum
E
1425-91, Standard Practice for Determining the
Acoustical Performance of Exterior Windows and
Doors
Lasers
Joseph T. Verdeyen
and
Thomas
A.
DeTemple
Elementary Relationships
41-2
The Line Shape Function
41-3
Amplification in a Quantum System
41-3
Laser Threshold
41-4
CW Oscillation
Transient Operation
Modelocking
Optical Cavities
41
-6
Hermite-Gaussian Beam Modes
Common Lasers
41-8
Semiconductor Lasers
41
-9
Nonlinear Optics
41-9
Second-Harmonic Generation
Pockels Effect
Stimulated Raman Emission
41
-8
41
-12
41
-1
41
-2
REFERENCE DATA FOR ENGINEERS
Technology Class
and
Common
Lens
Materials Window
The term “laser” is an acronym for Light Amplifica-
tion by Stimulated Emission of Radiation. The first
demonstration of an electronic amplifier using stimulat-
ed emission was in the microwave frequency domain
where the acronym “maser”
is
used. Stimulated emis-
sion has been observed in amorphous and single crystal
solids, semiconductors, and dyes and in atomic, molec-
ular, and ionized gases. Fig.
1
shows the spectral region
of
operation of the various types of lasers, a smoothed
version of atmospheric transmission, common detector
classes, and transmissive materials for various optical
functions. Since the operation of lasers is dependent on
quantum interactions, they are known as quantum
electronic devices.
VUV
I
UV
~
VISIBLE
I
INFRARED FAR INFRARED
I
I I
i
s,02
I
GLASS
~
SEM~CONDUCTOR/ALKALI
HALIDES
~EMICONDUCTOR/DIAMOND/PLASTICS
ELEMENTARY
RELATIONSHIPS
In order for a quantum electronic device to amplify
(or
attenuate) at a frequency
vZ1,
there must be a set of
allowed energy levels (or bands)
E2
or
E,
in the
material separated by the Planck spacing.
Common
Laser
Media
E2
-
E,
=
hvzl
(Planck’s law) (Eq.
1)
i-
GAS
FCRYSTAL~
I
p-
SEMICONDUCTOR
-1
t-
DYES-
I
I
An atom can make discontinuous jumps between
these energy levels, accompanied by either the emission
or absorption of a photon by one
of
three processes
(Fig.
2).
A.
B.
C.
Stimulated emission
occurs when an external
applied wave causes an atom to yield its poten-
tial energy given by Eq.
1
to the inducing wave.
This added energy (photon) propagates in the
same direction, at the same frequency and
phase, and in the same sense of polarization as
the stimulating field. This results
in
amplifica-
tion of the wave.
Absorption
occurs when the atom increases its
potential energy by
Eq.
1,
thereby decreasing
the wave energy by the same amount, and
results in attenuation.
Spontaneous emission
is the name for a
2+1
radiative transition for which there is no appar-
ent external influence and which yields a photon
at the frequency given by
Eq.
1
in any
of
4n
directions. It is the cause of noise in quantum
electronic amplifiers; however, discharge lamps
produce most
of
their radiation by this process.
The rate of production of electromagnetic energy by
the above processes
is
given
by
Spontaneous:
hv,,AzIN2
(Eq.
2)
Stimulated:
h~~,B~~N~g(v)l,/c’
(Eq.
3)
Absorption:
hv,
I
B,
N,
g(v)l,lc
’
(Eq
.
4)
Atmospheric
Transmissson
Ouer
Sea
Water
0
3
krn
Path
(Smoothed)
10
08
06
04
02
0
I
PHOTOELECTRON
i
COOLED PHOTOCONDUCTORS
77
K
4K
k
SCHOTTKY
DIODES
___7/
Ii
I-
P,HOTODIODES-
Common Detectors
1
1
I
I,,
I
I
,1,,111
I,
1
A
(pm)
0
1
10
10
100
1000
Y
(Hz)
315
314
313
312
3“
Fig.
1.
Regions
of
operation
of
various types
of
laser and examples
of
the technology required
in
the various domains.
LASERS
41
-3
where,
M
is the mass of the atom (or molecule),
T
is the temperature,
A
vD
is the full-width-at-half maximum
(FWHM)
At higher pressures and/or longer wavelengths (lower
frequency), elastic gas collisions broaden the transition,
and the homogeneous (Lorentzian) line shape function
becomes:
of the transition centered at
v21.
g(v)
=
Av~/~T[(v
-
~21)~
+
(Av,,/~)~]
(Eq. 10)
with
Av~
=
(1/2~)
[A2
+
k2
+
AI
+
kl
+
2~,,l,]
(Eq. 11)
In Eq. 11, the
A’s
are the total radiative decay rates of
levels
2
and
1,
the
k’s
are total quenching rates of the
two states, and
vcol,
is the elastic collision rate of the
active atoms or molecules with the other gas constitu-
ents. Usually, the elastic-collision term dominates, and
one refers to the resulting line profile as “pressure
broadened.
”
Seldom does one attempt
to
describe the line shape
of a transition in a solid or band-to-band transition in
semiconductors by elementary functions such as those
used in Eqs.
8
and 10. Rather, one depends on the
experimental determination of the line profile (e.g.,
C
above) and often approximates
g(v)
by
g(v)
=
l/Av
(Eq.
12)
where
Av
is the
FWHM
of the spontaneous emission
of
an optically thin line.
AMPLIFICATION
IN
A
QUANTUM SYSTEM
The net rate of increase of an externally applied
electromagnetic wave
Iv
(wattslarea) per unit of length
of active material is the difference between the stimulat-
ed rate of
Eq.
3 and the absorptive rate given by
Eq.
4:
dlJdz
=
hv21
B21
[N2
-
(gz/gi)
NiIg(v)
IV/c’
(Eq.
13)
If
N2
<
(g2/gl)
N,,
one has a “normal” population
with a positive “temperature” specified
by
Eq.
6;
such
a medium will attenuate an electromagnetic wave.
If
N2
>
(g2/gl)
N,,
one has an “inverted” population
with a “negative” temperature, and this medium will
amplify an electromagnetic wave. By combining Eqs.
5
and 13, one obtains the
small-signal gain coeficient
given by
A
=
A21
(A2/W
g(v)
[4
-
(gz/gi)
NII
y,(v)
=
(l/I,)(dI,,/dz)
(Eq.
14)
Fig.
2.
Radiative
processes between
two
levels
in an
atom.
(The letters
A,
B,
and
C
refer to stimulated emission,
absorption,
and
spontaneous emission
of
radiation.)
where,
iV2,(l)
is the density of atoms in energy state
2
(l),
AZI,
B2,,
and
BI2
are the Einstein coefficients,
g(v)
is the line shape function,
I,
is the intensity (wattdarea) of the externally
c’
=
c/n
is the velocity of propagation of the wave,
n
is
the index of refraction of the material.
applied wave,
The Einstein coefficients are interrelated.
A2II%21
=
8m3v3/c3
B21/%12
=
gllgz
(Eq.
5)
where
g2(,)
is the degeneracy of the state, Le., number
of quantum levels with the same energy.
At thermal equilibrium, the ratio of the population of
the two energy states,
E2
and
E,,
which are separated
by
Eq.
1
is given by the Boltzmann distribution:
THE
LINE SHAPE FUNCTION
The function
g(v)dv
has the following physical inter-
A. It is the fraction of the spontaneous emission
from state
2
to state
1
that appears in the
frequency interval
v
to
v
+
dv.
It is the relative strength of interaction between
the medium and the wave for stimulated emis-
sion or absorption.
It is the line profile
of
an optically thin transi-
tion.
Since it is a probability function, it obeys a normaliza-
tion condition
pretations:
B.
C.
I,”
g(v)
dv
=
1
0%.
7)
Transitions in the near IR through the
UV
in low-
pressure gases (a few Torr) are broadened by the
thermal motion of the atoms, which leads to a Doppler
broadened (Gaussian) line shape:
gD(v)
=
[(4
In
2)/7T]1’2(1/hVg)
eXp
((-4
In
2)
[(v
-
V~I)/AVDI~}
(Eq.
8)
with
AvD
=
(8kT/M~*)”~v21
(Eq.
9)
41
-4
REFERENCE
DATA
FOR ENGINEERS
where
A
=
A,/n.
The collection of terms multiplying
N2
is often referred to as the
stimulated emission cross
section
(i.e., “area”)
ase(v)
=
~21
(h2/8~)
g(v2i)
(Eq.
15)
The absorption cross-section
refers to the factors multi-
plying
N,
in Eq.
14.
(+abs(~)
=
(g2/g1)
rse(v)
0%.
16)
The small-signal gain of the medium of length
1
is given
by the expression
G,
=
exp
y,(v)l
(Eq.
17)
The subscript
“o”
in Eqs.
14
and
17
implies that the
intensity
of
the stimulating wave is sufficiently small
so
as to create negligible perturbation on the density of
atoms in the two states.
If the intensity is large enough, the populations,
N2,1,
in Eq.
14
are functions of the intensity,
I,,
of the
stimulating wave. For homogeneously broadened
CW
systems, the
saturated gain coejicient
is given by
U(v1
=
y,(v)/[l
+
(IJIS)
F
(41
4
(Ut,,)
dI,/dz
(Eq.
18)
where,
y,(v)
is the small-signal value,
&v)
is the line shape function normalized to unity
Is
is the saturation intensity given by
at line center,
I,
=
hV21/72
Cr,,(Vzl)
(Eq.
19)
In
Eq.
19,
is the decay rate of state
2
due to all
causes-radiation plus quenching, and
as,(vzI)
is the
stimulated-emission cross section given by
Eq.
15
evaluated at line center.
The overall gain,
G,
of the amplifier is thus a
function of the input intensity and is found by integrat-
ing
Eq.
18
from
0
to
1.
In
G
+
(Iin/[$)
g(v)
(G
-
1)
=
y,(v)l
0%.
20)
where
C
=
Iou,/Iin.
amplifier by stimulated emission is given by
The maximum intensity that can be extracted from an
(Io,,
-
[in)
=
~o(v21)Isl
(Eq.
21)
I
max
For inhomogeneously broadened transitions the satu-
rated law is modified:
y(v)
=
y,(v)/(l
+
Iv/Z5)’”
(Eq.
22)
where ail factors are as defined previously, and Eqs.
20
and
21
must be suitably modified.
LASER
THRESHOLD
Oscillation with an inverted population is possible
provided sufficient feedback is provided, usually in the
form
of
an open cavity. In general, the necessary
condition for oscillation is that net round-trip gain must
exceed
1
so
that an initially small wave can grow in
amplitude as it passes around the feedback path. For a
simple system such as that shown
in
Fig.
3,
the
threshold condition is
R,R,T~T;
exp
12
y,(v)/]
2
I
or
yo
2
(UZ)
In
+
(1/1)
ln
(l/To
Tb)
A
=
Ly
(Eq.
23)
In Fig.
3,
the power transmission coefficients for the
windows at the ends of the laser cell are
To
and
Tb,
and
those of the mirrors are
Ti
=
1
-
R,
and
T2
=
1
-
RZ.
Implicit in
Eq.
23
is the assumption that the population
inversion exists long enough to allow many transits back
and forth between mirrors.
If the transition is homogeneously broadened, then
oscillation will occur on a cavity mode that has the
highest gain-to-loss coefficient ratio
y,/a.
For
the
simple laser shown in Figs.
3
and
4,
this would
correspond to the cavity mode nearest line center. If the
transition is inhomogeneously broadened, multimode
oscillation is possible on those cavity modes at frequen-
cies
vq
such that
y0(vq)/a
>
1.
CW
Oscillation
For the simple laser shown in Fig.
3,
there are three
types of losses: internal discrete losses represented by
the imperfect transmission by the windows, a residual
loss
due to imperfections
in
the medium, and external
or
coupling losses by virtue of the transmission through
the mirrors. The photon flux inside the cavity builds up
to a large intensity
so
that the gain coefficient (Eq.
18
or
21)
saturates at the threshold value given by Eq.
23.
If
the mirrors are highly reflecting, little power escapes
the cavity; if the mirrors have too large a transmission
coefficient, the laser fails to oscillate. The transmission
coefficient of
M2
that provides maximum output power
from the laser illustrated in Fig.
3
is
T2
=
-L
+
(g,L)”*;
(TI
=
0)
(Eq.
24)
4
R1
Fig.
3.
A
simple
laser.
LASERS
a
(CAVITY
LOSSES
PRORATED OVER THE LENGTH)
41
-5
PUMP SHUTTER
Fig.
4.
Spectral
bandwidth
available
for
oscillation
where,
go
=
yo .
21
is the gain coefficient integrated over
the double pass through the active medium,
L
=
a,,,
*
21
is the integrated internal losses given
bY
The output intensity (through
M2)
is given by
I,,,
=
(Is/2)
*
[go/(L
+
T2)
-
11
T2
(Eq.
26)
For
T2
given by Eq.
24,
the output intensity is given by
the expression
I,,,
=
(IX/2)
.
[a
-
'4%]*
(Eq.
27)
Transient Operation
The previous discussion presumes steady-state or
CW laser oscillation. However, some lasers are operat-
ed in a pulsed mode with rather spectacular results, as
discussed below.
If the upper laser level has a lifetime that is long
compared to the pumping duration, then it
is
a candi-
date for Q-switching. The scheme is similar to that used
for radar modulators in which energy is slowly stored in
a capacitor, transmission line, or PFN and then is
rapidly discharged into the magnetron load.
In
the laser
scheme, the energy is stored in the upper quantum state
and is then extracted by stimulated emission.
The scheme is illustrated in Fig.
5.
The Q of the laser
cavity is spoiled (intentionally) during the time the
upper state is being populated by the external pump,
thus preventing lasing. When the upper state has
reached an equilibrium with the pump, the cavity Q is
restored to its high value. The population inversion is
larger than threshold for
CW
oscillation, and thus
stimulated emission rapidly amplifies the photons ini-
tially present in the cavity. This buildup occurs
so
rapidly that one can neglect any pumping on the time
scale
of
the output pulse; in fact, it is
so
rapid and
intense that the population inversion is driven below
threshold for CW oscillation.
(A)
The
optlcal
caulty
-
2t
/
(P~PULATION
I
X
~
INVERSION)
1
1
"Ih
0
LOW
LOSS
HIGHLOSS
t=O
(6)
Tlme
euoluuon
of
the
popdotion
difference
and
the
optlcol pulse.
+t
Fig.
5.
Q-switching
of
a simple laser.
(From
J.
T.
Verdeyen,
Laser Electronics.
Englewood
Clifs,
N.
J.:
Prentice-Hull,
Inc., 1981; Fig. 9-5.)
The parameters that characterize the Q-switched
n,
=
initial number of inverted atoms or
nfh
=
the inversion required for CW oscillation in
nf
=
the final inversion in population (see Fig.
5)
T
=
t/T,,
time divided by the passive cavity
pulse are:
molecules in the cavity
the high-Q cavity
lifetime
The peak power produced by such a laser is given by:
pp
=
k(hv/T,)
[(ni
-
nfh/2)
-
(n,h/2)
In
(ni/n,)]
(Eq.
28)
where
k
is the coupling factor
on
the order of
1/2.
The output energy of the Q-switch pulse is given by
and the final inversion
is
found from a solution to
n,h,
=
exp
[-(n,
-
nf)/n,,l
(Eq.
30)
Inasmuch as the maximum energy that could be extract-
ed is
n,
.
hv/2,
Eq.
29
can be written as:
W,,,
=
k
.
[(n,
-
nf)/n,]
(n,hvi2)
(Eq.
31)
The quantity
(n,
-
nf)/n,
is called the energy extraction
efficiency and is plotted in Fig.
6A.
The time evolution
of this Q-switch pulse requires a numerical solution of
the coupled differential equations.
A
typical solution
is
shown in Fig.
6B
for various values of the ratio
n,/nrh.
A
41
-6
REFERENCE DATA FOR ENGINEERS
spectral bandwidth available for oscillation (Fig.
4).
If
these modes can be locked together in a definite phase
sequence, then the time-domain representation of the
output consists of a series of pulses at a repetition rate
equal to the round-trip transit time of a photon in the
cavity.
Each individual pulse is very short; typically, 0.1 to
2
ns is common, with
30
fs (30
X
s)
being the
shortest presently recorded. The time evolution of these
pulses is related to the spectral distribution of modes by
the Fourier transform pair (see Chapter
7).
Rough rules
of thumb for the pulse characteristics are:
Peak power
=
N
X
average power
Pulse repetition rate
=
c'12d
Pulse width
=
(2dIc')lN
=
l/Av
(Eq. 33a)
(Eq.
33b)
(Eq. 33c)
where,
N
is the number of modes locked,
2dlc'
is the round-trip transit time,
Av
is the
FWHM
of the transition.
INVERSION RATIO,
ni/nrh
(A)
The enmctlon eficlency
for
a
Q-swltched laser.
OPTICAL CAVITIES
NORMALIZED TIME.
7
(E)
The time evolution
of
the Q-swltched
pulse,
the nomallzed
to
the passive
cavity
rlfetlrne.
Fig.
6.
Characteristic parameters
of
Q-switching.
(From
J.
T.
Verdeyen,
Laser Electronics.
Englewood
Cliffs,
N.J.:
Prentice-
Hall,
Inc.,
1981;
Figs.
9-7
and
9-8.)
rough estimate for the
FWHM
of the pulse can be found
by combining
Eqs.
28
and
31:
At112
=
w,ut/pp
(Eq.
32)
Modelocking
If the system is inhomogeneously broadened, the
laser may oscillate on many cavity modes within the
Optical cavities perform the essential function of
providing positive feedback for the quantum electronic
amplifier
so
as to obtain oscillation. The spontaneous
emission from the upper state of the quantum system is
reflected by the mirrors and amplified by the inverted
population. The cavity also selects the oscillation fre-
quency within the spectral bandwidth defined in Fig.
4.
The laser depends upon stimulated emission for amplifi-
cation, and that rate is a maximum at the resonant
frequencies of the cavity since the electromagnetic
energy is a maximum under these conditions.
The resonant buildup of electromagnetic energy in
any cavity is due to the near-perfect phase addition of
the various partial waves bouncing back and forth
between the mirrors. In general then, the resonant
frequency
of
any cavity is determined by
Round-trip phase shift
=
Integral multiple of
27
radians
(Eq.
34)
If one assumes a uniform plane wave for the normal
mode of the simple Fabry-Perot cavity shown in Fig.
7A,
then
Eq.
34 reduces to
vs
=
q
c'12d
(Eq.
35)
where,
c'
is the velocity of light in the medium between
the mirrors,
q
is an integer.
Typically
d
=
0.1
to
1.0
m
for solid-state and gas
lasers, and thus
c'/2d
=
150
MHz
to 1.5
GHz,
LASERS
41
-7
(A) Faby-Perot cavity
(B)
A stable optical CaUlty
(C)
An
unstable optlcal cauity.
Fig.
7.
Typical cavities used with lasers.
considerably smaller than the line width of many
transitions (approximately
5
to
100
GHz).
Most optical cavities use one or more focusing
elements in the manner indicated in Figs. 7B and 7C.
The cavity shown in Fig. 7B is
stable
in the sense that
the maximum excursion of a paraxial ray from the axis
of the cavity is bounded. However, a ray will “walk
OR”
the surfaces of the feedback mirrors of Fig. 7C;
hence, this cavity
is
classified as
unstable.
Simple
cavities such as those shown in Fig. 7
are
stable if
05g,g251
(Eq.
36)
where,
If the equality is not satisfied, the cavity is unstable.
The stability criterion given by
Eq.
36 is quite often
embodied in a graphical format such as shown in Fig. 8.
The dashed curves illustrate the equiloss contours for
the unstable cavity. The quantity
r
is related to the
mean fractional power
lost
per pass
by
r
=
1
-
L.
For
stable cavities, the diffraction loss (Le., loss around
edges
of
the mirrors) is very small; most of the loss of
electromagnetic energy is transmission through the
mirrors.
Unstable cavities have a very large mode volume and
thus can utilize the stimulated emission from a large
number of atoms. The losses from around the edges of
the mirrors are usually large compared to the loss from
direct coupling-usually by design. One can relate
r,
the mean fraction of the power that survives a single
pass through the cavity, to the cavity
g
parameters by
Fig.
8.
Graphical display
of
the stability
of
optical cavities.
where the choice of the sign is determined by the
requirement that
0
<
r
<
1.
This leads to the equiloss
contours shown in Fig. 8.
If all external sources
of
electromagnetic energy for
the resonant mode
of
a cavity are suddenly removed, the
energy will decay with time constant
rP,
called the
photon lifetime:
dWldt
=
-WIrp
(Eq.
39a)
7=
Fraction of energy lost per round trip
(Eq.
39b)
where
W
is
the energy stored in the cavity.
the expression
Time for a round trip
For the simple cavity shown in Fig. 7B, this leads to
rP
=
(2d/c)/(l
-
RIR,)
(Eq.
40)
where
all
losses except those due to imperfect mirror
reflectivities have been neglected.
The photon lifetime is directly related to the cavity
Q
by the equation
Energy stored
Energy lost in a cycle
Q=2.irx
m.
4w
or
(ES.
41b)
Q
=
mow/(-dwldt)
where
o,
is the (angular) resonant frequency
of
the
41
-8
REFERENCE DATA FOR ENGINEERS
cavity. The differential equation of
Eq.
41b is the same
as
F.q.
39a and relates
Q
and
T~
by
T~
=
Q/oo
(Eq.
42a)
The full-width at half maximum
(FWHM)
of
the cavity
resonance is given by the familiar formula
bo
=
oo/Q
or
Av=
v,/Q
(Eq.
42b)
Hence
AWT~
=
1
(Eq.
42c)
HERMITE-GAUSSIAN BEAM
MODES
Stable cavities such as that shown in Fig. 7B produce
a beam whose electric field is described by
E,,,(x,y,z)
=
E,(w0/w(2))
H,
(fiw) H,(a/w)
.
exp
[-(n2
+
y2)/w21
*
exp
[-jk(x2
+
y2)/2~(z)I
-
exp
{-j[kz
-
(1
+
rn
+
n)
tan-'
z/z,]}
(Es.
43)
where
H,
is
a Hermite polynomial and
k
=
dc.
The
mode indices,
rn
and
n,
refer to the variation of the
TEM field in the
x
and
y
directions similar to the
convention used for waveguides. The interrelationships
between the parameters in
Eq.
43 are listed below.
w2(z)
=
w:
[l
+
(2/2,)2]
R(z)
=
z
[l
+
(zo/z)2]
(Eq.
Ma)
(Eq.
ab)
zo
=
m:/A
(Eq.
44~)
0%.
444
The plane
z
=
0
is
defined to
be
the location where the
spot size,
w,
is a minimum and the wave-front curvature
becomes planar
(R
=
p).
The spot size of the beam will
expand by
fi
when it has propagated a distance
z,
from
z
=
0.
The interrelationships are plotted in Fig.
9
for the TEM,, mode.
Notice that in the far field the beam expands as if it
were a spherical wave
of
limited angular extent and
originating at
z
=
0.
Angle
e
is the angular divergence
of
the beam given by
H,(u)
=
(-1)" (exp
u2)
(d"/du")
exp
(-u2)
e
=
~AIW,
(Eq.
45)
and is an important laser characteristic.
The characteristic parameter,
z,,
and the location of
the plane
z
=
0
are related to the radius of curvature of
'
OFFIELD
'
OFTHEFIELD
/
Fig.
9.
The expansion
of
a
TEM,,, mode.
the mirrors and the spacing for the cavity shown in Fig.
7B by:
21
=
d(R2
-
d)/(Rl
+
R2
-
2d)
22
=
d(R1
-
d)/(R,
+
R2
-
2d)
(Q.
46b)
(Eq.
46~)
The resonant frequencies of the TEM,,, modes differ
slightly from those predicted by
Eq.
35:
v~,~,~
=
(c'/2d)
{q
+
[(l
+
rn
+
n)/7r]
c0s-I
(gl
g2)"21
(Eq.
47)
COMMON
LASERS
Gases, impurity doping in crystalline and amorphous
solids, direct bandgap semiconductors, and dyes dis-
solved in various solvents are the common materials
used for lasers. All have a common theoretical frame-
work, but the details of the excitation route differ from
one type to another as illustrated in Table
1.
The excitation can be either pulsed or CW, resulting
in the corresponding temporal characteristic of the laser
radiation. With the exception
of
the semiconductors,
the wall-plug efficiency of most lasers is typically less
than 1%; the CW C02 gas laser system is an exception
with efficiencies of
10-20%
being common. Therefore,
many very high power lasers are operated in a pulsed
mode
so
as
to
dissipate the waste heat associated with
the excitation. Many lasers, such as
N2
at 0.337 pm,
require pulsed operation since only a transient inversion
can be created.
Table 2 lists some common high-power pulsed lasers;
typical pulse widths and the optical energy produced
are shown. Table 3 lists common CW lasers. It should
be noted here that even a milliwatt
(lov3
W)
coherent
laser is potentially hazardous to the eye. Consequently,
wavelength-selective eyeglasses are recommended,
since even the backscatter of high-power lasers from
rough surfaces can exceed this level.
41
-9
TABLE 1. COMMON EXCITATION MECHANISMS
FOR
VARIOUS LASERS
Material Excitation Route
Gases
Impurity doping
in
solids
Electric discharge. Optical pumping,
chemical
reactions.
Optical
pumping usually
with
incoherent
flash
lamps,
halogen lamps,
or
semiconductor LEDs.
Semiconductor Carrier injection.
Dyes
Coaxial flash lamps,
N,
laser, excimer, or harmonics
of
YAG
(0.532
pm,
0.355
pm)
for
pulsed operation;
argon
or
krypton
ion
lasers
for
CW
operation.
The output of various dyes depends on the solvent,
the concentration of the dye, and the source of excita-
tion. This
is
illustrated in Fig. 10, where the relative
energy versus wavelength is plotted for the common
sources of excitation. Flash-lamp-pumped dyes produce
the highest energy per pulse, approximately
0.2
J,
and
the longest pulse width, approximately 0.2
ps,
but
typically at a rather slow repetition rate of a few pulses
per second. The N2 laser at
0.337
pm or the second and
third harmonics of Nd:YAG
are
used to produce high
power (tens of kilowatts) and short pulse widths (5 ns) at
moderate repetition rates of 10- 100 pps. Continuous-
wave dye lasers are usually excited with the lines from
an argon or krypton ion laser. While the conversion
efficiency of the dye
is
quite reasonable (10-25%), the
electrical efficiency of the pump laser is usually less
than
0.1%.
Titanium-doped sapphire lasers, pumped by argon
ion lasers, have tuning characteristics which
are
similar
to those of near-infrared dye lasers, but with a higher
output power. Typical tuning curves for this laser are
shown in Fig. 11.
Many of the far-infrared lasers, using various gases
such as CH,OH, are pumped by the C02 laser. Fig. 12
illustrates the range
of
powers and frequencies available
when such a pumping scheme is used.
SEMICONDUCTOR LASERS
Population inversion, and thus optical gain in the
sense of Eq. 14, may be obtained in a direct bandgap
semiconductor exhibiting radiative recombination of
electrons and holes provided there is, simultaneously in
space and time, a large density of electrons in the
conduction band and holes in the valence band such as
that
provided by the junction region
of
a diode under
heavy forward bias where large numbers
of
electrons
and holes exist.
Such a diode is shown in Fig.
13
for a homojunction
laser consisting
of,
for example, heavily doped regions
of
GaAs to
form
a pn junction. Under forward bias,
electrons are injected into the
p
region, leading to an
"inverted" population of width
1
pm or
so
perpendicu-
lar to the junction. The light
(L)
output is directly
proportional to the recombining carriers and thus is
directly related to the injected current
(I).
At low
currents, the device acts as a LED. Beyond a threshold
current, stimulated emission of recombination takes
place, and the slope of light output versus current shows
a dramatic increase. Due to the short recombination
lifetime (typically
lob9
second), one is able to ampli-
tude modulate the light output by simple current modu-
lation.
Semiconductor lasers that involve only one material
(e.g., GaAs) are classified as homojunctions; those that
utilize a different material on one side of the active
region are called single heterostructures (SH); if both
sides use different materials, the device is called a
double heterostructure (DH). Usually, the different
materials used to obtain a heterostructure have a differ-
ent bandgap and index of refraction than that of the
active region.
A
typical configuration for a double
heterostructure is shown in Fig. 14. The higher bandgap
confines the carriers, and change in index of refraction
guides the electromagnetic mode. Due to the small size
of the active region, the divergence angle of the mode
may be on the order of 10-20" and is different in the
planes parallel or perpendicular to the junction.
Fig. 15 shows the emission ranges of semiconductor
lasers obtained with various combinations
of
111-V
and
IV-VI materials. For a given material system, the
emission range corresponds to different crystal compo-
sitions and not the wavelength tuning of a single laser.
Typical threshold currents for heterostructure lasers are
in the range from
1
to
50
mA; output powers range from
1 to
50
mW. The very-long-wavelength diodes require
cryogenic cooling.
NONLINEAR OPTICS
As in the case of other electronic devices,
under
suitable conditions the ordinary linear behavior
of
transparent materials becomes nonlinear. These nonlin-
earities, which are caused by strong optical or low-
frequency electric fields, are used in the frequency and
amplitude control of light.
The various effects are described by a simple Taylor
series expansion of the microscopic polarization