8.
QUANTUM OPTICS
Interaction of light with matter.
In order to describe the interaction of light with matter it is necessary to move away from the
wave view of electromagnetic radiation and try to understand the ways in which quantised
electromagnetic radiation, photons, may interact with matter.
To do this consider the simplest possible situation where matter is to be described by a
quantum mechanical two level system consisting of the ground state and an excited state.
There are three basic processes by which photons can interact with such a two energy level
atomic (or molecular) system; absorption, spontaneous emission (fluorescence), and
stimulated emission. This last process is only predicted within quantum mechanics and is not
a classical process. A great deal of progress can be made in a simple manner by describing
each process by its Einstein coefficient , B12 , A21 and B21 , respectively and discovering
the relationship between the three coefficients.
Absorption, Fluorescence, Stimulated Emission
and the Einstein Coefficients.
By consideration of an atomic system consisting of two electronic quantum levels, the ground
state and the excited state (denoted by subscripts 1 and 2 respectively in what follows), in
equilibrium with a radiation field at temperature T, three different processes whereby an atom
can interact with light may be identified.
i) Absorption
The most familiar of the three in everyday life is the process of absorption whereby a photon
of appropriate frequency promotes an electron from the ground state to the excited state(s).
E2
N2
h21
N1
E1
The rate at which this process occurs per unit volume is proportional to;
248
i)
The number of occupied ground state atoms/molecules per unit volume, N1, (ie
the number of initial states) and
ii) The energy density in the electromagnetic field, (), at the transition frequency
(ie, the number of photons present per unit volume and available to take part in
the process).
The constant of proportionality is the Einstein coefficient B12.
The rate at which the absorption process occurs may be written in terms of the rate of
change of the population of level 1 as;
dN1
 B12   N1
dt
Losing an amount of energy h with each transition, the rate of decrease of energy density
per unit volume in the presence of this process alone can then be written;
d 
dN
 h 1  hB12   N1
dt
dt
NB. The minus sign on the RHS of the above equations exists as absorption involves
the loss of a ground state atom along with the loss of a photon from the
electromagnetic field.
Generally it is more useful to know the rate of decrease of intensity I with distance as this is
what is measured in a spectrophotometer where a light beam crosses a sample of known
thickness and the emergent intensity is measured. When light is of a frequency such that h =
E2 – E1 (with E2 being the excited state energy and E1 the ground state energy) some of the
light will be absorbed by the sample and it is the consequent loss in light intensity that the
spectrophotometer measures. By noting that for a radiation field (beam of light) propagating
at a velocity v,  and I are related by
I
  
v

I
dz
dt
 I
dt
dz
therefore
d dI

dt dz
The earlier equation can thus be rewritten;
dI 
n
I
 hB12  N1  hB12 N1I 
v
dz
c
249
dI 
n
 hB12 N1dz
I
c
and by integrating from 0 to L
 I (L ) 
n
  hB12 N1L
loge 
c
 I (0 ) 
finally;
I (L)  I(0) exp( L)
This final equation is known as Beers law where  is the absorption coefficient of the
system given by
  hB12
n
N1
c
Evidently knowledge of  from spectroscopy will lead to a value for B12
B12  
1 c 1
h n N1
It is necessary to note that in arriving at this equation the fiction was adopted that the two
level system was interacting with perfectly monochromatic radiation. In fact any energy levels
will have a spread of frequencies over which they may interact, . That is, the energy levels
are broadened by some external physical effect. For example
i)
In a crystal different atoms may lie in slightly different environments and local
fields will shift slightly the actual transition energy eg the Zeeman effect
(magnetic field) and the Ziman effect (electric field). This is similar to the
chemical shift seen in NMR absorption experiments where the local magnetic
field in combination with an applied magnetic field determines the transition
frequency.
Alternatively
ii) In a gas different atoms moving with different velocities (Maxwell-Boltzmann
distribution) will undergo a different Doppler shift
Or
iii) Excited electrons have a finite lifetime and the uncertainty principle requires a
corresponding uncertainty in the energies of the levels.
250
()
g()


If this level broadening is accounted for correctly an absorption coefficient ()d for a
specific frequency range lying between  and  + d is measured. In this circumstance the
equation for  is modified as:
()d  hB12
n
N1g  d
c
The broadening of the transition can be seen experimentally where a plot of () vs  will not
give a delta function but a broadened absorption.
g()d is known as the line shape function that gives the probability of a transition occuring
at a frequency that lies between  and  + d. This lineshape function is normalised such that

 g ( )d  1

The width of the transition is related to the lineshape function and at the central frequency of
the transition, 0 it is generally the case that
g ( 0 ) 
1

and
()  hB12
1 n
N1
 c
251
ii) Fluorescence
Fluorescence (spontaneous emission) is a mechanism whereby the atom will return to the
ground state having been previously excited by for example optical absorption.
N2
E2
h21
N1
E1
This process is spontaneous ie it occurs at random much like nuclear decay (the atom in an
excited state is like an unstable nucleus) and the rate at which it proceeds is proportional to
the number of excited atoms/molecules per unit volume with the constant of proportionality
being A21. Thus the rate of change of the excited state population is given by
dN 2
  A21N 2
dt
NB There is no photon involved in the initiation of this process and therefore the
energy density in the electromagnetic field, (), at the transition frequency (ie, the
number of photons present) is not a factor in determining the rate.
Suppose that at some instant, t = 0, we create a collection of atoms in their excited state and
then wait and see what occurs.
By re arrangement of the rate equation
dN 2
  A21dt
N2
and integrating between t =0 and t gives;
 N (t ) 
loge  2    A21t
 N 2 (0 ) 
finally
t 
N 2 (t )  N 2 (0) exp  A21t   N 2 (0) exp 
  
where  ( = A21-1) is the lifetime of the excited state against radiative recombination. It is
possible to measure  by exciting the atomic system with a pulse of absorbed light with pulse
duration TP less than  and recording the temporal decay of the fluorescence. Consider an
experiment where a laser pulse of pulsewidth, T P <<  , used to excite a molecule/atom into
252
level 2, sets up an initial state where the population is N 2 (0) . These excited electrons will
drop back to the ground state (level 1) over time and each loss of an atom from level 2 will
add a photon of h  E 2  E1 to the energy density of the radiation field,   at that
frequency. Using the rate equation for N2 ,
dN 2
  A21N 2 , the instantaneous rate of
dt
change of   is then given by;
d  t 
t 
  A21N 2 t h  A21N 2 (0) exp h
dt
  
In terms of fluorescence intensity using I    c ;
dI  t 
t 
 A21N 2 (0) exp hc
dt
  
And by integrating wrt t
t 
t 
I  t   A21N 2 (0) exp hc  N 2 (0) exp hc
  
  
By measuring the decay of the fluorescence after a fast pulse of exciting radiation it is then
1
straightforward to discover   A21
. N 2 (0) will depend on the intensity of the exciting
radiation and the quantum efficiency with which photons excite electrons to level 2 but it is not
necessary to know N 2 (0) in any event in order to discover the Einstein coefficient as the
1
exponential decay rate is sufficient to find   A21
.
The value of A21 may be found from quantum mechanical considerations to be
 16 3  3 
  2   2   2 
A21  
y
z
 3hc 3 e  x



where the component of the transition dipole moment x is given by
x 
x  
  2 x ex1 x dx
x  
 
 
where 1 and 2 are the wavefunctions of the lower and upper levels. For the integral and A21
to be non-zero the requirement is that the integrand be symmetric ie the parities of the
wavefunctions must differ, one wavefunction being even and the other odd. This requirement
is the origin of an optical selection rule.
253
iii) Stimulated Emission
The third process is a “non classical” process and is only predicted by quantum mechanics.
An atom/molecule in the excited state can be induced to make a transition to the ground
state, by interaction with a photon of the appropriate energy, causing emission of a further
photon. Furthermore the emitted photon will have the same frequency, phase, wavevector (ie
direction) and polarisation as the inducing photon. This process is the basis of light
amplification and laser action.
N2
h21
E2
h21
h21
N1
E1
The rate of loss of atoms from the excited state due to stimulated emission per unit volume is
proportional to
i) The number of occupied/excited state atoms/molecules per unit volume, N2, (ie
the number of initial states) and
ii) The energy density in the electromagnetic field,  , at the transition frequency
(ie, the number of photons present and available to take part in the process).
dN 2
 B 21N 2  
dt
Or
d 
 hB 21N 2  
dt
As for the case of absorption the relation between energy density and intensity may be used
to rewrite this equation as
dI 
n
 hB 21N 2I 
dz
c
dI 
n
 hB 21 N 2dz
I
c
NB Unlike in the case of absorption there is no minus sign on the RHS of the above
equations as the loss of an excited atom corresponds to the addition of a photon to
the electromagnetic field.
254
Summarising
1) Stimulated Emission
2)Spontaneous emission
Rate = B21N2
3)Absorption
A21N2
B12N1
The above diagram encapsulates all three processes and their rates.
NB processes 1) and 3) require the presence of a photon to proceed and this is
reflected in the presence of the energy density,  (which is related to the photon
density) , in their rates of transition.
Electromagnetic Mode Density & Blackbody Radiation
To continue further it is necessary to find relationships amongst the Einstein B and A
coefficients. This may be done by considering thermal equilibrium where the populations N1
and N2 must be independent of time. For example;
dN 2
 B12 N1   B21N 2   A21N 2  0
dt
In thermal equilibrium it is also required that the electromagnetic energy density,   , is that
of a blackbody.
Electromagnetic Mode Density.
To find the blackbody radiation distribution it is necessary to start by quantising the
electromagnetic waves, that is to say the electromagnetic waves lying in a frequency range
between  and  + d may not take on any frequency but are limited to certain allowed modes
from which any other wave may be constructed, the so called normal modes of the system. It
is possible to “count” the number of modes per unit volume in the frequency interval between
 and +d by considering the electromagnetic energy to be confined to a cube of side L
255
(which will later tend to infinity). An electromagnetic plane wave confined to this box will have
a spatial variation of the form;

A exp jk  r   A exp j k x x  k y y  k z z

Applying periodic boundary conditions to the cube the allowed wavevectors, k, can be
found from
A exp jk x 0  A  A exp jk x L   1
and similar for the other two components.




A exp jk z 0  A  A exp jk z L   1
A exp jk y 0  A  A exp jk y L  1
Therefore restrictions on kx (and ky , kz ) exist as follows;
k x L  n x 2 
kx  nx
2
L
k y L  n y 2 
k y  ny
2
L
k z L  n z 2 
k z  nz
2
L
where nx etc are integers 0, 1 2 etc and a set of three integers specify an electromagnetic
mode of the cube (NB there are two orthogonal polarisation states per set). Any particular
mode, as described by the three integers and the polarisation, is not describable in terms of
the other modes, they form an orthogonal set .
The following relationships between frequency and wavevector are used to find the mode
density:
2

 2 
k 2  k x2  k y2  k z2    n2x  n2y  n2z
 L 
2
2
2



 kc 
c 
 2          n2x  n2y  n2z
 2 
 2 
L
z

R
nz
With reference to the diagram
opposite, each mode represents
x
a point in the mesh over nx, ny and nz,
(ignoring for the moment a factor
of two for the two polarisations).
256
ny
nx
y
A sphere of radius R  n x2  n y2  n z2
will contain M  2 
4
R 3 modes
3
(with the additional factor of two to
account for the polarisations).
The number of modes, M, (the number of different ways in which an electromagnetic wave
can hold energy) can then be expressed in terms of  as
M
8  L 
 
3  c 
3
and the mode density is then
M
L3

8 
 
3 c 
3
Which is sensibly independent of the dimensions, L, of the artificial box that was used to
obtain this result.
The electromagnetic mode density per frequency interval in free space, m() can now
be found by differentiating
M
L3
wrt  ;
m  
d M
2
   8
d  L3 
c3
NB in a medium of refractive index n the mode density becomes
m   8n
3
2
c3
m d  8n 3
2
c3
d
Black Body Radiation.
The black body radiation distribution describes the unique emission spectrum of a
“perfectly emitting” body in thermal equilibrium with its environment at temperature T in terms
of the energy density in the frequency interval    + d. The early attempts to describe
theoretically the spectrum (Rayleigh-Jeans) failed badly at short wavelengths towards the UV
end of the spectrum, often referred to as the UV catostrophe. To achieve an accurate
description at all wavelengths a postulate, that radiation at any particular frequency can
only take up discrete energies given by U n    nh and cannot vary continuously in value,
257
was required. Here h is Plancks constant and n is the occupation number of the mode at
frequency  ie number of photons present in that particular quantum state recalling that
photons are spin zero particles or Bosons and any number may occupy the same quantum
state.
Statistical mechanics describes the probability of a mode having the energy Un in thermal
equilibrium and is given by;
  Un 

exp
k BT 

Pn 

  Un
 exp
n 0
 k BT
  nh 

exp
k BT 




  nh 


 exp
 n 0
 k BT 
This is a result from statistical physics but the Boltzman factor for a state with n photons is
recognizable in the numerator and the denominator is the sum of all such states called the
partition function.
The average energy of the mode of frequency  in thermal equilibrium is then given by;

  nh 

 nh  exp

k BT 
n 0

Wth   PnU n 

  nh 
n 0

 exp
n 0
 k BT 
as seen in Thermodynamics when discussing the kinetic theory and velocity/speed
distributions for example.
 h
With r  exp 
 k BT

 the denominator is an infinite geometric progression and


1
n
r 
1 r
n 0
And the numerator can be seen as;

h  nr n  hr
0
d  n
d
1  r 1  hr 2
 r  h r
dr 0
dr
1  r 
and the average energy of the mode is then
 h 

h exp 
k
T
hr
 B 
WTh   

1 r
 h 

1  exp 
k
T
 B 
 h
Multiplying top and bottom by exp
 k BT



258
WTh   
h
 h
exp
 k BT

  1

Finally it is now possible to find the blackbody energy density ,  bb d , of the radiation field in

the frequency interval  + d by multiplying by the mode density in that interval;
( )d 
h
 h
exp
 k BT

  1

8
2
c3
d
The RHS is made up of three terms and it is helpful to make these explicit in order to
understand the physics of the equation.






  2
1

 bb
d


(
h

)


 8 3 d 


h

c
 exp


 k    1

   

The first term is simply the energy per photon while the last term is the density of modes
around the frequency . The middle term is the Bose-Einstein distribution function






1
fBE h   

 exp h   1
k T  

 B  

which describes the probability that a state is occupied by a boson (particle of integer spin eg
photon with spin zero). So the energy density at frequency  is simply the product of the
energy per photon, the probability that a state at that frequency is occupied and the density of
states at that frequency.
 bb
 d  hfBE h m  d
a)
By differentiation of the blackbody distribution the maximum wavelength of emission is
discovered at a given temperature T;
MaxT = constant = 2.898x10-3mK (Wiens displacement law)
b)
By integration the total energy emitted by a body at temperature T may be found;
Utot = T4 (Stefans law where  = 5.67x10-8 W.m2.K4 )
259
At 300K (room temperature) Max = 9.6m. Detectors for night vision have to work at around
10m implying for photoconductive detectors a low energy gap semiconductor 0.12eV a
factor of ten lower than Si and only a factor five greater than kBT.
Einstein Coefficients and Thermodynamics.
Einstein noted that if thermal equilibrium was to be possible it is necessary that the three
coefficients, B12 , B21 and A21 are related. With the blackbody distribution it is now possible to
establish this relationship.
The following treatment pertains to a non-degenerate system where there is only one
quantum state per energy level. In this case the ratio of atoms in the excited state, N2 , to
those in the ground state, N1 is found in equilibrium by applying Boltzmann statistics.
 E  E2 
  h 
N2
  exp

 exp 1
N1
 k BT 
 k BT 
Or
  h 

N 2  N1 exp
 k BT 
Here Ei and Ni are the energy and population per unit volume of quantum level i.  is the
transition frequency =
E 2  E1
.
h
Degeneracy is the more usual case and what follows can easily be modified to take into
account degeneracy. If the degeneracy of the states 1 and 2 are g1 and g2 respecively
then the Boltzmann equation is straightforwardly modified to;
  h
N 2 g2

exp
N1 g1
 k BT
g
  h
N 2  2 N1 exp
g1
 k BT






The interaction of the two level system with radiation will result in transitions between the
excited and ground states. Furthermore, in thermal equilibrium the populations N1 and N2
must be independent of time. Therefore the number of “upward” transitions from ground
to excited state, (N1 N2) must equal the number of “downward” transitions from
excited to ground state, (N2 N1).
The equilibrium requirement that (N2 N1) = (N1 N2) then is expressed as;
B21 N2  A21N2  B12 N1
260
And using Boltzmann statistics to express N2 in terms of N1;
  h 
A21  B21    B12   N1
N1 exp
 k BT 

  h 
  h  
 A21  B12  B21 exp
 
exp
k
T
k
T
 B 
 B 

  h 
 and dividing by B21 ;
Multiply both sides by exp
 k BT 
 h
A21  B12

exp
B21  B21
 k BT
 
  1 
 
Finally rewriting the above equation with  as the subject and using the blackbody
distribution to describe  ;
A21
B21
 B12
 h


exp
B
 k BT
 21
 
  1

 


 8h 3 n 3

3 

c
   

 exp h   1 
k  

   

By inspection of the left and right hand sides of 8.40 and noting that the equation holds at all
frequencies it is seen that the Einstein coefficients are related as follows;
B12
1
B21
and
A21 8h 3 n 3 8h


B21
c3
3
The relation A21   1 may be used to rewrite the second of these
B21  B12 
c3
8h 3 n 3 
Or in terms of wavelength
B21  B12 
30
8hn 3 
261

3
8h
Optical Gain.
It is now possible to find the optical gain from earlier equations. To do this the net addition of
photons to the radiation field is found (stimulated emission – absorption).
So far by considering absorption and stimulated emission seperately;
dN1
dN 2
 B12 N1   
dt
dt
dN 2
 B21N 2  
dt
And including spontaneous emission;
dN 2
  A21N 2
dt
dN 2
 B21N 2   B12 N1   A21N 2
dt
When considering situations with a high photon density (eg. lasers) the third term on the RHS
will be much less than the other two terms and we therefore neglect it in what follows.
The net loss of atoms/molecules from level 2 (equal to the net addition of photons) is then;

dN 2
 B21N 2  B12 N1    B21 N 2  N1  
dt
And the net addition of energy to the radiation field is
d 
 B21 N 2  N1 h 
dt
Where B21  B12 has been used.
It has already been found that
d dI

dt dz
and
I
  
Therefore
dI 
n
 B21hI  N 2  N1 
dz
c
262
v
 I
n
c
dI 
n
 B21h N 2  N1 dz
I
c
Integrating gives
I  z  I  0expz
Where  is the optical gain coefficient and is given by;
  B21h
n
N2  N1
c
In general the transition is not monochromatic and to take into account the probablity that the
transition occurs at a particular frequency  of interest the lineshape function, g() must be
used. The equation is simply modified as follows (cf absorption);
    B21h
n
g  N 2  N1 
c
At the central frequency of the transition,  0 , the lineshape function is g  0  
1
where 

is the transition linewidth and thus;
 0   B21h
Using  0 
n 1
N2  N1 
c 
c

and
B21  B12 
30
8hn 3 

3
8h
the optical gain coefficient may also be written as;
 0  
20
8n 2 
N2  N1
Inspection of this equation shows that  is positive if N2 > N1 . This is an unusual situation and
is known as population inversion. To achieve a positive gain and therefore light
amplification, energy must be supplied to the two level system to get the atoms into the
excited state and achieve this population inversion.
263
Feedback & Oscillation Conditions.
With positive gain coefficient obtainable in the presence of a population inversion, light
amplification may occur in analagous fashion to the amplification of an electronic signal by an
amplifier.
Introduction, into a gain medium, of a light wave of the frequency at which positive gain is
available will result in the amplification of that light intensity, I(0), as it propagates through the
gain medium as described. So far the system is behaving as does any amplifier with a gain
A
I (L )
 expL 
I (0 )
What has been so far described is a light amplifer and not a laser!
Vin
Vout =AVin
A
AMPLIFIER
Vin + Vf
Vin
A
Vout = AF(Vin+Vf)
Vf = Vout

FEEDBACK NETWORK
In electronics an amplifier with gain A will simply amplify any input signal by a constant factor.
If the amplifier is provided with a mechanism to feedback a fraction of the output, , back to
the input then
Vf  Vout
Vout  AVin  Vf   AVin  Vout 
and
Vout 1 A  AVin
Vout
A

A
1  A F
Vin
264
In the case that 1 A  1 then AF  A and the feedback is negative.
If 1 A  1 then AF  A and the feedback is positive.
If  
1
the denominator on the RHS goes to zero, noise in the circuit will suffice to act as an
A
input signal and the amplifier plus feedback will break into spontaneous oscillation ie. there
will be output in the absence of any input. Remember in all of this the signals usually have an
amplitude and a phase and the phase relationship between Vf and Vin needs to be taken
into account.
As with any electronic amplifier having achieved gain it is possible to cause the optical
amplifier to break into oscilllation by providing positive feedback. With the addition of
positive feedback to the light amplifier an oscillator or laser has been created!
The most common way of implementing this feedback is to place the gain medium in an
optical cavity/resonator for example a Fabry Perot resonator as discussed previously in the
context of thin film interference.
The diagram below shows such a resonator and a consideration of this will enable the
conditions for oscillation to be discovered.
t1EI
EIexp(jkz)
t1EIexp(jk/L)
t1r2EIexp(2jk/L)
t1t2EIexp(jk/L)
t1r2EIexp(jk/L)
t1r1r2 EIexp(3jk/L)
t1r1r2 EIexp(2jk/L)
t1r1r22EIexp(4jk/L)
t1r1r22EIexp(3jk/L)
t1r12r22EIexp(4jk/L)
t1r12r22EIexp(5jk/L)


k/ = k + j(-)/2
R1, T1, r1, t1
R2, T2, r2, t2
t1t2r1r2 EIexp(3jk/L)
t1t2r12r22EIexp(5jk/L)
t1t2r13r23EIexp(7jk/L)
t1t2r14r24EIexp(9jk/L)
L
M2
MI
The quantities, t and r are the transmission and reflection coefficients for the field E or the
amplitude coefficients. T and R are the intensity transmission and reflection coefficients.
265
Once in the gain medium it is useful to describe the propagation of the light using a complex
wavevector, k   k  j

. This enables a succinct description of an oscillating wave that
2
also has an exponentially decaying character to account for losses and exponential growing
character to account for the gain. The decaying intensity is described by  where
I  I0 exp z 

  
E  E 0 exp  z 
 2 

 
E  E 0 exp z 
2 
And the growth described by
I  I0 expz
 
      
E  E 0  jk z   E 0 exp j  k  j 
 z
 2   
 
  
 
E  E 0 exp  jkz  exp  z  exp z 
 2 
2 
NB  is not the absorption coefficient found previously that is related to Beer’s law.
This absorption by the two level system has been naturally incorporated into the gain
coefficient 
    B21h
n
g  N 2  N1 
c
n
Where the term B21h g  N1 is the previous absorption coefficient.
c
In the case of most lasers the feedback is provided by mirrors to reflect some of the
electromagnetic energy back into the gain medium. The above diagram enables the
conditions for oscillation to be found. It shows an input wave with field E i exp jkz  at the LHS.
A fraction t1 of this enters the resonator through the left hand mirror at z = 0 and propagates
down the axis of the mirror pair with the complex wavevector k  . At the RHS some fraction,
t 2 , of the wave is transmitted and a fraction r2 reflects back into the gain medium to repeat
the process.
The total transmitted wave, E t
can be found by summing all the partial waves leaving the
right hand mirror (cf. thin film multiple reflections),

  r1r2 exp2 jk / Ln
/
E t  E i t1t 2 exp jk L
n 0
266

This is a geometric progression and using,

rn 
n 0
1
1 r
the ratio of the transmitted electric field to the incident electric field is given by;
 
 
Et
t1t 2 exp jk / L

E i 1  r1r2 exp 2 jk / L
This quantity goes to infinity when the denominator is zero. Physically this implies the
existence of an output, E t , in the absence of E i . in practice it means that the system will
break into spontaneous oscillation due to noise much as any electronic amplifier will
do if provided with positive feedback!
The oscillation condition is thus

1 r1r2 exp 2 jk / L

 
 
1  r1r2 exp 2 j  k  j
L   r1r2 exp2 jkL  exp   L
2  
 
This condition is in fact two seperately necessary conditions as it must be true for
both the real and imaginary parts separately :
1)
The Amplitude Condition requires the real part of the RHS to be equal to one ie
r1r2 exp   L  1
This may be rewritten, by taking logs, as
lnr1r2      L
There is thus a threshold gain, T , in order for oscillation
1
T    lnr1r2 
L
Physically this is a requirement that the round trip gain is equal to the round trip
losses.
The first term on the RHS represents parasitic losses that are to be avoided such as;
(i)
reabsorption,
(ii)
scattering,
(iii)
walkoff,
267
(iv)
absorption at the mirrors etc.
The coefficient  in this context is the distributed round trip loss coefficient. “Distributed”
because not all of the losses occur homogeneously throughout the round trip but at specific
places particularly the mirrors even though it is used as though the loss was distributed
evenly.
The second term on the RHS represents useful output which appears as a loss as far as
analysis of the resonator is concerned.
This threshold gain may be written in a number of different ways each useful in its own right.
For example the amplitude reflection coefficients r1 and r2 are not what is measured when
designing a mirror but rather the intensity reflection coeffcients, R1 and R 2
T   
1
lnR1R2 
2L
where the fact that R  r 2 has been used.
Alternatively the condition may be couched in terms of the transmission of the mirrors by
noting that the transmission of the mirror T  1  R (for a non-absorbing mirror) and using the
approximation 1 – x  -lnx for x  1 the equation may be recouched in terms of transmission
T   
1
1
ln R1  ln R2     1 (1  R1)  (1  R2 )
lnR1R2    
2L
2L
2L
T   
1
(T1  T2 )
2L
Using the previously found expression for the gain;
    B21h
n
g  N 2  N1 
c
the amplitude condition may be expressed as a condition on the population inversion with
a population inversion at threshold NT  N2  N1 T
T    B21h
n 1
1
NT   
lnR1R2 
c 
2L
and so;
NT 
1 c 
1

lnR1R2 
 
B21 nh 
2L

268
2) The Phase Condition requires that the imaginary part of the RHS of the equation
representing the oscillation condition is equal to one, ie
exp2 jkL  1
or
m  1, 2, 3.......
2kL  2m
Re-arranged,
k
2 m


L
ie only certain wavelengths will oscillate such that

2L
m
or of course
m

L
2
and

c
c
1

mm
 2nL
T
These are the axial modes of the laser resonator and physically they are such that after one
round trip the wave returns with the same phase. This means that successive waves can
constructively interfere. Another way of viewing this physically is that integer half
wavelengths fit between the resonator mirrors similarly to standing waves on a guitar
string. This is because of the boundary condition that the electric field must be zero at the
mirrors, ie a metal cannot support an electric field. The same would be true of any “perfect
reflector”. The axial modes are seperated in frequency by
 m 1   m   
c
 T 1
2nL
where T is the round trip time in the resonator.
CF. the Free Spectral Range for a Fabry Perot etalon that was found previously.
269
g

T = -(1/2L)lnR1R2

res

o
axial
The HeNe Laser an example.
The central frequency of the Helium Neon red transition is o = 4.741014 Hz. The line width,
, is 1.2 GHz and is a property of the Doppler broadening of the atomic transition
independent of any resonator. The separation of axial modes depends on the resonator
length and is 1/T where T is the “round trip time”. For a cavity of length 50cm axial = 0.30
GHz. The width of an axial mode, res is determined by the reflection coefficients (finesse) of
the resonator (see notes on thin film interference)
FWHM   res   m
2 1
c 2 1

m F 2L  F
F
4R
1  R 2
and for R=0.95% and for a cavity of length 20 cm res = 4.9 MHz.
This resonator will have 1.2/0.3 = 4 axial modes and these modes will be well defined as
axial/res = 60.
In an actual laser not all frequencies under the transition linewidth will satisfy the amplitude
oscillation condition;
T 
20 g  
2
8n 
N2  N1 T

and only those that do will have axial modes oscillating.
270
1
lnR1R2 
2L
Resonator Decay Time
I
R1
R2

L
An important consideration in the analysis of laser resonators are the losses and one of many
useful ways of describing these losses is through the resonator decay time or photon lifetime
of the cavity.
Consider a wave of intensity I undergoing one round trip in the resonator with the gain
abruptly "turned off" at time t = 0. It will then have a new intensity after the round trip
I   IR1R2 exp 2L 
and the loss in intensity is
I  I   I   I 1  R1R2 exp 2L 
and the fractional intensity loss is then
I
 1  R1R2 exp( 2L)
I
and this loss occurs in the round trip time
dt 
2L 2L

c
v
n
The resonator decay time, C , may be defined through the equation
dI
I

dt
C



dI dt

I
C
1 dI
1

dt I
C
Using the equations obtained earlier the resonator decay time is then found to be;
271
1
C
 1  R1R2 exp 2L 
c
2nL
For R1R 2 exp 2L   1 and using 1  x   ln x for x  1
1
c
1 
C
  
lnR1R 2 
2L

n
Comparing this with an earlier equation for threshold gain coefficient, found in the real part of
the amplitude condition, it is possible to express T in terms of C ;
T   
1
n
n 1
lnR1R2  
 B21h
NT
2L
cC
c 
The threshold population inversion may be written as;
NT 
Thus using B21  B12 
30
8hn 3 
NT 
1 
B21 hC
;
8hn 3  
n3
1 
 8

c
30 h C
20  C
Note that the threshold population inversion increases with decreasing cavity decay time and
with increasing transition linewidth, .
Note also that it is inversely proportional to the square of the wavelength hence it is far easier
to get lasers to work at red wavelengths. This explains why all the early examples of devices
based on optical gain work at longer wavelengths
cf Ruby Laser, HeNe, GaAs
semiconductor pn junction and of course the ammonia maser!
272
A further and frequently used way to define the losses of a resonant system is through the
quality factor, Q. The quality factor of the resonator is a measure of the sharpness/resolution
of a resonant cavity and is usually defined as
Q

 res
Where  is the resonant frequency and  res the width of the resonance. In the case of the
laser resonator under discussion this is a Fabry Perot resonator and the width of the
resonance has been discussed previously.
Another way of defining the quality factor more directly related to the losses is;
Q
S
 d
 2
dt
S
 d
 2
dt
I
dI

dt
Where εS is the energy stored in electromagnetic field of the resonator and  d
dt
is the
power dissipated.  is the resonant frequency of the cavity and I is the intensity proportional
to energy through the velocity of propagation of the electromagnetic energy. From the
definition of the cavity decay time C 
I
 dI
it is seen that Q can be alternatively
dt
expressed in terms of cavity lifetime
Q  C  2C
and so in terms of Q the threshold inversion may be written as
NT 
2
B21hQ
This equation shows that as Q is reduced NT ,and consequently N, may be driven to much
higher values thus allowing the temporary existence of a much higher gain should the losses
be subsequently and rapidly reduced. This is the basis of Q-Switching.
Q-switching is a technique for obtaining laser pulses of high intensity, brief duration (of the
order tC , the cavity decay time) and high energy. To achieve this the population inversion is
held unusually high through artificially high cavity losses thus inhibiting stimulated
emission.When those losses are suddenly reduced to their normal low value a "giant pulse"
results.
273
Shown above in figures a) to d) are the temporal variation of four parameters of a laser. It is a
flashlamp pumped laser and
a) shows the pump rate R (output of the pumping lamp).
b) shows the Q factor of the resonator (inversely proportional to the losses,
c) the resulting total population inversion and
d) the consequent laser output which after a rapid initial build up decays with the cavity
decay time C .
274
Four Level Lasers.
To arrive at an expression for and understanding of gain in an atomic/molecular system, a
two level system has been so far considered. While this is good enough to derive the gain of
a laser, in actual fact a two level system could not operate as a laser. This is because of the
prohibitive difficulty in attaining a population inversion and the impossibility of maintaining it
when it is recalled that a lasing transition takes the atom to the lower level reducing the
inversion by two. To avoid this 3 or 4 level systems are used.
Level 3
Pump Band
R2
Level 2
Laser transition upper level
h
W21 =B21N2
h
R1
h
Level 1
Laser transition lower level
Pump energy
W10 =A10N1
Level 0
Ground State
A three level laser would have the energy seperation of the lower lasing level and ground
state kBT or less and thus the lower lasing level would have a significant population at room
temperature thus making a population inversion less easy to achieve than for a four level
laser where the lower lasing level is much greater than k BT above the ground state and is
barely occupied at room temperature in thermal equilibrium. In the above diagram the energy
levels and transitions of a four level laser are shown schematically. The requirements are as
follows:
i)
The input “pump” energy is likely to be over a broad spectrum of energies and
therefore an upper pump level represented by a broad and closely spaced set
275
of levels is used to initially absorb the broadband pump energy in an efficient
manner and to take up the excited electrons.
ii)
Relaxation from the pump level, 3, to the upper lasing level, 2, should be rapid
and efficient while relaxation from 3 to 1 should be non existent (or at least
negligible). This ensures that the pump energy goes into creating a population
inversion, N2 > N1 .
iii)
The upper lasing level 2 must be connected to the lower lasing level 1 by a
strongly allowed radiative transition. ie. B21 (or equivalently A21) must be large.
iv)
The lower lasing level 1 must be
rapidly de-populated to the ground state
(something that is not available to a simple two level system) ie. A10 needs to be
large or another means of return to the ground state available.
The kinetics of the four level system shown schematically above and described in general
terms may be considered via the rate equations with R2 and R1 the pump rates into levels 2
and 1 respectively. The latter is undesirable and it needs to be reduced. The rate equations
give;
dN 2
 N 2 A21  W21 N 2  N1   R 2
dt
and
dN1
 N1A10  N 2 A21  W 21 N 2  N1   R1
dt
Note. For simplicity the spontaneous emission from level 2 to the ground state, A20 , is
subsumed in the R2 term and; the rate A10 here refers to the total rate of transition from 1 to 0
including radiative and non-radiative transitions and so is not an Einstein coefficient. Also
W21 (= B21) has been used in describing the stimulated emission rate.
In the steady state d/dt = 0
The two equations may then be manipulated to give an expression for the population
inversion
N2  N1 
R 2 1  ( A21 / A10 )(1  R1 / R 2 )
W21  A21
Examination of this reveals that for inversion to be at all possible it is necessary that A10 > A21
. ie the lifetime of the upper level (A21-1) is to be greater than that of the lower level (a more
stringent condition than iv described above). The rate of stimulated transition is inversely
276
proportional to the upper state lifetime (recall B21  A21 = 21-1 ) so A21 cannot be too small, ie
it is the lower level which should have a very short lifetime (A10 to be large)!
Physically this requirement is that the lower level is emptied at a rate faster than it fills
from the upper level in order to maintain the inversion, ie 10 <<21.
Threshold Pump Rate.
An effective pump rate may be defined for the purpose of analysis;
Reff  R2 1 ( A21 / A10 )(1 R1 / R2 )
Reff is increased by decreasing R1 (an unwanted side effect of the pump energy)
and by increasing A10 (the rate at which the lower level empties). The population inversion
can be rewritten using Reff as
N 2  N1 
R eff
W21  A21
above threshold
Below threshold there is effectively no stimulated emission and W21 = 0, then;
N 2  N1 
R eff
A21
below threshold
until the pump rate has increased to a value such that N 2  N1  N 2  N1 T and;
Reff  RT  N2  N1 T A21
at threshold
Gain Saturation.
Once Reff > RT, (N2 – N1) will not increase further but will remain clamped at a threshold
value, NT , and the gain will attain a value;
T ()  NT B21h
n 1
1

ln R1R2
c 
2L
The best way of viewing this is that the gain, (), which is proportional to (N2 – N1), becomes
clamped at some threshold value given by the round trip losses (as seen in the examination
of oscillation conditions). It cannot rise beyond the round trip losses as this would result in
runaway output. Rather, there will now be stimulated emission and with
Reff  NT W21  A21 
277
Reff  RT
it can be seen that any further increase in Reff above RT (by increasing the pumping power)
results not in an increase in (N2 – N1) but in an increase in W21 (= B21 ) , ie stimulated
emission.
This clamping phenomenom is known as gain saturation.
Power Generated in a Laser.
The power generated in the cavity is given straightforwardly by
PG  NT W21hV
where V is the volume occupied by the oscillating mode (ie the volume of atoms or molecules
with which the field interacts).
From the earlier equation
W21 
R eff
 A21
NT
then PG is
R

PG  NT h  eff  A21 V
 NT

Noting that the power generated due to fluorescence alone (ie spontaneous emission) is
given by
PS  N2 A21hV  NT A21hV
And using Reff  NT A21
R

PG  PS  eff  1
 RT

278
PS
PG
R
Power output from laser.
2RT
RT
Having arrived at an expression for the power generated within the laser medium it is
important to know how much of this appears as output. To do this consider the round trip
losses. Previously the losses were given by the right hand side of the amplitude condition (for
oscillation);
T ()  NT B21h
n 1
1

ln R1R2
c 
2L
As previously the losses may be divided up into parasitic losses and “useful output”,  and

1
ln R1R2 respectively.
2L
R1 and R2 will be close to 1 and using 1  x   ln x ;

1
1
1
1 R1  1 R2 
ln R1R2   ln R1  ln R2  
2L
2L
2L
The latter “useful losses” may then be written as
1
(T1  T2 ) and in most laser oscillators, to
2L
minimise the losses, one of the mirrors is 100% reflecting with zero transmission and the
output is from the second partially reflecting mirror, the output coupler, with transmission T.
Thus the useful losses may be written as
T
. Of the power generated (which will be
2L
T
proportional to the losses) only the fraction
2L
appears as emission and the power
T

2L
output is;
POut  PG
T
T
2L  P  R  1
S

 T
RT
 2L  T
2L
279
Optimum Output Coupling.
There is an optimum value for the transmission T of the output coupler. This is easily seen; if
T is 0 then there will be no output, all the radiation is confined to the resonator. If T becomes
too large then the losses are greater than the round trip gain attainable with the pumping
conditions prevailing and again there is no output.The actual power out of the laser is;
R

T
POut  PS 
 1
RT
 2L  T
The graph below shows the energy stored in the radiation field as T is increased,  and the
power out, POut . There is clearly some optimum value for the transmission.


POut
POut
TOPT
T
TOpt may be found by differentiating the power out with respect to T and equating this to
zero.

dPOut
d R
T
 PS
 1

dT
dT RT
 2L  T
To do this differentiation it is necessary to recognise that RT depends on T through the
amplitude condition as seen previously;
280
RT  N 2  N1 T A21
RT 
N2  N1 T
8 2 n 2
c2

1 c
T 

  

B21 nh 
2L 
T 

  

2L 

There is a more simple way of proceeding where it is noted that the gain constant,  , is
proportional to (N2 – N1) and also that below threshold with W21  0
N2  N1 
Reff
A21
This allows the gain to be written as

0
1  W 21
A21
where 0 is the unsaturated gain, (below threshold 0 and (N2 – N1) vary with Reff ).
With
PG hN 2  N1 W21V W21


PS
hN 2  N1 A21V
A21
It is possible to write

1
0
PG
PS
Once threshold is reached the gain is clamped (saturated) at T and this value may be used
for  and PG found


PG  PS  0  1
 T

The oscillation condition is T   
1
lnr1r2  with the round trip gain equal to the losses.
L
This may be written
expT L (1  fL )  1
where fL is the fraction of energy lost per pass. To see this consider a wave of unit intensity.
After one pass it will have intensity 1  f L  in the absence of gain. With an exponential gain
coefficient  the intensity will be exp T L 1  f L  and for oscillations to be sustained this must
be unchanged and equal to 1.
For fL <<1 (low losses)
281
 T L  fL
or
f
T  L
L
Using this and fL 
LI
2L
where LI  2L are the intrinsic losses to get

2L  T LI  T
  L

PG  PS  0  1
 LI  T

The fraction of the power coupled out is
T
and so the power out is
LI  T
  L
 T
PO  PS  0  1
 LI  T
 LI  T
Using PS  NT A21hV at threshold and N T 
L T
 T c
with T  I
the power out may
L
B21 nh
be written
PO 

 T
T c
A21hV  0  1
B21nh
 T
 LI  T
 c  LI  T
PO  
V 
 B21n  L
 T
  0 L
 1

 Li  T
 Li  T
  c    0 L
 T 
  
V 
 1 
  B21n   Li  T
 L 
The identities g(0) = -1 and A21 = --1 have been used
To find TOpt
dPO/dT is set equal to zero
TOpt   LI   0 LLI
and POpt may be found by using this value of TOpt in the equation for PO .
282
Real Lasers.
Having established how a laser can be made to work in practice in particular the
requirements for a four level laser;
i)
The input “upper pump level needs to be a broad and closely spaced set of
levels is used to initially take up the excited electrons.
ii)
Relaxation from the pump levels to the upper lasing level must be rapid while
relaxation from the pump level to the lower level should be non existent (or at
least negligible).
iii)
The upper lasing level 2 must be connected to the lower lasing level 1 by a
strongly allowed radiative transition.
iv)
The lower lasing level 1 must be
rapidly de-populated to the ground state
(something that is not available to a simple two level system).
it is now possible to look at a variety of actual lasers and see how these requirements are met
by the particular lasing medium involved.
i)
Dye Laser.
Dyes are complex and highly coloured organic molecules with tetravalent (four valence
bonds) carbon atoms covalently bonded to other carbon atoms, hydrogen, oxygen or nitrogen
being the other most common constituents. If a particular carbon atom is bonded to less than
four neighbours in the molecule there are spare valence electrons that are not strongly bound
in the molecular skeleton and are consequently delocalised over the molecule. This relative
uncertainty in the position of the electron as a consequence leaves the electron with low lying
(energetically) excitations excitable with near IR-visible-near UV light l(~ 300nm - 1000nm).
The strong colour is an indication that x is large, the same considerations applying to
absorption of light as for emission. Condition b) is then satisfied by the very definition as to
what constitutes a dye molecule. One of the most commonly used dye molecules,
Rhodamine 6G, and its absorption/fluorescence emission spectrum is shown. There are
carbon atoms at each vertex unless indicated otherwise and an hydrogen atom attached to
these vertices unless indicated otherwise . The double bonds (indicated by double lines)
occur where the carbon has only three neighbouring atoms and there is a "spare" non
bonding electron. In this example there are nine such electrons associated with the carbon
atoms.
283
Rhodamine 6G
Emission and absorption
spectra of Rhodamine 6G.
The emission spectrum is red shifted compared to the absorption spectrum for reasons that
will become clear. The two spectra are also approximate mirror images of one another.
The absorption is due to light exciting one of the pair of electrons in the ground, S0 ,
state of the molecule into a singlet excited state, S 1 . In the ground state the two electrons
have their spins antiparallel (Pauli exclusion principle). The total spin of the system, S, is zero
as the electrons contribute spin of +1/2 and -1/2. The total spin of the system, S, cannot
change during photon absorption and S = 0 after absorption with an excited electron of spin
+1/2 (-1/2) and ground state electron of spin -1/2 (+1/2). The degeneracy of the excited state
is given as (2S + 1) from quantum mechanics and is thus 1, hence the description, singlet
state. The ground and excited singlet states, S0 and S1 are more precisely represented as
an electronic energy level associated with the electron distribution over the molecule. There is
a manifold of closely spaced vibrational levels coupled to each electronic energy level
284
associated with the vibrational motions of the molecular skeleton and the energy level
structure is even more finely structured due to very closely spaced rotational energy levels.
Because this skeleton is complex the normal modes of vibration are many and closely spaced
as are the rotational levels. The electron may return to the ground, S 0 , state from S1 with
emission of a photon of lower energy than that which originally excited it in the process
known as fluorescence.
There is also a triplet state, T1 at energy less than S1 . This is inaccessable by photon
absorption as the spin of the electron in T 1 and the ground state, S0 , are parallel adding to 1.
The degeneracy, 2S + 1 = 3 and hence triplet. To get to the triplet state from S 1 requires the
spin of the electron to be reversed, a process possible only by intervention of some other
mechanism/collision in order to conserve total spin.
Such S1
 T1
transitions can occur but weakly and the event is called "intersystem
crossing". Once in the triplet state the excitation is long lived and the slow decay to S 0 is the
process known as phosphorescence when accompanied by emission of a photon.
Intersystem crossing is an undesirable effect as once the molecule is in the T 1 state, photon
absorption to the T2 state can occur and the pump energy may be effectively taken up in
these unwanted transitions. Absorption of a photon with transition from S0
to S1
as
represented in the above diagram tells only a part of the story. In fact absorption and a
change of electronic state changes the distribution, (r)2, of electronic charge over the
molecule. This affects the shape of the molecular skeleton and its vibrational modes. The
situation is more properly represented by a "configuration co-ordinate" diagram for the
molecule in its various electronic states.
S1
Relaxation
h
E
S0
Emission
Absorption
h
Relaxation
Q0
Q1
285
Q
In general Q represents the configuration (shape) of the molecule and given the complexity of
a dye molecule is a multi dimensional parameter. A simpler diatomic molecule could be
described by a configuration co-ordinate such that Q represents the internuclear seperation.
Considered in this light none of the physics of the more general case is lost and continuing in
this spirit.
In the diagram are shown variation of molecular energy, (elastic energy due to the
molecular shape plus electronic energy), for the ground and first singlet excited states. There
is an energy minimum or prefered configuration for the ground or excited state, Q 0 and Q1 ,
and these are in general different. Any small excursion, Q, from say Q0 when in the ground
state results in an increase of the elastic energy of the skeleton which is approximately
parabolic in Q (cf Hookes law). This leads to harmonic oscillator type behaviour and a set of
oscillator energy levels as shown. The lower vibrational levels will be occupied and the
electron wavefunction is localised around Q0. Absorption of a photon is an essentially
instantaneous process, the nuclei are 103 times heavier than the electron, and Q cannot
change instantaneously. On the diagram the transition is then to be represented by a vertical
line. The molecule will therefore finish up in one of its higher vibrational levels in S1 . Because
there is a vibronic manifold the pump energy h may be a spread in energy eg a flashlamp, c)
is thus satisfied. The molecule is now in a stressed configuration and will rapidly relax to its
equilibrium configuration, Q1 shedding energy as phonons (heat). This occurs in times far too
rapid than to allow immediate relaxation back to the S 0 level thus satisfying requirement d).
The molecule will then return to the ground state as before by a vertical transition ending up
in one of the higher vibrational levels of S0 and emitting a photon of lower energy than that
absorbed, (the redshift or Stokes shift described earlier). This higher level will be more than
kBT above the lowest vibrational level satisfying requirement a) and the molecule will again be
far from its equilibrium configuration. It will rapidly relax back to equilibrium in accordance
with requirement e)
All conditions are then satisfied and dyes dissolved in water or alcohol do indeed make
excellent lasers. A typical flashlamp pumped dye laser is shown below.
One reason why dye lasers are particularly useful is related to the broad emission spectrum.
This means that the actual laser wavelength of any particular dye may be tuned. The
resonator is made up of a partially reflecting mirror, from which is emitted the output, and a
diffraction grating with lines of spacing d etched into a reflecting surface. This has the
property that light incident at an angle, , will be reflected back upon itself when 2dcos =
m, m = 1.2,3,.......etc. Thus by selecting the angle , a wavelength that will be retroreflected
has been chosen and will have low losses while all other wavelengths will be reflected out of
286
the cavity. The laser mode will thus build up at this tuned wavelength. Furthermore, by the
simple expedient of changing dyes, further wavelengths may be accessed by the same basic
laser. The dye laser is thus highly tunable with a single dye and more so when a selection are
used. It is for this reason an extremely useful research tool where spectroscopy is to be
carried out.
Also to be noted in the design is that the dye is contained in a long narrow
transparent tube through which the dye is flowed. This is because frequently complex organic
molecules can undergo photochemistry and degrade. Either end of the tube is sealed by a
transparent window set at an angle known as the Brewster angle. This has the property that
its reflectivity for light polarised parallel to the plane of incidence (the plane of the paper in
this diagram) is zero while for the perpendicular polarisation it is finite. This has two effects.
Firstly, the perpendicular light is partially reflected out of the resonator and it thus "sees"
higher losses than the parallel polarisation. This has the effect of polarising the laser output in
a "survival of the fittest" mechanism where the parallel polarisation gets to use more of the
inversion reproducing photons with the same polarisation as itself.
Xenon
Flashlamp
Brewster Windows
Dye Cuvette
Diffraction
Grating
Dielectric Mirror
Dye out
Dye in
NB Lasers in general are not inherently polarised because of the effect of stimulated
emission but if they are it is because something in the design of the laser leads to a
higher loss for one polarisation than for another!
287
The second and designed effect of the Brewster windows and the reason they are used is to
cancel out the secondary resonators that would otherwise form between the mirror or grating
and the windows as the windows now have zero reflectivity for the parallel polarisation..
One last design feature for this particular laser is that the flashlamp and dye tube would be
held along the axis of an ellipsoidal and highly reflecting cavity with one at either focus of the
ellipse. In this way the flashlamp energy is efficiently coupled into the dye.
It is to be noted that there are many designs of dye laser and the above is one example. Dye
lasers may also be run continuous wave and pumped by other lasers.
ii)
Excimer Laser
Excimer lasers are large volume gas lasers utilising a mixture of a halogen and a noble gas.
They are used uniquely as high powered pulsed lasers with wavelengths ranging throughout
the UV from 190nm to 351nm the actual wavelength depending on the particular
Halogen/Noble gas mix.
An excimer is a molecular species which is unstable in the ground state. As an example
consider using the Noble gas/Halogen mixture Krypton/Chloride, Kr/Cl. If the molecule KrCl
were to form it should be ionically bonded (cf Na +Cl- ) as Kr+ Cl- where the Chlorine has
abstracted an electron from the Krypton and coulomb attraction binds the two atoms. In fact,
that this cannot happen is clear from energetic considerations:
i)
To ionise a Kr atom takes an energy input, I = +13.93eV, its ionisation energy.
ii) Adding an electron from the vacuum to Cl releases an energy, A = -3.75eV, its
electron affinity.
iii) The coulomb attraction between Kr+ and Cl- releases -8.0eV.
iv) The nuclear repulsive energy between Kr and Cl is +1.0eV.
Thus by adding these four energies it is found that to make Kr+Cl- from Kr and Cl requires a
net input of energy of +2.18eV per molecule. It is clearly unfavoured as an option for the two
atoms.
If however a Krypton atom in an excited state is used to start the process of excimer
formation the energy to ionise it will drop by an amount equivalent to the energy of the excited
state. The ionisation energy of Kr* = +5.0eV, a reduction of 8.93eV. All the other energies are
the same except the coulomb attraction which is now -7.0eV. The excited state molecule or
excimer is thus formed with a net release of 5.75eV.
288
Kr*
Kr*+Cl-
E
Cl
Relaxation
Excitation by
electric discharge
Fluorescence
Relaxation
Kr
+
Cl
Q
Again the energetic levels involved may be described using a configuration co-ordinate
diagram and now Q is simply the internuclear seperation. The diagram shows the molecular
energy as a function of Q for the excited and ground states. With Kr and Cl alone the curve
falls away as Q increases with no potential minimum indicating that the two atoms will tend to
stay apart in order to reduce their overall energy. If now Kr is excited to Kr * there is a potential
minimum at some value of seperation, Q . The initially seperated atoms will attract one
another and fall into the potential well as the excimer is formed. There will be vibrational
levels associated with the potential well as excursions Q either side of Q will result in a
restoring force, -dE/dQ, acting on the excimer thus causing oscillation.
The lifetime of the upper bound state is about 6ns before emission of light and a return to the
lower curve via a vertical transition. The two atoms will now rapidly dissociate down the curve
in about 100fs (1 femtosecond = 10-12 s). This rapid dissociation is the equivalent of condition
e) ie 6ns >> 100fs. The formation of the bound state is also rapid (condition iv)) once the Kr is
excited. The Kr is easily excited by a high voltage discharge in the gas mixture and
constitutes a broad range of energies for condition i) to be satisfied. Because of the short
excited state lifetime the excimer is used as a pulsed laser. Because of the high lasing
volumes achievable with a gas it is possible to attain high energy pulses. The table below
shows the halogen/noble gas combinations and their wavelengths. Not all combinations give
an excimer.
289
Xenon
Krypton
Argon
Fluorine
315nm
248nm
193nm
Chlorine
308nm
220nm
XXXXX
Bromine
XXXXX
XXXXX
XXXXX
The wavelengths covered are of great use as few lasers attain such short wavelengths. In
particular ArF with a wavelength of 193nm represents a photon energy of 6.42eV. This
combined with a high overall pulse energy gives rise to the process of ablation when a
focused ArF beam is incident on certain surfaces. The photon energy is high enough to
disrupt molecular bonds and the excess energy is transferred into kinetic energy of the
resulting fragments of macromolecules as they are ejected from the surface. This is a non
heat evolving process as the excess energy is carried away in these fragments. As such this
process known as ablation finds application in diverse processes such as dry lithography
where patterns may be etched directly into a photoresist over silicon without the reqirement of
intervening wet chemistry. Further more it may be used in surgical applications where
material is to be removed without damaging underlying tissue.
iii)
Carbon Dioxide laser.
The CO2 laser which is fundamentally different from the four lasers so far discussed. In the
previous four lasers the lasing occured between different electronic levels. In the CO 2 laser
the lasing is between vibrational levels and occurs in the far infrared.
The CO2 molecule is a linear molecule and it possesses three normal modes of vibration,
each with its own natural frequency. In decreasing frequency these modes are:
i)
The assymetric stretch mode (0,0,1),
ii)
The symmetric stretch mode (1,0,0)
iii)
The bending mode (0,1,0).
1

The energy of an oscillator of frequency, , takes up discrete values given by E   m  h ,
2

where m is an integer representing the degree of occupation of the mode. Thus an arbitrary
vibration of the CO2 molecule may be denoted (l,m,n) with l,m and n integers describing the
degree of excitation of each of the modes the ground state being (0,0,0). These modes are
illustrated below.
290
Oxygen
Carbon
Oxygen
Symmetric Mode
(0,0)
Bending Mode
(0,0)
Asymmetric Mode
(0.0,)
Vibrational modes of CO2
The upper lasing level is the (0,0,1) or assymetric stretch mode and the lower lasing
level is the (1,0,0) symmetric stretch mode giving 10.6m or the (0,2,0) mode at 9.6m. To
excite the (0,0,1) mode a CO2/N2 gas mixture is used and resonant transfer of energy from
nitrogen molecules to the CO2 is used. The N2 molecule has a vibrational energy close to
that of the (0,0,1) mode of CO2 and thus effectively passes on any vibrational energy that it
picks up after collisions with accelerated electrons in a high voltage discharge. The N 2 is
highly excited after collision but rapidly cascades down to collect in the lowest energy
vibrational state which is long lived and it is this state that resonantly transfers energy to the
CO2. There is thus a band of pump energy levels.
Helium is included in the gas mixture for two reasons. Firstly, collisions of CO 2 in the
lower lasing level with He will deactivate these molecules thus emptying that level as
required. Secondly, the He acts to cool the gas mixture by transfering energy in collisions with
the walls of the gas reservoir. This keeps the Doppler broadened linewidth, , low and the
gain which is proportional to --1, high.
291
The energy level scheme of a CO laser.
The CO2 laser is highly efficient reaching up to 30% efficiency. Further, large volumes of gas
may be used and the high mode volume leads to high output power. In order to maintain an
effective high voltage discharge when making the dimensions of the laser large it is
necessary to resort to the so called transverse excitation atmospheric or TEA laser design
where the discharge is perpendicular to the axis of the laser resonator unlike in the HeNe
where the discharge is along the resonator axis. Such a design is used in other large volume
gas lasers such as excimers.
Schematic diagram of the TEA CO laser design.
Because the laser works in the far infra red the materials for mirrors and windows must be
carefully selected. The mirrors would be of gold while an ideal window material is Zinc
292
Selenide upon which a semi transparent layer of gold would be evaporated to fabricate the
output coupler.
The CO laser is widely used in industry for its high output and its use in cutting and welding
metals.
iv)
Helium Neon laser
The HeNe laser is most commonly seen as a continuous wave laser operating at 633nm in
the red although other lines are available. The active participant in lasing is the Neon atom. It
is different from the other lasers so far reviewed in that the pumping band (level 3) is a level
of the Helium atom. Also the lower lasing level of the Neon atom has more levels below it
other than the ground state. The laser action is best described with reference to the Helium
and Neon energy level schemes shown below.
Electronic energy levels of Helium and Neon.
A relatively low voltage electrical discharge is maintained in a tube containing He and Ne at
low pressure which is closed off with Brewster windows. The electrons accelerating toward
the anode will collide with and excite the Helium atoms. The two excited Helium levels,
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1S
0
and
3S
0
, are relatively long lived. They are also very nearly isoenergetic with two sets of
levels of the Neon atom and are able to efficiently transfer their excitation energy to the upper
lasing levels of the Neon atoms (cf CO2 laser). It is this resonant transfer that effectively
satisfies d). The diagram shows a number of transitions to a closely grouped set of lower
lasing levels plus a low energy (long wavelength) transition to a lone level at 3.39m. The
lower lasing levels are depopulated by transitions to the 2p 53s levels and not the ground
state. There is then the potential for these "used" electrons to absorb some of the pumping
energy and return to the lower lasing level thereby increasing the threshold pump rate. To
avoid this the return of the Neon to the ground state is by collision of the Neon with the walls
of the gas tube. This is more effective the narrower the tube and consequently the gain is
larger in narrower tubes. The condition, e) is in this way partially satisfied.
The upper lasing, 2p55s, levels have a lifetime of = 10-7s while that of the lower, 2p53p level is
= 10-8s and condition e) is thus satisfied. Recalling that the gain of a laser, other things being
equal, is proportional to 2 it is to be expected that the long wavelength, 3.39m, transition
will dominate. This would indeed be the case except that in the design given here the
Brewster windows are made of glass which absorbs 3.39m while transmitting 0.6328m
radiation. To produce the 3.39m or 0.6328m lines instead of the others requires a 5:1
mixture He:Ne and at a pressure determined by the product of the pressure and the tube
diameter = 3 Torr.mm. The 6 transitions are very sharply defined and broadening of the
transition is through Doppler broadening of the high temperature gases.
He Ne laser with external mirrors and Brewster windows
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