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Lasers

1

Introduction
• A laser is a light source that produces a beam of highly coherent and nearly monochromatic
light as a result of cooperative emission from many atoms
...

• There are two essential components of a laser:
1
...

2
...

– Allows the propagation of photons in a restricted narrow frequency range and spatial
direction to build up
...

• Build up of laser radiation is achieved initially via spontaneous emission, and then, once the
photon population density in the cavity modes is sustained, via stimulated emission from the
gain medium in which population inversion has been attained
...


1

2

Radiation Process
• Consider a gas in a transparent container
...

• Atoms in the ground level can absorb photons and atoms in the exited level can emit photons
...


2
...


• In spontaneous emission, the direction and phase of the emitted photons are random
...
1)

spontaneous

where A21 is called the rate of spontaneous emission or Einstein A coefficient, with units of
s−1
...
2

Absorption

• The atoms can absorb photons of energy hν = Eb − Ea and move into the exited state
...
2)

• ρ(υ) is the energy density of photons with frequency ν such that hν = E2 − E1
...

• The rate equation can be written in the alternative form
dN2
= σ12 F (ν)N1
dt

(2
...

– F (ν) is the photon flux
...
3

Stimulated Emission

• Two photons result from the interation of a single incoming photon
...

• The rate of decay of the upper state population due to stimulated emission is given by
dN2
dt

= −B21 ρ(ν)N2

(2
...

• This can be re-written as

dN2
dt

= −σ21 F (ν)N2

(2
...


3

Relation Between the Einstein Coefficients
• Spontaneous emission, absorption, and stimulated emission are related radiation processes
...

• In thermal equilibrium the number of atoms in any quantum state is given by the Boltzmann
distribution
E1
N1 ∝ exp −
(3
...
2)

(3
...
4)

• In thermal equilibrium, the radiation power flux (energy per unit area per unit time) for a
black body is given by the Planck formula
B(ν) =

4πν 2 hν
hν
c2 e kT − 1

(3
...
6)

• The black body photon flux is given by
F (ν) =

• In thermal equilibrium, transitions from 1 to 2 must balance transitions from 2 to 1
...
7)

A21 N2
σ12 N1 − σ21 N2
A21
− σ21

σ12 N1
N2

A21

=
σ12 e

hν
kT

(3
...
6) and 3
...
9)

4πν 2
A21
= 2
σ12
c

(3
...


4

4

Spectral Broadening
• Broadening in laser physics is a physical phenomena that affects the lineshape of the laser
emission profile
...
It will be
broadened with respect to the ideal perfectly monochromatic emission
...

1
...

Examples:
• Natural lifetime broadening
– The uncertainty principle relates the lifetime of an exited state with the uncertainty
in its energy
...

• Pressure Broadening
– The collision of other particles with the emitting particle interrupts the emission
process, and by shortening the characteristic lifetime, the uncertainty in the energy
increases
...
1)

∆ν
2

g is the linespace function
g dν is the probability that a transition occurs between frequency ν and ν + dν
∆ν is the full width half maximum in frequency of the spectral line profile
...

The profile is normalised so that
∞

gL dν = 1

(4
...


5

2
...

• The addition of each line profile from all the atoms gives the total spectral line profile
...

– Each photon emitted will be red or blur shifted by the doppler effect, depending on
the relative velocities of the atom and the observer
...


5

The Laser Idea
• In stimulated emission, consider a thin element of media with flux F of photons impinging
...

• The flux coming out of the media element is
F + dF

(5
...

dF = g[σ21 F N2 − σ12 F N1 ]dz
= αF dz

(5
...
3)

α = g[σ21 N2 − σ12 N1 ] = σ(N2 − N1 )g

(5
...
5)

• α is the gain coefficient
...
6)

0

(5
...
8)

∴ Fout = F0 eαL

(5
...

• Loss occurs if α < 0
...


6

6

A Simple Laser
• A medium with α > 0 is placed between two mirrors, one of which is partially transparent
with a reflectivity R < 1
...

• The losses of amplified light are mainly due to leakage out of the partially transparent mirror
as desired, which forms the laser output
...
The threshold for lasing
is thus
Fout
= R1 R2 e2αth L = 1
(6
...

2αth L = ln(R1 R2 )

(6
...
3)

∴ αth = −
• This is the threshold gain coefficient
...

• In thermal equilibrium
hν
N2
= e− kT =⇒ N2 < N1
N1

(7
...

• The condition that N2 > N1 is called population inversion or the condition of negative
temperature (as kT < 0 to get N2 > N1 )
...

• It is impossible to use light to pump directly from 1-2 as the stimulated emission rate could
approach the pumping rate where N2 ≈ N1 and the medium would be transparent to the
pumping radiation
...

• Pumping using electrons is also much easier if intermediate stages are employed
...
1

3 Level

1
...

2
...

3
...

The decay from 3-2 usually occurs via radiationless transitions and the energy difference is
given to the gain medium
...
The lasing level decays to the ground level via radiative decay
...
Stimulated emission takes place when the population of the lasing level is larger than the
population of the terminal level, otherwise absorption exceeds stimulated emission and there
is a net loss of photons
...
2

4 Level

1
...
This is a four level
system
...
Exitation and lasing proceed as in the case of the 3 level system but the terminal level of the
lasing transitions is energetically above the ground level
3
...

4
...


8

8
...
1)
(8
...
3)

• Adding the equations:
0 = Rp − N1 A10 =⇒ N1 =

Rp
A10

(8
...
6)

(8
...
5)

−1

Rp 1 −

• Let τ ∝

Rp
A10

A21
A10

A21 + σgF (ν)
if A2 1 < A10

(8
...
9)

then
∆N > 0

if τ21 > τ10

(8
...
This will
destroy the population inversion
...
11)
R = Rp 1 −
A10
then
R
∆N =
(8
...
13)

∆N increases linearly with pumping rate
...
14)
(8
...

Rth = ∆Nth A21
(8
...


• Once the laser starts oscillating, ∆N is clamped at ∆Nth
...
17)

R
− A21
∆Nth

(8
...
19)

= A21

• R2 , the reflectivity of the output mirror, effects Rth
...

• The choice of R2 depends on how much energy is available
...
Monochromatic
• The double mirror arrangement of a laser forms a cavity
...

nλ
=L
(9
...

• The resonant frequencies of the laser can have a much narrower linewidth than the usual
linewidth of the 2-1 atomic transition
...
Coherence
• Must distinguish between spatial and temporal coherence
...

• Monochromatic sources emit light that can be represented as a sequence of harmonic
wave trains of finite length, each separated from the others by a discontinuous change
in phase
...

• A given source can be characterised by an average wave train lifetime τ0 called its
coherence time
...
2)

• In general the temporal coherence time τ0 is given by the reciprocal of the spectral width
∆ν
...
3)
∆ν
• Spatial (lateral) coherence: The correlations in phase between spatially distinct points
of the radiation field
...
The range of separation between two points over which there is significant
interference is called the coherence area Ac
...
Directionality
• Laser output is highly directional becuase of the laser cavity shape
...


• When the spatial coherence length = beam diameter
θd = β

λ
D

(9
...

• When the spatial coherence length traverse to the beam direction Lp is less than the
beam diameter
λ
θp = βp
βp = 1
(9
...
Brightness
• Defined as the power per unit area per unit solid angle, Watts per square meter per
steradian
...

• For small angles, the relation between a plane-angle θ and a solid angle of a cone Ω is
Ω=

π 2
θ
4

(9
...
Short Pulse Detection
• It is possible to produce a short pulsed laser using a technique called ’mode-locking’
...
1

Longitudinal Modes

• Longitudinal modes determine the emission spectrum of the laser
...

• There are nodes at the mirrors becuase they have high reflectivites
...

n

λn
=L
2

kn =

2πfn
2π
=
λn
v

(10
...
2)

(10
...
4)

• The cavity mode frequencies are separated by c/2Lnr , which is the inverse of the time for a
photon to do a round trip
...
2

Transverse Modes

• Form because of boundary conditions
...

• Transverse modes describe the intensity variation within a cross section slice of the laser
beam
...


• The T EM00 mode is the closest thing to a ray of light, it has a Gaussian radial distribution
of electric field
...

• We try to prevent other modes from oscillating, this is achieved by inserting apertures in the
cavity which can be lossy for the higher modes but not the T EM00 mode
...
5)

where r is the distance from the centre of the beam, w determines the size of the beam
...

• After a round trip, the intensity of the beam reduces from I0 to R1 R2 I0
...
1)

is the time for a photon to return to its starting point in the cavity
...
2)

dI
I(1 − R)v
=−
dt
2L

(11
...
4)

I
(1 − R)v
=−
t
I0
2L

(11
...
6)

2L
v(1 − R)

(11
...
8)

ln

I = I0 exp −
where
tphoton =
is the lifetime of a photon in the cavity
tphoton =

• This is the loss of intensity due to mirror losses
...
1)
(12
...
3)

2L
= 2πνtphoton
(1 − R)v

(12
...
5)

2

1

(12
...
7)

• Recall that the difference between two adjacent modes is v/2L
∆ν 1 = (νn+1 − νn )
2

1−R
2π

• ∆ν 1 << νn+1 − νn as R is close to unity
...
8)

12
...

• The frequency output is of the following form

• ∆ν is the gain bandwidth, the full width half maximum
...

• The number of modes that will be oscillating is equal to
Gain bandwidth
2L∆ν
=
M ode spacing
v

(12
...
5GHz
2x0
...
5x109
=3
3x108

(12
...

– Insert sharp frequency filter at ν0 (central frequency)
...


16

13

Hole Burning
• In lasing media with inhomogenous broadened gain lineshape, the modes which are lasing can
deplete the gain
...


• The gain profile develops holes
...
This only occurs in
inhomogenous broadened gain media
...


17

14

Time Dependence of Laser Output

14
...
A common technique is
Q-switching
...

• It initially makes the cavity lossy (low Q) and after the lasing level of the gain medium is
populated by pumping, the Q-switch is switched so that the cavity has a high Q (low loss)
...

• When the population inversion is near the peak, the Q is switched to a high value, photons
begin to build up in the laser cavity since the gain is well above the new threshold condition
...

The pulse then decays with a time constant equal to the photon lifetime in the cavity
...
1
...


• The linear polariser only transmits light with its E field oscillating along a perpendicular
axis
...

• If the light is allowed back there is a low loss
...


14
...

• Try to get as many longitudinal modes oscillating as possible but with all their phases locked
together
...
1)

∆ω is the angular frequency separation between the modes
∆ω = 2π(νn+1 − νn ) =

2πv
πv
=
2L
L

(14
...
3)

En exp[i(ω0 + n∆ω)t + φ]

(14
...
5)

n=0

= E0 eiω0 t [1 + ei∆ωt + (ei∆ωt )2 + · · · + (ei∆ωt )n + · · · + (ei∆ωt )N −1 ]
= E0 eiω0 t

1 − eiN ∆ωt
1 − ei∆ωt

(14
...
7)

• Recall that
I(t) ∝ E(t)E ∗ (t)
2
∴ I(t) ∝ E0

• I = Imax at t = 0
2
Imax ∝ E0

(14
...
9)

sin2 ( ∆ωt )
2

( N ∆ωt )2
2
( ∆ωt )2
2

t→0

2
Imax ∝ E0 N 2

• I = 0 at

N ∆ωt
2

(14
...
11)

=π
2π
π 2L
=
N ∆ω
N πv
tround trip
=
N

=⇒ t0 =

• I(t) is periodic, repeating every

∆ωt
2

(14
...
13)

=π

trepeat =

2L
2π
=
= tround
∆ω
v

• Pulses are emitted at time intervals of tround

20

trip
...
14)

• If N is large
N≈
=

∆ν
mode spacing

(14
...
16)

t0 =

tround
N

=

1 2L
N v

∴ t0 ≈

trip

(14
...
18)

1
∆ν

(14
...

• The coherence length of the laser pulses is
Lt = ct0
• Power
peak power =

pulse energy
t0

average power =

pulse energy
tseparation

average power
t0
=
peak power
tround

21

(14
...
21)

(14
...
23)
trip



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