Chapter 9

Threshold Requirements

Looking at Loss again

R1

R2

d

Loss/length = 1

Gain/length = (or )

I

I e-(1+)d

IR2 e-(1+)d

IR2R1 e-2(1+)d

All self-sustaining oscillators require

gain to exceed losses

= h r B/c (n1-n2) = -

Laser

medium

Gain Threshold

R2R1 e2( -1)d 1

Eq 9.1

Rewrite this as:

1 - ln[R2R1]/2d

Cavity stuff

Atom stuff

Cavity Lifetime

Lets make another defn so we can simplify: cavity lifetime

(also called cold cavity lifetime or photon lifetime). This is the time the intensity decreases to 1/e of its starting value when there is no gain in the cavity

In one round trip intensity decreases by R2R1 e2( -1)d

which takes 2d/c seconds (assumed cavity and gain material length are the same)

Assume intensity decrease can be approx. described as e-t/tc where tc is cold cavity lifetime

tc = 2d

c (21 d - ln [R1R2])

Eq 9.3

Population Difference Threshold

n2-n1

8r2 tsp

c3 g(r) tc

Since achieving n2>n1 is really hard, what should

a smart person do?

Example

Ruby rod from before except lets use a 5cm long rod

Between two 90% reflectivity mirrors

From Eq. 9.3 we get tc = 2.8ns, and therefore we find by substitution into Eq. 9.4 that:

n2-n1 2 x 1023 m-3

Typical Cr atom densities in ruby (about 1% replacement of Al)

are around 1.6 x 1025 m-3.

Resonator Transfer Function in presence of gain

1

0

2

1

0

What happens

after lots of

round trips?

Go to Phet Simulation

Talking about Gain

Also, Talk about two level situation

Chapter 10

Pumping Processes and Rate Equations

Two Level Atom

Narrow Band Radiation at r with density u, hit some atoms.

Rabs = n1c3A g(r) u(T)/(8 h r3)

= n1 W12

E1

E2

n1

n2

u(T)

Rem = n2c3A g(r) u(T)/(8 h r3)

= n2 W21

N.B. W12 = W21

n1 = W21 (n2-n1) + A n2

.

W12 n1

W21 n2

A n2

Solution for 2 level system

n2/n1 = W21/(W21 + A)

Cannot possibly work!

Three Level Atom

Narrow Band Radiation at p with density u.

E1

E3

n1 = W31 (n3-n1) + A31 n3 + A21 n2 + W21 (n2-n1)

.

W13 n1

W31 n3

A31 n3

A32 n3

hvp

A21 n2

W21 n2

W12 n1

Three Level Solution

n

N

=

W31-A21

W31+A21

1

1+ W12

3W31+2A32

A32 (W31+A21)

Below threshold essentially no laser light: W12 = 0

In this case pop difference, and hence gain, are indep. of W12

Intensity depends exponentially on distance

Lets look at saturation in more detail in 4 level laser

Example: pump power for Ruby

At threshold:

n

N

W31-A21

W31+A21

0

PD

is nearly

zero

No. of pumped atoms/sec/volume ~ W31 n1

Power (per unit volume) will be:

P = W31 n1 h p = A21 N/2 h p

For Ruby, A21 =1/ = 1/(3ms), N ~ 1.6 x 1019 cm-3

Pump wavelength ~ 480nm (frequency ~ 6 x 1014 Hz)

P = 1100 Wcm-3

Electrical thresholds are about 16 times higher than this

Four Level Atom

E1

E4

n1 = W41 (n4-n1) + A41 n4 + A21 n2

.

W14 n1

W41 n4

A41 n4

A43 n4

hvp

A21 n2

W32 n3

E3

A21 n2

E2

W23 n2

hvL

Smart: avoid using the ground state for one of the laser levels

If A21 > A43 then get PI for smallest pumping intensities

Four level solution

n

N

=

W31-A21

W31+A21

1

1+ W12

3W31+2A32

A32 (W31+A21)

n

N

=

W41

W41+A32

1

1+ W32

2W41+ A21

A21 (W41+A32)

Viz. 3 level:

Small signal behaviour

(below and near threshold)

Nd:YAG example

NB. 2 errors in notes

p. 125 - should be N ~ 6 x 1025 m-3 (0.5% doping)

p. 124 - should be Gain coeff/m on vertical axis of Fig 10.3

~ 1064nm, ~ 195 GHz, ~ 230s

N ~ 6 x 1025 m-3 , n0 ~ 1.82, p ~ 400 THz

R1 ~ 100%, R2 ~ 80%, l ~ 7cm 1 = 0

n ~ 2.5 x 1021 m-3

W41 ~ 0.2 s-1

P ~ W41 n1 h vp ~ W41 N h vp ~ 3.2 W/cm3

Gain dependence on output

W41 ~ 0.2 s-1

W41 ~ 0.4 s-1

W41 ~ 0.1 s-1

Threshold

No. of photons in mode

Power vs Current in LDs

Line Broadening

Chapter 11: Not for examination

Radiation from a bunch of excited atoms

FWHM = 1/(2 )

Collisions

Doppler Broadening