Chapter 9Threshold Requirements

Looking at Loss again

Lasermedium

R1 R2

d

Loss/length =α1

Gain/length =−αorκ

I

I e-(α1+αd

IR2 e-(α1+αdIR2R1 e-2(α1+αd

All self-sustaining oscillators require gain to exceed losses

α = h νr B/c (n1-n2) = -κ

Gain Threshold

R2R1 e2(κ -α1d ≥ 1 Eq 9.1

Rewrite this as:

κ ≥ α1 - ln[R2R1]/2dCavity stuffAtom stuff

Cavity LifetimeLets 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(κ -α1d

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 = 2dc (2α1 d - ln [R1R2])

Eq 9.3

Population Difference Threshold

n2-n1 ≥8πνr

2 tsp

c3 g(νr) tc

Since achieving n2>n1 is really hard, what should a smart person do?

ExampleRuby 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 GainAlso, Talk about two level situation

Chapter 10Pumping Processes and Rate Equations

Two Level AtomNarrow Band Radiation at νr with density u, hit some atoms.

Rabs = n1c3A g(νr) u(T)/(8 π h νr3)

= n1 W12

E1

E2

n1

n2 u(T)

Rem = n2c3A g(νr) u(T)/(8 π h νr3)

= n2 W21

N.B. W12 = W21

n1 = W21 (n2-n1) + A n2.

W12 n1

W21 n2A n2

Solution for 2 level system

n2/n1 = W21/(W21 + A)

Cannot possibly work!

Three Level AtomNarrow Band Radiation at νp with density u.

E1

E3

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

W13 n1

W31 n3A31 n3

A32 n3

hvp A21 n2 W21 n2

W12 n1

Three Level Solution

ΔnN

=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 RubyAt threshold:

ΔnN

≈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 νp = A21 N/2 h νp

For Ruby, A21 =1/τ = 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 n4A41 n4

A43 n4

hvpA21 n2 W32 n3

E3

A21 n2E2

W23 n2hvL

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

ΔnN

=W31-A21

W31+A21

1

1+ W12 3W31+2A32

A32 (W31+A21)

ΔnN

= W41

W41+A32

1

1+ W32 2W41+ A21

A21 (W41+A32)

Viz. 3 level:

Small signal behaviour (below and near threshold)

Nd:YAG exampleNB. 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

λ ~ 1064nm, Δν ~ 195 GHz, τ ~ 230µsN ~ 6 x 1025 m-3 , n0 ~ 1.82, νp ~ 400 THz

R1 ~ 100%, R2 ~ 80%, l ~ 7cm α1 = 0

Δn ~ 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

0.01 1 100 100000

0.25

0.5

0.75

1

1.25

1.5

W320.01 1 100 10000

Intensity (W/cm )2

W41 ~ 0.2 s-1

W41 ~ 0.4 s-1

W41 ~ 0.1 s-1

Threshold

No. of photons in mode

100

101

102

103

104

105

106

107

108

109

1010

Photon Number

0.12 3 4 5 6 7 8 9

12 3 4 5

Pump Intensity (as ratio with threshold)

Power vs Current in LDs

2.52.01.51.0 Pump Intensity (as ratio with threshold)

Line BroadeningChapter 11: Not for examination

Radiation from a bunch of excited atoms0.5

0.0

-0.5

Intensity (a.u)

2000150010005000 Time

τFWHM = 1/(2 π

τ)

Collisions40

30

20

10

0

Intensity (a.u)

1400120010008006004002000 Time (fs)

Doppler Broadening1.0

0.8

0.6

0.4

0.2

0.0

Intensity (a.u)