Laser Physics EAL 501

Lecture 6 Power & Frequency

• We said that there are two types of cavities• 1- Ring cavities

• Standing wave cavities• 1- Longitudenal modes due to standing waves

Do ring lasers have longitudenal modes?

nLc2

2-Transversal modes [Gaussian beams]The electric field distribution in the x-y plane

Satisfy the wave eq.The Fresnel-Kirchoff’s theory states that“if we know the field distribution at a plan 1 (x1,y1,z1) then the

distribution at any plane (x,y,z) is given by

We can write

Where The solution is

Higher order modes are given by

Gaussian BeamsThe solution of wave equation can be plane waves

But it is not real

q(z) is the complex beam parameter of the beam

Amplitude factor Transverse phase factor Longitudenal phase factorRayleigh range

Beam waist

Radius of curvature

Gain Saturation

We drive the rate equations

))(()( 12

12 NggNg

)1(21..

)1(21)1ln(

21

1

21

2121

221

rrL

lossescavity

rrLrrL

g

err

t

Lgt

• The steady state solution of the rate equation in the case I=0 gives us the threshold condition

• The steady state solution of the rate equation in the case after the threshold can be shown to be

Steady state under threshold solution is

In case of 3 level laser In case of 4 level laser

Solution above threshold in the case of two levels without pumping

Solution above threshold in the case of 3levels with pumping

Since g(ν)=σ(v)(N2-N1) then we can write

)()(1

)()(

sat

o

II

gg

)(2)(

)()(

21

21

21

PhI

NPPg

sat

To

Problem : find small signal gain and saturation intensity for 4 level laser

Hole burning1. Spatial hole burning2. Spectral hole burning

1- Spatial hole burning in the case of standing wave cavity there are points of zero intensity and points of max. intensity which depletes the gain much more rapidly2- Spectral hole burning in the case of inhomogeneous broadening the atoms that feed the laser modes will be depleted much more rapidly

Output power

Usually only one mirror produces output and if the mirror reflection is high

Where s is the scattering losses

The condition of steady state lasing

tgrrl

g )1(21)( 21

tsat

o gII

gg

/1)()(

It2

It is easy to find the value of transmission t to get the maximum output

So the optimum output intensity is