Difference between plane waves and

LASER

Lahiru De Silva

What is a plane wave?• A wave whose wave fronts (surfaces of constant phase) are parallel planes (plane wave

fronts) normal to propagation direction.• A plane wave's wave-fronts are equally spaced, a wavelength apart.• EM plane wave fronts propagate at the speed of light• Plane waves can be represented by the solution of “Wave equation”• Initially EM waves emitted from a point source have spherical wave fronts, but once

they extend to infinity they can be considered as plane wave frontsWave fronts

Propagation direction

3D wave equation• Waves can propagate any where in 3D space .The propagation of a plane EM

wave in 3D space is represented by the 3D electromagnetic wave equation

Where

• Solution for this wave equation is the plane wave represented by where

• Where K is wave vector • and r is Cartesian coordinate vector

2

2

2

2

2

22

zE

yE

xEE

22

2 0EEt

0( , , , ) exp[ ( )]E x y z t E i k r t

x y zk r k x k y k z

, ,x y zk k k k

, ,r x y z

)exp(0 iAE Maxwells equations

1D wave equation• 1 Dimensional wave equation is the simplest form of wave equation which shows

an electromagnetic (or any other) wave propagation in a scaler space.• Assuming wave propagates in Z direction

• 3D wave equation can be written as 1D wave equation obtained as

• Solution for 1D wave equation is

02

2

2

2

tE

zE

0and 0 2

2

2

2

yE

xE

tzkEtxf sin),( 0 )exp(0 iAE

What is a LASER?• A laser is a beam of light produced through a process of optical

amplification based on the stimulated emission of electromagnetic radiation.

• LASER = Light Amplification by Stimulated Emission of Radiation• Has only one wave length and moves in one direction• Properties:

• High monochromaticity • High coherence• Highly Collimated

• Equation that describes a LASER is known as a Gaussian Beam equation

)()(2)(

)()(0

22

2

22

)(zizR

yxiktkzizw

yx

eeeezwwAE

Principle of operation of a LASER• LASER work on the principle of stimulated emission • An atom is first “pumped” in to an excited state by a

photon emitted by flashing tube• When the excited electron is hit by another photon

having an energy equal to the energy gap, it stimulates the excited electron to fall back to lower energy level.

• The excited electron releases a photon having same phase and direction as the "stimulating" photon when coming back, and is called stimulated emission

• Inside a LASER tube, these photons are made to reflect back and forth by 2 mirrors intensifying the beam

• Once the beam is sufficiently intense it escapes through the semi reflective mirror as a beam

Derivation of the Gaussian equation -1• Wave equation is• Introducing a solution• Then we obtain wave equation as Helmholtz equation

• Considering variation of E in z direction as negligible• We obtain paraxial wave equation

22

2 0EEt

tiezyxEzyxE ),,(),,( 0

02 02

02

20

2

20

2

zEik

zE

yE

xE

02 020

2

20

2

zEik

yE

xE

020

2

zE

Derivation of the Gaussian equation -2• Solution to this paraxial wave equation is the Gaussian equation

• where

number wavek diverges) beamat which distance (z rangeRaleigh

0z fromfront thewaveof curvature of Radius)() w(z)0,z(at width Beam)( 0

RzzR

wzw

)()(2)(

)()(0

22

2

22

)(zizR

yxiktkzizw

yx

eeeezwwAE

)(tan)( 1

Rzzz

Wave front shape change

)()(2)(

)()(0

22

2

22

)(zizR

yxiktkzizw

yx

eeeezwwAE

Amplitude factor Plane wave nature Shift of wave to spherical wave front

Longitudinal phase factor

Features that make a LASER beam different from plane waves

1. Amplitude and beam width• A plane wave has its amplitude spread to infinity spread

in the transverse direction (x-y plane). • Since laser beam is more localized it can be assumed

to have high amplitude near propagating direction z and less amplitude when far away from z.

• Due to localized nature, we define a term called beam width as w(z).

• At beam width, the amplitude of beam is 1/e times that of the amplitude of beam at the z=0 axis.

Laser beam spot on wall

w

x

y

2. Wave front shape• Wave front radius is given by equation• Initially when z=0, it an be seen radius is infinite• Initially at beam waist radius is infinite. This means beam wave fronts show

plane wave nature• When Z increases, radius too increases meaning wave fronts show spherical

nature• This is opposite to plane waves which are spherical at the near field and planer

in far field.

Plane wave Gaussian wave

3. Beam divergence• For a certain distance ZR, LASER

beams follow plane wave characteristics.

• But after a that, LASER beams beam width diverges from being nearly linear.

• Angle of deviation of beam width is given by

Thank you