Lesson 2

P. Piot, PHYS 630 Fall 2008

Helmholtz Equation

Consider the function U to be complex and of theform:

Then the wave equation reduces to

where

!

U(r r ,t) = U(

r r )exp 2"#t( )

!

"2U(

r r ) + k

2U(

r r ) = 0

!

k "2#$

c=%

c Helmholtz equation

P. Piot, PHYS 630 Fall 2008

Plane wave

The wave

is a solution of the Helmholtz equations.

Consider the wavefront, e.g., the points located at a constant phase,usually defined as phase=2q.

For the present case the wavefronts are decribed by

which are equation of planes separated by .

The optical intensity is proportional to |U|2 and is |A|2 (a constant)

P. Piot, PHYS 630 Fall 2008

Spherical and paraboloidal waves A spherical wave is described by

and is solution of the Helmholtz equation.

In spherical coordinate, the Laplacian is given by

The wavefront are spherical shells

Considering give the paraboloidal wave:

-ikz

P. Piot, PHYS 630 Fall 2008

The paraxial Helmholtz equation Start with Helmholtz equation

Consider the wave

which is a plane wave (propagating along z) transversely modulatedby the complex amplitude A.

Assume the modulation is a slowly varying function of z (slowly heremean slow compared to the wavelength)

A variation of A can be written as

So that

Complexamplitude

Complexenvelope

P. Piot, PHYS 630 Fall 2008

The paraxial Helmholtz equation So

Expand the Laplacian

The longitudinal derivative is

Plug back in Helmholtz equation

Which finally gives the paraxial Helmholtz equation (PHE):

TransverseLaplacian

P. Piot, PHYS 630 Fall 2008

Gaussian Beams I The paraboloid wave is solution of the PHE

Doing the change give a shifted paraboloid wave (whichis still a solution of PHE)

If complex, the wave is of Gaussian type and we write

where z0 is the Rayleigh range

We also introduce

Wavefrontcurvature

Beam width

P. Piot, PHYS 630 Fall 2008

Gaussian Beams II R and W can be related to z and z0:

P. Piot, PHYS 630 Fall 2008

Gaussian Beams III Expliciting A in U gives

P. Piot, PHYS 630 Fall 2008

Gaussian Beams IV Introducing the phase we finally get

where

This equation describes a Gaussian beam.

P. Piot, PHYS 630 Fall 2008

Intensity distribution of a Gaussian Beam

The optical intensity is given by

z/z0

P. Piot, PHYS 630 Fall 2008

Intensity distribution Transverse intensity distribution at different z locations

Corresponding profiles

-4z0 -2z0

-z0 0 -4z0 -2z0

-z0 0

z/z0

P. Piot, PHYS 630 Fall 2008

Intensity distribution (cntd) On-axis intensity as a function of z is given by

z/z0

z/z0

P. Piot, PHYS 630 Fall 2008

Wavefront radius The curvature of the wavefront is given by

P. Piot, PHYS 630 Fall 2008

Beam width and divergence Beam width is given by

For large z

P. Piot, PHYS 630 Fall 2008

Depth of focus A depth of focus can be defined from the Rayleigh range

2z0!

2

P. Piot, PHYS 630 Fall 2008

Phase The argument as three terms

On axis (=0) the phase still has the Guoy shift

At z0 the Guoy shift is /4

Phase associatedto plane wave

Spherical distortion of the wavefront

Guoyphase shift

Varies from -/2 to +/2

P. Piot, PHYS 630 Fall 2008

Summary At z0

Beam radius is sqrt(2) the waist radius On-axis intensity is 1/2 of intensity at waist location The phase on beam axis is retarded by /4 compared to a plane

wave The radius of curvature is the smallest.

Near beam waist The beam may be approximated by a plane wave (phase ~kz).

Far from the beam wait The beam behaves like a spherical wave (except for the phase

excess introduced by the Guoy phase)