Transcript
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)
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