P. Piot, PHYS 630 – Fall 2008
Transmission of Gaussian beams I
• First consider the transmission through a thin lens (for sake of simplicity let’stake a plano-convex lens).
• What is the effect of a lens?– introduced a position-dependent
optical path length (OPL)
– Paraxial approximation
– Phase shift is
R
(x,y)
d0
d(x,y)
OPL
P. Piot, PHYS 630 – Fall 2008
Transmission of Gaussian beams II
• So the “transmittance” of the lens is
• Take a Gaussian beam centered at z=0 with waist radius W0 transmittedthrough a lens located at z.
• The transmittance indicates the radius of curvature is bent• At z we can write (assuming the lens is thin)
Phase of the incomingGaussian beam
Phase “kick” due to the lens
So we have:
P. Piot, PHYS 630 – Fall 2008
Transmission of Gaussian beams III
• Using results from homework I we have:
0 z z’
W0 W’0
2z’02z0
P. Piot, PHYS 630 – Fall 2008
Transmission of Gaussian beams IV
• Using the relations
• It is straightforward to find the relations between the incoming andtransmitted Gaussian beams:– Waist radius:– Waist locations:– Depth of focus:– Divergence:
• Where the magnification is defined as M is the magnification
Note that q’0W’0=q0W0=k/2
P. Piot, PHYS 630 – Fall 2008
Limit of Ray Optics
• Consider the limit
• The beam may be approximated by a spherical wave
• We also have so that
• The location of the waist is given by
– The maximum magnification is the ray optics limit– As r increases the deviation from ray optics grows and the
magnification decreases
P. Piot, PHYS 630 – Fall 2008
Beam focusing
• Consider the a incoming Gaussian beam with a lens located at itswaist. Use the previous formulae (with z=0)
z’~f
2z’0
z0
If depth of focus of incident beam is much larger than f
P. Piot, PHYS 630 – Fall 2008
Reflection from a spherical mirror
• The action of a spherical mirror with radius R is to reflect the beamand modify its phase by the factor -k(x2+y2)/R
• The reflected beam remains Gaussian with parameters
• Some special cases:– If R=∞ (planar mirror) then R1=R2– If R1= ∞ (waist on mirror) then R2=R/2– If R1=-R (incident wavefront has the same curvature as the
mirror), the incident and reflected wavefronts coincide.
P. Piot, PHYS 630 – Fall 2008
ABCD formalism for a Gaussian beam
• Consider a system such that
• The ratio x/x’ ~ can beseen as the radius of aspherical wavefront
• Generalizing to the complex parameter q:
x0’
x0
xx’
P. Piot, PHYS 630 – Fall 2008
Drift space
• Consider a drift space with length d
• Then q propagates as
• therefore
• The beam width and wavefront radius can be found from
P. Piot, PHYS 630 – Fall 2008
Hermite-Gaussian Beams
• Consider the complex envelope
• This is a solution of the paraxial Helmholtz equation
• Inserting we have
P. Piot, PHYS 630 – Fall 2008
Hermite-Gaussian Beams
• So we have
• recognizing
• We finally have
P. Piot, PHYS 630 – Fall 2008
Hermite-Gaussian Beams
• Doing the variable change
• so
• And requiring
gives
=-2n =-2m
P. Piot, PHYS 630 – Fall 2008
Hermite-Gaussian Beams
• The complex amplitude of a Hemite-Gaussian beam is finally
P. Piot, PHYS 630 – Fall 2008
Hermite Polynomials
• Recurrence relation is
• First few polynomials are
Multiply by a Gaussian
P. Piot, PHYS 630 – Fall 2008
Hermite-Gaussian Beams
Complex amplitude (arb. units)
P. Piot, PHYS 630 – Fall 2008
Hermite-Gaussian Beams
Complex amplitude (arb. units)
P. Piot, PHYS 630 – Fall 2008
Generation of Donut beams
Donut “beams” were proposedto serve as an acceleration
mechanism for chargedparticle beams
