Transmission of Gaussian beams I
• First consider the transmission through a thin lens (for sake of simplicity let’s take a plano-convex lens). • What is the effect of a lens? – introduced a position-dependent (x,y) d(x,y) optical path length (OPL) R
d – Paraxial approximation 0
– Phase shift is 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 transmitted through 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 incoming Gaussian beam So we have: Phase “kick” due to the lens
P. Piot, PHYS 630 – Fall 2008 Transmission of Gaussian beams III
• Using results from homework I we have:
0 z z’
W0 W’0
2z0 2z’0
P. Piot, PHYS 630 – Fall 2008 Transmission of Gaussian beams IV
• Using the relations
• It is straightforward to find the relations between the incoming and transmitted 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 its waist. Use the previous formulae (with z=0)
If depth of focus of incident beam is much larger than f
z0 z’~f
2z’0
P. Piot, PHYS 630 – Fall 2008 Reflection from a spherical mirror
• The action of a spherical mirror with radius R is to reflect the beam and 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
x0’
x0 • The ratio x/x’ ~ can be seen as the radius of a x spherical wavefront x’
• Generalizing to the complex parameter q:
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
=-2n =-2m
• so
• And requiring
gives
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 proposed to serve as an acceleration mechanism for charged particle beams
P. Piot, PHYS 630 – Fall 2008