P. Piot, PHYS 630 – Fall 2008Transmission 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-dependentoptical path length (OPL)– Paraxial approximation– Phase shift isR(x,y)d0d(x,y)OPLP. Piot, PHYS 630 – Fall 2008Transmission 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 beamPhase “kick” due to the lensSo we have:P. Piot, PHYS 630 – Fall 2008Transmission of Gaussian beams III• Using results from homework I we have:0z z’W0W’02z’02z0P. Piot, PHYS 630 – Fall 2008Transmission 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 asM is the magnificationNote that q’0W’0=q0W0=k/2P. Piot, PHYS 630 – Fall 2008Limit 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 themagnification decreasesP. Piot, PHYS 630 – Fall 2008Beam focusing• Consider the a incoming Gaussian beam with a lens located at itswaist. Use the previous formulae (with z=0)z’~f2z’0z0If depth of focus of incident beam is much larger than fP. Piot, PHYS 630 – Fall 2008Reflection 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 themirror), the incident and reflected wavefronts coincide.P. Piot, PHYS 630 – Fall 2008ABCD 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’x0xx’P. Piot, PHYS 630 – Fall 2008Drift space• Consider a drift space with length d• Then q propagates as• therefore• The beam width and wavefront radius can be found fromP. Piot, PHYS 630 – Fall 2008Hermite-Gaussian Beams• Consider the complex envelope• This is a solution of the paraxial Helmholtz equation• Inserting we haveP. Piot, PHYS 630 – Fall 2008Hermite-Gaussian Beams• So we have• recognizing• We finally haveP. Piot, PHYS 630 – Fall 2008Hermite-Gaussian Beams• Doing the variable change• so• And requiringgives=-2n =-2mP. Piot, PHYS 630 – Fall 2008Hermite-Gaussian Beams• The complex amplitude of a Hemite-Gaussian beam is finallyP. Piot, PHYS 630 – Fall 2008Hermite Polynomials• Recurrence relation is• First few polynomials areMultiply by a GaussianP. Piot, PHYS 630 – Fall 2008Hermite-Gaussian BeamsComplex amplitude (arb. units)P. Piot, PHYS 630 – Fall 2008Hermite-Gaussian BeamsComplex amplitude (arb. units)P. Piot, PHYS 630 – Fall 2008Generation of Donut beamsDonut “beams” were proposedto serve as an accelerationmechanism for chargedparticle
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