P. Piot, PHYS 630 – Fall 2008Plane wave revisited I• Consider a monochromatic wave with fields given by• Simple application of Maxwell’sequation gives• It follows that E ⊥ k ⊥ B . Since E,B are in a plane ⊥ k (the directionof propagation this is called a TEM wave. Note that the wavefront areplane so this is a plane wave. Cover of BYU optics textbookP. Piot, PHYS 630 – Fall 2008Plane wave revisited II• From Maxwell’s equationwe have• Defining the impedance of the medium• The complex Poynting vector is• So the optical intensity isP. Piot, PHYS 630 – Fall 2008Spherical wave revisited I• Consider the vector potentialwhere• Then the e.m. fields are given by• Work in spherical coordinates xzyP. Piot, PHYS 630 – Fall 2008Spherical wave revisited II• So we have• Taking the curl• And doing the approximation kr>>1 gives EBP. Piot, PHYS 630 – Fall 2008Spherical wave revisited III• In cartesian coordinate we can write• The electric field is approximately• If z large enough then which is ~ TEM plane wave Chose θ and φ to be ~ π/2Consider large zP. Piot, PHYS 630 – Fall 2008Gaussian beam• Doing the change z z+iz0 in the previous equation give the e.m. fieldassociated to a Gaussian beamwhere now U(r) is the complex amplitude associated to a Gaussianbeam (what we derived in the “Beams” lessons)P. Piot, PHYS 630 – Fall 2008“Vector” Beams• Can directly solve the wave equation (written in term of A)take A as• Energy flow• Momentum density• Angular momentumP. Piot, PHYS 630 – Fall 2008Bessel Beams• Helmholtz equation written in cylindrical coordinate is with solution• So the complex amplitude is• If taken as a field component Maxwell’s equations are notsatisfied.• So take U(r) as one of A component; take (the othe case are notso interesting):P. Piot, PHYS 630 – Fall 2008Bessel Beams• From A we can get B and E:these fields that define a Bessel beamP. Piot, PHYS 630 – Fall 2008Bessel Beams: diffraction-free beams!Durnin et al, Phys. Rev. Lett. 58(15), 1499 (1987)P. Piot, PHYS 630 – Fall 2008Bessel Beams: generationP. Piot, PHYS 630 – Fall 2008Laguerre-Gaussian (LG) BeamsHermite-Gaussian beamsArgument of the complex amplitude versus (x,y) Modulus of the complex amplitude versus (x,y) {m,n} n=1 n=2 n=3P. Piot, PHYS 630 – Fall 2008Beams with angular momentum Ihttp://www.physics.gla.ac.uk/Optics/projects/AM/P. Piot, PHYS 630 – Fall 2008Beams with angular momentum IIhttp://www.physics.gla.ac.uk/Optics/projects/AM/P. Piot, PHYS 630 – Fall 2008Beams with angular momentum III• Use a “fork” hologram to introducea space-dependent phase modulation• Hologram basically imparts a torqueon the beam and generate an angularmomentumP. Piot, PHYS 630 – Fall 2008Radially LG beam can yield sharper focusDorn et al, Phys. Rev. Lett. 91, 233901