PHYS 630: Homework Idue date: Thursday, September 11th, 2008 at class meeting.You are welcome to work together. If you partially use work from other (e.g.something you might have found in a book or a journal paper), you should properlycredit the author by citing the material used.1. Paraboloidal Beams Solution of the Helmholtz Equation (10 pts):Verify that the complex amplitude associated to a Paraboloidal beamU(−→r ) =Aze−ikze−ikx2+y22z.where A is a complex constant, satisfies the paraxial Helmholtz equation.2. Validity of the Paraxial Equation for a Gaussian Beam (10 pts):The complex envelop A(−→r ) of a Gaussian beam is an exact solution of theparaxial Helmholtz equation but its corresponding complex amplitude U(−→r ) =A(−→r ) exp(−ikz) is only an approximate solution of the Helmholtz equation.This is because the paraxial Helmholtz equation is itself approximate (slowlyvarying envelope approximate: ∂zA  kA). Show that if the divergence angleθ0of a Gaussian beam is small (θ0 1) the condition ∂zA  kA is satisfied.3. Elliptic Gaussian Beams (15 pts): The paraxial Helmholtz equation ad-mits a Gaussian beam solution with intensity I(x, y, 0) = |A0|2exp[−2(x2/W20x+y2/W20y)] in the z = 0 plane, with beam waist radii W0xand W0yin the x andy-direction respectively. The contour of constant intensity in (x,y) plane aretherefore ellipses instead of circles. Write expressions for the beam depth offocus, angular divergence, and radii of curvature in the x and y directions, asfunction of W0x, W0yand the wavelength λ. If W0x= 2W0ysketch the shapeof the beam spot in the z = 0 plane and in the far field (z much greater thanthe depth of focus).4. Determination of a Beam with Given Width and Curvature (15 pts):Assuming that the beam width W and curvature R are known at some pointon the beam axis. Find the location of the beam waist z and its width W0.15. Beam Relaying (15 pts): A Gaussian beam of width W0and wavelength λis repeatedly focused by a sequence of identical lenses, each with focal lengthf and separated by a distance d. Show that a necessary condition to have thefocused beam waist radius equal to the incident waist radius is d ≤ 4f.6. Passage of a Gaussian Beam through a Lens (15 pts): A Gaussianbeam is transmitted through a thin lens of focal length f. Show that thelocations of the waists of the incident and transmitted beams, z and z0arerelated viaz0f− 1 =z/f − 1(z/f − 1)2+ (z0/f)2.where z0is the Rayleigh length of the incident beam.7. Transmission of a Gaussian Beam through a Transparent Plate (10pts): Use the ABCD law to examine the transmission of a Gaussian beamfrom air, through a transparent plate of reflective index n and thickness d, andagain into air. Assume the beam axis is normal to the plate.8. “Donut” Beams (10 pts): We consider a wave which is a superimpositionof two Hermite-Gaussian beams of orders (1,0) and (0,1) of equal intensities.The two beams have independent and random phases so their intensities addwith no interference. Show that the total intensity is a donut-shaped circularlysymmetric function. Assuming that W0= 1 mm, find the radius of the circleof peak intensity and the radii of the two circles corresponding to the 1/e2times the peak intensity at the beam