Helmholtz Equation - Northern Illinois University

1 Helmholtz Equation Consider the function U to be complex and of the form: r r U( r ,t) = U( r )exp(2"#t ). Then the wave Equation reduces to 2 r 2 r ! " U( r ) + k U( r ) = 0. where 2#$ %. k" =. ! c c Helmholtz Equation ! P. Piot, PHYS 630 Fall 2008. Plane wave The wave is a solution of the Helmholtz equations . Consider the wavefront, , the points located at a constant phase, usually defined as phase=2 q. For the present case the wavefronts are decribed by which are Equation of planes separated by . The optical intensity is proportional to |U|2 and is |A|2 (a constant). P. Piot, PHYS 630 Fall 2008. Spherical and paraboloidal waves A spherical wave is described by and is solution of the Helmholtz Equation .

2 In spherical coordinate, the Laplacian is given by The wavefront are spherical shells Considering give the paraboloidal wave: -ikz P. Piot, PHYS 630 Fall 2008. The paraxial Helmholtz Equation Start with Helmholtz Equation Consider the wave Complex Complex envelope amplitude which is a plane wave (propagating along z) transversely modulated by the complex amplitude A. Assume the modulation is a slowly varying function of z ( slowly here mean slow compared to the wavelength). A variation of A can be written as So that P. Piot, PHYS 630 Fall 2008. The paraxial Helmholtz Equation So Expand the Laplacian Transverse Laplacian The longitudinal derivative is Plug back in Helmholtz Equation Which finally gives the paraxial Helmholtz Equation (PHE): P.

3 Piot, PHYS 630 Fall 2008. Gaussian Beams I. The paraboloid wave is solution of the PHE. Doing the change give a shifted paraboloid wave (which is still a solution of PHE). If complex, the wave is of Gaussian type and we write where z0 is the Rayleigh range We also introduce Wavefront Beam width curvature P. Piot, PHYS 630 Fall 2008. Gaussian Beams II. R and W can be related to z and z0: P. Piot, PHYS 630 Fall 2008. Gaussian Beams III. Expliciting A in U gives P. Piot, PHYS 630 Fall 2008. Gaussian Beams IV. Introducing the phase we finally get where This Equation describes a Gaussian beam. P. Piot, PHYS 630 Fall 2008.

4 Intensity distribution of a Gaussian Beam The optical intensity is given by z/z0. P. Piot, PHYS 630 Fall 2008. Intensity distribution Transverse intensity distribution at different z locations -4z0 -2z0. z/z0. -z0 0 -4z0 -2z0. -z0 0. Corresponding profiles . P. Piot, PHYS 630 Fall 2008. Intensity distribution (cnt'd). On-axis intensity as a function of z is given by z/z0. z/z0. P. Piot, PHYS 630 Fall 2008. Wavefront radius The curvature of the wavefront is given by P. Piot, PHYS 630 Fall 2008. Beam width and divergence Beam width is given by For large z P. Piot, PHYS 630 Fall 2008. Depth of focus A depth of focus can be defined from the Rayleigh range 2.

5 2z0. ! P. Piot, PHYS 630 Fall 2008. Phase The argument as three terms Spherical distortion of the wavefront Guoy Phase associated phase shift to plane wave On axis ( =0) the phase still has the Guoy shift . Varies from - /2 to + /2. At z0 the Guoy shift is /4. P. Piot, PHYS 630 Fall 2008. Summary At z0. Beam radius is sqrt(2) the waist radius On-axis intensity is 1/2 of intensity at waist location The phase on beam axis is retarded by /4 compared to a plane wave The radius of curvature is the smallest. Near beam waist The beam may be approximated by a plane wave (phase ~kz). Far from the beam wait The beam behaves like a spherical wave (except for the phase excess introduced by the Guoy phase).

6 P. Piot, PHYS 630 Fall 2008.