The 3D wave equation Plane wave Spherical wave MIT 2.71/2.710 03/11/09 wk6-b-13 Planar and Spherical Wavefronts Planar wavefront (plane wave): The wave phase is constant along a planar surface (the wavefront). As time evolves, the wavefronts propagate at the wave speed without changing; we say that the wavefronts are invariant to propagation in this case. Spherical wavefront (spherical wave): The wave phase is constant along a spherical surface (the wavefront). As time evolves, the wavefronts propagate at the wave speed and expand outwards while preserving the wave’s energy. MIT 2.71/2.710 03/11/09 wk6-b-14 Wavefronts, rays, and wave vectors Rays are: k 1) normals to the wavefront surfaces 2) trajectories of “particles of light” Wave vectors: At each point on the wavefront, we may assign a normal vector k This is known as the wave vector; k it magnitude k is the wave number and it is defined as k MIT 2.71/2.710 03/11/09 wk6-b-15 3D wave vector from the wave equation wavefront x kx k wave vector kz z ky y MIT 2.71/2.710 03/11/09 wk6 -b-16 3D wave vector and the Descartes sphere The wave vector represents the momentum of the wave. Consistent with Geometrical Optics, its magnitude is constrained to be proportional to the refractive index n (2π/λfree is a normalization factor) In wave optics, the Descartes sphere is also known as Ewald sphere or simply as the k-sphere. (Ewald sphere may be familiar to you from solid state physics) MIT 2.71/2.710 03/11/09 wk6-b-17 Spherical wave “point” source Outgoing rays Outgoing wavefronts (spherical) MIT 2.71/2.710 03/11/09 wk6-b-18 Dispersive waves Dispersion curves for glass Fig. 9X,Y in Jenkins, Francis A., and Harvey E. White. Fundamentals of Optics. 4th ed. New York, NY: McGraw-Hill, 1976. ISBN: 9780070323308. (c) McGraw-Hill. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. $! ' /011'23451 #26')781 $98')781 +08')781 #" -.* #, '#! #! ! '+ ! & guided light ! " Dispersion diagram for metal waveguide of width cut-off a=1µm frequency ' MIT 2.71/2.710 ! " #! #" $! ( !# 03/16/09 wk7-a- 1 %&!'#! ') * Superposition of waves at different frequencies Fig. 7.16a,b,c in Hecht, Eugene. Optics. Reading, MA: Addison-Wesley, 2001. ISBN: 9780805385663. (c) Addison-Wesley. All rights reserved. This content is excluded from ourCreative Commons license. For more information, see http://ocw.mit.edu/fairuse . MIT 2.71/2.710 03/16/09 wk7-a- 2 Group and phase velocity $! ' /011'23451 #26')781 $98')781 +08')781 #" -.* #, '#! #! ! '+ ! & ! " ' ! " #! #" $! ( !# %&!'#! ') * MIT 2.71/2.710 03/16/09 wk7-a- 3 Spatial frequencies We now turn to a monochromatic (single color) optical field. The field is often observed (or measured) at a planar surface along the optical axis z. The wavefront shape at the observation plane is, therefore, of particular interest. x observation planes x θ z Plane wave Spherical wave MIT 2.71/2.710 03/16/09 wk7-a- 4 Spatial frequencies Plane wave Spherical wave MIT 2.71/2.710 03/16/09 wk7-a- 5 Today • Electromagnetics – Electric (Coulomb) and magnetic forces – Gauss Law: electrical – Gauss Law: magnetic – Faraday’s Law – Ampère-Maxwell Law – Maxwell’s equations ⇒ Wave equation – Energy propagation • Poynting vector • average Poynting vector: intensity – Calculation of the intensity from phasors • Intensity MIT 2.71/2.710 03/18/09 wk7-b- 1 Electric and magnetic forces I r q q´ F + + dl free charges r F I´ Coulomb force Magnetic force (dielectric) permitivity (magnetic) permeability of free space of free space MIT 2.71/2.710 03/18/09 wk7-b- 2 Electric and magnetic fields Observation Generation F velocity E v static charge: electric E q + ⇒ field + electric field ⊗ electric magnetic B charge induction B v moving charge (electric current): + ⇒ Lorentz force + magnetic field MIT 2.71/2.710 03/18/09 wk7-b- 3 Gauss Law: electric field E E da da + + V A Gauss theorem ρ: charge density MIT 2.71/2.710 03/18/09 wk7-b- 4 Gauss Law: magnetic field da there are no magnetic B charges V A Gauss theorem MIT 2.71/2.710 03/18/09 wk7-b- 5 Faraday’s Law: electromotive force E dl B(t) (in/de)creasing A C Stokes theorem MIT 2.71/2.710 03/18/09 wk7-b- 6 Ampère-Maxwell Law: magnetic induction B B dl dl E I current capacitor A A C C Stokes theorem MIT 2.71/2.710 03/18/09 wk7-b- 7 Maxwell’s Equations (free space) Integral form Differential form MIT 2.71/2.710 03/18/09 wk7-b- 8 Wave Equation for electromagnetic waves MIT 2.71/2.710 03/18/09 wk7-b- 9 MIT OpenCourseWare http://ocw.mit.edu 2.71 / 2.710 Optics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. .