Electromagnetic Waves

1. , Electromagnetic Waves , Maxwell’s Equations for homogeneous media: 1) 2) 4) 3) Where: , are the electric and magnetic field vectors, is the charge density, j is the current (directed flow of charge), , are the dielectric permittivity and magnetic permeability, respectively, of free space, , are the ratio between the permittivity and permeability in a medium and space. (The relative permittivity and permeability.) and the del ( ) vector operator is defined (in Cartesian coordinates) as:

2. These equations are too general for our use, because: • We are not going to consider wave propagation in any media with a net charge density. (Hence, our conclusions may not be valid for propagation in a plasma, for instance.) • We are also not going to consider media with electric currents. • The magnetic permeability arises due to the interaction with atoms that have a net magnetic moment (unbalanced electron orbitals). At optical frequencies (~1014 Hz), no atoms can respond fast enough to afford significant interaction. (Electron orbitals can respond fast enough, however, so in general, )

3. Hence, setting 𝞀, j = 0, and , we get: Where we’ve made the substitution (1) These are Maxwell’s Equations for the conditions we are interested in for almost all optical design: No free charges. No currents. No magnetic interaction. The primary interaction is between the E-field and the orbital electrons in the dielectric media. (2) (3) (4)

4. The Electromagnetic Wave Equation Since these equations are simultaneously true, we can combine them: • Take the curl of both sides of the 3rd equation (Faraday’s Law): • Substitute for from the 4th equation (Ampere’s Law): • Applying the well known vector identity and (Gauss’s Law) , we arrive at the Electromagnetic Wave Equation for homogeneous media: An equivalent derivation shows that the magnetic field also obeys a wave equation.

5. Comparing the EM wave equation to the generic wave equation: We see that the velocity of electromagnetic waves, under our assumptions is: Where we have used the fact that the speed of light, And we have defined the index of refraction,

6. A useful solution to the wave equation is the “plane wave”: Or, for the case of an X-polarized wave moving down the Z-axis. For use in optical modeling, plane waves have several important characteristics: Plane waves have a constant amplitude (A) over all space and time – the only thing that changes with time and position is the phase. Hence plane waves are trivially easy to “propagate” – just make the appropriate phase change. We will show, when we cover wave propagation, that the Fourier Transform can decompose any Electromagnetic Wave field (defined on a plane) into an equivalent set of plane waves. This reduces the problem of simulating the propagation of the given field to simply re-phasing the constituent plane waves and adding them up at the desired location. A detailed analysis shows that plane waves travel in the direction perpendicular to their wavefronts. Hence, rays (defined as lines drawn perpendicular to the wavefronts) are perfect models for the propagation of plane waves.

7. The law of refraction (Snell’s Law) from plane waves: Consider a plane wave impinging on a plane interface with different indices of refraction (hence wave speeds) on each side. The wave changes speed, but not frequency, hence the phases (wavefronts) must meet from each side: Angles a1 and a2 are called the “angle of incidence” and “angle of refraction” respectively. n1 λ1 a1 a1 x a2 λ2 n2 a2 Note that: and Where is the free space wavelength

8. Looking at a close-up of the previous drawing: n1 n2 Hence,Snell’s Law is a consequence of the wave nature of light and the physical requirement for field continuity at sourceless points. Note: and ⇒ (Snell’s Law) Hence:

9. Total Internal Reflection (TIR) Consider a ray traveling from a medium of high index to one of low index: According to Snell’s Law, when Then And no light penetrates the second medium at all – all of the light must be reflected. This is called the ‘critical angle’ at or above which incident light will be totally reflected. This is the way that optical fibers contain light.

10. Fermat’s Principle Original: “Light travels along paths of minimum time” (1662) Modern: “Light travels paths that are “stationary” w.r.t. all nearby paths. Studing the following diagram shows why rays “obey” Fermat’s Principle: Waves travel ⊥ to wavefronts, hence rays must also. Any ray from ‘S’ to ‘P’ crosses the same number of wavefronts (hence takes equal time) and therefore obeys Fermat’s Principle Only one ray can go from ‘S’ to any point other than ‘P’. This ray is ⊥ to the wavefronts, hence takes the least time of any other path.