Classical Optics Prof. D. Rich Study of light: Wave / Particle duality of photons E = h Einstein: Photoelectric Effect h = 6.63 x 10-34 J·s Wave aspect of light stems from the unification of Through Maxwell’s equations in vacuum: Gauss’s Law: Magnetic Flux Law: (i.e. no magnetic monopoles) Faraday’s Law of Induction: Generalized Circuital (Ampere’s) Law:

Development of the idea of E-M wave propagation: Let Let’s use the differential vector identity: Let then

uE uB Take an integral  d 3r and use the Divergence theorem: 1 2 3 Rate at which total energy in V increases; Note that = rate at which energy flows out of V across the boundary . 2 Rate at which the Kinetic Energy of the particles change. 3 Rate at which Energy stored in the Fields increase; units in Joules/sec or Watts Power = Force Velocity Define: Poynting vector; points in the direction in which the fields E and B transport Energy. (Units W/m2)

We want to search for plane E-M wave solutions in a vacuum We want to search for plane E-M wave solutions in a vacuum. So, from Maxwell’s equations, using  = 0 and j = 0, we have Let This assumption leads to conditions on the E and B components: Fields in a sinusoidal electromagnetic wave. The E-Field is parallel to the x-axis and to the x-z plane and the B-Field is parallel to the y -axis and to the y-z plane. The propagation direction is along z.

0 0 Similarly, So, we can take Ez = Const. = 0 and Bz = Const. = 0 without a loss of generality. Also, from above

For (i) take /z and /t We arrive immediately to expressions of the 1D Wave Equation. We can now identify the constant representing the speed of light: (as predicted by Maxwell in the year 1861) The 3D expressions are as follows:

In general, the 3D wave equation has the form: For case (i) above i.e., a simple harmonic plane wave solution Insertion into the 1D wave eq. yields Note that k is the wavevector and is the direction of propagation; k without the arrow is the magnitude and is called the wavenumber.

Again use the result Thus, if then (in general) Consider a 1D propagation of an arbitrary wave of the form: f = f (z-vt) f(z) If z-vt = const. then f(const.) = const.1 z z1 t1 z2 t2 z3 t3 z 3> z 2> z1 t 3> t 2> t1

The same analysis can be performed of course for harmonic waves: If z-vt = const. then Ey(const.) = const.1 Ey z z1 t1 z2 t2 z 3> z 2> z1 z3 t3 t 3> t 2> t1 Same analysis can be applied in 3D for a plane wave: Surfaces on which the amplitude has a constant phase form a set of planes which are perpendicular to the propagation direction. For harmonic plane waves:

Planes are such that the phase defines a set of planes:

Use the vector identity using As shown before, it’s possible to express in 3D using a complex field representation:

With the complex representations, it is possible to derive explicit relations between E, B and k: Let’s examine the flow of energy again using the Poynting vector S:

Therefore, we can define irradiance as Average Energy Area·time In older texts (and in discussion) the term “intensity” is also used.

The energy per unit volume or energy density stored in the fields can be written as before Note, again a factor of ½ must be added for the time averages: The units Thus, similar to the Poynting vector, the E-M momentum P per unit volume that exerts a radiation pressure is given by:

Consider a slab of E-M radiation whose thickness is ct and cross-sectional area A: Vol.=Act k If the light is absorbed by an object, the momentum transfer is given by the impulse force·time: Area A ct Thus, the energy/vol. contained in the E-M propagation also represents the pressure exerted on an object. For example, if Eyo= 1 V/m, then <Pr> = 4.4 x 10-12 N/m2  10-17 atm. Note also that