PHYS 408Applied Optics (Lecture 4)JAN-APRIL 2016 EDITIONJEFF YOUNGAMPEL RM 113

Quick review of key points from last lecture

Transverse electromagnetic plane waves have the property that E0 and H0 and k are all mutually orthogonal

Plane waves are valid solutions of the Maxwell equations in uniform, isotropic dielectric media

They are characterized by wavefronts that propagate parallel to the wave’s k vector with a phase velocity =c0/n

The medium’s dielectric constant or refractive index also factors into the ratio of the E0 and H0 complex amplitudes as per;

The Poynting Vector and Field Energy DensitiesConsider the vector defined by the cross product of E and H. This is referred to as the Poynting Vector

What properties do you note about this vector, for a transverse electromagnetic plane wave as discussed last day?

- Direction?- Units?

Electromagnetic Energy con’t?

Bt

HDt

ES

To gain more insight

Discuss the physics of each term separately, starting with the middle one on the right hand side.

Units? Fundamental nature of P, and F=ma…

Using the relations to eliminate D and B.

Electromagnetic Energy con’tUse the divergence theorem to get an integral form of these equations.

Electromagnetic Energy con’t

How does one interpret this equation then? (using words)

Energy Density in the Electric and Magnetic Fields

Returning to earlier version

If you assume D=eE and B=mH, show that

Bt

HDt

ES

BHDEt

S21

21

What would you define as “the electric energy density”, and the “magnetic energy density”?

How do these differ from the electric and magnetic field energy densities?

The Poynting Vector for a Monchromatic Electromagnetic Wave

If you were to average this quantity over a timescale of say 25 fs, when the frequency of the field corresponded to green light, what would this equation simplify to?

How specific is this?

Does this apply to arbitrary macroscopically averaged electric and magnetic fields?Does this apply to only plane waves?

What are the functional arguments of S, E and H in this equation?

From last time, for transverse electromagnetic waves in a uniform medium

Using this form for E and H, what do the Maxwell Equations impose as a condition on the relationship between Eo and Ho?

k.E0= k.H0= 0

Find the time averaged Poynting Vector, and electric and magnetic energy densities

Con’t

Interpret with a diagram of a cylinder with uniform energy density of W

What are the time averaged electric and magnetic energy densities in this TEM case?

Other solutions to the Wave Equation

Easy to prove away from the origin!

What is k in this equation?

What do the wavefronts look like for this type of wave?

Spherical wavefronts

Approximations

If you are 1 m away from the source of a spherical wave with wavelength 500 nm, what fraction of a wavelength would the wavefront deviate from a planewave over a distance of 1 cm?