PHYS 408 Applied Optics (Lecture 4) Jan-April 2017 Edition Jeff Young AMPEL Rm 113

Quick review of key points from last lecture 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; Transverse electromagnetic plane waves have the property that E0 and H0 and k are all mutually orthogonal

How do we know that light waves transport energy? Implications? v=c Can cut/weld with it (lots of other answers) EM waves are just fields, so what must distinguish a powerful laser beam and a not-so-powerful laser beam? The energy density in some volume must be related to the amplitude of the fields somehow

The Poynting Vector and Field Energy Densities Consider 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 parallel to k From Lorentz force equation E has units of Force/q and B has units of Force*s/q/L, so S has units of F^2 s/q^2/L/mu_o mu_o has units Energy s^2/q^2/L, so S has units F^2 s/q^2/L * q^2 L/Energy/s^2 which is F^2/Energy/s Multiply top and bottom by L^2 Units of Energy/L^2/s “Energy per unit area per unit time” Direction? Units?

Electromagnetic Energy con’t ?

To gain more insight Using the relations to eliminate D and B. 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…

Electromagnetic Energy con’t Use 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 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? Why is script S defined in terms of the real, measurable fields, rather than left in terms of complex waves? Ans: when products of fields are involved (note ME’s don’t involve and products of fields), can get into big trouble assuming complex fields…only useful when dealing with linear PDEs)

How specific is this? What are the functional arguments of S, E and H in this equation? \vec r No, only harmonics ones No, more general than plane waves….HARMONIC macroscopically averaged waves is the limitation Does this apply to arbitrary macroscopically averaged electric and magnetic fields? Does this apply to only plane waves?

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 Get them to correct this to time-averaged Poynting vector, and to note it is position dependent in general. Again, an exercise in being careful of equations specific to certain situations! Tie in back to first slide (beam carries energy, can burn/weld etc.)

Con’t What are the time averaged electric and magnetic energy densities in this TEM case? Interpret with a diagram of a cylinder with uniform energy density of W

How do we know that light waves transport energy? Implications? Power =I times cross sectional area

Preview of next Day

Other solutions to the Wave Equation Preview of next day 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

From Lorentz force equation E has units of Force/q and B has units of Force*s/q/L, so S has units of F^2 s/q^2/L/mu_o mu_o has units Energy s^2/q^2/L, so S has units F^2 s/q^2/L * q^2 L/Energy/s^2 which is F^2/Energy/s Multiply top and bottom by L^2 Units of Energy/L^2/s “Energy per unit area per unit time”