1. 1 Chapter 3 Electromagnetic Theory, Photons and Light September 5,8 Electromagnetic waves 3.1 Basic laws of electromagnetic theory Lights are electromagnetic waves. Electric fields are generated by electric charges or time-varying magnetic fields. Magnetic fields are generated by electric currents or time-varying electric fields. Maxwell’s wave equation is derived from the following four laws (Maxwell’s equations). 3.1.1 Faraday’s induction law Electromotive force (old term, actually a voltage): B d Sd S d ld l E A time-varying magnetic field produces an electric field.

2. 2 3.1.2 Gauss’s law - electric Flux of electric field: For a point charge:,  0 is the permittivity of free space. Generally, For general material, the permittivity, where K E is the relative permittivity (dielectric constant). 3.1.3 Gauss’s law- magnetic There is no isolated magnetic monopoles:

3. 3 3.1.4 Ampere’s circuital law For electric currents:  0 is the permeability of free space. For general materials, the permeability, where K M is the relative permeability. J d Sd S d ld l B i B E C A1A1 A2A2 Moving charges are not the only source for a magnetic field. Example: in a charging capacitor, there is no current across area A 2 (bounded by C). Ampere’s law: A time-varying electric field produces a magnetic field.

4. 4 3.1.5 Maxwell’s equations Gaussian’s divergence theorem: Stokes’s theorem: Maxwell’s equations in differential form: (integrals in finite regions  derivatives at individual points) In free space,

5. 5 3.2 Electromagnetic waves Applying to free space Maxwell’s equations, we have the 3D wave equations: 3.2.1 Transverse waves For a plane EM wave propagating in vacuum in the x direction: For linearly polarized wave In free space the plane EM waves are transverse.

6. 6 Harmonic waves: Characteristics of the electromagnetic fields of a harmonic wave: 1)E and B are in phase, and are interdependent. 2)E and B are mutually perpendicular. 3)E × B points to the wave propagation direction. x

7. 7 Read: Ch3: 1-2 Homework: Ch3: 1,3,7 Due: September 12

8. 8 September 10 Energy and momentum 3.3 Energy and momentum 3.3.1 Poynting vector E-field and B-field store energy: Energy density (energy per unit volume) of any E- and B-field in free space: (The first equation can be obtained from a charging capacitor: E=q/   A, dW=El·dq). For light, applying E=cB, we have  The energy stream of light is shared equally between its E-field and B-field. The energy transport per unit time across per unit area: Assuming energy flows along the light propagation direction,  Poynting vector: is the power across a unit area whose normal is parallel to S.

9. 9 For a harmonic, linearly polarized plane wave: Time averaging: 3.3.2 Irradiance Irradiance (intensity): The average energy transport across a unit area in a unit time. In a medium Note The inverse square law: The irradiance from a point source is proportional to 1/r 2. Total power I·4  r 2 = constant, I  E 0 2  E 0  1/r.

10. 10 3.3.3 Photons The electromagnetic wave theory explains many things (propagation, interaction with matter, etc.). However, it cannot explain the emission and absorption of light by atoms (black body radiation, photoelectric effect, etc.). Planck’s assumption: Each oscillator could absorb and emit energy of h  where  is the oscillatory frequency. Einstein’s assumption: Light is a stream of photons, each photon has an energy of 3.3.3 Radiation pressure and momentum Maxwell’s theory shows radiation pressure P = energy density: (Work done = PAc  t = uAc  t  P=u) For light

11. 11 Momentum density of radiation ( p V ): Momentum of a photon (p): Vector momentum: The energy and momentum of photons are confirmed by Compton scattering.

12. 12 Read: Ch3: 3 Homework: Ch3: 8,14,16,19,27 Due: September 19

13. 13 September 12 Radiation 3.4 Radiation 3.4.1 Linearly accelerating charges 0 t1t1 t2t2 ct 2 c(t 2 -t 1 ) Constant speed Field lines of a moving charge With acceleration Assuming the E-field information propagates at speed c. Gauss’s law suggests that the field lines are curved when the charge is accelerated. The transverse component of the electric field will propagate outward.  A non-uniformly moving charge produces electromagnetic waves. 0 t1t1 t2t2 ct 2 c(t 2 -t 1 ) Analogy: A train emits smokes at speed c from 8 chimneys over 360º. What do the trajectories of the smoke look like when the train is: 1)still, 2)moving at a constant speed, 3)moving at a constant acceleration.

14. 14 Examples: 1.Synchrotron radiation. Electromagnetic radiation emitted by relativistic charged particles curving in magnetic or electric fields. Energy is mostly radiated perpendicular to the acceleration. 2.Electric dipole radiation. Far from the dipole (radiation zone): Irradiance: 1)Inverse square law, 2)Angular distribution (toroidal). 3)Frequency dependence. 4)Directions of E, B, and S. + - + B E S 

15. 15 3.4.4 The emission of light from atoms Bohr’s model of H atom: a0a0 E 0 (Ground state) E1E1 (Excited states) Pump Relaxation (  E = h ) E∞E∞

16. 16 Read: Ch3: 4 Homework: Ch3: 37 Due: September 19

17. 17 September 15,17 Dispersion 3.5 Light in bulk matter Phase speed in a dielectric (non-conducting material): Index of refraction (refractive index):. K E and K M are the relative permittivity and relative permeability. For nonmagnetic materials Dispersion: The phenomenon that the index of refraction is wavelength dependent. 3.5.1 Dispersion  How do we get  (  )? Lorentz model of determining n (  ): The behavior of a dielectric medium in an external field can be represented by the averaged contributions of a large number of molecules. Electric polarization: The electric dipole moment per unit volume induced by an external electric field. For most materials Examples: Orientational polarization, electronic polarization, ionic polarization.

18. 18 Atom = electron cloud + nucleus. How is an atom polarized ? Restoring force: Natural (resonant) frequency: Forced oscillator: Damping force: (does negative work)  Newton’s second law of motion: Solution: Electric polarization (= dipole moment density): + E  Dispersion equation: Frequency dependent frequency dependent n (  ):

19. 19 For a material with several transition frequencies: Oscillator strength: Quantum theory:  0 is the transition frequency. Re (n) -Im (n) Normal dispersion: n increases with frequency. Anomalous dispersion: n decreases with frequency. n'  Phase velocity n "  Absorption (or amplification)

20. 20 Sellmeier equation: An empirical relationship between refractive index n and wavelength for a particular transparent medium: CoefficientValue B1B1 1.03961212 B2B2 2.31792344×10 −1 B3B3 1.01046945 C1C1 6.00069867×10 −3 μm 2 C2C2 2.00179144×10 −2 μm 2 C3C3 1.03560653×10 2 μm 2 Example: BK7 glass Sellmeier equations work fine when the wavelength range of interests is far from the absorption of the material. Beauty of Sellmeier equations: are obtained analytically. Sellmeier equations are extremely helpful in designing various optics. Examples: 1) Control the polarization of lasers. 2) Control the phase and pulse duration of ultra- short laser pulses. 3) Phase-match in nonlinear optical processes.

21. 21 Read: Ch3: 5-7 Homework: Ch3: 45,46,48,57 Due: September 26