Chapter 3 Electromagnetic Theory, Photons and Light August 29, 31 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 S d l E A time-varying magnetic field produces an electric field.

3.1.2 Gauss’s law - electric 3.1.3 Gauss’s law- magnetic Flux of electric field: For a point charge: , e0 is the permittivity of free space. Generally, For general materials, the permittivity , where KE is the relative permittivity (dielectric constant). 3.1.3 Gauss’s law- magnetic There is no isolated magnetic monopoles:

3.1.4 Ampere’s circuital law J d S d l B For electric currents: m0 is the permeability of free space. For general materials, the permeability , where KM is the relative permeability. Moving charges are not the only source for a magnetic field. Example: in a charging capacitor, there is no current across area A2 (bounded by C). i B E C A1 A2 Ampere’s law: A time-varying electric field produces a magnetic field.

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,

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: In free space the plane EM waves are transverse. For a linearly polarized wave, choose

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

Read: Ch3: 1-2 Homework: Ch3: 1,3,4,8,12 Due: September 7

September 5 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 charging a capacitor: E=q/e0A, 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 in the light propagation direction, Poynting vector: is the power across a unit area whose normal is parallel to S.

3.3.2 Irradiance For a harmonic, linearly polarized plane wave: Irradiance (intensity): The average energy transport across a unit area in a unit time. Time averaging: In a medium Note The inverse square law: The irradiance from a point source is proportional to 1/r2. Total power I·4pr2 = constant I1/r2. I  E0 2 E01/r. Example 3.2,3.3

3.3.3 Radiation pressure and momentum 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 hn, where n 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 Let P be the radiation pressure. Work done = PAcDt = uAcDt  P=u (radiation pressure = energy density) For light

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

Read: Ch3: 3 Homework: Ch3: 13,14,22,24,27,35 Due: September 14

3.4.1 Linearly accelerating charges September 7 Radiation 3.4 Radiation 3.4.1 Linearly accelerating charges Field lines of a moving charge t1 t2 ct2 c(t2-t1) Analogy: A train emits smoke columns at speed c from 8 chimneys over 360º. What do the trajectories of the smoke look like when the train is: still, moving at a constant speed, moving at a constant acceleration? t1 t2 ct2 c(t2-t1) Constant speed 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.

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

3.4.4 The emission of light from atoms Bohr’s model of H atom: E0 (Ground state) E1 (Excited states) Pump Relaxation (DE = hn) E∞ a0

Read: Ch3: 4 Homework: Ch3: 46 Due: September 14

September 10 Dispersion 3.5 Light in bulk matter 3.5.1 Dispersion Phase speed in a dielectric (non-conducting material): Index of refraction (refractive index): . KE and KM 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 e (w)? Lorentz model of determining n (w): 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.

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 (w):

Quantum theory: w0 is the transition frequency. For a material with several transition frequencies: Oscillator strength: Normal dispersion: n increases with frequency. Anomalous dispersion: n decreases with frequency. Re (n) Im (n) n'  Phase velocity n"  Absorption (or amplification)

Beauty of Sellmeier equations: are obtained analytically. Sellmeier equation: An empirical relationship between refractive index n and wavelength l for a particular transparent medium: Sellmeier equations work fine when the wavelength range of interests is far from the absorption regions of the material. Beauty of Sellmeier equations: are obtained analytically. Sellmeier equations are enormously helpful in designing various optical elements. 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. Example: BK7 glass Coefficient Value B1 1.03961212 B2 2.31792344×10−1 B3 1.01046945 C1 6.00069867×10−3 μm2 C2 2.00179144×10−2 μm2 C3 1.03560653×102 μm2

Read: Ch3: 5-7 Homework: Ch3: 54,55,57,62,66 (66 is Optional) Due: September 21

Everyday comes only once. Pengqian Wang