Chapter 1 Electromagnetic Fields Lecture 1 Maxwells equations

Chapter 1 Electromagnetic Fields Lecture 1 Maxwell’s equations 1. 1 Maxwell’s equations and boundary conditions Maxwell’s equations describe how the electric field and the magnetic field are generated, and how they change in space and time. Lights are electromagnetic waves which obey Maxwell’s equations have three major equivalent forms. 1) General form of Maxwell’s equations: 1. It is also called Maxwell’s equations in vacuum. 2. It actually applies to all cases, either in vacuum or in a medium. It is thus the general form of Maxwell’s equations. 1

Definitions of the auxiliary fields: Our text is wrong in the definition of H. Bound charge and bound current density: Refer to an electrodynamics textbook. E. g. , Griffiths. E – Electric field H – Magnetic field D – Electric displacement We then come to 2) Maxwell’s equations in matter: 1. It contains no more physics than the general form, except that the charge density and the current density are decomposed into free and bound. 2. It will be extensively used in this course. B – Magnetic induction r – Electric charge density J – Electric current density e – Permittivity m – Permeability e 0 – Permittivity of vacuum m 0 – Permeability of vacuum P – Electric polarization We then drop the “f” subscripts. M – Magnetization 2

Material equations (Constitutive relations): In our course e and m are of essential importance. They characterize the materials. They are something you can explore throughout your life. 1. They are frequency dependent. 2. They are position dependent in an inhomogeneous medium. 3. They are tensors in an anisotropic medium. 4. They depend on the field strength (E and H) in a nonlinear medium. We then come to 3) Maxwell’s equations in a homogeneous isotropic medium: 1. They are valid only in a homogeneous isotropic medium. 2. They take the same general form, with the changes 3

Lecture 2 Energy of the electromagnetic fields Boundary conditions: Boundary conditions are the relations between the electromagnetic fields on two sides of the interface that separates two media. Gauss’ theorem: Stokes’ theorem: n – Unit normal to the surface s – Surface charge density K – Surface current density 4

1. 2 Poynting’s theorem and conservation laws Electromagnetic waves carry energy and momentum. Conservation of the energy of the electromagnetic fields requires: Energy flowing in through the boundary surface of a volume = Increase of the energy of the electromagnetic fields inside the volume + Work done by the electromagnetic fields on the electric charges inside the volume Define three quantities to be determined: S S – Energy flow through a unit area in a unit time U – Energy density of the electromagnetic fields U W – Work done by the electromagnetic fields on the electric charges in a unit volume W Considering energy transfer in a unit volume and in a unit time: Question: What are the expressions for S, U and W ? 5

Work done by the electromagnetic fields on the charges in a unit volume in a unit time: U – Energy density of the electromagnetic fields. Linear response assumed. Otherwise S – Poynting vector. Energy flow through a unit area in a unit time. Instantaneous intensity of light. Continuity equation. Poynting’s theorem. 6

Lecture 3 Wave equations 1. 3 Complex function formulism Light is most often described by sinusoidal time-varying fields: Complex representation: Let represent the field, and we always mean its real part. Complex representations have no problems with linear mathematical operations: It does not work for the product of fields: The correct way is: This method is needed in discussing the energy of light field and in nonlinear optics. 7

Time averaging of two sinusoidal products with the same frequency: Examples: Time-averaged Poynting vector and the energy density: Conventions: We then drop the tidal sign, but we need to be careful in 1) Multiplying two fields, and 2) Multiplying a field with a scalar or tensor which is possibly a complex number. 8

1. 4 Wave equations and monochromatic plane waves Each of the field vectors of light oscillates in space and time, which satisfies a wave equation. Assume the medium is isotropic (e and m are scalars), linear (e and m are independent of fields), and there is no free charge or free current (r =0, J =0). The Maxwell’s equations and the materials equations are mathematically symmetric in the exchanges of E −H, B D, and e −m. 9

Further suppose the medium is homogeneous (e and m does not change in space under a translational shift), the wave equations are A partial, linear, second order, homogeneous differential equation Solution to the wave equations: Plane waves Wavelength Period 10

Vector nature of the fields of electromagnetic waves (in an isotropic medium): In an anisotropic medium (e and m are tensors), only D and B are perpendicular to k. Time-averaged Poynting vector (intensity of light) and energy density: 11

Lecture 4 Propagation of a laser pulse 1. 5 Propagation of a laser pulse; group velocity Intense ultra-short laser pulses (Dt ~ 10 -15 second, femto-second) are particularly important in exploring the dynamic structure of atoms and molecules, and their interaction with light. Bandwidth of a laser pulse: A laser pulse can be treated as a sum of many plane waves with different wavelengths. The pulse has a bandwidth in frequency domain, which satisfies the uncertainty rule: In the language of Fourier transform, at a fixed point in space 12

Example: A Gaussian laser pulse The Fourier transform of a Gaussian pulse is again a Gaussian function. All E(w) are in phase. This is a transform-limited pulse. 13

Lecture 5 Group velocity: A laser pulse normally has a carrier and an envelope. Group velocity is the velocity at which the envelope of the laser pulse travels. t =0 vg vp t =t 14

More about group velocity: Group velocity dispersion: • Group velocity dispersion • broadens or compresses a laser pulse. distorts a laser pulse. 15

Lecture 6 Dispersion 1. 6 Dispersion and the Sellmeier equations How do we get n (l)? Electric polarization: The electric dipole moment per unit volume induced by an external electric field. For an isotropic and linear material, Atom = electron cloud + nucleus. How is an atom polarized ? Restoring force: Natural (resonant) frequency: + External force: Damping force: E (does negative work) Newton’s second law of motion: 16

Solution: x 0 is frequency dependent Electric polarization (and thus e and n ) is frequency dependent n(w). Electric polarization (= dipole moment density): Dispersion equation: 17

Quantum theory: w 0 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) n' Phase velocity n" Absorption (or amplification) Im (n) 18

Sellmeier equation: An empirical relationship between refractive index n and wavelength l for a transparent medium: • Sellmeier equations work fine when the wavelength range of interests is far from the absorption of the material. • The beauty of Sellmeier equations: are obtained analytically. • Sellmeier equations are extremely helpful in designing various optical devices. 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: BK 7 glass Coefficient Value B 1 1. 03961212 B 2 2. 31792344× 10− 1 B 3 1. 01046945 C 1 6. 00069867× 10− 3 μm 2 C 2 2. 00179144× 10− 2 μm 2 C 3 1. 03560653× 102 μm 2 19