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**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. Maxwell’s equations have three major equivalent forms. 1) General form of Maxwell’s equations: It is also called Maxwell’s equations in vacuum. It actually applies to all cases, either in vacuum or in a medium. It is thus called the general form of Maxwell’s equations.

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**Definitions of the auxiliary fields:**

Our text was wrong in the definition of H. Bound charge and bound current density: E – Electric field H – Magnetic field D – Electric displacement 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 M – Magnetization Refer to an electrodynamics textbook. E.g., Griffiths. We then come to 2) Maxwell’s equations in matter: It contains no more physics than the general form, except that the charge density and the current density are decomposed into free and bound. It will be extensively used in our course. We then drop the “f” subscripts.

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**Material equations (Constitutive relations):**

Note: e and m are of essential importance in our course. They characterize the materials and are something you can explore all through your life. They are position dependent in an inhomogeneous medium. They are tensors in an anisotropic medium. They depend on the field strength (E and H) in a nonlinear medium. We then come to 3) Maxwell’s equations in a homogeneous medium: It is valid only in a homogeneous medium. It takes the same general form, with the changes

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**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 the two media. Gauss’ theorem: Stokes’ theorem: n – Unit normal to the surface s – Surface charge density K – Surface current density

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**1.2 Poynting’s theorem and conservation laws**

Electromagnetic waves carry energy and momentum. Conservation of the energy of the electromagnetic fields requires: Work done by the electromagnetic fields on the electric charges inside the volume Increase of the energy of the electromagnetic fields inside the volume Energy flowing in through the boundary surface of a volume = + Considering energy transfer in a unit volume and in a unit time: U S W S – Energy flow through a unit area in a unit time U – Energy density of the electromagnetic fields W – Work done by the electromagnetic fields on the electric charges in a unit volume Question: What are the expressions for S, U and W ?

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**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.

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**Lecture 3 Wave equations**

1.3 Complex function formulism Lights are 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 is often used in nonlinear optics.

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**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 multiplying fields, and multiplying a field with a scalar or tensor which are possibly complex numbers.

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**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.

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**Solution to the wave equations: Plane waves**

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

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**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:

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**Lecture 4 Propagation of a laser pulse**

1.5 Propagation of a laser pulse; group velocity Intense ultra-short laser pulses (Dt ~ second, femto-second) are extremely 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 colors. 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

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**Example: A Gaussian laser pulse**

The Fourier transform of a Gaussian pulse is again a Gaussian distribution. All E(w) are in phase. This is a transform-limited pulse.

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**Lecture 5 Group velocity**

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 t =t vg vp

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**More about group velocity:**

Group velocity dispersion: Group velocity dispersion broadens or compresses a laser pulse. distorts a laser pulse.

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**1.6 Dispersion and the Sellmeier equations**

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: (does negative work) Newton’s second law of motion: + E

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Solution: x0 is frequency dependent Electric polarization (and thus e and n ) is frequency dependent n(w). Electric polarization (= dipole moment density): Dispersion equation:

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**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)

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**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 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. Example: BK7 glass Coefficient Value B1 B2 ×10−1 B3 C1 ×10−3 μm2 C2 ×10−2 μm2 C3 ×102 μm2

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Prof. David R. Jackson Adapted from notes by Prof. Stuart A. Long Notes 4 Maxwell’s Equations ECE 3317.

Prof. David R. Jackson Adapted from notes by Prof. Stuart A. Long Notes 4 Maxwell’s Equations ECE 3317.

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