Introduction to Electromagnetics; Maxwell's Equations and Derivation of the Wave Equation for Light; Polarization Justificat
The 3D wave equation
Plane wave Spherical wave
MIT 2.71/2.710 03/11/09 wk6-b-13 Planar and Spherical Wavefronts
Planar wavefront (plane wave):
The wave phase is constant along a planar surface (the wavefront).
As time evolves, the wavefronts propagate at the wave speed without changing; we say that the wavefronts are invariant to propagation in this case.
Spherical wavefront (spherical wave):
The wave phase is constant along a spherical surface (the wavefront).
As time evolves, the wavefronts propagate at the wave speed and expand outwards while preserving the wave’s energy.
MIT 2.71/2.710 03/11/09 wk6-b-14 Wavefronts, rays, and wave vectors
Rays are:
k 1) normals to the wavefront surfaces 2) trajectories of “particles of light”
Wave vectors:
At each point on the wavefront, we may assign a normal vector k
This is known as the wave vector; k it magnitude k is the wave number and it is defined as
k
MIT 2.71/2.710 03/11/09 wk6-b-15 3D wave vector from the wave equation
x wavefront
kx k
wave vector kz z ky y
MIT 2.71/2.710 03/11/09 wk6 -b-16 3D wave vector and the Descartes sphere
The wave vector represents the momentum of the wave. Consistent with Geometrical Optics, its magnitude is constrained to be proportional to the refractive index n (2π/λfree is a normalization factor)
In wave optics, the Descartes sphere is also known as Ewald sphere or simply as the k-sphere. (Ewald sphere may be familiar to you from solid state physics)
MIT 2.71/2.710 03/11/09 wk6-b-17 Spherical wave
“point” source Outgoing rays
Outgoing wavefronts (spherical)
MIT 2.71/2.710 03/11/09 wk6-b-18 Dispersive waves
Dispersion curves for glass Fig. 9X,Y in Jenkins, Francis A., and Harvey E. White. Fundamentals of Optics. 4th ed. New York, NY: McGraw-Hill, 1976. ISBN: 9780070323308. (c) McGraw-Hill. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. $! ' /011'23451 #26')781 $98')781 +08')781 #" -.* #,
'#! #! ! '+ ! & guided light ! " Dispersion diagram for metal waveguide of width cut-off a=1µm frequency ' MIT 2.71/2.710 ! " #! #" $! ( !# 03/16/09 wk7-a- 1 %&!'#! ') * Superposition of waves at different frequencies
Fig. 7.16a,b,c in Hecht, Eugene. Optics. Reading, MA: Addison-Wesley, 2001. ISBN: 9780805385663. (c) Addison-Wesley. All rights reserved. This content is excluded from ourCreative Commons license. For more information, see http://ocw.mit.edu/fairuse .
MIT 2.71/2.710 03/16/09 wk7-a- 2 Group and phase velocity
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MIT 2.71/2.710 03/16/09 wk7-a- 3 Spatial frequencies
We now turn to a monochromatic (single color) optical field. The field is often observed (or measured) at a planar surface along the optical axis z. The wavefront shape at the observation plane is, therefore, of particular interest. x observation planes x θ
z
Plane wave Spherical wave
MIT 2.71/2.710 03/16/09 wk7-a- 4 Spatial frequencies
Plane wave Spherical wave
MIT 2.71/2.710 03/16/09 wk7-a- 5 Today
• Electromagnetics – Electric (Coulomb) and magnetic forces – Gauss Law: electrical – Gauss Law: magnetic – Faraday’s Law – Ampère-Maxwell Law – Maxwell’s equations ⇒ Wave equation – Energy propagation • Poynting vector • average Poynting vector: intensity – Calculation of the intensity from phasors • Intensity
MIT 2.71/2.710 03/18/09 wk7-b- 1 Electric and magnetic forces
I
r q q´ F + + dl free charges r F
I´ Coulomb force Magnetic force
(dielectric) permitivity (magnetic) permeability of free space of free space
MIT 2.71/2.710 03/18/09 wk7-b- 2 Electric and magnetic fields
Observation Generation
F velocity E v static charge: electric E q + ⇒ field + electric field ⊗ electric magnetic B charge induction
B v moving charge (electric current): + ⇒ Lorentz force + magnetic field
MIT 2.71/2.710 03/18/09 wk7-b- 3 Gauss Law: electric field
E E da da + + V
A
Gauss theorem
ρ: charge density
MIT 2.71/2.710 03/18/09 wk7-b- 4 Gauss Law: magnetic field
da there are no magnetic B charges V
A
Gauss theorem
MIT 2.71/2.710 03/18/09 wk7-b- 5 Faraday’s Law: electromotive force
E dl B(t) (in/de)creasing
A C
Stokes theorem
MIT 2.71/2.710 03/18/09 wk7-b- 6 Ampère-Maxwell Law: magnetic induction
B B dl dl
E I
current capacitor A A C C
Stokes theorem
MIT 2.71/2.710 03/18/09 wk7-b- 7 Maxwell’s Equations (free space)
Integral form Differential form
MIT 2.71/2.710 03/18/09 wk7-b- 8 Wave Equation for electromagnetic waves
MIT 2.71/2.710 03/18/09 wk7-b- 9 MIT OpenCourseWare http://ocw.mit.edu
2.71 / 2.710 Optics Spring 2009
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