The 3D wave equation

Plane wave Spherical waveMIT 2.71/2.710 03/11/09 wk6-b-13

Planar and Spherical Wavefronts

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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.

Wavefronts, rays, and wave vectors

kRays are:

1) normals to the wavefront surfaces

2) trajectories of “particles of light”

Wave vectors:

k

k

At each point on the wavefront, we may assign a normal vector k

This is known as the wave vector; it magnitude k is the wave number and it is defined as

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3D wave vector from the wave equation

k kx

ky kz

x

y

zwave vector

wavefront

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3D wave vector and the Descartes sphere

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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)

Spherical wave

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“point” source

Outgoing rays

Outgoing wavefronts (spherical)

Dispersive waves

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guided light

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Dispersion diagram for metal waveguide of width

a=1µm

Dispersion curves for glass

cut-offfrequency

(c) McGraw-Hill. All rights reserved. This contentis excluded from our Creative Commons license.For more information, see http://ocw.mit.edu/fairuse.

Fig. 9X,Y in Jenkins, Francis A., and Harvey E. White.Fundamentals of Optics. 4th ed. New York, NY:McGraw-Hill, 1976. ISBN: 9780070323308.

MIT 2.71/2.710 03/16/09 wk7-a-

Superposition of waves at different frequencies

2

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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

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 waveMIT 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

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Electric and magnetic forces

+ + r

free charges

q q´ F dl

Fr

I

I´

Coulomb force Magnetic force

(dielectric) permitivity (magnetic) permeabilityof free space of free space

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Electric and magnetic fields

Observation Generation

+ E

+

F velocityv static charge:

electric E q ⇒ field electric field

⊗ electricmagnetic B chargeinduction

moving charge (electric current):

v

+ +

B

⇒Lorentz force magnetic field

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Gauss Law: electric field

+ E da

E

+

A

V

da

Gauss theorem

ρ: charge density

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Gauss Law: magnetic field

B

A

V

there are no magnetic charges

da

Gauss theorem

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Faraday’s Law: electromotive force

dl E

B(t) (in/de)creasing

C

A

Stokes theorem

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Ampère-Maxwell Law: magnetic induction

B Bdl

I

CA

current capacitor

dl

Stokes theorem

A C

E

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Maxwell’s Equations (free space)

Integral form Differential form

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Wave Equation for electromagnetic waves

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