Review of waves
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Review of Waves. Properties of electromagnetic waves in vacuum : Waves propagate through vacuum (no medium is required like sound waves) All frequencies have the same propagation speed, c in vacuum.

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Review of Waves

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Review of waves

Review of Waves


Review of waves

Properties of electromagnetic waves in vacuum:

Waves propagate through vacuum (no medium is required like sound waves)

All frequencies have the same propagation speed, c in vacuum.

Electric and magnetic fields are oriented transverse to the direction of propagation. (transverse waves)

Waves carry both energy and momentum.


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E and B fields in waves and Right Hand Rule:

f+ solution

Wave propagates in E x B direction

f- solution


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Special Case Sinusoidal Waves

Wavenumber and wavelength

These two contain the same information


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Special Case Sinusoidal Waves

Introduce

Different ways of saying the same thing:


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Polarizations

We picked this combination of fields: Ey - Bz

Could have picked this combination of fields: Ez - By

These are called plane polarized. Fields lie in plane


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Which of the following are valid EM waves?

1.

Yes

No

A.Yes

B. No

2.


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3.

A.Yes

B. No

4.

What direction is this wave propagating in?

Y

Z

X

None of above


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z

Direction of propagation

Ex

.

Wave Vector

By , Bz

y


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Energy Density and Intensity of EM Waves

Energy density associated with electric and magnetic fields

For a wave:

Thus:

Units: J/m3

Energy density in electric and magnetic fields are equal for a wave in vacuum.


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We now want to expand the picture in the following way:

EM waves propagate in 3D not just 1D as we have considered.

- Diffraction - waves coming from a finite source spread out.

EM waves propagate through material and are modified.

- Dispersion - waves are slowed down by media, different

frequency waves travel with different speeds

- Reflection - waves encounter boundaries between media.

Some energy is reflected.

- Refraction - wave trajectories are bent when crossing from

one medium to another.

EM waves can take multiple paths and arrive at the same point.

- Interference - contributions from different paths add or

cancel.


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Waves emanating from a point source


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Wave crests

When can one consider waves to be like particles following a trajectory?

Motion of crests

Direction of power flow

• Wave model: study solution of Maxwell equations.

Most complete classical description. Called physical optics.

• Ray model: approximate propagation of light as that of particles following specific paths or “rays”. Called geometric optics.

• Quantum optics: Light actually comes in chunks called photons


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What is the difference?

Diffraction.

Comparable to opening size

Much smaller than opening size


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Diffraction


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Propagation of light through dielectric media

In a dielectric the Electric field causes molecules/atoms to become polarized.

Molecules align with field

Medium becomes “polarized”


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What happens when the electric field oscillates in time?

A current is induced.


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How is the Ampere-Maxwell Law modified?

Dielectric

For S2 there is both displacement current and real current


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Real current given by dielectric constant

Electric flux

Dielectric constant

In a dielectric material

Bottom Line:

Dielectric constant

Replace

by


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Static dielectric constants

To complicate matters, dielectric constant depends on frequency.

Dielectric function depends on frequency


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Consequences for EM Plane waves

Propagation speed changes in a material

Refraction

Ratio of E to B changes

Reflection

For sinusoidal waves the following is still true


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Index of refraction


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For sinusoidal waves the following is still true

Frequency is the same in both media

Wavelength changes


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A light wave travels through three transparent materials of equal thickness. Rank in order, from the largest to smallest, the indices of refraction n1, n2, and n3.

  • n1 > n2 > n3

  • n2 > n1 > n3

  • n3 > n1 > n2

  • n3 > n2 > n1

  • n1 = n2 = n3


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Example: Optical fiber

d

Fused silica: n=1.48

Vacuum wavelength l = 1550 nm

Wavelength in fiber l/n = 1045nm

D = 10 mm

P=1mW inside fiber

I=P/A=?

E=?

In vacuum

Rays bounce back and forth


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Reflection from surface

incident

transmitted

reflected

Region 1

Region 2

Reflection coefficient

What if

“Index matched”


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Ey, Bz

How to calculate reflection

At x=0 fields are continuous

x

region 1

0

region 2

For x<0

incident

For x>0

reflected

transmitted


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Continuity of Ey at x=0

x<0

x>0

Continuity of Bz at x=0

x<0

x>0

Solve for reflected & transmitted pulses


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Example: what fraction of power is reflected at boundary between glass and air?

Reflection coefficient

Air: n1=1.0003Glass: n2 =1.5

ratio of reflected to incident electric field amplitude

I=Power/Area

ratio of reflected to incident intensity


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Suppose the wave was incident on the boundary from the glass side

incident

transmitted

reflected

Region 1

Region 2

n1=1.5

n2=1.0

What is reflection coefficient?


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incident

transmitted

reflected

Region 1

Region 2

n1=1.5

n2=1.0

What is reflection coefficient?

r=0

r =.2

C. r =-.2

D. r =5


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Region 2 has higher velocity

Region 2 has lower velocity


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What if

Same as for reflection from a conductor


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Interference in 1 Dimension

incident

Incident and reflected fields add (superposition)

reflected

incident

reflected

Incident and reflected waves will interfere, changing the peak electric field at different points


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Case #1: no reflection r=0.

No interference,

E plotted versus x for several values of t

E plotted versus t for a single value of x.

Same peak value independent of x

Traveling wave


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Case #2: total reflection r= -1.

Anti-nodes

E plotted versus x for several values of t

nodes

How far apart are the nodes?

Anti-nodes?

What is peak E?

Case #3: total reflection r= +1.

E plotted versus x for several values of t


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= VSWR

Emax

Emin

Case #4: partial reflection r= -0.5

E plotted versus x for several values of t

How far apart are the minima?

The Maxima?

What is peak E?

When reflection is not total there are still local maxima and minima.

Voltage Standing Wave Ratio

Pronounced “vizwarr”


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Half Wavelength Window

Eliminates Reflection

incident

incident

transmitted

reflected

Region 1

Region 2

Region 1’

D

When dielectric #2 is an integer number

of half-wavelengths thick, no reflection


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CVD diamond window

Manufactured by elementsix

http://www.e6.com/wps/wcm/connect/E6_Content_EN/Home


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Summary

Waves are modified by dielectric constant of medium, k.

All our Maxwell equations are valid provided we replace

3. Speed of waves is lowered. (Index of refraction - n)

4. Frequency of wave does not change in going from one medium to another. Wavelength does.

5. Waves are reflected at the boundary between two media.

(Reflection coefficient)


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= VSWR

6. Reflected waves interfere with incident waves.

Distance between interference maxima/minima

Ratio of maximum peak field to minimum peak field

(Voltage standing wave ratio)


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Solution of the Wave equation

Where f+,- are any two functions you like,

and

is a property of space.

Represent forward and backward propagating wave (pulses). They depend on how the waves were launched


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Shape of pulse determined by source of wave.

Speed of pulse determined by medium


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What is the magnetic field of the wave?

Notice minus sign


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Ey (V/m)

The graph at the top is the history graph at x = 4 m of a wave traveling to the right at a speed of 2 m/s. Which is the history graph of this wave at x = 0 m?


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What is the frequency of this traveling wave?

  • 0.1 Hz

  • 0.2 Hz

  • 2 Hz

  • 5 Hz

  • 10 Hz


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What is the phase difference between the crest of a wave and the adjacent trough?

  • 0

  • π

  • π/4

  • π/2

  • 3 π/2


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