VLE 3 Wave optics
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Mathematical basics of the Maxwell’s equations
Divergence: is the tendency in every point to flow towards (divergence <0) or away (divergence >0) to another point close by. Is the divergence zero the field has no sources.
Rotation: is the double angular velocity in each point o a field |
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a moving point would have (speed and axis of the rotation). |
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Is the rotation zero the field is irroational. |
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(Nabla-Operator) |
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VyzVzxVx y
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Maxwell’s equations
The basic principle of the electromagnetic waves are the Maxwell’s equations.
The equations for a linear, homogenous, isotropic, non moving medium are:
E
0
B 0
E Bt
B 0 J 0 0 E t
E – electric field strength B – magnetic flux density j – total current density– total charge density
= 8.854187817620... × 10−12 (F·m−1) – permittivity of free space (electric constant)= 4π×10−7 V·s/(A·m) – permeability of free space (magnetic constant)
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Electromagnetic wave equation
Derivation of the wave equation in free space
First the curl operator is applied to eq. (4) :
B 0 0 t E
Eq. (3) is used to express E
B 0 0 2 B t 2
Using the simplification:
v v 2v
One can rewrite:
B B 2 B
In Cartesian coordinates:
B 2 B 2 B 2 B 2 Bx2 y2 z2
(5)
(6)
(7)
(8)
(9)
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Electromagnetic wave equation (2)
From equation (2) it follows that |
B 0 |
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B 0 0 |
2 B |
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(10) |
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Similar approach can be taken to derive an equation for the electric field:
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(11) |
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E |
t B |
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Using equation (4) the previous expression can be rewritten as:
E 0 0 |
2 E |
(12) |
t 2 |
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Using the same trick and remembering that = 0 :
E E 2 E 2 E |
(13) |
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Combining everything in one equation : |
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2 E |
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E 0 |
(14) |
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t 2 |
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Electromagnetic wave equation (3)
Wave equation for electromagnetic waves:
2 B |
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2 B |
(15) |
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2 E 0 |
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2 E |
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(cf. three-dimensional wave equation: 2 |
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Speed of light
Comparing equations (14), (15), and (17)
0 0 1
2
With being the speed of light in vacuum. New notation for speed of light: c
c 1 
0 0 299 792 458 m
s 3 108 m / s
For electromagnetic waves, the electric field and the magnetic field are nearly always orthogonal.
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Electromagnetic waves
Electric field
λ = wavelength (distance between two succeeding peak crests)
Distance
Magnetic field
Velocity of light
ν = frequency
(number of cycles that pass a certain point per second)
source: http://www.fe-lexikon.info, 14.10.2008
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Electromagnetic Spectrum
Visible spectrum (light)
Infrared
Cosmic |
Gamma |
hard |
medium |
soft |
Infrared |
Terahertz |
Radar MW-oven |
USW |
Medium wave |
high |
medium |
low |
radiation |
radiation |
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X-rays |
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radiatio |
radiation |
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Short wave |
long wave |
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frequent |
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Ultraviolet |
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Microwaves |
Radio |
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alternating currents |
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radiation |
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Wave lenght [m]
Frequency [Hz]
Source: www.chemie-im-alltag.de, 07.10.09
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wave frequency; |
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2,9979 108 m / sec |
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108 m / sec |
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2 - angular frequency; |
medium |
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2,9979 |
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
4. Superposition, interference
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Superpositionsprinzip, beat
Principle of superposition: Covering two (or more) waves Basic principle: 1 und 2 are both solutions of the wave equation,
thus, ( 1 + 2) is a solution of the wave equation, too.
Are two waves at the same place, they superimpose additively:
→ Interference
Wave with the same wavelength have in dependence of the phase
Amplification (constructive interference) or
Weakening (destructive interference).
With waves of different frequency beats are generated.
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany