VLE 3 Wave optics
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Waves (6)
Phase velocity: |
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→ Speed, the wave front is moving
Complex representation of the wave equation:
x,t Aei t kx Aei
→ Real part: |
Re Aei t kx Acos t kx |
→ Imaginary part: |
Im Aei t kx Asin t kx |
Both representation are suited as description of a harmonic wave.
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
2. Three dimensional wave description
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Examples of three dimensional waves
Plane wave
Spherical wave
Circular wave
source: http://www.matheplanet.com, Stand 18.05.2009
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Plane wave (1)
The simplest example of a three dimensional wave is the plane wave Condition: All planes with the same phase are making a sea of planes
→ are normally perpendicular to the movement direction
Mathematical description of a plane which is perpendicular to the vector k and intersects with the point (x0,y0,z0) is described as follows:
Position vector r xex yey zez
The vector starts in a arbitrary point 0 and ends in the point (x,y,z).
Similar description:
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r r0 x x0 ex y y0 ey z z0 ez |
So: |
r r0 k . 0 |
With this the vector r r0 is perpendicular to a plane
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Plane wave (2)
With |
k kx ex ky ey kz |
ez |
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kx x x0 k y y y0 kz z z0 0 |
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kx x k y y kz z a |
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With |
a kx x0 ky y0 kz z0 |
const |
The shortest equation of plane perpendicular to k is:
k r a konst
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Plane wave (3)
A set of planes with sinusoidal variations r :
r Asin k r
r Acos k r
Or r Aeik r
For every of these terms r is constant above every plane which is defined by |
k r konst. |
→ Function repeats itself in a distance of in direction of k |
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Plane waves (4)
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Only a few planes are shown |
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Planes with the same color shall |
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have the same amplitude |
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Periodicity: |
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k |
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Andk being called wave vector |
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k / k is the unit vector parallel |
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to k
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Wave vector and wave front
Scientific notation:
Aei k r Aei k r k / k Aei k r ei k
Hence the following has to be true:
ei k |
1 ei 2 |
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Therefore: |
k 2 |
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The vector k |
is the wave vector and k is the wave number |
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Introduction of the time dependence (same as one dimensional wave):
r, t Aei k r t
with A, , k constant
A plane, connecting all points in time with the same phase is the: wave front.
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Three dimensional wave equation
Three dimensional wave equation:
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With k |
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Simplified representation with the nabla operator: |
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
3. Electromagnetic wave
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Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany