What is Omega in waves

refraction


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Subsections


(See Tipler, Physik [Tip94, 1032])

As each Huygens' elementary wave a periodic oscillation with a given frequency represents, the frequency does not change when moving from one medium to the second. Since the speed of propagation is smaller, applies to the wavelength

(3.5)

In a medium with a Refractive index the wavelength is smaller. So has red light in glass the wavelength .

Geometry of Refraction

We now consider the path that light travels inside a medium. We take into account that the speed in the medium is around the Refractive index is smaller. From the right triangle we know that


Next is

So it applies
(3.8)

We cut with and put and and get that.





materials

(PDF)

With this law there is only always a solution if is. Otherwise there is the angle of the Total reflection. When the incident light beam from the optically denser medium against the interface normal makes the angle and the angle of the resulting light beam against the interface normal in the optically thinner medium is, the law of refraction just has a real solution.

(3.9)

For angles greater than light is totally reflected from the optically thinner medium. The reflection happens at a depth of about within the optically thinner medium.

Total reflection

(See Tipler, Physik [Tip94, 1035]) (See Gerthsen, Physik [GV95, 485])

Transport of light in a step index fiber

If light is coupled into the optical fiber at an angle close to the axis of the optical fiber, then the light will with Total reflection transported. Only light that hits the fiber core within the acceptance angle is transported further. When the fiber is bent, some of the light leaves the fiber: bends in the fiber increase the losses.

When the fiber core has the diameter is the effective path from the angle dependent on the axis. The hypotenuse is long, the direct route would be . The relative change in length is

(3.10)

The transit time therefore depends on how the light travels through an optical fiber. In addition, dispersion occurs. is on all glasses
(3.11)

Therefore, the running time for the different colors is also different. There is is also
(3.12)


Dispersion

(See Tipler, Physik [Tip94, 1038])

In general, the phase velocity of a wave depends on the frequency and the medium. For light this means that every color has its own speed of propagation.

Left: beam path through a prism. Right: dispersion of some materials

Through the dispersion of the light, which means that the refractive index depends on the wavelength, the different colors are refracted differently. Every time light passes through the air-matter interface, different colors are refracted differently. This has the following effects:

  • Chromatic aberration in lenses (color fringing)
  • the possibility of using a prism as a spectral apparatus
  • the divergence of signals in optical fibers
  • the Rainbow

Spring model for the dispersion according to (see Känzig, Mechanik und Wellenlehre [Kän78, 292]).

We consider a longitudinal wave on a spring-mass system. The equation of motion for the nth mass is

(3.13)

analogous to the equation for an inner pendulum with coupled pendulums. At very low frequencies all masses oscillate in phase: as with the coupled pendulums, the movement of all masses in the same direction gives the lowest frequency, which here, since we assume an infinite number of masses, is zero. The maximum frequency is obtained when two adjacent masses oscillate in opposite directions. A higher oscillation frequency is not possible. The minimum wavelength is and accordingly . We sit and receive
(3.14)

We put as a temporary solution for at: . Since the oscillation is only defined for discrete positions, we replace and received as a final solution
(3.15)

Inserted into the equation of motion we get






Dispersion relation for spring chains with two different atoms.

If a spring chain is formed with a regular arrangement of two unequal masses, there is an optical branch to the known from the previous embodiments. In addition, there are frequencies for which there is no real one -Vector there. These frequencies (or above also these energies) no propagating waves are possible. If there are both longitudinal and transverse waves, the dispersion relation shows not one but three branches of acoustic phonons.

Gravity waves in deep water have the dispersive relationship

(3.17)

One consequence is that very long waves are very fast (e.g. tsunamis)

  • Then

A pulse or a wave group consists of waves of neighboring frequencies. Analogous to modulation3.2 a pulse consists of an envelope and a phase, which does not carry any information. A longer calculation [Kän78] shows that the resulting wave function consists of harmonic wave as well as the modulation . The resulting wave is

(3.18)





In our spring-mass system, if is. That is, the pulse that carries the information is stationary. If is not constant, the shape of the pulse changes because the different frequency components spread at different speeds.

possible solutions

  • Dispersion compensation. It is complex and is mainly used in short-pulse laser systems.
  • Operation of the system at a wavelength at which the dispersion is minimal, i.e. is as constant as possible. This is used in optical communication (wavelengths 1300 nm and 1500 nm).
  • The data rate is set to lower values, so the pulses are broadened and the errors caused by the dispersion are minimized. Up to a reduction in the transmitted data rate by a factor of 2, the loss of speed can usually be minimized by using compression algorithms.


Next page:The Fermat principle Upwards:light Previous page:reflection
This version is out of date. You can find a more up-to-date version at http://wwwex.physik.uni-ulm.de/lehre/gk3a-2003Othmar Marti
Experimental physics
Ulm University