VLE 3 Wave optics
.pdfOptics and optical technologies
Lecture part 03:
Wave optics
1
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Additional literature
•E. Hecht: Optik. Oldenburg Verlag
•F. Pedrotti; L. Pedrotti: Optik für Ingenieure. Springer Verlag. 2005 (www.springerlink.com)
•B. Saleh: Grundlagen der Photonik. Wiley VCH. 2008
2
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Overview
1.Basics of wave motion
2.Three dimensional wave description
3.Electro magnetic waves
4.Superposition, interference
5.Interference for measurement instrumentation
6.Fresnel-Huygens-Principle
7.Diffraction basics
8.Fresnelund Fraunhofer-diffraction
9.Double slit and grating
3
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
1. Basics of wave motion
4
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Types of waves
Transverse waves
Propagation direction
Oscillation direction
e.g. Water waves
source: www.gymnasium-parsberg.de, 07.10.09
Longitudinal wave
Propagation direction
source: www.baunetzwissen.de, 07.10.09
Oscillation direction e.g. Sound waves
source: http://leifi.physik.uni-muenchen.de, 07.10.09
5
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Waves (1)
In general: |
|
|
3,5 |
|
|
|
|
A wave is a moving |
|
|
3 |
v = 1 m/s |
|
|
|
Distortion |
|
t=0s |
|
|
|
|
|
|
|
|
|
|
|
||
|
t=1s |
|
|
|
|
|
|
|
|
2,5 |
|
|
|
|
|
|
|
t=2s |
|
|
|
|
|
Assumption: moves with constant |
|
|
|
|
|
||
t=3s |
2 |
|
|
|
|
||
Speed in direction |
(x,t) |
|
|
|
|
|
|
|
|
|
|
|
|
||
f x,t |
|
|
1,5 |
|
|
|
|
|
|
|
|
|
|
||
|
|
1 |
|
|
|
|
|
One dimensional wave function: |
|
|
|
|
|
||
|
0,5 |
|
|
|
|
||
(x, t) f (x t) |
|
|
|
|
|
||
|
0 |
|
|
|
|
||
|
|
|
|
|
|
|
|
|
-2 |
-1 |
0 |
1 |
2 |
3 |
4 |
|
|
|
|
x |
|
|
|
Example: x 3 / 10x2 |
1 f x |
x, t 3 / 10 x t 2 |
1 |
|
|
||
Moving of the wave in positive x-direction: |
|
|
→Representation in the diagram for = 1 m/s at different points in time, shape of the wave does not change
→For (x+ t) the wave would move in negative x-direction.
6
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Waves (2)
Representation as a differential equation:
2 |
|
1 2 |
→ One-dimensional wave equation |
|
x2 |
2 |
t2 |
|
Harmonic wave: Sine - / cosine function
x,t f x t Asin k x t
with A – Amplitude, k – wavenumber
→ Solution of the wave equation
Waves are periodic in time and space.
The periodicity in space is called wavelength
→ x,t x ,t
This means for a harmonic wave that the argument changes by 2
→ |
sin k x |
|
t |
|
sin k x |
|
|
|
t |
|
sin k x |
|
t |
|
|
||||||
|
|
|
|
|
|
|
2 |
|
|||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
Hence it follows: |
|
k |
|
2 |
|
→ |
k 2 |
|
with k being called wave number |
||||||||||||
|
|
|
7
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Waves (3)
|
|
|
|
|
|
x/ |
|
) = A sin |
|
|
|
|
|
|
|
|
(x) = A sin (2 |
|
|
|
|
|
|||
|
1 |
|
|
|
|
|
|
|
|
|
|
|
(*A) |
0,5 |
|
|
|
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
-0,5 0 |
1 |
2 |
3 |
4 |
|
|
5 |
6 |
7 |
8 |
9 |
|
-1 |
= 2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(* ) |
|
|
|
|
The argument of the sine-function is called phase φ.
At = 0, , 2 , 3 , … (x) = 0
→ also: (x) = 0 at = 0, /2, , 3 /2, …
8
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Waves (4)
Temporal period
→x,t x,t
→sin k x t sin k x t sin k x t 2
With its: : |
k or2 |
The period is equal to the time The inverse is the frequency
→1
2 |
→ |
|
|
||
|
2 |
|
|
|
per wave
Further derived quantities: angular velocity
2 2 wave number
1
Equivalent representation of a harmonic wave: |
Asin kx t |
9
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany
Waves (5)
Phase angle: |
|
x,t |
|
Asin kx |
t |
|
|
|
kx t
→ at x,t t 0,x 0 0,0 0
General case: x,t Asin kx t
with – original phase (phase constant)
→ x,t kx t
Change of the phase over time:
|
|
|
→ is equal to the angular velocity |
|
|
||
|
t x |
|
|
Chang of the phase over distance:
|
|
k |
→ is equal to the wavenumber k |
|
|
||
|
x t |
|
|
10
Prof. M. Schmidt
Institute of Photonic Technologies, Univ. Erlangen, Germany