(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
where
(8) |
From this two approximation are applied so as to enable a relationship to a fourier transform to be made.
The general form of the expansion is
(9) |
Now we rearrange r by removing z fromt he square root as to match up with the general expresion
(10) |
So now we can presides with the expansion, taking only the first 3 terms as relevant
(11) |
Now this new r is substituted into huygens-Fresnel, qith the exponinent this is a simple thing but int he fraction r the contribution from the terms involving x and y are so small that they become in consequential. all this leads to the following equation which is the general form of the Fresnel approximation
(12) |
(13) |
This means that this part of the expanded exponent can be ignored to give
(14) |
The condition which should be satisfied so that this approximation can be used is
(15) |
where D is the linear dimension of the aperture.
A Fourier transform is normally an analytical calculation, but a computer does it discretely gemma smells, this two approaches can not be exactly related, therefore a correction factor must be found to enable an exact comparison and further more enable the computer to do the right calculation. this is done as follows.
(16) |
equationing releven parts
(17) |
(18) |
log both sides
(19) |
replacing k
(20) |
cancel down
(21) |
equating for:
(22) |
rearranged
(23) |
(24) |
rearranged
(25) |
we also know that
(26) |
(27) |
so
(28) |
canceling m and p
(29) |
(30) |
(31) |
There are element that will be constantly reapering through out this equation, i.e.
This is the start wave field put in:
(32) |
This is the end intensity when the calculation is done by hand:
(33) |
where
(34) |
This is the start wave field put in:
(35) |
This is the end intensity when the calculation is done by hand:
(36) |
where we use the Fresnel integrals
(37) |
(38) |
(39) |
(40) |
Another interesting piece of information is the Fresnel Number:
(41) |
As can be seen in the example below as the Fresnel number increases the distance decreases, also it can be seen how as the distance gets closer to the one used for a Fraunhofer approximation the shapes become the same.For this example all the wavelengths are 60mm and the widths is 5m.
* fresnelRect.py.txt: Code for Fresnel Rectangular Apperture
In this case the code has been split into two classes
When adding a lens it is just a case of ajusting the phase of the wave field. This is done by multipying the wave feild by the a phase factor before doing a normal fresnel approxiamtion.
(42) |
Where f is the focal length
NB The lens is still having problems and therefore still being worked on
Latex rendering error!! dvi file was not created.
I | Attachment | History | Action | Size | Date | Who | Comment |
---|---|---|---|---|---|---|---|
png | 2M.png | r1 | manage | 28.7 K | 11 Aug 2009 - 16:07 | GemmaSmith | This is a plot of the second moment, or in other words its a plot of the outside edge of constuctive interference |
png | Ciri.png | r1 | manage | 53.8 K | 07 Aug 2009 - 16:02 | GemmaSmith | 2-D Plot of Circular Function |
png | Gau.png | r1 | manage | 61.0 K | 07 Aug 2009 - 16:03 | GemmaSmith | 2-D Plot of Gaussian Function |
txt | Gaussian.py.txt | r1 | manage | 0.1 K | 07 Aug 2009 - 16:07 | GemmaSmith | Gaussian Function |
txt | Lens.py.txt | r1 | manage | 1.1 K | 11 Aug 2009 - 16:48 | GemmaSmith | Code for Lens |
png | Rect.png | r1 | manage | 46.9 K | 07 Aug 2009 - 16:05 | GemmaSmith | 2-D Plot of Rectangular Function |
txt | circular.py.txt | r1 | manage | 0.2 K | 07 Aug 2009 - 15:50 | GemmaSmith | Circular Function |
png | fra.png | r1 | manage | 43.0 K | 07 Aug 2009 - 16:45 | GemmaSmith | 1-D Plot for a Franuhofer Calculation, with: *wavelength = 60mm *z = 4km *w = 5m |
txt | fraunhoferRect.py.txt | r1 | manage | 0.8 K | 07 Aug 2009 - 17:02 | GemmaSmith | Code for Fraunhofer Rectangular Apperture |
png | fre0.1.png | r1 | manage | 55.2 K | 07 Aug 2009 - 16:58 | GemmaSmith | 1-D Plot for a Fresnel Calculation, with: *z = 4km *Fresnel Number = 0.1 |
png | fre1.png | r1 | manage | 50.0 K | 07 Aug 2009 - 16:48 | GemmaSmith | 1-D Plot for a Fresnel Calculation, with: *z = 400m *Fresnel Number = 1 |
png | fre10.png | r1 | manage | 72.1 K | 07 Aug 2009 - 16:59 | GemmaSmith | 1-D Plot for a Fresnel Calculation, with: *z = 40m *Fresnel Number = 10 |
png | fre4.png | r1 | manage | 57.7 K | 07 Aug 2009 - 17:00 | GemmaSmith | 1-D Plot for a Fresnel Calculation, with: *z = 100m *Fresnel Number = 4 |
txt | fresnelRect.py.txt | r1 | manage | 1.4 K | 07 Aug 2009 - 17:03 | GemmaSmith | Code for Fresnel Rectangular Apperture |
png | gua_fra.PNG | r1 | manage | 15.4 K | 10 Aug 2009 - 17:00 | GemmaSmith | 2-D Plot of gaussian through fraunhofer |
png | image.png | r1 | manage | 207.4 K | 11 Aug 2009 - 16:01 | GemmaSmith | Plots just after lens |
png | intensity.png | r1 | manage | 102.8 K | 11 Aug 2009 - 16:04 | GemmaSmith | This are intensity plot along z NB - this distance in z is at the top of each graph |
txt | intensity.py.txt | r1 | manage | 3.7 K | 11 Aug 2009 - 14:21 | GemmaSmith | Intensity Class Code |
txt | propagate.py.txt | r1 | manage | 1.3 K | 11 Aug 2009 - 14:21 | GemmaSmith | Propagation Class Code |
txt | rectangular.py.txt | r1 | manage | 0.2 K | 07 Aug 2009 - 15:51 | GemmaSmith | Rectangluar Function |
txt | rum.py.txt | r1 | manage | 1.3 K | 11 Aug 2009 - 14:11 | GemmaSmith | Test Code using Classes system and in 2-D |
png | testcirfra.png | r1 | manage | 31.6 K | 11 Aug 2009 - 13:55 | GemmaSmith | 2-D Plot of Circular Function through Fraunhofer |
png | testcirfre.png | r1 | manage | 223.8 K | 11 Aug 2009 - 13:52 | GemmaSmith | 2-D Plot of Circular Function through Fresnel |
png | testfreRec.png | r1 | manage | 330.4 K | 11 Aug 2009 - 15:00 | GemmaSmith | 2-D Plot of Rectangular Function through Fresnel |
png | testgaufra.png | r1 | manage | 27.4 K | 11 Aug 2009 - 14:02 | GemmaSmith | 2-D Plot of Gaussian Function through Fraunhofer |
png | testgaufre.png | r1 | manage | 307.0 K | 11 Aug 2009 - 14:03 | GemmaSmith | 2-D Plot of Gaussian Function through Fresnel |
png | testrecfra.png | r1 | manage | 35.1 K | 11 Aug 2009 - 13:56 | GemmaSmith | 2-D Plot of Rectangular Function through Fraunhofer |