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Old Feb 11th 2009, 02:09 AM   #1
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Optimization of S/N-ratio using Beer-Lambert and Inverse Square Law

Hello!

I am writing my master's thesis and have run into a problem. It is somewhat embarrassing because it feels like such a simple problem and I should be able to solve it by myself but I just seem to have gotten stuck in the same (faulty) thinking and need a push in the right direction.

My problem:

I need to calculate the maximal signal-to-noise-ratio for a setup consisting of a narrow-bandwidth light source located at a distance from a equally narrow-bandwidth detector. In between is a gaseous medium which partially absorbs electromagnetic radiation at the wavelength of the source.

According to the Beer-Lambert Law the intensity reaching the detector is decreasing exponentially in proportion to the distance traveled through the medium. Beer Lambert law - Wikipedia, the free encyclopedia

Also, according to the Inverse Square Law, the intensity of the radiation received by the detector drops proportionally to the square of the distance between source and detector. Inverse-square law - Wikipedia, the free encyclopedia

The maximum S/N-ratio is defined as the highest measured intensity drop due to absorption by the medium (compared to a reference signal) and thus increases with the distance due to the B-LL but at the same time decreases due to the ISL.

The light source beam has a spread of no more than 10 degrees off axis. From this follows that the lower distance boundary is about 6 cm since that is the distance at which all of the intensity is received by the detector. Further from the detector some light hits outside of it.

I hope someone takes an interest in this problem and can point me to the solution. I have omitted my own calculations at this point so as to get completely fresh ideas but will be happy to answer any questions you might have.

Thank you!
G
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Old Feb 12th 2009, 01:06 AM   #2
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I see the implicit question. Does Beers law include the loss of signal due to the inverse square law as well as due to absorbtion. After all, the article says it is an empirical law. I don't claim to understand this yet.

Most definitions of S/N that I have seen involve the ratio of the power of signals. Is the "maximum S/N ratio" actually defined as an intensity drop? Are you saying it is
(intensity of signal at detector - intensity of signal at source)/ (intensity of reference wave at detector - intensity of reference wave at source) ?

I don't understand the idea of a "reference wave". Does that represent "noise"?
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Old Feb 12th 2009, 02:00 AM   #3
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Thank you for your reply, I will try to answer your questions.

I do not think Beer includes the Inverse Square Law since nothing demands the radiation sent through the medium is expanding. If the beams are all parallel the Inverse Square Law does not apply, right? Then the intensity decrease is caused only by photons being absorbed by the medium.

I mean my signal strength equals the absolute intensity drop due to absorption of photons (as compared to an identical reference beam traveling through a non-absorbent medium). Sorry if I used confusing terms. By measuring the drop in intensity the concentration of absorbing gas can be calculated, which is what my work is about.

In essence:
- The longer the path through the medium, the larger the relative intensity drop because of absorption.
- The longer the path through the medium, the smaller the absolute intensity drop because of beam spread.

Hope this clarified things.
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Old Feb 12th 2009, 08:00 PM   #4
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No, that doesn't clarify what your definition of S/N is. I am particularly puzzled by the statement "The longer the path through the medium, the smaller the absolute intensity drop because of beam spread".

As to S/N being a ratio ,suppose there is no absorption and the drop in intensity at distance r is accounted for by multiplying some "at the source" level of intensity by a factor of $\displaystyle \frac {k}{r^2} $, That multiplication wouldn't change a ratio of intensities.

If we are talking about a ratio of "intensity drops" then we have something like:
$\displaystyle S/N = \frac { I_s - I_s \frac {k}{r^2}} {I_b - I_b \frac {k}{r^2} } $
$\displaystyle = \frac { I_s ( 1 - \frac {k}{r^2})} {I_b ( 1 - \frac {k}{r^2}) } $
$\displaystyle = \frac {I_s}{I_b} $
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Old Apr 22nd 2009, 09:57 PM   #5
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A simple solution is to keep the light source at the focus of a convex lens and generate a parallel beam which is then sent thru; the medium as this is what is done in professional/ industrial equipment. Else if a laser of suitable wavelength is available that may be used.
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