What units should be used for the cauchy dispersion formula?

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Discussion Overview

The discussion revolves around the appropriate units to use for the Cauchy dispersion formula, specifically regarding the parameters involved in the equation and their conversions. Participants explore the implications of unit conversions on the calculation of the index of refraction (IOR) using the formula.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant presents the Cauchy dispersion formula and their initial values for parameters A and B, expressing uncertainty about the results obtained.
  • Another participant suggests that the conversion of the dispersion coefficient B from square micrometers to square meters should involve multiplying by 10000002 instead of 1000000.
  • A participant questions the need to convert B to square meters if it is already in square micrometers, noting that the resulting calculation should be unit-less when the wavelength is also in micrometers.
  • There is a mention of a potential misunderstanding regarding the units of the C parameter, with one participant expressing confusion about whether it should be in square meters or square micrometers.
  • Another participant acknowledges a previous oversight regarding the source of the values used in their calculations, indicating a realization about the units involved.
  • One participant raises a suspicion that some sources may use meters for the wavelength, which could lead to confusion about the units for the C parameter.

Areas of Agreement / Disagreement

Participants express differing views on the correct units for the dispersion coefficient and the wavelength, leading to a lack of consensus on the appropriate conversions needed for the Cauchy formula.

Contextual Notes

There are unresolved questions about the assumptions regarding the units of the parameters and the conversions necessary for accurate calculations. The discussion highlights the dependency on the definitions of the units used in the formula.

spectral
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Hi everybody,

I would like to use the 'cauchy dispersion formula', ie (http://en.wikipedia.org/wiki/Cauchy's_equation):[/PLAIN]

eta = A + B / w²

Where :
eta is the resulting IOR
A is the base IOR
B is the dispersion coefficient expressed in squarre meter (m²)
w is wavelength expressed in micro-meters

I try to apply this formula this way :
B = 0.013 in micro-meters,
A = 1.72

So first,
1) I convert the B parameter from micro-meter to m² : B = B / 1000000
2) I convert my wavelength from nm to micro-meter : wv = 500 / 1000

But the result I got seems incorrect, I got no dispersion !
Maybe there is a problem with my units ?
 
Last edited by a moderator:
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spectral said:
1) I convert the B parameter from micro-meter to m² : B = B * 1000000

I think you mean to convert B's units from m2 to (μm)2. For that, you have to multiply by 10000002, not 1000000.
 
Thanks for your answer,

But based on the formula here :http://en.wikipedia.org/wiki/Cauchy's_equation[/PLAIN]

the C parameters (I use B instead) is expressed in micro-meters !

So, if C = 0.013 in micro-meters I have to convert it to meters to use the Cauchy formula...
... then I "divide" by 1000000² ?

Or there is something wrong in what I do or my units ?

Thanks
 
Last edited by a moderator:
By example, I should have :

eta = 1.72 + (0.013 / 1000000²) / 0.500²
 
OK, I see now. I didn't look at the Wikipedia page to see where the numbers came from before.

If your B (their C) is already in units of (μm)2, and you've converted your wavelength λ to μm, then the quantity B / λ2 = 0.013 / 0.52 has units of (μm)2/(μm)2, i.e. it's unit-less, as it should be because the index of refraction is unit-less. Why do you want to convert B to m2?
 
Last edited:
I see...

It is just that I have see somewhere that the C parameter has to be in m²... but it seems false !

Thanks for your help
 
spectral said:
I have see somewhere that the C parameter has to be in m²...

I suspect that they do their calculations using λ in meters.
 

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