Relationship between Light Transmission and Index

[Falstaff]

I have come up with an expression (for normal incidence) which relates these parameters, and which takes into account the multiple reflections between the top and bottom surfaces of a substrate:

T = 2n / (n^2 + 1),

where T is the Transmission and n is the Index.

If this is correct, then it follows that I can try to determine a value for n if I have measured T.

With a little manipulation I get that:

n = a + ((a^2 + 1)^0.5),

where a = 1/T.

My problem is, that while this works (more or less) for a CR39 lens, it does not work very well for higher index materials. I cannot, at the moment see why.

I have tried using plano lenses, but perhaps the surfaces are not exactly parallel, and some of the reflected light does not follow a predictable path.

Can anyone help, or suggest a more reliable method for finding the index of a material?

[MikeAurelius]

Chromatic abberation IMO. My guess would be that using "white light" instead of a single wavelength will throw off the calculation because of the differing focus points for the various wavelengths.

Why not use a calculation based on curvatures and back into the index of refraction that way?

For example, a lens with a plano base that reads -10, and the back surface clocks at -7.50. Index of refraction is 1.70. ( .53/.70 = .75).

[MakeOptics]

I have come up with an expression (for normal incidence) which relates these parameters, and which takes into account the multiple reflections between the top and bottom surfaces of a substrate:

T = 2n / (n^2 + 1),

where T is the Transmission and n is the Index.

If this is correct, then it follows that I can try to determine a value for n if I have measured T.

With a little manipulation I get that:

n = a + ((a^2 + 1)^0.5),

where a = 1/T.

My problem is, that while this works (more or less) for a CR39 lens, it does not work very well for higher index materials. I cannot, at the moment see why.

I have tried using plano lenses, but perhaps the surfaces are not exactly parallel, and some of the reflected light does not follow a predictable path.

Can anyone help, or suggest a more reliable method for finding the index of a material?

Although a relationship does exist measuring the index this way is futile since the transmittance of a material is going to be the measure of the percentage of light that gets through he lens after reflectance and absorption. Since various MAR's will reduce reflectance to different degrees you could potentially calculate a higher index lens as being a lower index lens if it had an MAR coating.

[MakeOptics]

Also consider since the index is dependent upon the wavelength, you may want to measure transmittance using the reference wavelength of your country rather than a mean transmittance value since most materials and coatings will transmit different wavelengths to varying degrees.

[optical24/7]

Chromatic abberation IMO. My guess would be that using "white light" instead of a single wavelength will throw off the calculation because of the differing focus points for the various wavelengths.

Why not use a calculation based on curvatures and back into the index of refraction that way?

For example, a lens with a plano base that reads -10, and the back surface clocks at -7.50. Index of refraction is 1.70. ( .53/.70 = .75).

or;

n = .53 X (true power/clock power) + 1

[Darryl Meister]

I have come up with an expression (for normal incidence) which relates these parameters, and which takes into account the multiple reflections between the top and bottom surfaces of a substrate

I'd have to double-check, but at first glance your equation looks like the correct expression obtained from an infinite series summation of Fresnel's equation.

With a little manipulation I get that

Your manipulated equation rearranged to solve for n, on the other hand, is not correct; you may have a typo or something here. As the correct expression, I get:



That said, this equation is very sensitive to accurate measurements of lens transmittance. A 1% error in the transmittance measurement will equate to an error of about 0.04 in refractive index, making the result useless for most practical applications.

And, of course, you would have to ensure that a number of measurement conditions are satisfied, including 1) transmitted light incident at a normal angle of incidence, 2) no loss of material transparency due to haze, bluing agents, coatings, et cetera, and 3) measurement at the intended reference wavelength.

Can anyone help, or suggest a more reliable method for finding the index of a material?

If your work relies on a truly accurate and precise measurement of refractive index, you may need to invest in an Abbe refractometer or similar device.

Otherwise, as the others have pointed out, you can also try calculating the refractive index by measuring the surface curvatures and back vertex power of a high minus lens with a plano front curve.

Best regards,
Darryl

[Darryl Meister]

As the correct expression, I get

It looks like your radicand should read T^2 - 1, instead of T^2 + 1.

I should add that, while this makes for an interesting technical exercise, you will probably get your best answer by contacting the supplier of the original lens material...

Best regards,
Darryl

[Falstaff]

Many thanks. There is indeed a typo on my original post. It should have read n = a + (a^2 - 1)^0.5, which, I think (and hope) is equivalent to the expression given above.

Perhaps I should have added the reason for the enquiry. A colleague (albeit in another lab) has been working on a theoretical way of allowing for the differences in transmission of tinted lenses before and after coating. He is trying to calculate suitable target transmissions for the tinters, who apply tints before hardcoating. When these lenses go on to be AR coated, it is to be hoped that they will then come out correctly. For this he needs to know, amongst other things, how the transmission depends on the index of the substrate, and how this changes when various coatings are applied. When I stopped to actually measure some transmissions of uncoated plano lenses, I found a discrepancy between the expected transmissions that I could not account for. This being so, I took it to mean that there was something I have not understood. There was no particular intention to use the mathematical expression as an accurate measurement tool, though I am interested in techniques for this, as the use of spectrophotometers here often given rise to unlikely values.

Perhaps this theoretical way of tinting is doomed to failure anyway, and is best left to the craft of the tinters themselves. However, I thank those that have come up with some very relevant, and interesting, points.

Best Regards,

Falstaff

[Darryl Meister]

For this he needs to know, amongst other things, how the transmission depends on the index of the substrate... When I stopped to actually measure some transmissions of uncoated plano lenses, I found a discrepancy between the expected transmissions that I could not account for.

If you are primarily interested in determining the expected transmittance of a lens with a given refractive, then the first equation based upon Fresnel's equaiton is probably as good any any that I am aware of, but you need to remember that there are a several limitations involved.

Also keep in mind that "transmittance" in Fresnel's equation is a function of wavelength. So, unless you are measuring the specific reference wavelength with your spectrophotometer, either 587.56 nm for helium d or 546.07 nm for mercury e, your results will not correlate to the specified refractive index of the material (although both reference wavelengths are fairly close to the peak of the luminous sensitivity function anyway).

Best regards,
Darryl