Hi,

I am not sure but I think equating lens formula and Cauchy's Equation should solve your problem. As we all know that Cuachy's equation gives relationship between refractive index and wavelength of light for a particular transparent material and again lens formula gives a relationship between focal length and refractive index, so eliminating refractive index term from both equation should give you a mathematical relation between the wavelength and the focal distance.

Sincerely,

Upkar

Hi,

I am not sure but I think equating lens formula and Cauchy's Equation should solve your problem. As we all know that Cuachy's equation gives relationship between refractive index and wavelength of light for a particular transparent material and again lens formula gives a relationship between focal length and refractive index, so eliminating refractive index term from both equation should give you a mathematical relation between the wavelength and the focal distance.

Sincerely,

Upkar

Thank you Upkar! I am using at the moment the Sellmeier's equation (because I have the coefficients I need), and as you said, I combined it with the lens equation. The problems raise with the (1/R1 - 1/R2) term in the final formula that comes from the lens equation. I work with a bi-convex lens of CaF2, and I do not really know how to calculate R1 and R2. I know they are the radii of curvature of the lens, but I cannot find a relation for them which can help me. The focal length is 500mm, the diameter 50.8 mm and the thickness at the centre of the lens is 4.1mm. I guess that it exists a relation between R1, R2 and the focal length

sincerely

Andrea

Hi,

If you suppone a change in wavelength:

d_lambda=lambda_F-lambda_c

corresponding to a change in focal length:

d_f=f_F-f_c

At first order you will have:

d_f=delta(f)/(delta(lamba)*d_lambda

delta=partial derivative

so you will have:

f_F-f_C=delta(f)/(delta(lamba)*(lambda_F-lambda_c)

that for a thin lens reduce to:

delta(f)/(delta(lamba)=-f/(n-1)*delta(n)/(delta(lamba)

Cheers

Well, the focal length you provide is EFL (effective focal length). Namely the distance from the last principal plane to the focal point. Generally it is common for a positive lens to take the second principal plane to be ~1/3 of the lens thickness from the back edge of the lens. So you can use that to estimate your curvatures. The better way, so, is just to measure the curvature of both sides. One optical way to do it would be just to shine a collimated light at the lens and to measure the divergence of the reflection from the front surface. Then just use an equation for focal length of a curved mirror to calculate the curvature. Do it for the second side as well, and you got your R-s. It is really not a difficult measurement, and should not take you more than couple of hours to make the entire setup.

Dear Andrea,

I have not come across any mathematical equation which has relationship between refractive index of the lens and wavelength. But if we know the spot size/beam waist at the focal point of the lens and incident beam size, we can calculate the focal length of the lens as a function of wavelength with fallowing relation. I hope this can help and we use it often.

All the best..

http://www.rp-photonics.com/focal_length.html

All of this only works of course for a singlet lens since any doublet or other compound lens may imply that the lens is achromatized, or color corrected, in which case any simple formula is moot. From a thin lens case, you can still estimate the new EFL as long as you know it at one wavelength and you know the index of refraction at that wavelength and the new one you want to calculate. Since you are using the same lens and the radii of curvature have not changed from one wavelength to the other, then you can cancel that out of the equation and simply show that the ratio of lens power from one wavelenth to the other is just [n(lambda1)-1]/[n(lambda2)-1] will be the multiplicative change in lens focal length.

alan

Well, thank you all for the really exhaustive answers!!!

I managed, at the end, to find in a cataloge the wavelength at which the lens was characterised by the supplier. With this data I calculate the n by using the Sellmeier equation and then the R by means of the lensmaker's equation. At this point I could plot the focal length in function of the wavelength thanks again to the lensmaker's equation.

Regards

Andrea

Are you using an len or a mirror? You can obtain a mathematical expression by using the F-number and the lens equation......

After use the diffraction limit : lambda/d.......

Other way would be doing the experiment, in other words to generate a collimated beam and to measure the place where the light converges. Do this for 3 lambdas, to fit an mathematical function and finally to extrapolate for IR

Hi

If I were you, I'd choose two ways: either compute or simulate.

Compute is about claculating the focal length with the lens formula using Sellmeiaer's formula for changing the refractive index. you get a number of different focal lengths which enable you to estimate your longitudinal and transversal chromatic aberration (over monochromatic light means you have aberration).

Simulate is about using Zemax or the equivalent in order to compute the spot diagram at focus and search for minimum rms across the wavelengths of interest.

I'd only simulate if large apertures are involved: otherwise the paraxial lens formula (with thickness!) should be enough.