Dear Sahil

You can use the same focal length for both wavelenghts. Which kind of problems are you thinking to solve by changing the focal length?

Best regards

J. Carlos

For effective input radiation into the fiber you need to use a short-focus lens. The minimum focal length of the lens is determined by the difference between the refractive indices of the core and shell. For these wavelengths the difference is very small, so you  can use the same lens. But keep in mind that at these wavelengths fiber will not be a single-mode one.

Are you coupling from a collimated beam or another optical fibre?  For lens coupling from a collimated Gaussian beam, the mode field diameter at 1/e2 intensity is

d = 4 f wavelength / (pi D)

where f is the lens focal length, D is the 1/e2 Gaussian beam width, and the lens aperture is preferably 1.5 times D or larger to minimize vignetting.

If the required spot diameter d is constant, and the diameters of the collimated beams are identical, then the lens focal lengths should vary inversely with the wavelength. This will not necessarily give optimum coupling.

If you are coupling into a single mode fibre, be aware that the fundamental mode field diameter also varies with wavelength.  The diameter typically decreases monontonically with wavelength.  The rate of change is slow beyond cut-off where the fibre is multi-mode, increasing at longer wavelengths as the normalised frequency V approaches unity

V = 2 pi a NA / wavelength

where a is the fibre core radius and NA is the core-cladding numerical aperture.

For a step index fibre, a simple and convenient approximation for the 1/e2 mode field diameter is        d = 2 a / sqrt(ln(V))

Note that this approximation fails both for large V >> 3 and for small V ~ 1, where the mode field distribution is no longer close to Gaussian in profile.  Marcuse's approximation is also widely used, and may be more accurate under some conditions  https://www.rp-photonics.com/mode_radius.html

What kind of fibre are you using and what is the cut-off wavelength?  Is it designed for single mode operation at these wavelengths?  With standard single mode fibre, such as Corning SMF28+, you will be operating beyond cut-off at both wavelengths. 

https://en.wikipedia.org/wiki/Gaussian_beam#Beam_waist

https://www.newport.com/n/gaussian-beam-optics

Thank you Mr. Robinson

I cannot actually tell the type of fiber since I am bound by a NDA. But it is graded index and  multi-mode at both the frequencies. Hence I am using structured light (lower order modes). I know there are no exact formulas for them so I am just approximating it with the Gaussian ones. Using the equation mentioned by you above I used a 20x objective lens with 9mm FL but I think there are aberrations which are causing bad coupling. So I plan to use aspheric lenses. I hope I am going right with these assumption?

Also I am coupling light from a collimated beam