Given the input and output focus waist sizes (w0 and w2, respectively) which are separated by a distance L this calculator will determine where the two lenses (with focal lengths f1 and f2) should be placed. For this calculator, you must enter a positive, non-zero number for the wavelength, w0, w2, and L. It is often the case that two arbitrarly chosen lenses cannot perform the mode-matching as shown above. Therefore, multiple input boxes are given for each lens. This calculator tool will try every combination of f1 and f2 lenses and return every combination which is capable of performing the mode-matching. For the focal lengths, at least one number must be entered for each lens -- up to three boxes can be left empty in each row. Positive (converging lens) and negative (diverging lens) values are acceptable. The only non-acceptable value for a focal length is zero. All units are milimeters, except the wavelength which is in nanometers. When all values are entered, click the "Run!" button. Note that this computation can take up to 30 seconds to finish. It is not uncommon for there to be no solutions, in which case, see the troubleshooting secton below.

Troubleshooting

If there are no solutions, here are some things you can try:

Pick lenses so that f1/f2 is approximately equal to w0/w2.

Pick lenses with focal length around L/4.

Pick lenses with focal length around pi*w0*w2/(wavelength) (all in same lenght units).

Chang the length L so that it is closer to pi*w0*w2/(wavelength) (all in same lenght units).

Try several of these together.

Background

It is common in an optics lab to need to couple light efficiently from one optical component into another (for example an optical fiber and an optical resonator). The process of optimizing this coupling is called mode-matching and it is typically implemented with lenses. If you are using plano-convex lenses, the plano side should be on the same side as the tighter of the two focuses, to minimize spherical abberation. This calculator has been tested experimentally and been used to achieve better than 95% mode matching efficiency between optical fibers and fabry-perot resonators. This computation is based on the seminal paper "Laser Beams and Resonators" by Kogelnik and Li, which is accessable here.