In an optics lab you often need to couple light efficiently from one component into another — for example, from an optical fiber into a resonator. Optimizing that coupling is called mode-matching, and it is typically done with lenses. The calculators below have been verified experimentally and used to achieve better than 95% mode-matching efficiency between optical fibers and Fabry–Pérot resonators. They are based on the seminal paper "Laser Beams and Resonators" by Kogelnik and Li, available here.
User Inputs
Wavelength [nm]
w1 [mm]
w2 [mm]
f [mm]
Minimum f [mm]
Computation Outputs
d1 [mm]
d2 [mm]
L [mm]
Short Solution
Long Solution
Enter the wavelength and the input (w1) and output (w2) waists. The waist size is defined as the transverse distance from the center of the beam at which the electric field falls to 1/e of its on-axis value. For a given pair of waists there is a minimum focal length that can perform the mode-matching; once the wavelength and waists are entered, this value appears automatically in the "Minimum f" box. Next, enter a focal length greater than that minimum and press Enter (Windows) or Return (Mac). Two solutions appear in the "Computation Outputs" box. The "Short Solution" sometimes contains negative (i.e. unphysical) values, in which case it should not be used. The results update automatically whenever you change an input and press Enter/Return.
User Inputs
Wavelength [nm]
w0 [mm]
w2 [mm]
L [mm]
f1 [mm]
f2 [mm]
Computation Outputs
| f1 [mm] | f2 [mm] | d0 [mm] | d1 [mm] | d2 [mm] |
|---|
Given the input and output waist sizes (w0 and w2) separated by a distance L, this calculator determines where the two lenses (with focal lengths f1 and f2) should be placed. As above, the waist size is the transverse distance from the center of the beam at which the electric field falls to 1/e of its on-axis value. You must enter a positive, non-zero number for the wavelength, w0, w2, and L. Two arbitrarily chosen lenses often cannot perform the mode-matching, so several focal-length boxes are provided for each lens. The calculator tries every combination of f1 and f2 and returns every pair capable of performing the mode-matching. At least one focal length must be entered per lens; up to three boxes can be left empty in each row. Positive (converging) and negative (diverging) values are both acceptable — only zero is not. All lengths are in millimeters, except the wavelength, which is in nanometers. When all values are entered, click the "Calculate" button. It is not uncommon for there to be no solutions, in which case see the troubleshooting section below.
If there are no solutions, here are some things to try: