Since the mode field diameter is directly proportional to the calculated coupling focal length, the calculations concerning the overlap are also true for different focel lengths f1 and f2 and a= f1 / f2.
Mismatch / NA Mismatch
Fiber Mismatch or combining fibers with different NAs
For both single-mode and polarization-maintaining fibers, the effective numerical aperture NAe2 and mode field diameter MFD may vary by up to 10% from the specified values, simply arising from manufacturing tolerances. Selected fibers with characterized values are available on request.
The theoretical coupling efficiency η (overlap integral between two Gaussian intensity distributions) is still close to η = 1 even when mode field diameter of an actual fiber differs from the theoretical value.
The linear relationship between mode field diameter MFD and effective numerical apterture NAe2 means that this is also valid for a mismatch in the values for NAe2.
Example
For two fibers with exemplary effective numerical apertures NAe2 of 0.07 and 0.08 respectively, the overlap is still η = 0.982
Mismatch of focal lengths
Example
If the calculated coupling focal length ideally should be 22 mm, but the closest available optics has a focal length of 20 mm, the calculated overlap is still 0.991.
Detailed Calculations: Overlap of two single-mode fibers
When coupling radiation from one fiber to another fiber with different numerical aperture NA, the maximum coupling efficiency η, which can be obtained for a physical contact theoretically, is reduced. The maximum coupling efficiency η (or overlap) for a fiber-to-fiber coupling with a symmetric optics is the same. The coupling (overlap) is
{!{!{ \eta = \frac{\left|\int\int U_1(x,y)\cdot U_2^{*}(x,y)\,dx\,dy\right|^2}{\int\int\left|U_1(x,y)\right|^2dx\,dy\cdot \int\int\left|U_2(x,y)\right|^2dx\,dy} (1)}!}!}using the mode fields Ui(x,y) of the two different fibers and (x,y) coordinates in the plane of the fiber end face.
Assuming that the mode field of a (weakly guiding) single-mode fiber can be approximated by a Gaussian, the mode fields are
with the mode field diameters MFDi of the two fibers.
Assuming furthermore that the two mode fields are centered to each other, the coupling efficiency using equation (2) in integral (1) is:
{!{!{ \eta = \left[2\frac{MFD_1\cdot MFD_2}{MFD_1^2+ MFD_2^2}\right]^2 (3). }!}!}When the mismatch of the two mode field diameters is defined as the ratio
{!{!{ a = \frac{MFD_1}{MFD_2} (4) }!}!}
equation (3) can be expressed as
{!{!{ \eta=4\frac{a^2}{(1+a^2)^2} (5) }!}!}
Since the numerical aperture NA of a single-mode fiber is inversely proportional to its mode field diameter MFD and since {!{!{η(a)=η(1/a)}!}!} equation (5) is valid even when using the numerical apertures NAi in (4) instead of the mode field diameters,
{!{!{ a=\frac{NA_1}{NA_2}.}!}!}
The figure at the top of the page shows the coupling efficiency η for different values for a.