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 f_{1} and f_{2} and a= f_{1 }/ f_{2}.

## Mismatch / NA Mismatch

### Fiber Mismatch or combining fibers with different NAs

For both single-mode and polarization-maintaining fibers, the effective numerical aperture NAe^{2} 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 NAe^{2} means that this is also valid for a mismatch in the values for NAe^{2}.

### Example

For two fibers with exemplary effective numerical apertures NAe^{2 }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 U

_{i}(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 MFD_{i} 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 NA

_{i }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.