The single-mode fiber coupling calculation (available under Analysis > Calculations > Fiber Coupling Efficiency) provides a more powerful capability for fibers with Gaussian-shaped modes. It performs two calculations: an energy-transport calculation and a mode-matching calculation. The system efficiency (S) is the sum of the energy collected by the entrance pupil which passes through the optical system, accounting for both the vignetting and transmission of the optics (if polarization is used), divided by the sum of all the energy which radiates from the source fiber:

where F_s(x,y) is the source fiber amplitude function and the integral in the numerator is only done over the entrance pupil of the optical system, and t(x,y) is the amplitude transmission function of the optics. The transmission is affected by bulk absorption and optical coatings if use polarization is checked on.

Aberrations in the optical system introduce phase errors which will affect the coupling into the fiber. Maximum coupling efficiency is achieved when the mode of the wavefront converging towards the receiving fiber perfectly matches the mode of the fiber in both amplitude and phase at all points in the wavefront. This is defined mathematically as a normalized overlap integral between the fiber and wavefront amplitude:

where F_r(x,y) is the function describing the receiving fiber complex amplitude, W(x,y) is the function describing the complex amplitude of the wavefront from the exit pupil of the optical system, and the ' symbol represents complex conjugate. Note that these functions are complex valued, so this is a coherent overlap integral.

T has a maximum possible value of 1.0, and will decrease if there is any mismatch between the fiber amplitude and phase and the wavefront amplitude and phase.

ZEMAX computes the values S and T. The total power coupling efficiency is the product of these numbers. A theoretical maximum coupling efficiency is also computed; this value is based upon ignoring the aberrations but accounting for all vignetting, transmission, and other amplitude mismatches between the modes.

Now in this calculation, the source and receiver modes are defined by their Gaussian NA, which is defined as the refractive index n of the object or image space surfaces times the sine of the half-angle to the 1/e² power point. This angle can be computed in one of two ways:

• From the divergence angle of the Gaussian beam calculation, using the mode field diameter to define the beam waist (as calculated on the previous page)
• From the 1% power NA given in the Corning datasheet, and computing the 1/e² power point from that

The appropriate value for NA is 0.09 for both receiver and source fibers, and so the calculation is set up as follows:

which gives the results:

We may use the FICL operand to optimize the coupling efficiency with the following one-line merit function:

And running a few cycles of optimization gives a fiber/lens thickness of 0.110 mm (was 0.117 with the simple Gaussian calculation) and the detailed results:

Note the following:

• The system efficiency has not changed significantly, as this is set by the apertures of the surface and the size of the modes, which do not change much for this slight refocus
• The receiver efficiency has improved as the refocus makes the source fiber mode, after transmission through the optical system, a better match to the receiver fiber mode
• The maximum efficiency indicates the improvement that can be made by e.g. adding aspherics, extra surfaces etc. In this case, the efficiency is about as high as it can be

The file at this point is saved in the attached archive as after FICL optimization.zmx.