Optical Communications, 2nd Edition

4.8 OPTIMAL COLLECTION OF MUL TIMODE SIGNAL POWER

The analysis in Section 4.7 assumed equal signal power in all spatial modes of a multimode source. The receiver with coherent combining maximized SNR by opening the field of view to collect all modes. Suppose now that the source mode powers P_si were not equal in all modes but instead had a spatial distribution of its power over the set of receiver modes. Opening the receiver view may no longer produce optimal power collection.

Let us first assume that the receiver had the capability of separately detecting each mode and applying arbitrary photodetection gain to each mode. Let g_i be the gain (nonrandom) applied to the ith mode during photodetection. Because the gain multiplies the detected field power, the combined outputs of the individual detectors would have the SNR given by

If we rewrite the numerator sum as

we can apply the Schwarz inequality,

This establishes that Eq. (4.8.1) has the bound

Furthermore, the bound occurs only when the equality m the Schwarz inequality holds,

or when

where C is any scale factor. This represents ...