Why are "square law" devices important?

It's a particular classification of nonlinear behavior that has important applications.

In the same way as you can consider, say, a BJT as linear over a limited range, you can consider something like a diode as square law over a limited range. That simplification allows you to analyze functions such as RF detectors analytically.

See, for example, this Agilent paper "Square Law and Linear Detection".

Electronic things that follow square laws:

• power loss with distance over the air aka "Friis Loss" for RF due to power is spread by broadcast over beam width arc path at a distance is proportional to \$r^2\$

  • the same is true with optical communication, sound and other signal sources like WiFi when there are no reflections of obstacles.

• power loss in conductors from Ohm's Law, \$V=IR\$ we get \$Pd = VI = V^2/R = I^2R\$

• diode impedance = voltage/current for small ac signals with Vdc bias before saturation at rated current, then it becomes linear
• reverse diode capacitance vs voltage. C is maxed at 0V and reduces by k*V² at rated reverse Vr and C(0V) is a function of rated power and 1/Rs the linear saturation resistance in forward bias. Varicaps are controlled diodes for specific square law VCO tuning with C ratios given at 2 voltages.
• because of diode square law when used in negative feedback can be used as "analog signal "multipliers" or converting the voltage to power.

Fundamentally, based on the geometry of a 2D circle \$C=\pi R^2\$ then we have cubic laws based on 3D geometry of a sphere and higher orders that describe laws of nature.

Square Law devices make excellent RF mixers. A perfect square law device produces only the Sum and the Difference outputs.