squarelaw

Square-law refers to a class of nonlinear relationships and devices in which the output is proportional to the square of the input. In electronics and optics, a square-law transfer characteristic means y ≈ k x^2 over a defined operating range. This quadratic behavior makes square-law devices useful for detecting or processing signal power rather than signal amplitude.

In RF engineering, square-law detectors use nonlinear elements such as diodes or transistors to produce a DC output proportional to the average input power. They are commonly used for envelope detection of small-amplitude AM signals, where the instantaneous voltage is small enough for the output to grow roughly with the square of the input. At higher input levels, the response typically transitions to a linear or saturated regime, reducing the square-law accuracy.

Square-law behavior also appears in mixing and downconversion, where quadratic nonlinearities generate sum and difference frequencies, enabling certain direct-conversion or homebrew demodulation schemes. In optics and photodetection, square-law detectors describe devices whose output current is proportional to the incident optical power (which is proportional to the square of the optical field amplitude in free space), making them suitable for general power detection and optical communication receivers.

In semiconductor physics, long-channel MOSFETs historically exhibited Id ∝ (Vgs − Vth)^2 in the square-law region, though modern short-channel devices often deviate from the ideal quadratic law. Overall, square-law concepts are foundational in understanding nonlinear detection, downconversion, and power sensing across multiple technologies.