Fundamentals of Photonics, 2 Volume Set, 3rd Edition

APPENDIX B LINEAR SYSTEMS

This appendix provides a review of the essential characteristics of one-and two-dimensional linear systems.

B.1 ONE-DIMENSIONAL LINEAR SYSTEMS

Consider a system whose input and output are the functions f₁(t) and f₂(t), respectively. The system is characterized by a rule that relates the output to the input. In general, the rule may take the form of a simple mathematical operation such as f₂(t) = log[f₁(t)], an integral transform, or a differential equation. An example is a harmonic oscillator that undergoes a displacement f₂(t) in response to a time-varying force f₁(t).

Linear Systems

A system is said to be linear if it satisfies the principle of superposition, i.e., if its response to the sum of any two inputs is the sum of its responses to each of the inputs separately. The output at time t is, in general, a weighted superposition of the input contributions at different times τ,

(B.1-1)

where h(t; τ) is a weighting function representing the contribution of the input at time τ to the output at time t. If the input is an impulse at time τ, so that f₁(t) = δ(t − τ), then (B.1-1) yields f₂(t) = h(t; τ). Thus h(t; τ) is the impulse response function of the system (also known as the Green's Function).

Linear Shift-Invariant Systems

A linear system is said to be time-invariant or shift-invariant if, when its input is shifted in time, its output shifts ...