Wave-number domain analysis for determining the response of linear space-invariant time-varying systems

System response analysis is a powerful method for analyzing linear time-invariant (LTI) systems. In this work, we have demonstrated that a system response approach can also be applied to analyze the so-called linear space-invariant (LSI) but time-varying problems, which represent a dual of the conventional LTI problems. In this proposed approach, we perform a Fourier transform of the electric field distribution on the space coordinate, rather than in time, and express it in the wave-number domain. Specifically, we express any input signal and its corresponding output in the wave-number domain. Then, the transfer function for the LSI time-varying system can be extracted as a one-time computation by evaluating the ratio of the output signal to the input signal in the wave-number domain. Once the transfer function is extracted, the output response to any input with an arbitrary temporal profile can be computed instantaneously. Furthermore, for a system with a complicated temporal profile, the proposed method allows us to decompose it into several simpler subsystems that appear sequentially in time. The transfer function of that complicated system can be expressed as the product of those of the individual subsystems, such that it can be evaluated more efficiently.