Space-time duality and the theory of temporal imaging

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Abstract
There exists a beautiful duality between the equations that describe the paraxial diffraction of beams confined in space and the dispersion of narrow-band pulses in dielectrics. We will see that this duality leads naturally to the conclusion that a quadratic phase modulation in time is the analog of a thin lens in space. Therefore, by a suitable combination of dispersion and quadratic phase modulation (now a "time lens"), we can synthesize the time-domain analog of an imaging system. Such a temporal-imaging system can magnify time waveforms in the same manner as conventional spatial-imaging systems magnify scenes. We analyze this space-time duality and derive expressions for the focal length and f-number of a time lens. In addition, the principles of temporal imaging are developed and we derive time-domain analogs for the imaging condition, magnification ratio, and impulse response of a temporal-imaging system.