Signal-to-noise-ratio limit to the depth-of-field extension for imaging systems with an arbitrary pupil function

We discuss the limit of the depth-of-field (DOF) extension for an imaging system using aspheric surfaces. In particular we consider an imaging system with an arbitrary pupil function and present the rigorous tradeoff between the DOF of the system and the spectral signal-to-noise ratio (SNR) over an extended DOF, to our knowledge for the first time. In doing so we use the relation between the conservation of ambiguity and modulation-transfer function (MTF) on one hand and the relation between the spectral SNR and MTF on the other. Using this, we rigorously derive the expression for an upper bound for the minimum spectral SNR, i.e., the limit of spectral SNR improvement. This leads to the introduction of our spectral SNR conservation principle. We also draw the relation between our result and the conservation of brightness theorem and establish that our result is the spectral version of the brightness conservation theorem.