Effect of noise on modulation amplitude and phase in frequency-domain diffusive imaging

We theoretically investigate the effect of noise on frequency-domain heterodyne and/or homodyne measurements of intensity-modulated beams propagating through diffusive media, such as a photon density wave. We assumed that the attenuated amplitude and delayed phase are estimated by taking the Fourier transform of the noisy, modulated output data. We show that the estimated amplitude and phase are biased when the number of output photons is small. We also show that the use of image intensifiers for photon amplification in heterodyne or homodyne measurements increases the amount of biases. Especially, it turns out that the biased estimation is independent of AC-dependent noise in sinusoidal heterodyne or homodyne outputs. Finally, the developed theory indicates that the previously known variance model of modulation amplitude and phase is not valid in low light situations. Monte-Carlo simulations with varied numbers of input photons verify our theoretical trends of the bias.