Noise characteristics of heterodyne/homodyne frequency-domain measurements

Abstract

We theoretically develop and experimentally validate the noise characteristics of heterodyne and/or homodyne measurements that are widely used in frequency-domain diffusive imaging. The mean and covariance of the modulated heterodyne output are derived by adapting the random amplification of a temporal point process. A multinomial selection rule is applied to the result of the temporal noise analysis to additionally model the spatial distribution of intensified photons measured by a charge-coupled device (CCD), which shows that the photon detection efficiency of CCD pixels plays an important role in the noise property of detected photons. The approach of using a multinomial probability law is validated from experimental results. Also, experimentally measured characteristics of means and variances of homodyne outputs are in agreement with the developed theory. The developed noise model can be applied to all photon amplification processes.

Fig. 1

(a) Simple scheme for the spatial distribution of secondary photons. Experimentally measured profiles [(b) 2-D and (c) 1-D] of averaged normalized spatial covariance. In (c), circles, squares, and dots indicate exposure times of 60, 120, and 240 ms, respectively. In (d), the temporal autocovariance measured at a single CCD pixel is indicated.

Fig. 2

(a) Examples of measured homodyne profile. (b) Means and (c) variances of measured dc and ac. Labels on right $y$-axes in (b) and (c) indicate ratios of ${\overline{M}}_{\mathrm{AC}}/{\overline{M}}_{\mathrm{DC}}$ and ${\sigma }_{\mathrm{AC}1}^2/{\sigma }_{\mathrm{AC}2}^2$, respectively. (d) Varied $m_p$ changes not dc variance, but ac variance of homodyne outputs. The $x$-axes for (b) and (c) indicate relative MCP voltages.