## Abstract

The simulation of beam propagation is used to examine the uncertainty inherent to the process of optical power measurement with a practical heterodyne receiver because of the presence of refractive turbulence. Phase-compensated heterodyne receivers offer the potential for overcoming the limitations imposed by the atmosphere by the partial correction of turbulence-induced wave-front phase aberrations. However, wave-front amplitude fluctuations can limit the compensation process and diminish the achievable heterodyne performance.

© 2008 Optical Society of America

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### Equations (11)

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(1)
$$\mathrm{\Phi}\left(\mathit{v}\right)=\sum _{j=1}^{\infty}{a}_{j}{Z}_{j}\left(\mathit{v}\right)$$
(2)
$${\mathrm{\Phi}}_{c}\left(\mathit{v}\right)=\sum _{j=1}^{J}{a}_{j}{Z}_{j}\left(\mathit{v}\right)$$
(3)
$${\mathrm{\Phi}}_{J}\left(\mathit{v}\right)=\mathrm{\Phi}\left(\mathit{v}\right)-{\mathrm{\Phi}}_{c}\left(\mathit{v}\right)=\sum _{j=J}^{\infty}{a}_{j}{Z}_{j}\left(\mathit{v}\right)$$
(4)
$$\frac{\partial}{\partial {a}_{j}}\int W\left(\mathit{v}\right){\left[\mathrm{\Phi}\left(\mathit{v}\right)-\sum _{j=1}^{J}{a}_{j}{Z}_{j}\left(\mathit{v}\right)\right]}^{2}\mathit{d}\mathit{v}=0$$
(5)
$$\mathrm{\int}W\left(\mathit{v}\right){Z}_{i}\left(\mathit{v}\right){Z}_{j}\left(\mathit{v}\right)\mathit{d}\mathit{v}={\delta}_{\mathit{ij}}$$
(6)
$${a}_{j}=\frac{\mathrm{\int}W\left(\mathit{v}\right){Z}_{j}\left(\mathit{v}\right)\mathrm{\Phi}\left(\mathit{v}\right)\mathit{d}\mathit{v}}{\int W\left(\mathit{v}\right){Z}_{j}^{2}\left(\mathit{v}\right)\mathit{d}\mathit{v}}$$
(7)
$$P=\u3008{[\mathrm{\int}W\left(\mathit{v}\right){U}_{S}\left(\mathit{v}\right){U}_{\mathit{LO}}^{*}\left(\mathit{v}\right)\mathit{d}\mathit{v}]}^{2}\u3009$$
(8)
$$P=\iint W\left({\mathit{v}}_{1}\right)W\left({\mathit{v}}_{2}\right){M}_{S}({\mathit{v}}_{1},{\mathit{v}}_{2}){{M}_{\mathit{LO}}}^{*}({\mathit{v}}_{1},{\mathit{v}}_{2})d{\mathit{v}}_{1}d{\mathit{v}}_{2}$$
(9)
$${M}_{S}({\mathit{v}}_{1},{\mathit{v}}_{2})=\u3008{U}_{s}\left({\mathit{v}}_{1}\right){U}_{S}^{*}\left({\mathit{v}}_{2}\right)\u3009$$
(10)
$${M}_{\mathit{LO}}({\mathit{v}}_{1},{\mathit{v}}_{2})={U}_{\mathit{LO}}\left({\mathit{v}}_{1}\right){\mathit{U}}_{\mathit{LO}}^{*}\left({\mathit{v}}_{2}\right).$$
(11)
$${U}_{S}\left(\mathbf{v}\right)={A}_{S}\left(\mathbf{v}\right)\mathrm{exp}\left[-j\mathrm{\Phi}\left(\mathbf{v}\right)\right]\mathrm{exp}\left[j{\mathbf{\Phi}}_{c}\left(\mathbf{v}\right)\right]={A}_{S}\left(\mathbf{v}\right)\mathrm{exp}\left[-j{\mathrm{\Phi}}_{J}\left(\mathbf{v}\right)\right]$$