## Abstract

We show that the minimal phase of the temporal coherence function *γ*(τ) of stationary light having a partially-coherent symmetric spectral peak can be computed as a relative logarithmic Hilbert transform of its amplitude with respect to its asymptotic behavior. The procedure is applied to experimental data from amplified spontaneous emission broadband sources in the 1.55 µm band with subpicosecond coherence times, providing examples of degrees of coherence with both minimal and non-minimal phase. In the latter case, the Blaschke phase is retrieved and the position of the Blaschke zeros determined.

© 2008 Optical Society of America

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### Equations (8)

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(1)
$$\gamma \left(\tau \right)={\int}_{{v}_{a}}^{{v}_{b}}s\left(v\right)\mathrm{exp}\left(-2\pi iv\tau \right)d\tau .$$
(2)
$${\gamma}_{b}\left(z\right)={\int}_{-\Delta v}^{0}s\left(v+{v}_{b}\right)\mathrm{exp}\left(-2\pi ivz\right)dv=\eta \left[1+\frac{{\gamma}_{L}\left(z\right)}{C}\right],$$
(3)
$${\varphi}_{H}\left(\tau \right)=H\left[\mathrm{log}\mid {\gamma}_{b}\left(\tau \right)\mid \right]=\frac{2\tau}{\pi}\underset{0}{\overset{\infty}{\int}}\frac{\mathrm{log}\mid {\gamma}_{b}(\tau \text{'})\mid}{{\tau}^{2}-\tau {\text{'}}^{2}}d\tau \text{'}.$$
(4)
$${\gamma}_{b}\left(z\right)\prod _{k=1}^{N}\left(\frac{z-{a}_{k}^{*}}{z-{a}_{k}}\frac{z+{a}_{k}}{z+{a}_{k}^{*}}\right)=\mid \gamma \left(z\right)\mid \mathrm{exp}\left[i{\varphi}_{H}\left(z\right)\right].$$
(5)
$$\varphi \left(\tau \right)=\mathrm{arg}\phantom{\rule{.2em}{0ex}}\gamma \left(\tau \right)={\varphi}_{H}\left(\tau \right)+{\varphi}_{B}\left(\tau \right)-2\pi {v}_{b}\tau ,$$
(6)
$${\varphi}_{B}\left(\tau \right)=\sum _{k=1}^{N}\mathrm{arg}\left(\frac{\tau -{a}_{k}}{\tau -{a}_{k}^{*}}\frac{\tau +{a}_{k}^{*}}{\tau +{a}_{k}}\right)$$
(7)
$${\varphi}_{R}\left(\tau \right)=\mathrm{arg}{\gamma}_{R}\left(\tau \right)=\Delta \left(\tau \right)-2\pi {v}_{b}\tau ,$$
(8)
$$\varphi \left(\tau \right)=H\left[\mathrm{log}\left(\frac{\mid \gamma \left(\tau \right)\mid}{\mid {\gamma}_{R}\left(\tau \right)\mid}\right)\right]+{\varphi}_{B}\left(\tau \right)-2\pi {v}_{b}\tau ,$$