Quantum ellipsometry using correlated-photon beams

We report ellipsometric measurements made on semiconductor samples using photon-correlated beams produced by the process of spontaneous parametric down-conversion. Such a source yields higher accuracy than its quantum-limited conventional counterpart. We also show that our approach has the added advantage of not requiring an external reference sample for calibration.

Figure 1

Correlated-photon ellipsometry. $\mathrm{SU}{\left(2\right)}_{1,2}$ represents the allowed polarization devices used, $P$ a linear polarizer, $A$ a linear polarization analyzer, and $D_{1,2}$ single-photon detectors with efficiencies ${\eta }_{1,2}$. The sample $S$ is characterized by the ellipsometric parameters $\psi$ and $\Delta$, as defined in the text.

Figure 2

Coincidence interference patterns obtained from a $\mathrm{Si}$ sample. The labels $H$, $V$, and RHC correspond to a horizontal, vertical, and right-hand-circular input polarization state of the light. Each sinusoidal pattern represents the sample’s response to the particular polarization state. ${\theta }_2$ is the angle of the axis of the half-wave retarder placed after the sample, with respect to the horizontal direction. The solid sinusoidal curves correspond to the expected theoretical fit obtained from Eq. (6).

Figure 3

Coincidence interference patterns obtained from a $\mathrm{GaAs}$ sample. The labels $H$, $V$, and RHC correspond to a horizontal, vertical, and right-hand-circular input polarization state of the light. Each sinusoidal pattern represents the sample’s response to the particular polarization state. ${\theta }_2$ is the angle of the axis of the half-wave retarder placed after the sample, with respect to the horizontal direction. The solid sinusoidal curves correspond to the expected theoretical fit obtained from Eq. (6).

Figure 4

Comparison of the signal-to-noise ratio, as a function of the effective intensity reflectance $R$, for correlated-photon beams (solid curves) and the conventional system (dashed curves) for two values of the quantum efficiency: (a) $\eta ={\eta }_1={\eta }_2=1$; (b) $\eta ={\eta }_1={\eta }_2=0.6$.

Figure 5

The relative error $e_{\psi }$ in the ellipsometric parameter $\psi$ for correlated-photon beams (solid curves) and the conventional system (dashed curves) for two values of the quantum efficiency: (a) $\eta ={\eta }_1={\eta }_2=1$; (b) $\eta ={\eta }_1={\eta }_2=0.6$.

Figure 6

The relative error $e_{\Delta }$ in the ellipsometric parameter $\Delta$ for $\psi =45°$ for correlated-photon beams (solid curves) and the conventional system (dashed curves) for two values of the quantum efficiency: (a) $\eta ={\eta }_1={\eta }_2=1$; (b) $\eta ={\eta }_1={\eta }_2=0.6$.