## Abstract

A measured displacement resolution of <3 nm is demonstrated with a
common cathode differential photodetector combined with a laser-diode optical
source and a fiber-optic collimator. Resolution, standard deviation, and
differences between maxima and minima values for the residuals of the
least-squares fit suggest that a coherent laser-diode source temporally
correlates photoelectron flux between adjacent detector segments, suggesting
reduced signal variance and associated electronic (shot) noise. For otherwise
similar systems, the laser-diode source provides approximately an order of
magnitude reduction in standard deviation compared with a light-emitting-diode
source, which implies an equivalently improved measured (including standard
deviation) resolution. Combined variances for correlated and uncorrelated
detectors and their measured variances are outlined. The measured resolution
is a sum of both the (ideal) mathematical variance based on the detector noise
(millivolts) divided by the system sensitivity (millivolts per nanometer, and
the standard deviation of the noise (nanometers).

© 1997 Optical Society of America

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### Equations (24)

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(1)
$$\u3008\mathrm{\varphi}\u3009=\u3008n\u3009/\mathrm{\tau}.$$
(2)
$$\u3008I\u3009=e\mathrm{\eta}\u3008n\u3009/\mathrm{\tau},$$
(3)
$$=e\mathrm{\eta}\u3008\mathrm{\varphi}\u3009,$$
(4)
$${\mathrm{\sigma}}^{2}=\u3008n\u3009.$$
(5)
$$\u3008\mathrm{\varphi}\u3009={\mathrm{\sigma}}^{2}/\mathrm{\tau},$$
(6)
$$\u3008I\u3009=e\mathrm{\eta}{\mathrm{\sigma}}^{2}/\mathrm{\tau}.$$
(7)
$${\u3008{i}_{\mathit{SN}}\u3009}^{2}=2e\u3008I\u3009/\mathrm{\tau}$$
(8)
$$=2\mathrm{\eta}{\left(e\mathrm{\sigma}/\mathrm{\tau}\right)}^{2}.$$
(9)
$${R}_{0}=\text{inherent detector photoelectric variance}\left(\text{noise in mV}\right)/\mathrm{sensitivity}\left(\mathrm{mV}/\mathrm{\mu}\mathrm{m}\right)=\mathrm{\sigma}_{d}{}^{2}/S,$$
(10)
$$R=\mathrm{photoelectron\; variance}/\text{square mean current}=\mathrm{\sigma}_{i}{}^{2}/{\u3008i\u3009}^{2}=1/\mathrm{\eta}\u3008n\u3009.$$
(11)
$${V}^{2}=V_{1}{}^{2}+V_{2}{}^{2}.$$
(12)
$$\mathrm{\sigma}_{i}{}^{2}=\u3008{\left({V}_{i}-\u3008{V}_{i}\u3009\right)}^{2}\u3009.$$
(13)
$${\mathrm{\sigma}}^{2}=\mathrm{\sigma}_{1}{}^{2}+\mathrm{\sigma}_{2}{}^{2}.$$
(14)
$${V}_{2}=\mathrm{\alpha}{V}_{1},$$
(15)
$${V}^{2}=\left(1+{\mathrm{\alpha}}^{2}\right)V_{1}{}^{2},$$
(16)
$${\mathrm{\sigma}}^{2}=\left(1+{\mathrm{\alpha}}^{2}\right)\u3008{\left({V}_{1}-\u3008{V}_{1}\u3009\right)}^{2}\u3009.$$
(17)
$$V={V}_{1}+{V}_{2}.$$
(18)
$${\mathrm{\sigma}}^{2}=\mathrm{\sigma}_{1}{}^{2}+\mathrm{\sigma}_{2}{}^{2}+2{C}_{12},$$
(19)
$${C}_{12}=\left({V}_{1}-\u3008{V}_{1}\u3009\right)\left({V}_{2}-\u3008{V}_{2}\u3009\right).$$
(20)
$$V=\left(1+\mathrm{\alpha}\right){V}_{1},$$
(21)
$${\mathrm{\sigma}}^{2}={\left(1+\mathrm{\alpha}\right)}^{2}\mathrm{\sigma}_{1}{}^{2}.$$
(22)
$${R}_{m}=\mathrm{\sigma}_{c}{}^{2}/S,$$
(23)
$$\mathrm{\sigma}_{c}{}^{2}=\mathrm{\sigma}_{d}{}^{2}+\mathrm{\sigma}S.$$
(24)
$${R}_{m}=\mathrm{\sigma}_{d}{}^{2}/S+\left(1+\mathrm{\alpha}\right){\mathrm{\sigma}}_{1}.$$