Correction of non-linearities in quadrature phase differential interferometry

Abstract Quadrature phase differential interferometry is a technique of choice for measuring arbitrary large displacements with record resolution. The so-called Heydemann corrections are generally employed to take account of the unavoidable imperfections due to the optics or the electronics, and a linearisation of the signals is usually performed when the displacement to be determined is the superposition of a quasi-static component and a small harmonic component. Here we show that this standard analysis can lead to non-linearities of two types. The first type of non-linearities is due to optical and electronic imperfections that were not reported in the past and not included in the Heydemann corrections. These imperfections appear as amplitude difference and phase shift between each pair of intensity signals, and produce periodic non-linearities of nanometer amplitude when the intensity signals are combined to obtain contrats, i.e. quantities that are insensitive to intensity fluctuations of the light source. The second type of non-linearities is due to large amplitudes of harmonic oscillation, which become significant when the oscillations are typically larger than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>20</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mrow> <mml:mi>nm</mml:mi> </mml:mrow> </mml:mrow> </mml:math> for a red-emitting laser source, and can lead to an underestimation of the measured amplitude of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>∼</mml:mo> </mml:mrow> <mml:mn>20</mml:mn> <mml:mi mathvariant="normal">%</mml:mi> </mml:mrow> </mml:math> . In order to correct these non-linearities, new procedures of calibration and analysis are presented, implemented and successfully tested over a wide range of quasi-static velocities, frequencies and amplitudes of harmonic oscillation. The proposed corrections reduce non-linearities below the noise level, making them smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>100</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mrow> <mml:mi>pm</mml:mi> </mml:mrow> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>10</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mrow> <mml:mi>pm</mml:mi> </mml:mrow> </mml:mrow> </mml:math> for the quasi-static and harmonic displacement, respectively.