Nonlinearity and frequency-path modelling of three-longitudinal-mode nanometric displacement measurement system S. Olyaee and S. Mohammad Nejad Abstract: The effects of periodic nonlinearity, intermodulation distortion, phase detection uncer- tainty and electrical cross-talk in a three-longitudinal-mode laser heterodyne Doppler interferom- eter (TLMI) displacement measurement system are presented. A frequency-path model consisting of three frequency components and two paths is also designed. As a result, there will be six frequency-path elements and 21 distinct interference terms. The periodic nonlinearity in the TLMI is mathematically modelled and simulated. Simulation results are compared with the typical two-mode heterodyne interferometer to confirm the designed model performance. The authors demonstrate that the nonlinearity of TLMI can be greatly reduced using the same com- pensation method as used in a two-frequency interferometer proposed by Hou et al. 1 Introduction A high precision displacement measurement is necessary in many applications such as photolithography, transducer calibration, geodesy, semiconductor fabrication, precision cutting, shape measurement and robotic systems. The fabri- cation of semiconductor chips requires lithographic stepper machines in order to measure high accuracy displacement [1, 2]. The minimum feature of the integrated circuits can be decreased by improvement in accuracy in the displace- ment measurement. The most important requirements for future photolithography and masking process such as accu- racy of nano-displacement are annually reported by ITRS [3]. Two frequency laser interferometers are being widely used as useful instruments for nanometric displacement measurements. On the other hand, heterodyne method such as Doppler-interferometry in comparison to homodyne method provides more signal-to-noise ratio and easier alignment. To reach higher resolution and accuracy in the displace- ment measurement, two-mode laser can be replaced by a stabilised three-longitudinal-mode laser. In the two-mode interferometer, one inter-mode beat frequency is produced, whereas in three-mode interferometer three primary beat frequencies and a secondary beat frequency appear. Yokoyama et al. [4] designed a three-longitudinal-mode interferometer with 0.044 nm resolution, assuming the phase detection resolution of 0.18. Although the three- longitudinal-mode interferometers have a higher resolution compared to two-longitudinal-mode type due to the inter- mode beat frequency reduction, maximum measurable velocity is dramatically reduced. However, limitation of the velocity in the displacement measurement can be elimi- nated by a design proposed in [5]. The main errors affecting the displacement accuracy can be categorised into three groups. The first group related to electronic and instrumentation section includes laser fre- quency instability, phase detection error and data age uncer- tainty [2]. Many efforts are still being devoted to improve the performance of laser frequency stabiliser and phase detector [6–15]. The second group consists of environ- mental effects. The accuracy of refractive index determi- nation, turbulences and thermal instability are the environmental parameters affecting the accuracy of the dis- placement [16–18]. Finally, the third group is related to misalignment and deviations in the optical setup and com- ponents such as polariser, polarising beam splitter (PBS) and laser head. Instability in the mechanical instruments, cosine error and Abbe error are directly related to the setup configuration. Non-orthogonality of the polarising radiations, elliptical polarisation and imperfect alignment of the laser head and other components produce periodic nonlinearity error [19–27]. The periodic nonlinearity in the two-frequency interferometer is modelled and presented by Cosijns et al. [20]. In this research, a new three-longitudinal-mode laser het- erodyne Doppler-interferometer (TLMI) system is pre- sented in which the periodic nonlinearity, intermodulation distortion, electrical cross-talk and phase detection error are considerably reduced. A frequency-path model [26] for this system is also designed. In addition, the periodic nonlinearity error due to the optical setup is calculated and mathematically modelled. 2 Frequency-path model of the TLMI 2.1 Principles The number of modes in the Doppler broadened gain curve can be determined by the free spectral range (FSR) or mode spacing of a He–Ne laser as FSR ¼ c/2nL, where L is the cavity length, c is the speed of light in vacuum, and n is the refractive index of medium. The gain profile of # The Institution of Engineering and Technology 2007 doi:10.1049/iet-opt:20060107 Paper first received 13th July 2006 and in revised form 26th March 2007 The authors are with the Department of Electrical Engineering, Iran University of Science and Technology (IUST), Narmak 16846, Tehran, Iran E-mail: s_olyaee@srttu.edu IET Optoelectron., 2007, 1, (5), pp. 211–220 211