## 1 Introduction  
Long-stroke and nanopositioning planar stages with nanometer-scale resolution and accuracy are essential in various precision systems, such as electronic device manufacturing equipment and scanning-type measuring instruments [1; 2]. Stacked-type planar stages are conventionally constructed with two linear stages by stacking one on top of the other [3; 4]. However, the stacked-type planar stages suffer from Abbe error, because the reference axis cannot be in line with the functional axis in the working space; thus, the stacked-type planar stages are not proper from the viewpoints of both positioning accuracy and positioning speed. Gantry stages [5], consisting of two \(x\)-axes and a \(y\)-axis, provide high capacity and high stiffness. Their travel accuracy, however, is significantly influenced by the parallelism of two \(x\)-axes and the Abbe error is also large due to the large Abbe offset. Compared with the stacked-type planar stage and the gantry stage, coplanar stages can eliminate the Abbe error by allowing the \(x\)- and \(y\)-motions along a common plane [6; 7; 8; 9]. The coplanar stage needs to be equipped with a high-precision position sensor for closed-loop control of each axis' motion. Linear encoders are regarded as more stable than laser interferometers for ultra-precision displacement measurement, as the mechanical grid is less sensitive to the ambient condition than the laser wavelength [10]. However, due to the geometric design of linear encoders, it is difficult to follow the Abbe principle when being installed into the coplanar stage as the feedback sensor. Commercial laser interferometers, made of He-Ne lasers featuring a long measurement range, fast speed, and nanometric resolution, are widely used for the calibration tool of linear stages. However, their sizes are too bulky to be embedded into the small nanopositioning stage as a feedback sensor. A linear displacement grating interferometer (LDGI), which employs the polarization technique to obtain low-noised sinusoidal waveforms, was developed to detect the \(x\)- and \(y\)-motions [6; 7]. However, the stage motion has inevitable geometrical angular errors that would cause pulse counting errors of the LDGI. A novel miniature laser diode interferometer based on the Michelson interferometer principle was, thus, developed for precision displacement feedback in each axis of a nanopositioning stage by the authors' group [11]. However, although the laser diode has the advantage of small size, it suffers from low stability in output power and wavelength, which is the length unit of laser interferometer, thereby limiting its applications in length measurements. In addition, the novel miniature laser diode interferometer cannot measure angular errors of the moving stage. It is known that angular errors induce Abbe errors in displacement measurements, which influence the displacement feedback accuracy of the laser diode interferometer. In order to achieve a high-precision closed-loop displacement feedback control of the nanopositioning stage, it is essential to ensure the accuracy of the laser diode wavelength and compensate for the angular-error-induced Abbe error. There are many methods for correcting the laser wavelength in room conditions. An air sensor, which is used to detect the current temperature, relative humidity, and atmospheric pressure to correct the refractive index of air through an empirical equation with a reference wavelength in vacuum, is commonly required in commercial laser interferometers. The empirical equation is usually taken from the Edlen equation [12] or its modified equations [13; 14]. However, on the one hand, the empirical equation was obtained based on the He-Ne laser, which has different properties from the laser diode. On the other hand, the empirical method is an indirect measurement, which is subject to the sensitivity and accuracy of the air sensor. Therefore, the correction of laser diode wavelength cannot directly adopt existing empirical equations.