A theoretical investigation of generalized grating imaging and its application to optical encoders

• We perform an analysis of generalized grating imaging using a scalar Fresnel approach.
• Expressions for exact intensity distribution, location and contrast of the pseudoimages are derived.
• We apply generalized grating imaging to a reflective optical encoder using divergent illumination.
• Harmonic distortion of the pseudoimages is studied and methods to eliminate the harmonics are given.

Since optical encoders based on generalized grating imaging are more compact and robust than conventional Moiré encoders, they are preferable to be installed in industrial systems. To some extent, the measurement resolution and accuracy are limited by harmonic distortion of the encoder signals. In this work, an analysis of the pseudoimaging formation in a system including one grating illuminated by a point light source is firstly performed using a scalar Fresnel approach. Then, we analyze an optical encoder with a double grating configuration by considering the slits of the index grating as an array of point light source and considering the pseudoimages at the observation plane as superposition of the pseudoimages formed by each point light source. Conditions for the locations and contrast of the pseudoimages are derived. It shows that sufficient signal amplitude can be obtained at Talbot planes allowing a relative large gap tolerance. In particular, the harmonic distortion of the pseudoimages is analyzed and conditions for eliminating or suppressing the harmonic components are studied. The simulation results indicate that the n-th order harmonic component can be eliminated by setting the size of the slits of the index grating to be 2p1/n, where p1 is the period of the scale grating. Moreover, the dominant 3rd order harmonic component can also be suppressed by setting the aperture ratio of the scale grating to be 72.2%.