Subtle Sources of Error in Laser Trackers Due to Dispersion in the Internal Optical Elements

It is well known that the speed of light depends on the index of refraction of the medium in which the light is propagating. It is also well known that in a dispersive medium, the speed of an amplitude modulated wavefront depends on the group refractive index, i.e., slightly slower than the carrier light. Corrections for the group refractive index in air are typically made for temperature, humidity, and pressure—without which measurements could be in error by tens of parts per million. The internal instrument optical elements are also subject to dispersive effects, which have heretofore been ignored in the literature—and presumably in the design. Note that this is probably because no commercially available optical design software package models amplitude modulated wavefronts. A thought experiment will illustrate the problem.

From Fermat’s principal, a plane wave intersecting a converging lens bends the wave to converge at a focal point. The lens is shaped such that the propagation time to the focal point is the same for all rays. For example, a ray passing through the outer radius of the lens passes through a thinner section of glass but must propagate a longer distance to the focus. A ray passing through the center of the lens passes through a thicker section of glass but propagates through a shorter distance to the focus. However, for optical amplitude modulated (OAM) light, the modulated wavefront, which has two sidebands that propagate at slightly different speeds in a dispersive medium, does not reach the focus at the same time! In other words, there is a slight phase shift in the modulated wavefront between the beam passing through the center of the lens, and the beam passing through the outer radius of the lens. This makes the net phase of the modulated waverfont, as received by a detector at the focal point, dependent on the beam geometry—which most likely depends on distance, due to divergence of the beam. At close range, the majority of the received beam passes through the center of the lens, due to the small beam size. At long range, the received beam passes through the entire lens, due to the expanded beam filling the lens. This source of error can be misinterpreted as being due to distance or power level, when in fact it is the optical design. Spherical, or cat’s eye, retroreflectors are also subject to the same source of error. A simple test to measure the errors is proposed.

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About The Author

David H. Parker

David H. Parker is the president of Parker Intellectual Property Enterprises LLC. With an master’s degree in physics and a bachelor’s degree in electrical engineering from Auburn University, Parker has more than 40 years of engineering experience, more than 30 years of metrology experience, and more than 10 years of patent-prosecution experience. Parker is a registered patent agent, a registered professional engineer, and holds 13 U.S. patents, with two U.S. patents pending.