As one kind of anamorphic laser triangulation displacement sensor, rotationally symmetric triangulation (RST) has a lot of advantages comparing to traditional ones. Laser speckle and CCD noise are also two fundamental uncertainty factors in this kind of sensors. The analytic expression of centroid uncertainty limit from speckle in RST sensor is derived in this paper. It is shown that the uncertainty limit in RST, and also in anamorphic triangulation, is dependent on the solid angle subtended by entrance pupil as seen from the illuminated laser spot, as well as the laser wavelength. Because it's very easy to get a bigger entrance pupil in RST than in traditional laser triangulation, the centroid uncertainty limit from speckle in RST is much smaller. The CCD noise in RST and anamorphic laser triangulation is dependent on photon shot noise, dark current, photo-response non-uniformity, and cross talk. The centroid uncertainty limit from CCD noise in this case is described by 1-D Cramer-Rao lower bound, and is also smaller than in traditional triangulation. By using more CCD pixels, RST is less sensitive to noise level. Results of simulation and experiments verify the result of deriving.
Classical triangulation sensors are wildly used but they have some typical drawbacks. The measurement result depends always on the angular orientation of the sensor what can be especially troublesome at steps or gaps. To eliminate this disadvantage of the classical triangulation we introduced in  a new kind of optical triangulation - the rotationally symmetric triangulation sensor. Therefore the measurement result depends not any longer on the angular orientation of the sensor. This is achieved by imaging the scattered light from an illuminated object point to a centered and sharp ring on a low cost area detector. The diameter of the ring is proportional to the distance of the object.
The theoretical limit of the measurement uncertainty of the rotationally symmetric triangulation sensor is 3 to 4 times lower than the limit of the classical triangulation  for comparable and application oriented designs, because a complete ring is used for distance evaluation instead of only a point.
In this contribution we show for the first time a design and a corresponding hardware which is completely realized by two toriodal formed aspherical plastic lenses. These lenses can be manufactured by injection molding for approximately the same costs than ordinary aspherical plastic lenses. So it is possible to realize this new sensor for the same price than a classical triangulation sensor but with higher accuracy and a much better robustness.
For the rotationally symmetric triangulation sensor a standard 2D detector is used, the same detector like in standard vision systems. Additionally it is stressed that close to the axis of toriodal lenses is enough available design space to add a second optical system to image the object. The toriodal lenses allow to realize a retrofocus typ of imaging system without increasing the number of optical elements. However, in the middle of the lenses the surfaces are used for imaging and on the outer section they are used for triangulation. These two multipurpose optical elements can still be manufactured by injection moulding. To summarize, we show a low cost system with only one standard 2D detector and two aspheric lenses that realizes two tasks, i.e. imaging the object on the detector and distance measurement by rotationally symmetric triangulation.
In this paper a distance measurement sensor is introduced, equipped with two integrated optical systems, the first one for rotationally symmetric triangulation and the second one for imaging the object while using only one 2D detector for both purposes. Rotationally symmetric triangulation, introduced in , eliminates some disadvantages of classical triangulation sensors, especially at steps or strong curvatures of the object, wherefore the measurement result depends not any longer on the angular orientation of the sensor. This is achieved by imaging the scattered light from an illuminated object point to a centered and sharp ring on a low cost area detector. The diameter of the ring is proportional to the distance of the object. The optical system consists of two off axis aspheric reflecting surfaces. This system allows for integrating a second optical system in order to capture images of the object at the same 2D detector. A mock-up was realized for the first time which consists of the reflecting optics for triangulation manufactured by diamond turning. A commercially available appropriate small lens system for imaging was mechanically integrated in the reflecting optics. Alternatively, some designs of retrofocus lens system for larger field of views were investigated. The optical designs allow overlying the image of the object and the ring for distance measurement in the same plane. In this plane a CCD detector is mounted, centered to the optical axis for both channels. A fast algorithm for the evaluation of the ring is implemented. The characteristics, i.e. the ring diameter versus object distance shows very linear behavior. For illumination of the object point for distance measurement, the beam of a red laser diode system is reflected by a wavelength bandpath filter on the axis of the optical system in. Additionally, the surface of the object is illuminated by LED's in the green spectrum. The LED's are located on the outside rim of the reflecting optics. The scattered LED light is transmitted by the before mentioned bandpath filter and is captured by the imaging lens. A simultaneous mode, in which the ring for distance measurement is superimposed to the image of the object, and a time multiplexing mode were implemented thus demonstrating the flexibility and performance options of this approach.
Toroidal elements are special aspheric elements with a missing axial section. Such elements consist of several refractive and/or reflective optical surfaces which are generally tilted with respect to a base ray. This base ray replaces the optical axis in ordinary centered systems. Toroidal elements can be efficiently applied e.g. in LED illumination systems or in optical metrology systems. For these elements there is a lack of design principles, only very few approaches like the Coddington equations are known. In this paper an efficient method is presented that facilitates the design when the requirement or knowledge of the orientation of the image plane is necessary, i.e. where a generalized Scheimpflug condition is needed. In more general terms, the method results in imaging properties of second order expansion, but the method itself is linear. Therefore, the complexity of the design process is considerably reduced. Additionally it is shown how the individual surfaces of the toroidal element can be easily aspherized for sharp imaging omitting tedious optimization. The strength of the design method is demonstrated for a novel application where a complex toroidal element is required for rotationally symmetric triangulation integrated in a vision systems and for a high aperture illumination element based on TIR for LEDs.