An instrument for angle measurement based on laser feedback has been designed. The measurement technique is based on the principle that when a wave plate placed into a feedback cavity rotates, its phase retardation varies. Phase retardation is a function of the rotating angle of the wave plate. Hence, the angle can be converted to phase retardation. The phase retardation is measured at certain characteristic points identified in the laser outputting curve that are then modulated by laser feedback. The angle of a rotating object can be measured if it is connected to the wave plate. The main advantages of this instrument are: high resolution, compact, flexible, low cost, effective power, and fast response.
©2013 Optical Society of America
High-precise small-angle measurement is very important in many application areas, such as robots, manufacturing automation, coordinate measuring machines, and control of adaptive optical system. Angle measurements are traditionally employed using autocollimators [1,2] and interferometers [3,4]. Although both traditional methods provide high resolution, devices based on these principles are usually large, making the devices hard to integrate with machines for online measurement in many cases. A new method of angle measurement, namely that based on the inter-reflection effect, is proposed to fill this gap . The basic idea of this method is to use a differential detection scheme to largely reduce the inherent nonlinearity of the reflectance versus the angle of incidence so that the angular displacement of the laser beam can be accurately measured by the detection of the reflectance. For this method, in high measurement accuracy areas, elongated critical-angle prisms are unfortunately custom components and are not readily obtainable. These also require tight tolerances and, as a result, are expensive. Several other techniques have been proposed for angle measurement. These methods are based on single crystals of magnetic garnet , pattern projection techniques  double-exposure speckle photography , speckle shearing interferometry , photorefractive speckle correlation , Fourier transform , fringe  and circular Dammann gratings .
The methods listed above cannot simultaneously meet high angular resolution and low cost. In this paper, we propose an angle measurement method based on laser feedback. The advantages of this instrument are: high resolution, low cost, compact, flexible, effective power, and fast response.
Laser feedback was first observed in 1963 . Since then, it has been used in the fields of velocity , displacement , absolute distance  and vibration measurement . Our team began researching this phenomenon about 20 years ago. Some progress has been made experimentally and theoretically [19–22]. We have designed various instruments to perform displacement , phase retardation  and internal stress measurements .
2. Setup and principle
2.1 Measurement setup
The angle measurement instrument is shown in Fig. 1 . A half-intracavity, single mode, linearly polarized He-Ne laser is used as light source. The ratio of gaseous pressure in the laser is He:Ne = 9:1 and Ne20:Ne22 = 1:1. The working wavelength is 632.8 nm. The laser cavity is made up of antireflection window film, mirror M1, and mirror M2 with reflectivities of 99.8 and 98.8%, respectively. The cavity length is 150 mm.
The feedback cavity is used to reflect the laser back into the laser resonator and it is made up of M2 and feedback mirror ME, with the wave plate S between them. The feedback cavity length is 100 mm. ME has reflectivity of 15% and is used to reflect the laser beam back into the laser. A piezoelectric transducer (PZT) is used to tune, push and pull ME.
The laser outputting is detected by detector D and converted to voltage signal. Then, the voltage signal is observed by Oscilloscope and recorded in Computer by data acquisition card. AMP supplies triangular wave voltage and drives PZT. The maximum voltage applied to PZT is 100 V, which makes PZT a displacement of 0.5 μm.
When the length of the feedback cavity is scanned by PZT, laser-intensity transfer occurs, (see Fig. 2 ). There, the curve plots the laser intensity outputted from photo detector D. This is different from that of conventional laser feedback. There are dips at B and F point while the conventional laser feedback curves are similar to cosine form. The distance between points A and D or E and H is one period of λ/2. Through detecting the polarization states, we find the polarization states between AB and CD or EF and GH are mutually orthogonal. At points B and F, the polarization varies form one direction to its orthogonal direction. Thus, the points B and F are polarization flipping points.
The relationship between polarization flipping point and phase retardation magnitude is analyzed as Fig. 3 . The top curves in Fig. 3 are o light and e light intensity curves without laser feedback. The second curve is o light and e light intensity curves with laser feedback. The third curve is polarization states of laser outputting. The lowest curve is the voltage applized to PZT. In the feedback cavity, laser passes through the wave plate twice, so, the retardation of o light and e light in Fig. 3 is 2δ, whereδ is phase retardation of the wave plate. According to Fig. 3, the relationship between flipping point and phase retardation is calculated:
The measurement accuracy of this setup for phase retardation is 0.5° and the repeatability is 0.05°. The long term stability is 0.19°.
When the wave plate S placed in the feedback cavity is rotated, the phase retardation of the wave plate is changed. Thus, the variation of phase retardation reflects the rotation angle of the wave plate. The variation of phase retardation is the comprehensive effects of the thickness and refractive variation, and interference influence.
Firstly, the changes of thickness and refractive index are considered. The refractive index of ordinary (o) light and extraordinary (e) light varies when the wave plate is rotating. The phase retardation is given by:
where δ1 represents the phase retardation when the wave plate rotates without interference effect, δ0 represents the phase retardation when the rotating angle is zero, d is the thickness of the wave plate, λ is the laser wavelength, θ is the rotating angle, θo and θe are the angles of refraction of o light and e light, respectively, when the rotating angle is θ, no and ne refractive indices of o light and e light, respectively, when the rotating angle is zero, and noo and nee refractive indices of o light and e light, respectively, when the rotating angle is θ. If the tilt axis is perpendicular to the ray axis, noo and nee is given by:
The experimental results of phase retardation depending thickness and refractive index are shown in Fig. 4 .
Actually, multi-beam interference will appear in the transmitted light. Here, we use the secondary refraction from the inner surface of the wave plate to analyze the influence of interference on phase retardation.
The upper and lower surfaces of the wave plate are highly parallel. The refracted light of the same polarization directions will interfere with each other. From the refractive index ellipsoid, we can analyze the effects of secondary refraction. The secondary refraction is as shown in Fig. 5 . From the refractive index formula, the relations among θ, θo, and θe are given:
Based on the refractive index ellipsoid, noo and nee is given by:
whereas, based on the Fresnel formula, we obtain the various reflectivities and transmittances of o light and e light denoted by ro, re, to, and te:
The light rays passing through the wave plate will interfere with each other. The light field is given by:
The phase retardation including the interference, thickness variation and refractive index variation can be rewritten as:
In our experiment, simultaneous measurements were taken of the phase retardation using the laser feedback instrument and rotating angles using the high-precise goniometer. The experimental data for a six-degree angular displacement are shown in Fig. 6 . The phase retardation oscillates as the wave plate rotates; its variation is similar to a cosine waveform with attenuating peaks.
In practical applications of angle measurements, the wave plate is first placed in the feedback cavity, as shown in Fig. 1, with the wave plate surface aligned perpendicular to the ray direction. Next, the rotating object is connected to the wave plate. The phase retardation of wave plate is measured in real time as the object is rotating. The angular resolution of this laser feedback instrument is 0.00003° over the range ± 15°.
In this paper, an instrument for angle measurement is introduced. For this technique, the angle is converted to a phase retardation, which is measured using laser feedback. The measurement instrument is compact, flexible, low cost, and has effective power and a fast response. It is a useful tool for precision measurement and can also be used for establishing calibration and metrological standards.
This work is supported by the Key Program of the National Natural Science Foundation of China (NSFC) (No. 61036016) and Scientific and Technological Achievements Transformation and Industrialization Project by the Beijing Municipal Education Commission.
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