Angle measurement based on the internal-reflection effect: a new method

Two-Axial Measurement of the Angular Microdeflection of a Laser Beam Using One Single-Axis Sensor

The majority of current methods for measuring the angular deflection of a laser beam enable measurement only in one selected plane. However, there are tasks in which measurements of laser beam deflections in 3D are required. In this paper, we present a way of enabling two-axial measurements of the deflection of a beam based on a single-axis sensor. The key idea is to direct a laser beam, alternately, into one of two arms of a measurement system. In the first arm, the beam is transmitted directly to the angular sensor, while in the second, the beam is directed to the sensor via a special optical element that rotates the plane of the beam deflection; in other words, this element changes the deflection in the horizontal plane into a deflection in the vertical plane, and vice versa. To alternate the path of the beam, a variable phase retarder and a polarising beamsplitter are used. The proposed technique was experimentally verified, and the results confirm its effectiveness.

Integrated Optical Deformation Measurement with TIR Prism Rods

In this paper, a novel optical measurement principle for deformation, especially torsion, is presented. A laser beam is guided via total internal reflection (TIR) in a prism rod. Every single reflection causes an increasing change in the beam path, which can be measured by its effect on the outcoupling position of the laser. With a diameter of the prism rod of 10 mm and a length of 120 mm, the system achieves torsion sensitivities between 350 µm/° and more than 7000 µm/°, depending on the actual torsion angle φ. A decency level of sensitivity is defined for comparison, which is exceeded by a factor of ~55 at φ=0. The presented principle of TIR prism rods can be adapted to measure different load cases. Using two laser beams, bending and torsion can be distinguished and combined load cases analyzed. The resulting system can be integrated into machine elements, such as screws, to perform condition monitoring on mechanically loaded components.

An efficient analysis of oblique reflection of airborne ultrasound beams from thin membranes for gas sensing

The use of the angular spectrum method (ASM) to simulate the reflection of airborne ultrasound beams from a thin membrane separating air from a mixture of air and another gas is examined. The main advantage of this method is its high computing speed and efficiency for practical design calculations, suitable for sensing applications. The implemented ASM code is validated against custom Rayleigh integral code in a pure propagation simulation. In addition, ultrasound beam reflection calculations using ASM with finite element numerical results and experimental measurements are compared, finding good agreement in both cases. Then, ASM is used to estimate the sensitivity of specular reflection signals to variations in the composition of the incidence medium as a function of the angle of incidence. Conditions for which a reflection signal using inexpensive commercial ultrasound emitter/receiver at 40 kHz, in a simple configuration, offer a high enough sensitivity suitable for monitoring air quality indoors are found.

Theoretical Investigation for Angle Measurement Based on Femtosecond Maker Fringe

This paper proposes to utilize a femtosecond Maker fringe for angular measurement to expand the measurement range by using the characteristic of the multiple visible peaks in the Maker fringe. Four different z-cut nonlinear materials and three different polarization combinations of SHG were considered in the study, and various theoretical results are calculated for both the intensity-based angle measurement and the frequency-domain angle measurement. As a result, the p-s polarization shows a significant angle dependence in the range of ±20° compared with the other polarization. In addition, the BBO and KDP are superior to the other investigated materials because of the relatively higher sensitivity and visibility. The refractive index difference was introduced in this paper, and it was applied to explain the angle measurement performance of the Maker fringe successfully.

Machine vision-based high-precision and robust focus detection for femtosecond laser machining

We propose a machine vision-based focus detection method (MVFD) for femtosecond laser machining. By analyzing the laser focus pattern, the defocus direction and distance are obtained simultaneously. The proposed technique presents high precision with an average error of 0.047 µm and a root mean square error (RMSE) of 0.055 µm. Moreover, the method is robust and is less affected by the tilted sample. For the curved surface sample, the average error and RMSE are 0.093 and 0.145 µm, respectively. Thus, the proposed focus detection method can be easily combined with laser processing equipment, which is widely used in large-range and high-precision femtosecond laser processing.

An Optical Frequency Domain Angle Measurement Method Based on Second Harmonic Generation

This paper proposes a new optical angle measurement method in the optical frequency domain based on second harmonic generation with a mode-locked femtosecond laser source by making use of the unique characteristic of the high peak power and wide spectral range of the femtosecond laser pulses. To get a wide measurable range of angle measurement, a theoretical calculation for several nonlinear optical crystals is performed. As a result, LiNbO₃ crystal is employed in the proposed method. In the experiment, the validity of the use of a parabolic mirror is also demonstrated, where the chromatic aberration of the focusing beam caused the localization of second harmonic generation in our previous research. Moreover, an experimental demonstration is also carried out for the proposed angle measurement method. The measurable range of 10,000 arc-seconds is achieved.

Optical Sensors for Multi-Axis Angle and Displacement Measurement Using Grating Reflectors

In dimensional metrology it is necessary to carry out multi-axis angle and displacement measurement for high-precision positioning. Although the state-of-the-art linear displacement sensors have sub-nanometric measurement resolution, it is not easy to suppress the increase of measurement uncertainty when being applied for multi-axis angle and displacement measurement due to the Abbe errors and the influences of sensor misalignment. In this review article, the state-of-the-art multi-axis optical sensors, such as the three-axis autocollimator, the three-axis planar encoder, and the six-degree-of-freedom planar encoder based on a planar scale grating are introduced. With the employment of grating reflectors, measurement of multi-axis translational and angular displacement can be carried out while employing a single laser beam. Fabrication methods of a large-area planar scale grating based on a single-point diamond cutting with the fast tool servo technique and the interference lithography are also presented, followed by the description of the evaluation method of the large-area planar scale grating based on the Fizeau interferometer.

Double-grating with multiple diffractions enabled small angle measurement

We demonstrate a small angle measurement setup enabled by the double-grating configuration, where the multiple diffractions are used to magnify the deflection angle of the optical beam. Such small angle measurement setup has characteristics of high sensitivity and compact size. The use of two unparallel blazed gratings with a special included angle can realize multiple diffractions for the incident light, leading to the realization of deflection angle amplification. Experimental results verify that small angle can be accurately characterized and the angular resolution of measurement can be improved more than 40 times by inserting the double-grating configuration into the conventional auto-collimation angle measurement setup. Therefore, the micro-radian angle can be accurately measured with our proposed compact characterization setup.

Interference sensor for ultra-precision measurement of laser beam angular deflection

A new interference sensor for ultra-precise measurement of laser beam angular deflection is proposed. The angular tilt of the measuring device, in relation to the beam axis, can also be measured. The theoretical basis of the angle evaluation presented is based upon analyzing the fringe period, as well as the suggested optical setup and design of the device. Calculating algorithms for the evaluation of the beam angle change, the influence of the fringe phase on angle measurement, and the metrological feasibilities are also detailed herein. The main source of measurement errors is explored, and the influence of the sensor's positioning procedure in relation to the laser beam on the repeatability of the processing function is statistically analyzed. The results show that the sensor resolution reaches 25 nrad with an accuracy better than 1% in the measuring range 350 µrad. Finally, the potential applications of the interference sensor are described.

Full-circle range and microradian resolution angle measurement using the orthogonal mirror self-mixing interferometry

The self-mixing technique based on the traditional reflecting mirror has been demonstrated with great merit for angle sensing applications. In order to solve the problems of the narrow measurement angle range and low resolution in traditional angle measurement method, we proposed an angle measurement system using orthogonal mirror self-mixing interferometry combine an orthogonal mirror with designed mechanical linkage. It overcomes the shortcomings of traditional angle measurement methods and realized the angle measurement with microradian resolution in a full-circle range of 0 rad to 2π rad. In the experiment, the measurement resolution can reach to 5.27 µrad and the absolute error can lower to ± 0.011µrad, which satisfies the requirements of most high accuracy angle measurement.

Air wedge with variable refractive index for precise laser beam steering in a small range

This paper provides the results of work on a new method for metrological laser beam steering for the purpose of angular stabilization. Frequency stabilized metrological lasers are used for interference measurements of length and angle. Angular deviation of the beam may cause cosine errors when measuring displacements. This paper presents a method for metrological laser beam deflection with an air wedge of variable refractive index. The change in the index is achieved by changing the air pressure in the wedge volume. The implemented system produced a range of angular displacements of the 0.1 mrad (10^-4  rad; 100 μrad) beam with a resolution of 10^-7  rad.

Full-field transmission-type angle-deviation optical microscope with reflectivity-height transformation

This full-field transmission-type three-dimensional (3D) optical microscope is constructed based on the angle deviation method (ADM) and the algorithm of reflectivity-height transformation (RHT). The surface height is proportional to the deviation angle of light passing through the object. The angle deviation and surface height can be measured based on the reflectivity closed to the critical angle using a parallelogram prism and two CCDs.

Small-angle measurement with highly sensitive total-internal-reflection heterodyne interferometer

In this paper, a high-sensitivity total-internal-reflection (TIR) heterodyne interferometer is proposed for measuring small angles. In the proposed interferometer, a half-wave plate and two quarter-wave plates that exhibit specific optic-axis azimuths are combined to form a phase shifter. When a rhomboid prism is placed between the phase shifter and an analyzer that exhibits suitable transmission-axis azimuth, it shifts and enhances the phase difference of the s- and p-polarization states at double TIR. The enhanced phase difference is dependent on the incident angle; thus small angles can be easily and accurately measured by estimating the phase difference. The experimental results demonstrate the feasibility of this method. Angular resolution and sensitivity levels superior to 1.2×10⁻⁴ deg (2.1×10⁻⁶ rad) and 100 (deg/deg), respectively, were attainable in a dynamic range of 0.5 deg.

Interference method for ultra-precision measurement and compensation of laser beam angular deflection

A new interferometric method for ultra-precise measurement of laser-beam angular deflection is proposed. The angular tilt of a measuring device in relation to the beam axis also can be measured. The method is based on interference fringe period analysis in the selected plane of measurement. The theoretical basis and experimental verification of the method are presented. It is shown that by using the proposed technique, it is possible to measure the laser beam angular deflection or instability with ultrahigh resolution reaching single nanoradians. The proposed method allows the measurement and further compensation of laser beam deflections in a very compact design.

Maintaining a stationary laser footprint during angular scan in internal-reflection experiments

In internal-reflection metrology using prisms, the prism is usually mounted on a rotation/translation stage to enable adjusting angle and location of the laser footprint on the surface. If a visual inspection method is used to find the laser footprint, the task becomes impossible if invisible radiation in the near infrared is employed. In addition, it may be desirable to perform angular scan experiments with a stationary footprint on the surface during scans, or even to automatically probe specific points on an extended prism face for predetermined incidence angles. In this paper, a formulation is developed to determine the required translation along the prism face to allow maintaining the laser footprint stationary under a given rotation. A web-based app developed under the scope of this work demonstrates the applicability of the approach for silica, BK7 and SF2 glasses, in the wavelength range from 500 to 1500 nm and for an arbitrary geometry of the glass prism.

Littrow angle based autocollimation method for precision online monitoring three-dimensional angular drifts of chirped-pulse compression-gratings

A simple optical method for online precisely monitoring three-dimensional (3D) angular drifts of chirped-pulse compression-gratings in chirped-pulse amplification (CPA) lasers is proposed. Monitoring objects include all dimensional angular drifts of a grating, especially the grating groove in-plane-rotation. Monitoring contents contain direction and amount. And monitoring results are obtained by detecting the direction and the amount of 2D relative offsets of three focal spots formed by a ternary monitoring beam line. The theoretical model of the proposed method is set up using a matrix ray-tracing method, and a simplified monitoring equation is concluded based on numerical simulations. And the validity of this method is verified by a simple demonstration experiment. Besides, monitoring error and reliability discussion of the method are presented, as well.

Angle measurement with laser feedback instrument

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.

Nonscanning three-dimensional optical microscope based on the reflectivity-height transformation for biological measurements

We propose a nonscanning three-dimensional (3D) optical microscope based on reflectivity-height transformation in applications of biological and transparent plate measurements. The reflectivity of a prism can be transformed to the surface height of the specimen based on geometrical optics and the principle of internal reflection. Thus, the pattern of reflectivity is representative of the surface profile. Using charge-coupled device cameras to obtain the two-dimensional image patterns and combining with its reflectivity pattern, the 3D profile can be generated. The lateral resolution is determined by the diffraction limit, and the vertical resolution is better than several nanometers according to the incident angle and polarization used.

Angular amplification by a diffraction grating for chiro-optical measurements

The angles at which a light beam gets diffracted by a grating depend strongly on the direction of incidence for diffraction angles close to a right angle. Accordingly, it is possible to amplify small beam deflections by placing a grating at an optimal orientation to the light path. We use this principle to amplify small beam deviations arising out of a light beam refracting at the interface of an optically active medium, and demonstrate a new technique of enhancing the limit of detection of chiro-optical measurements.

Compact sensor for measuring two-dimensional tilt using a two-dimensional transmission grating and the Talbot effect

This paper proposes a tilt sensor that measures the small two-dimensional tilt of a plane reflective object using the Talbot effect. It is an extension of a previously proposed one-dimensional tilt sensor. The light beam reflected from the object impinges on a hexagonal grating, and the intensity of the diffracted wave is detected on an image sensor located at a Talbot distance from the grating. The diffraction intensity displaces due to the tilt of the object. The displacement is calculated by the Fourier transform method to obtain the two-dimensional tilt. This sensor is very simple and compact. The principle of the sensor is explained for a grating with a general pattern. An experiment using a hexagonal grating shows its validity. Discussions are given for making it more practical.

Measurement of angle and axis of rotation in a carousel interferometer: a detailed analysis

A detailed analysis of a carousel interferometer is presented for the measurement of an unknown angle and axis of rotation. The technique exploits a set of compensator glass plates and a right-angle prism that is placed in each of the two arms of the interferometer. The two sets are placed at the same rotational stage, while the end mirrors of the interferometer are static. When rotation takes place, individual and relative optical path differences are generated in the two beams of the interferometer. The generated phase differences contribute toward finding the angle and axis of rotation. The analysis is presented for any initial position of the interferometer, i.e., the radial vector from the axis of rotation to the apex of one of the prisms used. The results show the slight variations in the error and nonlinearity when different parameters are manipulated. Moreover, the trade-off between the maximum size of the prisms and the radial distances are also presented.

Angular displacement and deformation analyses using a speckle-based wavefront sensor

Wavefronts incident on a random phase plate are reconstructed via phase retrieval utilizing axially displaced speckle intensity measurements and the wave propagation equation. Retrieved phases and phase subtraction facilitate the investigations of wavefronts from test objects before and after undergoing a small rotation or deformation without sign ambiguity. Angular displacement (Deltatheta) between incident planar wavefronts is determined from the light source vacuum wavelength (lambda) divided by the fringe spacing (Lambda). Fourier analysis of the wavefront phase difference yields a peak frequency that is inversely proportional to Lambda, and the sign gives the direction of rotation. Numerical simulations confirm the experimental results. In the experiments, the smallest Deltatheta measured is 0.031 degrees . The technique also permits deformation analysis of a reflecting test object under thermal loading. The technique offers simple, high resolution, noncontact, and whole field evaluation of three-dimensional objects before and after undergoing rotation or deformation.

Optical measurements of angle and axis of rotation

A carousel interferometer is designed to find the unknown angle and axis of rotation simultaneously. A set consisting of a compensator glass plate and a right-angle prism is placed in each of the two arms of the interferometer while located at the same rotational stage. When rotation takes place individual and relative optical path differences are generated in the beams that contribute toward finding out the angle and axis of rotation. Computer simulation shows that error remains less than a nanometer for a 1 m length of unknown radius and rotation range of +/-15 degrees .

Low-coherent light-source angular interferometer using a square prism and the angular-scanning technique

An interferometer based on using low-coherent light source, a square prism, and the angular-scanning technique is proposed for absolute angular-displacement determinations. An angular displacement of the square prism shifts the correlogram, which is modulated by an envelope function, of the interference signal of the beams passing through the prism. This angle can thus be discovered by detecting the shifting of the envelope peak. A setup constructed to validate the interferometer is used. The results of using this setup are then presented.

Use of the circular Dammann grating in angle measurement

We describe a novel method of angle measurement by borrowing the concept of the circular Dammann grating (CDG). A three-order CDG is employed in this experiment. The displacement of the tilted angle can be determined accurately by measuring the projection from the distorted CDG image. This method is controlled only by the initial radius of the image and the converging ratio of the lens. Compared with conventional techniques, this technique has the advantages of a simple design with superior resolution to within 1 degree, low cost, and compactness. A theoretical analysis together with experimental results is presented.

Measurement of two-dimensional small angle deviation with a prism interferometer

A new technique for the measurement of two-dimensional small angular deviation is presented. A compound prism, which effectively produces a combination of two right-angled prisms in orthogonal directions, and plane reference surfaces have been utilized for the measurement of the orthogonal components of the angular tilt of an incident plane wavefront. Each orthogonal component of the angular tilt is separately measured from the angular rotation of the resultant wedge fringes between two plane wavefronts generated due to splitting of the incident plane wavefront by the corresponding set of right-angled prism and plane reference surface. The technique is shown to have high sensitivity for the measurement of small angle deviation. A monolithic prism interferometer, which is practically insensitive to vibration, is also proposed. Results obtained for the measurement of a known tilt angle are presented.

Interferometric evaluation of angular displacements using phase retrieval

Phase retrieval is carried out using sequential intensity measurements of a volume speckle field and a wave propagation equation. Retrieved phases and phase subtraction facilitate the analysis of wavefronts before and after undergoing a small rotation. Angular displacement between incident planar wavefronts is determined from the unwrapped phase difference, phase diffuser aperture diameter, and the light source wavelength. Numerical simulations confirm the experimental results.

Three-dimensional angle measurement based on propagation vector analysis of digital holography

We propose what we believe to be a novel method for highly accurate three-dimensional (3D) angle measurement based on propagation vector analysis of digital holography. Three-dimensional rotations in space can be achieved by use of a CCD camera and a multifacet object, which reflects an incident wave into different directions. The propagation vectors of the reflected waves from the object can be extracted by analyzing the object spectrum of the recorded hologram. Any small rotation of the object will induce a change in the propagation vectors in space, which can then be used for 3D angle measurement. Experimental results are presented to verify the idea.

Angle amplification for nanoradian measurements

A method to amplify the rotation angle of a mirror, based on multiple reflections between two quasi-parallel mirrors, is presented. The method allows rotations of fractions of nanoradians to be measured with a simple setup. The working principle, the experimental setup, and the results are presented.

Optical approach to angular displacement measurement based on attenuated total reflection

An optical approach for angular displacement measurement (ADM) based on the attenuated total reflection technique is presented. As a laser beam is incident upon a planar optical waveguide, an m line is obtained by scanning the incident angle. Theoretical analysis shows that the m line sharply shifts with a tiny variation of the thickness of the waveguided layer. And the specific schemes for ADM, which are based on the angular interrogation and the intensity measurement, are analyzed. The calculated result of sensitivity demonstrates that the intensity measurement is more efficient than the angular interrogation. Furthermore, small incident angles indicate higher sensitivity to the angular displacement than relatively large incident angles for the intensity measurement.

Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry

A new instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry is presented. With this instrument we can achieve a small rotation angle only by measuring the variation in phase difference between s- and p-polarization states. To improve its sensitivity we increase the number of total internal reflections by using two parallelogram prisms instead of two right-angle prisms. The angular resolution of the new instrument is better than 2.2 x 10(-6) rad over the measurement range -2.12 degrees < or = theta < or = 2.12 degrees for 20 total-internal reflections. The experimental results and the theoretical curve are in good agreement.

Optimization of linearity by use of a glass plate in carousel interferometers

Carousel interferometers are compact swinging interferometers, but their popularity is limited because of significant nonlinearity in rotational measurements. Introducing a glass plate in a rotating interferometer and optimizing its thickness has improved the linearity drastically. Computer simulations show reductions in nonlinearity of several orders of magnitude.

High-resolution optical angle sensors: approaching the diffraction limit to the sensitivity

We carry out a detailed analysis of angle-sensitive devices based on the critical-angle effect. We consider their use in measuring small angular deflections of a laser beam. We establish the diffraction limit to the sensitivity for optical-angle sensors based on reflection and transmission of a laser beam. We find that this limit is identical to that of the triangulation scheme when using a position-sensitive detector or the autocollimation scheme. We analyze the main proposals to date of optical-angle sensors based on the critical-angle effect, focusing on their maximum sensitivity and their polarization dependence in practical conditions. We propose and analyze theoretically a novel and simple angle-sensitive device for sensing optical-beam deflections with very low polarization dependence and a maximum sensitivity close to the diffraction limit when used with typical laser beams. We discuss the basic principles for designing this type of device, provide numerical results, and point out a convenient fabrication procedure.

Simple three-dimensional laser angle sensor for three-dimensional small-angle measurement

A simple three-dimensional (3D) laser angle sensor for 3D measurement of small angles based on the diffraction theorem and on ray optics analysis is presented. The possibility of using position-sensitive detectors and a reflective diffraction grating to develop a 3D angle sensor was investigated and a prototype 3D laser angle sensor was designed and built. The system is composed of a laser diode, two position-sensitive detectors, and a reflective diffraction grating. The diffraction grating, mounted upon the rotational center of a 3D rotational stage, divides an incident laser beam into several diffracted rays, and two position-sensitive detectors are set up for detecting the positions of +/-1st-order diffracted rays. According to the optical path relationship between the three angular motions and the output coordinates of the two position-sensitive detectors, the 3D angles can be obtained through kinematic analysis. The experimental results show the feasibility of the proposed 3D laser angular sensor. Use of this system as an instrument for high-resolution measurement of small-angle rotation is proposed.

High-resolution two-dimensional angle measurement technique based on fringe analysis

A novel angle-measurement technique based on fringe analysis for phase-measuring profilometry is proposed. A two-dimensional (2-D) angle between two mirror surfaces is determined by least-squares fitting of a plane to the 2-D distribution of the phase difference introduced by the 2-D tilt angle. To evaluate the performance of the proposed technique, numerical simulations that use the Fourier-transform technique and the phase-shift technique for fringe analysis were performed, and the results are compared. A 2-D angle-measurement interferometer based on a Mirau interference microscope was developed that demonstrated the validity of the proposed principle. It is shown by simulation and experiment that the proposed 2-D angle-measurement technique can achieve both a wide measurement range and a high angular resolution simultaneously.

Holographic common-path interferometer for angular displacement measurements with spatial phase stepping and extended measurement range

A novel technique for extending the unambiguous measurement range for differential measurements of angular deflections is presented. The technique utilizes a common-path interferometer that simultaneously probes the out-of-plane displacement of three points on the object surface. The system is based on a single laser diode, and all the optical functions of the system are implemented in a dedicated holographic optical element (HOE). The HOE automatically provides spatially phase-stepped interference signals for real-time phase measurement. It is therefore not necessary to employ any polarizing optics or active elements to introduce the phase stepping. The common-path scheme combined with the HOE provides a system that is inherently stable, since the HOE operates as both transmitter and receiver in the system. The system is compact, is robust, and has the potential for being mass-produced at a low cost and is thus well suited for industrial use, such as in commercial vibrometers. The technique is demonstrated in a system for measuring angular deflections of a plane mirror. The technique, however, is not restricted to this use alone and can easily be configured to probe other types of surface displacements, e.g., the deflection of a diaphragm. In the present configuration, the system can measure angular deflections with a sensitivity of 2.5 x 10(-7) rad over a measurement range that is approximately 3.5 x 10(-3) rad, i.e., a dynamic range of approximately 1:14,000. Furthermore, the system can easily be reconfigured for a desired angular sensitivity and measurement range.

Total internal reflection for precision small-angle measurement

A method for precision small-angle measurement is proposed. This method is based on the total-internal-reflection effect of a light beam at a pair of glass prisms. Angular displacement of the light beam is measured when the intensity change of the reflected beam is detected as a result of the relative phase shift between the s- and the p-polarized beams. An initial phase shift between the s- and the p-polarized components is introduced to increase measurement sensitivity. For increased measurement linearity and reduced effect of laser power fluctuation on the output, a differential method is used in which the light beam is split equally into two beams, each reflected at a prism and detected by a photodiode. The output is obtained as the difference of the two detected intensities divided by their sum. A prototype device was built, which demonstrated a nonlinearity error of 1.3% in a measurement range of ?0.6 degrees or 0.4% in ?0.3 degrees . The peak-to-peak noise level was found to be at approximately 0.5 arc sec. This noise level can be reduced further and resolution increased by a reduction of the measurement range.

Reflectivity of a gaussian beam near the critical angle with external optically absorbing media

One can use the angle-modulated reflectance of a gaussian beam near the critical angle to sense with high resolution the index of refraction of the external medium. We analyze in detail the reflectivity of a gaussian beam near the critical angle and its dependence on the optical absorption of the external medium. The given formulation is relatively simple and is useful in discerning the effects of the various parameters involved on the reflectivity and its differentials with respect to the angle of incidence. The results presented can be readily used for the quantitative design of novel sensors based on modulated reflectance near the critical angle. We provide a simple algebraic expression for the loss of sensitivity of modulated reflectance near the critical angle as the sample's absorption coefficient increases. We find that, in a typical case, the sensitivity has decreased to approximately half its value for transparent samples when the absorption coefficient has increased to 25 cm(-1). We conclude that modulated reflectance near the critical angle remains a competitive technique for monitoring the index of refraction of an external medium with an absorption coefficient of as much as 120 cm(-1). We compared experimentally obtained curves of the first differential of the reflectivity with respect to the angle of incidence with theory and found good agreement.

Small-angle measurement based on surface-plasmon resonance and the use of magneto-optical modulation

A new, to our knowledge, optical method for small-angle measurement based on surface-plasmon resonance (SPR) is presented. In this method the high sensitivity of the phase of SPR to the angle of incidence is employed to improve the resolution of the measurement of the angle. Small-angle measurement is performed by the monitoring of the phase shift resulting from the minute change of the angle of incidence with the use of magneto-optical modulation. The validity of this method is demonstrated, and a measurement resolution of 0.2 arc sec is achieved experimentally.

Use of thin films for high-sensitivity angle measurement

We propose to use thin films to provide a drastic improvement of measurement sensitivity in the recently developed small-angle measurement method, namely, angle measurement based on the internal-reflection effect. By designing the thin films (single layer or multiple layers) so that they provide an antireflection effect in the vicinity of the critical angle, we show that the sensitivity of angle measurement can be increased exponentially with the increase of the number of thin-film layers. This method provides a new means of designing angle sensors with increased sensitivities without having to increase the number of reflections and therefore the physical size and the required fabrication accuracy of the reflection prisms. We describe the design of the thin films for this particular application and the analysis of measurement sensitivity and range as determined by the material and the number of layers of the thin films. Selection of the optimal initial angle for high linearity performance is also discussed.

Improved angle interferometer based on total internal reflection

We describe an improved interferometer for angle measurement based on the internal reflection effect. The improvement is achieved by eliminating the influence of wave-plate rotation on the measurement. In the proposed angle interferometer the wave plate is fixed and placed between a rhomb assembly and a retroreflector. This scheme not only allows the angle interferometer to keep the optical configuration compact but also doubles the resolution and can measure the pitch and yaw of moving objects. Both a theoretical analysis and an experimental verification have been conducted on the interferometer. The results indicate that the performance of the modified angle interferometer is greatly improved, especially when the rotation angle is large. The nonlinearity error of the measurement equation is also addressed.

Measuring optical power transmission near the critical angle for sensing beam deflection

We investigate an internal-transmission method for measuring microdeflections of an optical beam as a potential tool for the development of new compact and stable optical sensors. We calculate the detection limits of the internal-transmission method when an ideal coherent optical source and an ideal quasi-monochromatic thermallike source are used. The proposed method is compared with an internal-reflection method previously studied. It is found theoretically and verified experimentally that the transmission method may have better resolution than the reflection method. We also compare the calculated sensitivity as a function of the angle of incidence with experimental results for both methods.

Small-angle measurement by use of a single prism

A new method of angle measurement based on the internal reflection effect is proposed that uses a single right-angle prism. We measure the angular displacement between a laser beam and the prism by detecting the changes in reflectance as a function of the angle of incidence. We achieve high linearity of measurement by taking the inverse of reflectance as the output. The inverse of reflectance is obtained from the intensities of the reflected and the transmitted beams measured by two photodiodes. Experiments with a prototype device have demonstrated that angle measurement with a range of ?500 arc sec, a nonlinearity error of ?0.1%, and a resolution of 0.1 arc sec can be readily achieved. The measurement range can be further increased with some sacrifice of linearity.

Interferometer for small-angle measurement based on total internal reflection

We describe a new method for angle measurement based on the internal-reflection effect and heterodyne interferometry. A novel prism assembly is designed that can always parallel retroreflect the incoming light beams so the optical configuration is compact. As a differential common-path optical configuration is integrated into the design, the linearity of the method is greatly improved. Details of theoretical analysis of the method and experimental verification of the principle are presented. The resolution can be better than 0.3 arc sec. The experimental results and further improvements of the proposed method are also addressed.

Nanoradian angle sensor and in situ self-calibration

We describe a high-precision angle sensor and a new calibration system for calibrating the linearity error of an angle sensor under in situ conditions without the need for accurate reference instruments. To achieve accuracy in the nanoradian (nrad) order, we elongated the critical-angle prism to increase the sensitivity, and we added a thermal controller to the angle sensor. This served to suppress thermal drift of the light source to improve stability. The proposed angle sensor demonstrated a stability of 3 nrad over 1 min, and the maximum repeatability error of the calibration curve determined by the in situ self-calibration method was approximately 35 nrad with a measurement range of 500 murad.

Improved technique for measuring small angles

Based on the total-internal-reflection effect and heterodyne interferometry, an improved technique for measuring small angles is proposed. This technique not only expands the measurement range but it also improves measurement performances. Its validity is demonstrated.

High accuracy, wide range, rotation angle measurement by the use of two parallel interference patterns

Based on the idea of measuring small rotation angles with a parallel interference pattern (PIP), a method is developed to measure large rotation angles accurately. Two parallel PIP's that have different periods are used to measure a rotation angle of an object. The measurement made with a small-period PIP provides a high accuracy, and the measurement made with a large-period PIP provides a wide range. An accurate measurement for wide-range angles is made by combining the two measured values. The accuracy of the phase detection is determined by the periods of two PIP's. Rotation angles from approximately -30 to 30 arc min can be measured with an accuracy of 0.2 arc sec. Analytical results are supported by experimental results.

High-resolution angular and displacement sensing based on the excitation of surface plasma waves

The possibility of building angular and displacement sensors based on the phenomenon of attenuated total reflection (ATR) is explored both numerically and experimentally. ATR occurs when a surface wave is excited by an incoming TM electromagnetic wave through a resonant phase-matching process, as in the Kretschmann coupling scheme. The reflected intensity strongly depends on the angle of incidence of the beam. We first show some computations of the sensitivity and the linearity of an ATR-based sensor, then proceed to the experiment, illustrating how an angular resolution of the order of 0.1 arc sec can be obtained with moderate effort. Finally we show how the sensor, combined with a simple optical arrangement, can be used to detect and measure nanometric displacements, as those provided by piezoelectric actuators.