DE102017002419A1 - Method and device for detecting a rotation angle and a direction of rotation of an object by means of spatial phase shift speckle interferometry - Google Patents

Abstract

Method and apparatus for detecting an angle of rotation and a direction of rotation of an object by means of spatial phase shift speckle interferometry The angle of rotation and the direction of rotation of an object are recorded simultaneously online with a simple and compact spatially phase shifting unit in-plane speckle interferometer. The elaborate method of unwrapping is not necessary for the rotation measurement, which simplifies the data processing algorithms considerably.

Description

Technical area

The present invention is in the field of optical metrology by means of in-plane speckle interferometry for rotational measurement.

State of the art

A very accurate and precise in-plane measurement of rotation is an essential part of geometric metrology and is in great demand in a variety of applications [1] [2]. Since the 1970s, various optical systems have been proposed for in-plane rotation measurement, such as Speckle Photography [3] and Electronic Speckle Pattern Interferometry (ESPI) [4]. Their common principle, however, is based on the digital image correlation of the speckle pattern before and after rotation, which has lost the original advantage of an interferometer system, ie a high accuracy of measurement. Another problem was that the determination of the direction of rotation required the algorithms for the multi-stage phase-stepping and phase-unwrapping, which hindered the real-time measurement and increased the difficulty in the data processing [5]. In recent years, several systems have been described with specific optical elements for in-plane rotation measurement. In 2010 have W. Li et al. presented a measuring system based on a calibration pattern with spot array [6] , One year later, an ESPI system with a diffractive holographic optical element [7] by R. Jallapuram et al. proposed. In 2014, a measuring system with transmission gratings and a heterodyne interferometer [8] by S. Tang et al. presents. However, all special optical elements must be placed in the plane of angular displacement, or even installed on the surface of the object, which hinders non-contact measurements. In [16] a method for on-line rotation measurement by means of in-plane speckle interferometry is described, but with a much worse resolution than in this patent application.

In [17] a method for on-line rotation measurement by means of in-plane speckle interferometry is described, with a similar structure as in this patent application, but with much higher expenditure due to phase shifting.

Quoted literature

1. [1] G. Hussain, and M. Ikram, Optical Measurements of Angle and Axis of Rotation Optics Letters 33 (21), 2419-2421 (2008) ,
2. [2] H. Kim, Y. Yamakawa, T. Senoo, and M. Ishikawa, "Visual Encoder: Robust and Precise Measurement Method of Rotation Angle via High Speed RGB Vision" Optics Express 24 (12), 13375-13386 (2016) ,
3. [3] J. Briers, and I. Angus, "Speckle Photography Applied to the Measurement of In-plane Rotation" Optica Acta 26 (10), 1229-1231 (1979). ,
4. [4] A. Nassim, L. Joannes, and A. Cornet, "In plane rotation analysis by two-wavelength electronic speckle interferometry" Applied Optics 38 (12), 2467-2470 (1999).
5. [5] J. Goodman, "Speckle Phenomena in Optics: Theory and Applications" Roberts & Company Publishers (2010) ,
6. [6] W. Li, J. Jin, X. Li, and B. Li, "Method of Rotation Angle Measurement in Machine Vision Based on Calibration Pattern with Spot Arrays" Applied Optics 49 (6), 1001-1006 (2010) ,
7. [7] R. Jallapuram, C. Healy, E. Mihaylova, and V. Toal "In-plane sensitive electronic speckle pattern interferometer using a diffractive holographic optical element" American Journal of Physics 79, 389-391 (2011) ,
8. [8th] Tang, Z. Wang, J. Gao, and J. Guo, "Measurement method for rolling angular displacement using a diffraction gratings and a heterodyne interferometer" Review of Scientific Instruments 85, 045110 (2012) ,
9. [9] A. Koch, M. Ruprecht, O. Toedter, and G. Häusler, "Optical Measurement Technique on Technical Surfaces" expert Verlag (1998) ,
10. [10] E. Berger, W. Linden, V. Dose, M. Ruprecht, A. Koch, "Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation" Applied Optics 36 (29), 7455-7460 (1997) ,
11. [11] E. Mihaylova, M. Whelan, and V. Toal, "Simple phase-shifting lateral shearing interferometer" Optics Letters 29 (11), 1264-1266 (2004) ,
12. [12] J. Valera, and J. Jones, "Phase-stepping in fiber-based speckle shearing interferometry" Optics Letters 19 (15), 1161-1163 (1994) ,
13. [13] M. Lu, S. Wang, L. Aulbach, and A. Koch, "Simultaneous Displacement and Slope Measurement in ESPI Using Adjustable Aperture Multiplexing" Applied Optics 55 (22), 5868-5875 (in production) (2016 ).
14. [14] K. Gasvik, "Optical Metrology Third Edition," John Wiley & Sons, Ltd. (2002) ,
15. [15] L. Aulbach, A. Koch, "Generation of high-resolution dynamic wavefronts for speckle-based measurements of complex surface shapes" Proc. SPIE 9660, 96600Y (2015) ,
16. [16] One-wavelength in-plane rotation analysis in electronic speckle pattern interferometry, Optics and Lasers in Engineering 35 (1), 2001.
17. [17] Spatial phase shifting for pure in-plane displacement and displacement-derivative measurements in electronic speckle pattern interferometry (ESPI), Applied Optics 36 (23), 1997.

Presentation of the invention and ways for carrying out the invention

Compared to existing techniques, the method of this patent application offers several advantages. First, the dynamic phase determination from a single interferogram using an adjustable spatial carrier frequency allows measurements to be taken in real time. Second, in addition to the rotation angle, the direction of rotation can be detected simultaneously by the sign of the result. Third, because the phase unwrapping process is not needed, the data processing algorithms become quite simple, but with high accuracy and precision. Fourth, completely non-contact measuring system with simple and compact optical design. Apart from the requirement of an optically rough surface of the measurement object, there are no further special requirements with respect to the measurement object. No special optical elements are needed. In addition, the measuring system is robust against environmental disturbances. Due to the relationship between the in-plane rotation and the phase change distribution of speckle patterns, this pure interferometric approach provides high measurement accuracy and precision. By obtaining the phase distribution from only a single interferogram, the results are achieved in real time by introducing a spatial carrier frequency. At the same time, the direction of rotation can be determined by the sign of the results. In addition, the need to perform unwrapping of the phase images is eliminated, which greatly simplifies the data processing procedure and therefore increases measurement efficiency. In addition, no additional specific optical elements are required, therefore, this measuring system is compact and performs completely non-destructive measurements.

auf den Empfindlichkeitsvektor $\overrightarrow{k}\left(\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\right)$

[9] [10]. Gleichung (1), mit Gleichung (2). l_x, l_y und l_z sind die absoluten Werte der x-, y- und z-Komponenten des Verschiebungsvektors; $\overrightarrow{u},\overrightarrow{v}\text{ und }\overrightarrow{w}$

bezeichnen die Einheitsvektoren entlang der positiven x-, y- und z-Achse. Die Phasenänderung, die durch die willkürliche Drehung des Objekts verursacht wird, kann bestimmt werden durch: (3), mit (4). 1 shows the optical configuration of the ESPI measuring system. An expanded and collimated laser beam 113 illuminates the test object 101, which has an optically rough surface. The diffused light 100, the object beams, is observed from two directions. After passing through the individual apertures 103 and 104, these two object beams are reflected by two plane mirrors 107 and 105 or 106 and the reflector 108. Then they pass through the imaging lens 109 and form an interferometric speckle pattern on the image sensor 110. This measuring system offers the possibility of introducing an adjustable lateral displacement between two object beams 100 by aligning the planar mirrors 105/106 and 107, ie a spatial phase shifting (PLC)). The object beams reflected from 106 (mirror 1 in position 2) and 107 (mirror 2) completely overlap. However, by turning the mirror 1 to position 1 (106), the object beams laterally move by Δs 112 in the negative x direction, hereinafter referred to as Δx. An in-plane rotation of the object surface 101 brings an additional phase difference between two object waves. To detect the angle of rotation, the phase change distribution must be quantified. The general expression for the phase change distribution Δφ (x, y, z) is proportional to the projection of the displacement vector $\overrightarrow{l}\left(x.y.z\right)$

on the sensitivity vector $\overrightarrow{k}\left(\overrightarrow{u}.\overrightarrow{v}.\overrightarrow{w}\right)$

[9] [10]. Equation (1), with equation (2). l _x , l _y and l _z are the absolute values of the x, y and z components of the displacement vector; $\overrightarrow{u}.\overrightarrow{v}\text{and}\overrightarrow{w}$

denote the unit vectors along the positive x, y and z axes. The phase change caused by the arbitrary rotation of the object can be determined by: (3), (4).

bezeichnet den Wellen vektor des Einfallsstrahls 113, während ${\overrightarrow{k}}_{21}\text{ und }{\overrightarrow{k}}_{22}$

die Wellenvektoren der beiden Objektstrahlen 100 darstellen; α und β sind die Einfallswinkel. Here, λ is the wavelength of the laser 102; ${\overrightarrow{k}}_1$

denotes the wave vector of the incident beam 113 while ${\overrightarrow{k}}_{21}\text{and}{\overrightarrow{k}}_{22}$

represent the wave vectors of the two object beams 100; α and β are the angles of incidence.

After substituting the expressions of the wave vectors (Equation 4) into Equation 3, the phase change distribution can be rewritten as follows: [11] [12] (5). ∂l _{x / ∂} x and ∂l _z / ∂x denote the corresponding partial derivatives.

parallel zur Ebene der Objektoberfläche. In diesem Zustand ist das Messsystem nicht mehr empfindlich gegenüber die Out-of-plane-Verschiebungen. Weiterhin gibt es im Fall einer reinen In-plane-Drehung keine Out-of-plane-Komponente. Somit kann die Gleichung 5 vereinfacht werden zu: (6).Obviously, the phase change distribution has the terms of in-plane shift and out-of-plane shift and the terms of its first-order derivatives at the same time. By setting the two observation directions symmetrical to the surface normal, ie α = β = θ, the sensitivity vector is located $\overrightarrow{k}\left(\overrightarrow{u}.\overrightarrow{v}.\overrightarrow{w}\right)$

parallel to the plane of the object surface. In this state, the measuring system is no longer sensitive to the out-of-plane shifts. Furthermore, in the case of pure in-plane rotation, there is no out-of-plane component. Thus, Equation 5 can be simplified to: (6).

Since the object beams propagate through the respective individual apertures 103 and 104 in the separate light paths, an adjustable spatial carrier frequency f ₀ (f _0x , f _0y ) is introduced into the measuring system. The phase change distribution can then be determined dynamically in the frequency domain. The direction of the carrier frequency depends on the relative position between the individual apertures and their absolute value depends on the distance between the apertures of the apertures 103 and 104 [13]. The intensity of the interferometric speckle pattern I (x, y) is given by: [14] [15] (7), with (8).

I ₁ (x, y) and I ₂ (x, y) are the intensities of the two object beams; φ (x, y) denotes the phase difference between them. Using the Fast Fourier Transform, the intensity image is transmitted from the spatial domain to the frequency domain: (9).

The first term, which is at the origin, represents the background intensity of the image. The last two terms correspond to the spectral pair of interference speckle patterns separated by the spatial carrier frequency from the zeroth order terms. The desired spectrum with surface information is filtered out and represented by the inverse Fourier transformation in order to obtain the phase distribution in the spatial domain: (10). Similarly, the phase mapping after object rotation can be obtained. By subtracting the phase map after the rotation φ _after from the phase map before the rotation φ _before , the phase change distribution is determined by: (11).

In this way, the generated spatial carrier frequency enables dynamic measurements since the phase map can be extracted from a single interferogram and only two interferograms are needed to evaluate the phase change due to surface deformations.

Based on the geometric relationship, the in-plane shift components I _x and I _y caused by the rotational motion of the object can be represented mathematically by: (12)
where (x, y) and (x ', y') represent the coordinates of the examined point before and after the rotation; W denotes the in-plane rotation angle. It should be noted that the formulas with angle addition and subtraction are applicable for counterclockwise and clockwise rotation, respectively. With the trigonometric formulas, Equation 12 can be further simplified to: (13).

The above equations show that the displacements caused by the rotation in the plane relate only to the coordinate of the observation point and its rotation angle. Since the angle of rotation is identical over the entire field of observation, both sides of the equations can be derived partially with respect to x or y.

If the derivative direction is perpendicular to the sensitivity vector, the first-order derivatives of the in-plane rotation are as follows: (14).

Similarly, if the derivative direction is parallel to the sensitivity vector, the first order derivatives are expressed by: (15).

Since the first-order partial derivatives of the displacement due to the in-plane rotation are represented only in the function of the rotation angle, the rotation caused in the plane can Expression of the phase change distribution (Equation 6) can be represented as a linear function with respect to the in-plane displacement as follows: (16).

In the above equation and in the following derivation process, the arguments of the functions specifying the observation point are omitted for the sake of simplicity. K and D are the pitch coefficient and the intercept, respectively. (17). l _i denotes the in-plane shift whose index i is determined by the direction of the sensitivity vector in the measuring system. When the sensitivity vector points in the x (or y) direction, l _{i is} equal to l _x (or l _y ); l _i '(Ω) is the derivative of the first order of the deformation, which refers to the rotation in the plane. The direction of the shift gradient is based on the direction of the lateral shift Δs between two object beams in the camera plane. When Δs is along the x (or y) axis, l _i '∂l _i / ∂x (or ∂l _i / ∂y), and Δs is identical to Δx (or Δy).

Since the in-plane rotation angle Ω has only an influence on the point of intersection, we can obtain the first order derivative of the displacement directly from the measured phase change distribution by differentiating both sides of equation 16: (18). By substituting equation 14 into equation 18, the in-plane rotation angle of the object can be determined. Meanwhile, the sign of the results indicates the direction of rotation. Assuming that the sensitivity vector occurs along the x-axis and the phase change distribution is differentiated with respect to y, the angle of rotation can be calculated by the following equation: (19). A positive (or negative) result represents rotation in the plane in a clockwise (or counterclockwise) direction. Due to in-plane rotation, the first order derivative of the unwrapped phase change distribution over the entire object surface is constant. Therefore, the in-plane angle of rotation can be theoretically evaluated by using only two adjacent distinct values on the phase change map.

Given the wavelength of the laser λ, the angle of incidence θ can be measured as the lateral displacement between the object waves Δy and the phase change distribution Δφ. The in-plane rotation angle can be evaluated according to the following formula instead of equation 19: (20).

Δd represents the actual length measured by each pixel of the camera. Δφ may be the phase distribution in either the unwrapped phase change map or in the continuous part of the wrapped phase change map. In this way, the need to unwrap the phase change card is eliminated, and the measurement efficiency is thus improved. Regarding the distribution of the in-plane rotation angles, evaluated by applying the equation 19: Any angle value from this distribution can also be interpreted as the amount of results evaluated with the phase changes of every two adjacent points using Equation 20. In practice, the measured in-plane rotation angle is not constant because the first order derivative of the phase change distribution is not completely equal everywhere due to the measurement uncertainty.

2 represents a further embodiment.

The in-plane angle measurement can also be done by including a reference light path in the speckle interferometer in 1 be implemented. The optical configuration is in 2 specified. The two object paths are identical to the in-plane rotation measuring system in 1 , The beam splitter 112 splits the laser beam into two orthogonal beams 003 and 118. The transmitted beam 003 illuminates the measurement object 000 while the reflected beam 118 illuminates the reference object 114. Both objects have a visually rough surface. The reference light beam 119 passes through the lens 116 and the single aperture 117 and interferes with the two object beams 100 on the detector 110 after reflection by the mirror 115 and the beam splitter 113. By tilting 115 or 113, the position of the reference speckle pattern be adjusted on the detector. In this way, a three-beam interferometer is constructed and the phase change distributions between each of the two light beams due to movements of the object can be calculated according to Equation 1. The sensitivity _{vector of} the two object _beams k _{obj1 / obj2} and the sensitivity _vectors between reference beam and object _beams k _{ref / obj1} and k _{ref / obj2} can be expressed by: Equation (21) where k _l denotes the unit vector in the illumination _direction ; k ₂₁ and k ₂₂ are unit vectors in the respective observation direction. The object moves in any direction and can be described by equation (2). Thus, the corresponding phase change distributions can be calculated by: equations (22), (23) and (24), where l _x 'is the first-order derivative of l _x in the direction of the shear distance s 111; θ denotes the angle between the direction of illumination and observation. Based on equation (23) and equation (24), the displacement of the measurement object in the plane and out of plane can be evaluated by subtraction and addition _{formulas of} the phase changes Δφ _{ref / obj1} and Δφ _{ref / obj2} by: equations (25) and (26 ). By equation (14) and Equation (15), the rotation angle can be calculated by: Equation (27a) or Equation (28a). Similarly, if the sensitivity _vector k _{shows obj1 / obj2} in the y-direction, the rotation angle refers to the first-order derivative of the displacement along the y-axis: equation (27b) or equation (28b). It should be noted that in addition to the in-plane rotation angle, the out-of-plane displacement parameter may also be determined using this measurement system.

3 represents a further embodiment.

Under the condition that the two waves have orthogonal linear or circular polarizations, a rotating half wave plate or a quarter wave plate can produce the phase delay. Based on this principle of operation, a three-channel polarization interferometer can be used to perform spatial phase shift measurements to obtain in-plane rotation. The optical configuration is in 3 shown. In order to make the polarization of the two object beams orthogonal, two conditions must be satisfied: the axes of the two half-wave plates 201 and 202 in the two object paths are inclined by 45 ° to each other; the incident beam 003 must be linearly polarized. The orthogonally polarized beams 100 continue to propagate to the three-channel direct-phase measuring module. In channel 1, the light beam passes directly through a linear polarizer 208 whose transmission axis is 45 ° to the polarization direction of the incident beam and enters the imaging plane of detector 214. In channels 2 and 3, two quarter wave plates 203 and 204 are used to to shift the phase by π / 2 or π before the light beams reach the detectors 215 and 216. Thus, the recorded three interferograms I ₁ (x, y), I ₂ (x, y) and I ₃ (x, y) can be expressed by Equation (29). A (x, y) is the background intensity, while B (x, y) is the modulation intensity. Based on Equation 29, the phase distribution before and after the rotation can be determined by Equation (30). Subsequently, the phase change map due to the rotation can be calculated by using equation 11 and the rotation angle by equation 19. Likewise, the three-channel polarization phase-shift interferometer can be further extended to four channels or even more.

4 represents a further embodiment.

$$\overrightarrow{l}\left(\text{x},\text{y},\text{z}\right)=l_x\cdot \overrightarrow{u}+l_y\cdot \overrightarrow{v}+l_y\cdot \overrightarrow{w}$$

$$\mathrm{\Delta \varphi }\left(x,y,z\right)=\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{21}\right)\cdot \overrightarrow{l}\left(x,y,z\right)-\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{22}\right)\cdot \overrightarrow{l}\left(\text{x}\mathrm{+\Delta }\text{x},\text{y},\text{z}\right)$$

$${\overrightarrow{k}}_1=\frac{2\pi }{\lambda }\cdot \left(-\overrightarrow{w}\right)$$

$${\overrightarrow{k}}_{21}=\frac{2\pi }{\lambda }\cdot \left(\text{sin}\alpha \cdot \overrightarrow{u}+\text{cos}\alpha \cdot \overrightarrow{w}\right)$$

$${\overrightarrow{k}}_{22}=\frac{2\pi }{\lambda }\cdot \left(\mathrm{-}\text{sin}\beta \cdot \overrightarrow{u}+\text{cos}\beta \cdot \overrightarrow{w}\right)$$

$$\begin{array}{c}\mathrm{\Delta }\varphi \left(\text{x},\text{z}\right)=-\frac{2\pi }{\lambda }\left\{\left[\left(\text{sin}\alpha +\text{sin}\beta \right)\cdot l_x+\left(\text{cos}\alpha +\text{cos}\beta \right)\cdot l_z\right]\right\}\\ +\left[\text{sin}\beta \cdot \frac{\partial l_x}{{\partial }_x}-\left(1+\text{cos}\beta \right)\cdot \frac{\partial l_x}{{\partial }_x}\right]\mathrm{\Delta }\text{x}\end{array}$$

$$\mathrm{\Delta }\varphi \left(x\right)=-\frac{2\pi }{\lambda }\left(2\text{sin}\theta \cdot l_x+\text{sin}\theta \cdot \frac{\partial l_x}{{\partial }_x}\cdot \mathrm{\Delta }\text{x}\right)$$

$$I\left(x,y\right)=a\left(x,y\right)+c\left(x,y\right)e^{12\pi \left(f_{0x}x+f_{0y}y\right)}+c\left(x,y\right)\text{*}{e}^{-i2\pi \left(f_{0x}x+f_{0y}y\right)}$$

$$a\left(x,y\right)=I_1\left(x,y\right)+I_2\left(x,y\right)$$

$$c\left(x,y\right)=\sqrt{I_1\left(x,y\right)+I_2\left(x,y\right)}{e}^{i\varphi \left(x,y\right)}$$

$$\mathcal{J}\left(f_x,f_y\right)=\mathcal{A}\left(f_x,f_y\right)+\mathcal{C}\left(f_x-f_{0x},f_y-f_{0y}\right)+\mathcal{C}\text{*}\left(f_x-f_{0x},f_y-f_{0y}\right)$$

$$\varphi ={\text{tan}}^{-1}\frac{Im\left[\mathcal{C}\left(f_x-f_{0x},f_y-f_{0y}\right)\right]}{Re\left[\mathcal{C}\left(f_x-f_{0x},f_y-f_{0y}\right)\right]}$$

$$\mathrm{\Delta }\varphi \mathrm{=}{\varphi }_{after}-{\varphi }_{before}$$

$$l_x=x\text{'}-x=\sqrt{x^2+y^2}cos\left(atan2\frac{y}{x}\pm \Omega \right)-x$$

$$l_y=y\text{'}-y=\sqrt{x^2+y^2}sin\left(atan2\frac{y}{x}\pm \Omega \right)-y$$

$$l_x=xcos\Omega \mp ysin\Omega -x$$

$$l_y=ycos\Omega \mp xsin\Omega -y$$

$$\frac{\partial l_x}{\partial y}=\mp sin\Omega$$

$$\frac{\partial l_y}{\partial x}=\mp sin\Omega$$

$$\frac{\partial l_x}{\partial x}=cos\Omega -1$$

$$\frac{\partial l_y}{\partial y}=cos\Omega -1$$

$$\Delta \varphi =-\frac{2\pi }{\lambda }\left\{2\text{sin}\theta \cdot l_i+\text{sin}\theta \cdot l_i\text{'}\left(\Omega \right)\cdot \Delta s\right\}=K\cdot l_i+D$$

$$K=-\frac{4\pi }{\lambda }\text{sin}\theta$$

$$D=-\frac{2\pi }{\lambda }\cdot \text{sin}\theta \cdot l_i\text{'}\left(\Omega \right)\cdot \Delta s$$

$$l_i\text{'}=-\frac{\lambda }{4\pi \cdot \text{sin}\theta }\cdot \Delta \varphi \text{'}$$

$$\Omega ={\text{sin}}^{-1}\left[\frac{\lambda }{4\pi \cdot \text{sin}\theta }\cdot \frac{\partial \left(\Delta \varphi \right)}{\partial \text{y}}\right]$$

$$\Omega ={\text{sin}}^{-1}\left\{\frac{\lambda }{4\pi \cdot \text{sin}\theta }\cdot \frac{\Delta \mathrm{\varphi }\left(x,y+1\right)-\Delta \mathrm{\varphi }\left(x,y\right)}{\Delta d}\right\}$$

$${\overrightarrow{k}}_{obj1/obj2}=\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{21}\right)-\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{22}\right)={\overrightarrow{k}}_{22}-{\overrightarrow{k}}_{21}$$

$${\overrightarrow{k}}_{ref/obj1}={\overrightarrow{k}}_1-{\overrightarrow{k}}_{21}$$

$${\overrightarrow{k}}_{ref/obj2}={\overrightarrow{k}}_1-{\overrightarrow{k}}_{22}$$

$$\Delta {\phi }_{obj1/obj2}={\overrightarrow{k}}_{obj1/obj2}\cdot \Delta \overrightarrow{d}=\frac{4\pi }{\lambda }\text{sin}\theta \cdot {\text{l}}_x\text{'}\cdot s$$

$$\Delta {\phi }_{ref/obj1}={\overrightarrow{k}}_{ref/obj1}\cdot \Delta \overrightarrow{d}=\frac{2\pi }{\lambda }\left[\text{sin}\theta \cdot {\text{l}}_x+\left(1+\text{cos}\theta \right)\cdot l_z\right]$$

$$\Delta {\phi }_{ref/obj2}={\overrightarrow{k}}_{ref/obj2}\cdot \Delta \overrightarrow{d}=\frac{2\pi }{\lambda }\left[-\text{sin}\theta \cdot {\text{l}}_x+\left(1+\text{cos}\theta \right)\cdot l_z\right]$$

$$l_x=\frac{\lambda }{4\pi \text{ sin}\theta }\left(\Delta {\phi }_{ref/obj1}-\Delta {\phi }_{ref/obj2}\right)$$

$$l_z=\frac{\lambda }{4\pi \left(1+cos\theta \right)}\left(\Delta {\phi }_{ref/obj1}+\Delta {\phi }_{ref/obj2}\right)$$

$$\Omega =\pm {\text{sin}}^{-1}\left(\frac{\partial l_x}{\partial y}\right)$$

$$\Omega ={\text{cos}}^{-1}\left(\frac{\partial l_x}{\partial x}+1\right)$$

$$\Omega =\pm {\text{sin}}^{-1}\left(\frac{\partial l_y}{\partial x}\right)$$

$$\Omega ={\text{cos}}^{-1}\left(\frac{\partial l_y}{\partial y}+1\right)$$

$$I_1\left(x,y\right)=A\left(x,y\right)+B\left(x,y\right)$$

$$I_2\left(x,y\right)=A\left(x,y\right)+B\left(x,y\right)\text{cos}\left[\varphi \left(x,y\right)+\frac{\pi }{2}\right]$$

$$I_3\left(x,y\right)=A\left(x,y\right)+B\left(x,y\right)\text{cos}\left[\varphi \left(x,y\right)+\pi \right]$$

$$\varphi \left(x,y\right)={\text{tan}}^{-1}\frac{2I_1-I_2-I_3}{I_2-I_3}$$

$$I_4\left(x,y\right)=A\left(x,y\right)+B\left(x,y\right)\text{cos}\left[\varphi \left(x,y\right)+\frac{3\pi }{2}\right]$$

$$\varphi \left(x,y\right)={\text{tan}}^{-1}\frac{I_4-I_2}{I_1-I_3}$$

The interferometer with a phase shift element, z. A pixel-shaped phase mask, is capable of independently introducing a discrete phase shift at each pixel since each pixel of the phase mask has a unique phase delay. For this reason, it is able to recover the phase information and requires only a single speckle pattern for instant measurements. A spatially adjacent pixel block is considered as a unit cell that is repeated over the entire phase mask. If there are at least three phase shifts within a unit cell, the phase map can be recovered from the recorded interferograms using conventional phase shift algorithms. As in 4 (b) 1, an elementary cell comprises 2 × 2 pixels with the phase shifts of 0, π / 2, π or 3π / 2. Thus, the four-step algorithm can be used to reconstruct the phase information. The optical configuration of a pixelated phase shift element interferometer is shown in FIG 4 (a) shown. The polarization interferometer generates the linearly polarized beam whose polarization states of the two object beams are orthogonal to each other. Therefore, the axes of the two half-wave plates 201 and 202 in the two object paths must be inclined by 45 ° to each other, and the incident beam 003 must be linearly polarized. After passing through the phase shift element 300, the phase shifts are coded pixel by pixel in the optical wave, which are subsequently detected by the detector 110. In this way, the recorded interferogram can be divided into four subinterferograms with phase shifts of 0, π / 2, π and 3π / 2. The first three interferograms under different phase shifts can be expressed by equation (29) and the fourth is determined from equation (31). The four step phase shift algorithm can be used to determine the phase distribution either before or after a rotational movement of the DUT: Equation (32). Thereafter, the phase change map due to the rotation can be determined using equation (11) and the rotation angle can be determined by equation (19). $$\mathrm{\Delta }\phi \left(x.y.z\right)=\overrightarrow{k}\left(\overrightarrow{u}.\overrightarrow{v}.\overrightarrow{w}\right)\cdot \overrightarrow{l}\left(x.y.z\right)$$

$$\overrightarrow{l}\left(\text{x}.\text{y}.\text{z}\right)=l_x\cdot \overrightarrow{u}+l_y\cdot \overrightarrow{v}+l_y\cdot \overrightarrow{w}$$

$$\mathrm{\Delta \phi }\left(x.y.z\right)=\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{21}\right)\cdot \overrightarrow{l}\left(x.y.z\right)-\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{22}\right)\cdot \overrightarrow{l}\left(\text{x}\mathrm{+\; \Delta }\text{x}.\text{y}.\text{z}\right)$$

$${\overrightarrow{k}}_1=\frac{2\pi }{\lambda }\cdot \left(-\overrightarrow{w}\right)$$

$${\overrightarrow{k}}_{21}=\frac{2\pi }{\lambda }\cdot \left(\text{sin}\alpha \cdot \overrightarrow{u}+\text{cos}\alpha \cdot \overrightarrow{w}\right)$$

$${\overrightarrow{k}}_{22}=\frac{2\pi }{\lambda }\cdot \left(\mathrm{-}\text{sin}\beta \cdot \overrightarrow{u}+\text{cos}\beta \cdot \overrightarrow{w}\right)$$

$$\begin{array}{c}\mathrm{\Delta }\phi \left(\text{x}.\text{z}\right)=-\frac{2\pi }{\lambda }\left\{\left[\left(\text{sin}\alpha +\text{sin}\beta \right)\cdot l_x+\left(\text{cos}\alpha +\text{cos}\beta \right)\cdot l_z\right]\right\}\\ +\left[\text{sin}\beta \cdot \frac{\partial l_x}{{\partial }_x}-\left(1+\text{cos}\beta \right)\cdot \frac{\partial l_x}{{\partial }_x}\right]\mathrm{\Delta }\text{x}\end{array}$$

$$\mathrm{\Delta }\phi \left(x\right)=-\frac{2\pi }{\lambda }\left(2\text{sin}\theta \cdot l_x+\text{sin}\theta \cdot \frac{\partial l_x}{{\partial }_x}\cdot \mathrm{\Delta }\text{x}\right)$$

$$I\left(x.y\right)=a\left(x.y\right)+c\left(x.y\right)e^{12\pi \left(f_{0x}x+f_{0y}y\right)}+c\left(x.y\right)\text{*}{e}^{-i2\pi \left(f_{0x}x+f_{0y}y\right)}$$

$$a\left(x.y\right)=I_1\left(x.y\right)+I_2\left(x.y\right)$$

$$c\left(x.y\right)=\sqrt{I_1\left(x.y\right)+I_2\left(x.y\right)}{e}^{i\phi \left(x.y\right)}$$

$$\mathcal{J}\left(f_x.f_y\right)=\mathcal{A}\left(f_x.f_y\right)+\mathcal{C}\left(f_x-f_{0x}.f_y-f_{0y}\right)+\mathcal{C}\text{*}\left(f_x-f_{0x}.f_y-f_{0y}\right)$$

$$\phi ={\text{tan}}^{-1}\frac{Im\left[\mathcal{C}\left(f_x-f_{0x}.f_y-f_{0y}\right)\right]}{Re\left[\mathcal{C}\left(f_x-f_{0x}.f_y-f_{0y}\right)\right]}$$

$$\mathrm{\Delta }\phi \mathrm{=}{\phi }_{after}-{\phi }_{befOre}$$

$$l_x=x\text{'}-x=\sqrt{x^2+y^2}cOs\left(atan2\frac{y}{x}\pm \Omega \right)-x$$

$$l_y=y\text{'}-y=\sqrt{x^2+y^2}sin\left(atan2\frac{y}{x}\pm \Omega \right)-y$$

$$l_x=xcOs\Omega \mp ysin\Omega -x$$

$$l_y=ycOs\Omega \mp xsin\Omega -y$$

$$\frac{\partial l_x}{\partial y}=\mp sin\Omega$$

$$\frac{\partial l_y}{\partial x}=\mp sin\Omega$$

$$\frac{\partial l_x}{\partial x}=cOs\Omega -1$$

$$\frac{\partial l_y}{\partial y}=cOs\Omega -1$$

$$\Delta \phi =-\frac{2\pi }{\lambda }\left\{2\text{sin}\theta \cdot l_i+\text{sin}\theta \cdot l_i\text{'}\left(\Omega \right)\cdot \Delta s\right\}=K\cdot l_i+D$$

$$K=-\frac{4\pi }{\lambda }\text{sin}\theta$$

$$D=-\frac{2\pi }{\lambda }\cdot \text{sin}\theta \cdot l_i\text{'}\left(\Omega \right)\cdot \Delta s$$

$$l_i\text{'}=-\frac{\lambda }{4\pi \cdot \text{sin}\theta }\cdot \Delta \phi \text{'}$$

$$\Omega ={\text{sin}}^{-1}\left[\frac{\lambda }{4\pi \cdot \text{sin}\theta }\cdot \frac{\partial \left(\Delta \phi \right)}{\partial \text{y}}\right]$$

$$\Omega ={\text{sin}}^{-1}\left\{\frac{\lambda }{4\pi \cdot \text{sin}\theta }\cdot \frac{\Delta \mathrm{\phi }\left(x.y+1\right)-\Delta \mathrm{\phi }\left(x.y\right)}{\Delta d}\right\}$$

$${\overrightarrow{k}}_{Obj1/Obj2}=\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{21}\right)-\left({\overrightarrow{k}}_1-{\overrightarrow{k}}_{22}\right)={\overrightarrow{k}}_{22}-{\overrightarrow{k}}_{21}$$

$${\overrightarrow{k}}_{ref/Obj1}={\overrightarrow{k}}_1-{\overrightarrow{k}}_{21}$$

$${\overrightarrow{k}}_{ref/Obj2}={\overrightarrow{k}}_1-{\overrightarrow{k}}_{22}$$

$$\Delta {\phi }_{Obj1/Obj2}={\overrightarrow{k}}_{Obj1/Obj2}\cdot \Delta \overrightarrow{d}=\frac{4\pi }{\lambda }\text{sin}\theta \cdot {\text{l}}_x\text{'}\cdot s$$

$$\Delta {\phi }_{ref/Obj1}={\overrightarrow{k}}_{ref/Obj1}\cdot \Delta \overrightarrow{d}=\frac{2\pi }{\lambda }\left[\text{sin}\theta \cdot {\text{l}}_x+\left(1+\text{cos}\theta \right)\cdot l_z\right]$$

$$\Delta {\phi }_{ref/Obj2}={\overrightarrow{k}}_{ref/Obj2}\cdot \Delta \overrightarrow{d}=\frac{2\pi }{\lambda }\left[-\text{sin}\theta \cdot {\text{l}}_x+\left(1+\text{cos}\theta \right)\cdot l_z\right]$$

$$l_x=\frac{\lambda }{4\pi \text{sin}\theta }\left(\Delta {\phi }_{ref/Obj1}-\Delta {\phi }_{ref/Obj2}\right)$$

$$l_z=\frac{\lambda }{4\pi \left(1+cOs\theta \right)}\left(\Delta {\phi }_{ref/Obj1}+\Delta {\phi }_{ref/Obj2}\right)$$

$$\Omega =\pm {\text{sin}}^{-1}\left(\frac{\partial l_x}{\partial y}\right)$$

$$\Omega ={\text{cos}}^{-1}\left(\frac{\partial l_x}{\partial x}+1\right)$$

$$\Omega =\pm {\text{sin}}^{-1}\left(\frac{\partial l_y}{\partial x}\right)$$

$$\Omega ={\text{cos}}^{-1}\left(\frac{\partial l_y}{\partial y}+1\right)$$

$$I_1\left(x.y\right)=A\left(x.y\right)+B\left(x.y\right)$$

$$I_2\left(x.y\right)=A\left(x.y\right)+B\left(x.y\right)\text{cos}\left[\phi \left(x.y\right)+\frac{\pi }{2}\right]$$

$$I_3\left(x.y\right)=A\left(x.y\right)+B\left(x.y\right)\text{cos}\left[\phi \left(x.y\right)+\pi \right]$$

$$\phi \left(x.y\right)={\text{tan}}^{-1}\frac{2I_1-I_2-I_3}{I_2-{\mathrm{<figure-callout\; id="3"\; label="I"\; filenames="DE102017002419A1\_0056.png"\; state="\{\{state\}\}">I</figure-callout>}}_3}$$

$$I_4\left(x.y\right)=A\left(x.y\right)+B\left(x.y\right)\text{cos}\left[\phi \left(x.y\right)+\frac{3\pi }{2}\right]$$

$$\phi \left(x.y\right)={\text{tan}}^{-1}\frac{I_4-I_2}{I_1-{\mathrm{<figure-callout\; id="3"\; label="I"\; filenames="DE102017002419A1\_0056.png"\; state="\{\{state\}\}">I</figure-callout>}}_3}$$

list of figures

• 1 : Speckle interferometer in which the carrier frequency is generated by the aperture
• 2 : Speckle interferometer with aperture generated carrier frequency and reference light beam
• 3 : Multi-channel speckle interferometer
• 4 : Speckle interferometer with phase shift element

LIST OF REFERENCE NUMBERS

000:

measurement object

001:

laser

002:

Lens or lens system

003:

laser beam

100:

light

101:

Lens or lens system

102:

Lens or lens system

103:

Aperture 1

104:

Aperture 2

105:

Mirror 1 in position 1

106:

Mirror 1 in position 2

107:

Mirror 2

108:

reflector

109:

Lens or lens system

110:

camera

111:

Shift Δs

112:

beamsplitter

113:

beamsplitter

114:

reference object

115:

mirror

116:

Lens or lens system

117:

Aperture 3

118:

laser beam

119:

reference light

201:

λ / 2 plate

202:

λ / 2 plate

203:

λ / 4 plate or rotating λ / 2 plate

204:

λ / 4 plate or rotating λ / 2 plate

205:

beamsplitter

206:

beamsplitter

207:

mirror

208:

polarizer

209:

polarizer

210:

polarizer

211:

Lens or lens system

212:

Lens or lens system

213:

Lens or lens system

214:

camera

215:

camera

216:

camera

217:

Shift Δs

218:

Shift Δs

219:

Shift Δs

300:

Phase shift element, z. B. phase mask

QUOTES INCLUDE IN THE DESCRIPTION

This list of the documents listed by the applicant has been generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Cited non-patent literature

• W. Li et al. presented a measuring system based on a calibration pattern with a spot array [6]. [0002]
• G. Hussain, and M. Ikram, "Optical Measurements of Angle and Axis of Rotation" Optics Letters 33 (21), 2419-2421 (2008) [0003]
• H. Kim, Y. Yamakawa, T. Senoo, and M. Ishikawa, "Visual Encoder: Robust and Precise Measurement Method of Rotation Angle via High Speed RGB Vision" Optics Express 24 (12), 13375-13386 (2016) [0003]
• J. Briers, and I. Angus, "Speckle Photography Applied to the Measurement of In-plane Rotation" Optica Acta 26 (10), 1229-1231 (1979) [0003]
• J. Goodman, "Speckle Phenomena in Optics: Theory and Applications" Roberts & Company Publishers (2010) [0003]
• W. Li, J. Jin, X. Li, and B. Li, "Method of Rotation Angle Measurement in Machine Vision Based on Calibration Pattern with Spot Arrays" Applied Optics 49 (6), 1001-1006 (2010) [0003]
• R. Jallapuram, C. Healy, E. Mihaylova, and V. Toal "In-plane sensitive electronic speckle pattern interferometer using a diffractive holographic optical element." American Journal of Physics 79, 389-391 (2011) [0003]
• Tang, Z. Wang, J. Gao, and J. Guo, "Measurement method for rolling angular displacement using a diffraction gratings and a heterodyne interferometer" Review of Scientific Instruments 85, 045110 (2012) [0003]
• A. Koch, M. Ruprecht, O. Toedter, and G. Häusler, "Optical Measurement Technique on Technical Surfaces" expert Verlag (1998) [0003]
• E. Berger, W. Linden, V. Dose, M. Ruprecht, A. Koch, "Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation" Applied Optics 36 (29), 7455-7460 (1997) [ 0003]
• E. Mihaylova, M. Whelan, and V. Toal, "Simple phase-shifting lateral shearing interferometer" Optics Letters 29 (11), 1264-1266 (2004) [0003]
• J. Valera, and J. Jones, "Phase-stepping in fiber-based speckle shearing interferometry" Optics Letters 19 (15), 1161-1163 (1994) [0003]
• M. Lu, S. Wang, L. Aulbach, and A. Koch, "Simultaneous Displacement and Slope Measurement in ESPI Using Adjustable Aperture Multiplexing" Applied Optics 55 (22), 5868-5875 (in production) (2016 [0003]
• K. Gasvik, "Optical Metrology Third Edition," John Wiley & Sons, Ltd. (2002) [0003]
• L. Aulbach, A. Koch, "Generation of high-resolution dynamic wavefronts for speckle-based measurements of complex surface shapes" Proc. SPIE 9660, 96600Y (2015) [0003]

Claims (10)

Method for the non-contact optical detection of a rotation angle and a direction of rotation of an object using an ESPI Speckle Pattern Interferometer (ESPI) having a spatial phase shifting (SPS) option, comprising the steps of: providing an ESPI speckle interferometer with a laser light source and a camera detection unit according to 1 - recording a first interferogram in a first angular position (reference angular position) of the object, the phase image being calculated by Fourier transformation of the interferogram (formula 9) with subsequent angle calculation from the real and imaginary parts (formula 10). Method according to Claim 1 in which a second interferogram is recorded in a second angular position of the object and a second phase image is recorded according to the same method as in Claim 1 is calculated, - a phase change map by subtracting the reference phase image after Claim 1 is calculated from the second phase image. Method according to Claim 2 , wherein - the rotation angle of the object, ie the difference angle between the second angular position after Claim 2 and the reference angular position Claim 1 is computed over adjacent pixels of the phase change image and Formula 20. The direction of rotation is derived from the sign of the rotation angle. Method and device for the non-contact optical detection of a rotation angle and a direction of rotation of an object with the aid of an ESPI Speckle Pattern Interferometer (ESPI) having a spatial phase shifting (SPS) option, comprising the steps: - Providing an ESPI speckle interferometer with a laser light source and a camera detection unit according to 2 , - Recording a first interferogram in a first angular position (reference angular position) of the object and recording a second interferogram in a second angular position of the object. Method according to Claim 4 in which - the phase changes are calculated according to equations (22), (23) and (24), - the displacements are calculated according to equations (25) and (26). Method according to Claim 5 in which - the angle of rotation of the object, ie the difference angle between the second angular position and the reference angular position after Claim 4 , the direction of rotation and the out-of-plane shift are calculated by equations (27a), (28a), (27b) and (28b). Method and device for the non-contact optical detection of a rotation angle and a direction of rotation of an object with the aid of an ESPI Speckle Pattern Interferometer (ESPI) having a spatial phase shifting (SPS) option, comprising the steps: - Providing an ESPI speckle interferometer with a laser light source and a camera detection unit according to 3 - recording a first interferogram in a first angular position (reference angular position) of the object, the phase image being calculated by equation (30). Method according to Claim 7 in which a second interferogram is recorded in a second angular position of the object and a second phase image is recorded according to the same method as in Claim 7 is calculated, - a phase change map by subtracting the reference phase image after Claim 7 is calculated from the second phase image. Method according to Claim 8 , wherein - the rotation angle of the object, ie the difference angle between the second angular position after Claim 8 and the reference angular position Claim 7 , is calculated via equation (11). The direction of rotation is derived from the sign from equation (19). Method and device for the non-contact optical detection of a rotation angle and a direction of rotation of an object with the aid of an ESPI Speckle Pattern Interferometer (ESPI) having a spatial phase shifting (SPS) option, comprising the steps: - Providing an ESPI speckle interferometer with a laser light source and a camera detection unit according to 4 - recording a first interferogram in a first angular position (reference angular position) of the object, the phase image being calculated by equation (30), - a second interferogram being recorded in a second angular position of the object and a second phase image being taken by equation (30) is calculated, - the rotation angle of the object, ie the difference angle between the second angular position and the reference angular position, is calculated via equation (11). The direction of rotation is derived from the sign from equation (19).

Images