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Interferometry is a technique in which waves, usually electromagnetic waves , are superimposed , causing the phenomenon of interference , which is used to extract information. Interferometers are widely used in science and industry for the measurement of small displacements, refractive index changes and surface irregularities. In most interferometers, light from a single source is split into two beams that travel in different optical paths , which are then combined again to produce interference; however, under some circumstances, two incoherent sources can also be made to interfere.
The consequence of this effect is a change in measurement accuracy. Our work provides a theoretical analysis of the influence of aberrations, which are induced by the change in the object position, on the accuracy of optical measuring systems.
Equations were derived for determination of the relative measurement error for monochromatic and polychromatic light using the dependence of the third-order aberrations on the object position. Both geometrical and diffraction theory is used for the analysis. The described effect is not removable in principle and it is necessary to take account to it in high accuracy measurements.
Errors can be reduced by a proper design of optical measuring systems. The proposed analysis can be used for measurement corrections. The accuracy of optical instruments that are used for measurements of distances and dimensions in science and engineering depends on a distance from the measurement device to the measured object. Photogrammetric systems, optical measuring instruments based on CCD sensors, fringe projection systems for 3D shape measurement, theodolites, level instruments, microscopes and projection measurement systems are mainly used for such kind of measurements [ 1 , 2 ].
These instruments are designed by the manufacturer so that they provide optimum imaging properties only for a specific distance which is characteristic for different measurement instruments. It is well known that aberrations of optical systems depend on the distance from the optical system to the object [ 3 — 9 ]. This problem is analyzed in detail by Buchdal [ 4 ]. Moreover, it is also described by Herzberger [ 6 , 7 ], Wynne [ 8 ], and Walther [ 9 ].
The measurement error originates from the dependence of aberrations of the optical system of the measurement device on the distance from the device to the object. Aberrations of the optical system affect the measurement accuracy of the system, which can differ substantially from the optimum accuracy given by the manufacturer.
It is presented that this effect can be reduced by a proper design of optical measuring systems. The aim of our work is not a derivation of exact analytical formulas for errors due to varying object distance.
Such formulas cannot be obtained in a simple analytical form. For example, with the use of the fifth order aberration theory [ 4 , 7 ] one can derive very complicated equations, which can be hardly used in practice. It is important to estimate measurement errors with respect to varying object distance in practice, i. Optical measuring systems, such as autocollimators, theodolites, leveling instruments, photogrammetry systems, topography measurement systems, use low numerical apertures and have usually a small field of view, which makes possible to apply the third-order aberration theory for approximate analysis of such systems.
Our work deals with a theoretical description of the mentioned effect in terms of geometrical and diffraction theory of optical imaging on the basis of the third-order aberration theory Seidel aberration theory [ 3 — 8 ]. The change in the object position with respect to the measurement instrument leads to the change in aberrations of the optical system and this change affects negatively the measurement accuracy.
The aim of this work is to derive simple formulas for approximate estimation of errors of optical measurement systems using the Seidel aberration theory both for monochromatic and polychromatic light. Derived formulas can be used to estimate measurement errors for objects that are located in different distances from the optical measuring system, which is a very important task from a practical point of view.
As far as we know such relations had never been published. It is possible to carry out approximate corrections of the measured data based on the derived equations in order to obtain higher measurement accuracy.
Consider the problem of influence of the change in the object position on imaging properties of a general rotationally symmetrical optical system. Figure 1 shows imaging of two different planes by the optical system. It causes the deviation of imaging properties. The relationship between the size of the object and its image will not be further linear.
If we use such optical system for measurement the mentioned effect causes the measurement error, which cannot be removed. If the optical system is aberration free for a specific position of the object, then it has aberrations for other object positions and the image has lower quality. The described problem will be analyzed using the theory of third-order aberrations [ 3 — 8 ] which enables to obtain the solution in a simple analytical form.
Consider the rotationally symmetrical optical system. The object can be represented in a general case as the image created by the preceding optical system. The case, when object and image surfaces are not planar, is described e.
Equations 2 represent generally valid formulas for calculation of aberration coefficients of the optical system corresponding to varying distance of the object. Moreover, Eq. From previous relations, it is evident that if aberration coefficients of the optical system are known for one value of magnification, than we can calculate aberrations coefficients of the optical system for any other value of magnification.
If we write Eq. One can see from the previous equation the advantage of the matrix form of formulas for aberration coefficients. The matrix B has to be calculated once for all and one can use it for different values of magnification.
The matrix form is also very useful for zoom lens design [ 10 ]. It can be shown that previous formulas are generally valid within the validity of the third-order aberration theory and do not depend on the composition of the optical system [ 3 — 9 ].
We can clearly see from previous equations that aberrations of the optical system change in case of the varying object position. The optical system is called aberration-free for a given value of magnification object position if all aberration coefficients are zero for this magnification object position , i.
Furthermore, we will focus on a special case which is interesting both from the theoretical and practical point of view.
Assume now that we have the optical system, which is aberration-free for imaging the object at infinity and for the selected spectral range, i. If we use Eq. Equations 4 and 5 are very interesting because these formulas show that aberration properties aberration coefficients of the considered optical system aberration-free for the object at infinity depend only on the focal length f' , the angular magnification g , and the angular magnification g P between pupils of the optical system.
As one can see from Eq. Other aberration coefficients have no influence on the position of the centroid because integrals 6 and 7 of terms corresponding to aberration coefficients S I o and S I I I o are zero.
However, aberration coefficients affect the radius of gyration over the exit pupil. This problem is treated in more detail in [ 11 ]. Concerning distortion one can clearly see from Eq. In measurement practice, optical systems with a telecentric path of the principal ray object-side, image-side and double-sides telecentric lenses are frequently used. Telecentric optical systems are treated in detail in [ 12 ].
Formulas 9 and 10 are very useful for estimation of measurement errors caused by varying object distance. These errors cannot be explained using the paraxial approximation. We had used the third-order aberration theory and obtained resulting formulas that have shown the magnitude of errors, both generally and for a special case of optical system without aberrations. In the last case optical system without aberrations for objects at infinity derived equations present approximate estimation of measurement accuracy obtained with optical instruments.
Presented formulas extend knowledge in the field of optical metrology and other areas, such as geodesy, photogrammetry, fringe projection methods, etc. It is clear from the presented results that these errors have physical character and cannot be removed, but their influence can be reduced by a proper design of optical measuring systems.
This can be done as follows. We can write Eqs. The procedure is the same as in the Ref [ 10 ]. The problem is also solved in a different way by Walther [ 9 ] who used mock ray tracing and numerical optimization in his work.
Using Eq. It is well known that aberrations of the optical system vary with the wavelength of light, i. In order to analyze the influence of chromatic aberration of the optical system on measurement accuracy for different object positions object located at finite distance from the optical system , we have to differentiate Eq. Assume now that we have an optical system, which is chromatic aberration-free for imaging the object at infinity and for the selected spectral range, i.
Then, we obtain from previous relations. By substitution of Eq. If chromatic aberrations in pupils of the optical system are corrected, i.
In such case the optical system is achromatic and the measurement accuracy is not dependent on wavelength. The wave aberration can be then calculated by integration, i. Considering wave properties of light and the finite size of optical systems, the image of the point in the object plane is the diffraction pattern in the image plane.
The response of the optical system to the point signal is called the point spread function PSF [ 3 , 10 , 11 , 13 — 19 ]. The shape of the PSF, i. If wave aberration changes due to the variation of the object position, the shape and position of the peak of point spread function will also change and measurement errors occur. Assuming the case that aberrations are small, then we can take as a criterion of the quality of optical systems for imaging the point object the normalized intensity in the peak of the diffraction pattern Strehl definition, Strehl ratio [ 10 , 11 , 20 ] that is defined as the ratio of the maximum of the point spread function of the real optical system to the maximum of the point spread function of the diffraction limited system optical system without aberrations.
With respect to the Strehl definition, we consider the optical system to be equivalent to the diffraction limited system if the Strehl definition is higher than 0.
Wave aberrations can be expressed e. In our work we will use Seidel aberration polynomials for further analysis. Aberration coefficients W 11 and W 20 are unrestrained parameters that express the coordinates of the centre of the reference sphere. The position of the optimum image point [ 11 ], i. The solution of these equations can be expressed as [ 11 ]. Formula 29 is valid only for the optimum image point.
It holds. The detailed calculation of aberration coefficients is described in Ref [ 3 — 5 ]. We can see that one needs to know aberration coefficients W 11 and W 31 for analysis of the error of measurement with the optical system.
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Haynes ManualsThe Haynes Author : Toru Yoshizawa editor Description:The field of optical metrology offers a wealth of both practical and theoretical accomplishments, and can cite any number of academic papers recording such. However, while several books covering specific areas of optical metrology do exist, until the pages herein were researched, written, and compiled, the field lacked for a comprehensive handbook, one providing an overview of optical metrology that covers practical applications as well as fundamentals. Carefully designed to make information accessible to beginners without sacrificing academic rigor, the Handbook of Optical Metrology: Principles and Applications discusses fundamental principles and techniques before exploring practical applications.
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Handbook of optical metrology: principles and applications [1ed.], , admin | January 30, | Mathematics | No.