1. Introduction

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. The described effect is not removable in principle and it is necessary to take account to it in high accuracy measurements. 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.e. found out if measurement errors are tenths of millimeters, millimeters or centimeters. 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.

2. Change in aberrations with object position

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. Suppose that the optical system is aberration-free for imaging of the plane ξ_A, then an arbitrary point A of this plane is imaged as the point A´ in the plane ξ′_A. Choose now the different plane ξ_B. An arbitrary point B of this plane will not be imaged by the optical system as the point B´, but as the circle of least confusion d_B according to aberrations of the optical system, which originate from the fact that the planes ξ_A and ξ_B are not identical. 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.

 

Fig. 1 Imaging of two different planes by optical system.

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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. Aberration properties for light of a specific wavelength are given by its third-order aberration coefficients S_I, S_II, S_III, S_IV, S_V, and S_VI, where S_I is the coefficient of spherical aberration, S_II is the coefficient of coma, S_III is the coefficient of astigmatism, S_IV is the Petzval’s sum, S_V is the coefficient of distortion, and S_VI is the coefficient of spherical aberration in pupils. We obtain the following equations [3–6] for transverse ray aberrations within the validity of the third-order aberration theory

The coefficients $a_{ij}$ are given by the following formulas

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. (3) for two different values of magnification $g=g_1$ and $g=g_2$, we obtain for the vectors of Seidel aberration coefficients the following simple formula

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]. Properties of the optical system are then fully specified by its focal length f′ and third-order aberration coefficients $S_I^{\infty },S_{II}^{\infty },S_{III}^{\infty },S_{IV}^{\infty },S_V^{\infty },S_{VI}^{\infty }$, which characterize imaging properties of the optical system for imaging of the object at infinity. Aberration coefficients S_I, S_II, S_III, S_IV, S_V are calculated for the following input values

Furthermore, we will focus on a special case which is interesting both from the theoretical and practical point of view. We will analyze the changes of aberrations with respect to the change in object’s position (change in magnification of the optical system) for the optical system which is aberration-free for objects at infinity. 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.e.

If we use Eq. (1) and set $\delta x=\delta y=0$, then the transverse ray aberrations $\text{δ}{x}^{\prime }$,$\text{δ}{y}^{\prime }$, which are induced by the change in position of the object, can be expressed as

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. In order to determine the shift of “geometric-optical energy centre of the circle of confusion” we calculate mean values of transverse ray aberrations. The mean values (centroid of the spots) $〈\delta x^{\prime }〉$ and $〈\delta y^{\prime }〉$ of the transverse ray aberrations $\delta x^{\prime }$and $\delta y^{\prime }$ (Eq. (3) are given by [8]

As one can see from Eq. (7) the position of the centroid is affected only by coma coefficient in our case. 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 $S_{III}^o$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. (3) that aberration coefficient S_V is zero for all positions of the object. The expression

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. Firstly, we choose the range of angular magnification $g\in 〈g_{\min },g_{\max }〉$ for which we want to optimize the optical system. We can write Eqs. (1) and (2) for several values of angular magnification $g\in 〈g_{\min },g_{\max }〉$. Then, using the least squares method we can calculate the values of the aberrations coefficients $S_I^{\infty },S_{II}^{\infty },S_{III}^{\infty },S_{IV}^{\infty },S_V^{\infty },S_{VI}^{\infty }$ that minimize aberrations in the given range of the change of magnification g. Using aberrations coefficients $S_I^{\infty },S_{II}^{\infty },S_{III}^{\infty },S_{IV}^{\infty },S_V^{\infty },S_{VI}^{\infty }$ we can calculate parameters of the optical system. These parameters then serve as a starting state (predesign) for further optimization of the optical design of the optical system using optical design software, such as OSLO, ZEMAX, CODE V, OPTALIX. 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. Equations (9) and (10) are also interesting from a theoretical point of view because they say that if the optical system is aberration-free ($S_I^{\infty }=0$, $S_{II}^{\infty }=0$, $S_{III}^{\infty }=0$, $S_{IV}^{\infty }=0$, $S_V^{\infty }=0$, $S_{VI}^{\infty }=0$) for the object at infinity, then the relative measurement error does not depend (within the validity of the third-order aberration theory) on the type and composition of the considered optical system, but it depends only on its transverse magnification m, transverse magnification m_P between pupils, and f-number F ₀. Using Eq. (9) we can calculate the ratio of the mean value (centroid) of transverse ray aberration to the image size in dependence on the transverse magnification m of the optical system.

3. Influence of wavelength on measurement accuracy

Let us now deal with the influence of the wavelength of light λ on the accuracy of measurement. It is well known that aberrations of the optical system vary with the wavelength of light, i.e. the optical system has chromatic aberration [3–5,7]. Assume that we have the optical system, which is aberration-free for imaging the object at infinity and for the selected spectral range $\lambda \in 〈{\lambda }_1,{\lambda }_2〉$ (e.g. visible spectral range), i.e.

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. (10). After some calculations we obtain

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.e. $C_I^{\infty }=C_{II}^{\infty }=0$ ($\delta {s^{\prime }}_{\lambda }^{\infty }=0$,$\delta {y^{\prime }}_{\lambda }^{\infty }=0$). Then, we obtain from previous relations

By substitution of Eq. (15) into Eqs. (13) and (14) we have

We can derive for the chromatic aberration $\text{d}{y}_{P\lambda }^{\prime }$ in pupils

By substitution of Eq. (19) into Eq. (18) we obtain

By substitution of Eq. (17) and Eq. (20) into Eq. (11) we obtain provided that $C_I^{\infty }=C_{II}^{\infty }=0$ the following relation

Equation (21) enables to analyze the influence of chromatic aberrations of the optical system (aberration corrected for the object at infinity: $\delta {s^{\prime }}_{\lambda }^{\infty }=0$, $\delta {y^{\prime }}_{\lambda }^{\infty }=0$) on measurement accuracy for different object positions (object located in a finite distance from the optical system). If chromatic aberrations in pupils of the optical system are corrected, i.e. $C_{IP}^{\infty }=C_{IIP}^{\infty }=0$, then we obtain from Eq. (21) that $\text{d}{\varphi }_{\lambda }=0$. In such case the optical system is achromatic and the measurement accuracy is not dependent on wavelength.

The relationship between the wave aberration and the ray aberration is given by [3–5]

4. Dependence of image quality on object position change

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.e. the energy distribution in the diffraction pattern, depends on the position of the point in the object plane and on the distance from the optical system to the object plane. 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). It is well known [10,11,13–15] that the Strehl ratio of the optical system with small aberrations can be expressed in the form

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.e. the point with the minimum E ₀ and maximum Strehl definition, can be calculated from the necessary conditions for the extremum of function E ₀

The solution of these equations can be expressed as [11]

Formula (29) is valid only for the optimum image point. By substitution of Eq. (27) for coefficients $W_{11},W_{31},W_{20},W_{40}$ into Eq. (29) one can calculate the position (coordinates s ₀, y ₀), where the Strehl definition is maximum with respect to the paraxial 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. Using formula (29) we have

Using formulas (4), (5), (8), (30) and (31), we obtain for the relative measurement error

Formulas (32) and (33) are showing the influence of the shift of the point spread function central peak of the optical system under consideration with respect to the position of the point spread function central peak of the ideal optical system. Equation (30) is the resulting formula for the relative measurement error that is calculated on the basis of the diffraction theory of optical imaging.

We can compare values of function φ(m,m_P) that was calculated on the basis of the geometrical theory of optical imaging and values of function ψ(m,m_p) that was obtained using the diffraction theory of optical imaging. We can see that the less accurate geometrical theory gives larger values of relative measurement errors than more accurate diffraction theory of optical imaging. We obtain by comparison of both functions

One can see that the geometrical theory of optical imaging gives 50% higher estimate of relative measurement errors than the diffraction theory of optical imaging regardless of the value of the transverse magnification m of the optical system.

5. Examples and analysis

We will show how to use the results obtained in the previous section. We focus on optical measuring systems e.g. in survey engineering, where theodolites and level instruments are most widely used for measurements. If we carry out measurements of objects that are situated at very long distances in comparison with the focal length of the objective lens of the measurement instrument, then the value of the transverse magnification m and similarly the measurement error induced by the change in the object position is practically zero. On the other hand, if one makes measurements of objects that are situated near the measurement instrument, then the value of the transverse magnification is nonzero, and the measurement error increases. Such a situation is very frequent, for example, in survey engineering during measurement of various structural parts and buildings, in photogrammetry, etc. when a high measurement accuracy is required. Objects are situated near the measurement instrument if their distance to the optical system is smaller than the “practical infinity”. The “practical infinity” corresponds to such distance of the axial object point which produces defocus aberration of W = λ/4 (Rayleigh quarter-wavelength rule). Using Eq. (27) and Newton's conjugate distance equation we obtain for the “practical infinity” the following formula

For example, the value of “practical infinity” lies at distance $s_{\infty }\approx 366\text{\hspace{0.17em} m}$ in front of the optical system with the focal length$f^{\prime }=100\text{\hspace{0.17em} mm}$, f-number $F=5$, and wavelength $\lambda =0.546\mu \text{m}$. Consider that we want to determine the shortest distance $s_{\epsilon }$ from the optical system to the object which satisfies the condition that the relative measurement error $\epsilon =\left|〈\delta y^{\prime }〉/y^{\prime }\right|$ is smaller than some given value. Assume that the transverse magnification in pupils is $m_P=1$ (the most frequent case in practice). Using formulas (9), (10), and Newton's conjugate distance equation, we obtain

In order to give some impression about the magnitude of aberration induced measurement errors we will calculate the relative measurement errors for the optical system, which is aberration-free ($S_I^{\infty }=0,S_{II}^{\infty }=0,S_{III}^{\infty }=0,S_{IV}^{\infty }=0,S_V^{\infty }=0,S_{VI}^{\infty }=0$) for an object at infinity. Let the optical system has the f-number F ₀ = 5 and the transverse magnification in pupils of the optical system is m_P = 1. The focal length of the optical system is $f^{\prime }=300\text{\hspace{0.17em} mm}$and the image size is $y^{\prime }=-5\text{\hspace{0.17em} mm}$.

Example 1

Firstly, we determine the relative measurement error that is obtained using the geometrical theory of optical imaging for values of the transverse magnification m = - 0.5 and m = - 1. The relative measurement error ${\epsilon }_g(m,m_P)$ expressed in percents can be calculated according to Eq. (9)

We obtain for the first case $〈\delta y^{\prime }〉=0.0085\text{\hspace{0.17em} mm}$, ${\epsilon }_g(-0.5,1)=-0.17\%$, and for the second case $〈\delta y^{\prime }〉=0.0125\text{\hspace{0.17em} mm}$, ${\epsilon }_g(-1,1)=-0.25\%$. Figure 2 presents spot diagrams for different values of transverse magnification m. Transverse ray aberrations dx and dy in Fig. 2 are expressed in millimeters.

 

Fig. 2 Spot diagrams for different values of transverse magnification m.

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Example 2

Further, we can calculate the relative measurement error that is obtained using the diffraction theory of optical imaging for the same optical system and same values of the transverse magnification m = - 0.5 and m = - 1. The relative measurement error ${\epsilon }_d(m,m_P)$ expressed in percents can be calculated according to Eq. (32)

We obtain for the first case ${\epsilon }_d(-0.5,1)=-0.11\%$, and for the second case ${\epsilon }_d(-1,1)=-0.16\%$. We can clearly see from previous examples that the measurement error is not insignificant for high accuracy measurements.

Example 3

We will show how accurately describes the third-order theory a real example of a collimator objective lens. Consider two element achromatic objective lens with the focal length f' = 100 mm and the f-number F = 10. The design parameters of the objective lens are given in Table 1 , where r is the radius of curvature and d is the axial thickness. Table 2 presents values of longitudinal spherical aberration ($\Delta s_{III}^{\prime }$,$\Delta s_{exact}^{\prime }$), tangential coma (K_III, K_exact) and astigmatism ($as{t}_{III}$,$as{t}_{exact}$), calculated using third-order aberration coefficients S_I, S_II, S_III (index ″III″) and the exact calculation by ray tracing (index ″exact″). As one can see from Table 2 the third-order theory gives sufficiently accurate results that are practically the same as the exact values obtained by ray tracing.

Table 1. Achromatic Objective Lens (f' = 100 mm, F = 10)

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Table 2. Aberrations – Exact and Approximate Calculation Example 4

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Furthermore, we will analyze the accuracy of error calculation using Eq. (9) by a comparison with exact calculations using ZEMAX software. Considering the optical system with parameters described in Table 3 . This optical system has very small residual aberrations and one can consider the system practically as diffraction limited.

Table 3. Parameters of the Optical System

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Assume now that the optical system forms images of objects in different distances from the optical system with the transverse magnification m. The paraxial image height was chosen $y^{\prime }=10\text{\hspace{0.17em} mm}$. Table 4 presents calculations using Eq. (9) and ZEMAX software. As one can see from Table 4, Eq. (9) gives results which differ from the exact calculation using ZEMAX only 7.6% on average. This can be considered as a good agreement. The third-order aberration theory provides sufficiently accurate results for estimation of measurement errors due to the varying distance of the measured object from the optical system, even in the case when we do not know detailed parameters of the optical system.

Table 4. Comparison of Results Between Derived Equations and Exact Calculations with ZEMAX Software

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6. Summary

Our work was focused on the theoretical analysis of the influence of aberrations on the accuracy of optical measuring systems. Using the Seidel aberrations theory we derived simple equations for approximate determination of measurement errors of optical measurement systems, both for monochromatic and polychromatic light. The derived formulas enable to estimate the errors during measurements of objects that are situated at different distances from the considered optical measurement system. The change in the object position affects aberrations and the image quality of the measuring optical system, which is directly related to the measurement accuracy. These aberrations cannot be explained using the paraxial approximation and thus we had used the third-order theory of aberration. We obtained approximate formulas, which have shown the magnitude of measurement errors, both generally and for a special case of optical system without aberrations for the object at infinity ($S_I^{\infty }=0,S_{II}^{\infty }=0,S_{III}^{\infty }=0,S_{IV}^{\infty }=0,S_V^{\infty }=0,S_{VI}^{\infty }=0$). In the last case (aberration-free optical system) derived equations present an approximate estimation of measurement accuracy with optical instruments for different object positions. The mentioned effect was theoretically described in terms of geometrical and diffraction theory of optical imaging. From a theoretical point of view we obtained interesting results. Namely, that in the case of the aberration-free optical system ($S_I^{\infty }=0,S_{II}^{\infty }=0,S_{III}^{\infty }=0,S_{IV}^{\infty }=0,S_V^{\infty }=0,S_{VI}^{\infty }=0$) for the object at infinity, the relative measurement error does not depend (within the validity of the third-order aberration theory) on the type and composition of the considered optical system, but it depends only on its transverse magnification m, transverse magnification mP between pupils, and f-number F0. Derived formulas extend knowledge in the field of optical metrology and other areas, such as photogrammetry, optical measuring instruments based on CCD sensors, fringe projection systems for 3D shape measurement, theodolites, level instruments, etc. It was shown on the example of real optical system that measurement errors calculated using derived formulas in formula (9) differ from the exact calculation with ZEMAX software only 7.6% on average. It is clear from the presented results that these errors have physical character and cannot be removed in principle, but their influence can be reduced by appropriate design of optical measuring systems.

Acknowledgments