## Estimation of accuracy of optical measuring systems with respect to object distance |

Optics Express, Vol. 19, Issue 15, pp. 14300-14314 (2011)

http://dx.doi.org/10.1364/OE.19.014300

Acrobat PDF (1198 KB)

### Abstract

It is well-known that the change in the distance from the optical system to the object affects the image quality. Optical measurement systems, which are aberration-free for a specified position of the measured object, are then limited by induced aberrations for other object positions due to the dependence of aberrations on the varying object position. 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.

© 2011 OSA

## 1. Introduction

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

9. A. Walther, “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A **6**(3), 415–422 (1989). [CrossRef]

7. M. Herzberger, “Theory of image rrrors of the fifth order in rotationally symmetrical systems. I,” J. Opt. Soc. Am. **29**(9), 395–406 (1939). [CrossRef]

9. A. Walther, “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A **6**(3), 415–422 (1989). [CrossRef]

7. M. Herzberger, “Theory of image rrrors of the fifth order in rotationally symmetrical systems. I,” J. Opt. Soc. Am. **29**(9), 395–406 (1939). [CrossRef]

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

## 2. Change in aberrations with object position

_{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*according to aberrations of the optical system, which originate from the fact that the planes ξ

_{B}_{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.

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

*S*, and

_{I}, S_{II}, S_{III}, S_{IV}, S_{V}*S*, where

_{VI}*S*is the coefficient of spherical aberration,

_{I}*S*is the coefficient of coma,

_{II}*S*is the coefficient of astigmatism,

_{III}*S*is the Petzval’s sum,

_{IV}*S*is the coefficient of distortion, and

_{V}*S*is the coefficient of spherical aberration in pupils. We obtain the following equations [3

_{VI}3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

*x*and δ

*y*describe transverse ray aberrations of the preceding optical system in the object plane of the optical system under consideration. We can set δ

*x*= 0 and δ

*y*= 0 if any optical system is not located in front of the considered optical system. The coefficients in previous formulas are given by

*m*is the transverse magnification of the optical system,

*g*is the angular magnification of the optical system,

*g*is the angular magnification of the optical system in pupils,

_{P}*p*

_{1}is the distance from the object to the entrance pupil,

*x*

_{P}_{1},

*y*

_{P}_{1}are the coordinates of the intersection of the ray with the plane of entrance pupil,

*y*is the size of the object, and

*n*,

*n'*are indices of refraction of object and image space. The case, when object and image surfaces are not planar, is described e.g. in Ref.6. Coefficients

*S*,

_{I}*S*,

_{II}*S*,

_{III}*S*,

_{IV}*S*, which describe imaging properties of the optical system for an arbitrary position of the object (arbitrary transverse magnification

_{V}*m*), can be expressed using the third-order aberration coefficients

*m*= 0), and the coefficient of spherical aberration in pupils

*S*,

_{I}*S*,

_{II}*S*,

_{III}*S*,

_{IV}*S*[3

_{V}3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

*f′*is the focal length of the optical system. Equations (2) represent generally valid formulas for calculation of aberration coefficients of the optical system corresponding to varying distance of the object. Moreover, Eq. (2) can be modified into the form where the dependence on the magnification (position of the object) is explicitly expressed in contrast to previously published papers. We obtain the following linear system:where

**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

10. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. **47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

9. A. Walther, “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A **6**(3), 415–422 (1989). [CrossRef]

*f′*and third-order aberration coefficients

*S*,

_{I}*S*,

_{II}*S*,

_{III}*S*,

_{IV}*S*are calculated for the following input valuesand aberration coefficients

_{V}*s*

_{1}is the distance from the first surface of the optical system to the object plane, and

*s*

_{P}_{1}is the distance from the first surface of the optical system to the entrance pupil. 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.e. in our case we have

*A*

_{X}= x_{P}_{1}/

*p*

_{1}and

*A*

_{Y}= y_{P}_{1}/

*p*

_{1}are numerical apertures (in air) in the object space,

*w*is the angle of field of view (tan

*w*=

*y/p*

_{1}), and

*n = n'*= 1 (the most common situation in practice). The coefficients

*f'*, the angular magnification

*g*, and the angular magnification

*g*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)

_{P}*r*, φ) are polar coordinates in entrance pupil plane,

*R*is entrance pupil radius, and

11. J. B. Develis, “Comparison of methods for image evaluation,” J. Opt. Soc. Am. **55**(2), 165–173 (1965). [CrossRef]

*S*is zero for all positions of the object. The expressiongives the maximum numerical aperture of the optical system in the object space, and

_{V}*F*

_{0}is the

*f*-number of the optical system for the object at infinity. We can writewhere

*y*is the size of the object,

*s*

_{1}is the object distance, and

*s*is the position of the entrance pupil. We obtain for the relative error of measurementwhere we denoted

_{P}*m*is the transverse magnification of the optical system (

*m*= 1/

*g*),

*m*is the transverse magnification in pupils of the optical system (

_{P}*m*= 1/

_{P}*g*), and

_{P}*y´*is the image size. 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]. In case of image-side telecentric optical system (

*g*. Using aberrations coefficients

10. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. **47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

**6**(3), 415–422 (1989). [CrossRef]

*m*, transverse magnification

*m*between pupils, and

_{P}*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

*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

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

7. M. Herzberger, “Theory of image rrrors of the fifth order in rotationally symmetrical systems. I,” J. Opt. Soc. Am. **29**(9), 395–406 (1939). [CrossRef]

*m*, and

*m*. Further, we focus on calculation of the lateral chromatic aberration

_{P}*C*is the coefficient of longitudinal chromatic aberration,

_{I}*C*is the coefficient of lateral chromatic aberration, and

_{II}3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

*X*and

*Y*are normalized coordinates of the intersection of the ray with the reference sphere in the image space,

*F*is the

*f*-number of the optical system,

*A*and

_{X}*A*are numerical apertures in the direction

_{Y}*x*and

*y*in the object space. The wave aberration can be then calculated by integration, i.e.where the transverse ray aberrations

## 4. Dependence of image quality on object position change

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

10. A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. **47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

11. J. B. Develis, “Comparison of methods for image evaluation,” J. Opt. Soc. Am. **55**(2), 165–173 (1965). [CrossRef]

19. V. N. Mahajan, “Zernike polynomials and optical aberrations,” Appl. Opt. **34**(34), 8060–8062 (1995). [CrossRef] [PubMed]

**47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

11. J. B. Develis, “Comparison of methods for image evaluation,” J. Opt. Soc. Am. **55**(2), 165–173 (1965). [CrossRef]

20. A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. **2**, 07008 (2007). [CrossRef]

**47**(32), 6088–6098 (2008). [CrossRef] [PubMed]

**55**(2), 165–173 (1965). [CrossRef]

15. W. B. King, “Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. **58**(5), 655–661 (1968). [CrossRef]

*E*

_{0}is the variance of wave aberration andwhere

*W*is wave aberration,

*k*

_{0}= 2π/λ

_{0}, and λ

_{0}is the wavelength of light in the vacuum. 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 0.8 (

**55**(2), 165–173 (1965). [CrossRef]

16. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. **72**(9), 1258–1266 (1982). [CrossRef]

19. V. N. Mahajan, “Zernike polynomials and optical aberrations,” Appl. Opt. **34**(34), 8060–8062 (1995). [CrossRef] [PubMed]

**6**(3), 415–422 (1989). [CrossRef]

*W*of the third order can be expressed aswhere

*r*and φ are polar coordinates at the exit pupil sphere of the rotationally symmetric optical system,

*W*

_{11}is the coefficient of tilt,

*W*

_{20}is the coefficient of defocus,

*W*

_{40}is the coefficient of the third-order spherical aberration,

*W*

_{31}is the coefficient of the third-order coma, and

*W*

_{22}is the coefficient of the third-order astigmatism. Aberration coefficients can be calculated from the following formulas [3

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

**55**(2), 165–173 (1965). [CrossRef]

*s*

_{0}is longitudinal defocus,

*y*

_{0}is the transverse defocus, δ

*y*´ is distortion, δ

_{z}*s*´ is spherical aberration for the aperture ray that passes through the edge of the entrance pupil, δ

_{K}*s*´ is meridional astigmatism, δ

_{t}*s*´ is sagittal astigmatism,

_{s}*F*is the

*f*-number of the optical system, and

*n*´ is the index of refraction in the image space. By substitution of Eq. (26) into Eq. (25) we obtain

*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

**55**(2), 165–173 (1965). [CrossRef]

*E*

_{0}and maximum Strehl definition, can be calculated from the necessary conditions for the extremum of function

*E*

_{0}

**55**(2), 165–173 (1965). [CrossRef]

*s*

_{0},

*y*

_{0}), where the Strehl definition is maximum with respect to the paraxial image point. It holds

3. A. Miks, *Applied Optics* (Czech Technical University Press, 2009). [PubMed]

*W*

_{11}and

*W*

_{31}for analysis of the error of measurement with the optical system. Using formula (29) we havewhere

*m,m*) that was calculated on the basis of the geometrical theory of optical imaging and values of function ψ(

_{P}*m,m*) 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

_{p}*m*of the optical system.

## 5. Examples and analysis

*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

*f*-number

*f*-number

*F*

_{0}= 5 and the transverse magnification in pupils of the optical system is

*m*= 1. The focal length of the optical system is

_{P}## Example 1

*m*= - 0.5 and

*m*= - 1. The relative measurement error

## Example 2

*m*= - 0.5 and

*m*= - 1. The relative measurement error

## Example 3

*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 (

*K*,

_{III}*K*) and astigmatism (

_{exact}*S*,

_{I}*S*,

_{II}*S*(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.

_{III}*m*. The paraxial image height was chosen

## 6. Summary

## Acknowledgments

## References and links

1. | N. Suga, |

2. | T. Yoshizawa, |

3. | A. Miks, |

4. | H. A. Buchdahl, |

5. | W. T. Welford, |

6. | M. Herzberger, |

7. | M. Herzberger, “Theory of image rrrors of the fifth order in rotationally symmetrical systems. I,” J. Opt. Soc. Am. |

8. | C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. |

9. | A. Walther, “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A |

10. | A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. |

11. | J. B. Develis, “Comparison of methods for image evaluation,” J. Opt. Soc. Am. |

12. | http://www.schneiderkreuznach.com/pdf/div/optical_measurement_techniques_with_telecentric_lenses.pdf |

13. | E. L. O'Neill, |

14. | J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function ” in |

15. | W. B. King, “Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. |

16. | V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. |

17. | V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. |

18. | V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. |

19. | V. N. Mahajan, “Zernike polynomials and optical aberrations,” Appl. Opt. |

20. | A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(110.3000) Imaging systems : Image quality assessment

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.4640) Instrumentation, measurement, and metrology : Optical instruments

(150.3045) Machine vision : Industrial optical metrology

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 10, 2011

Revised Manuscript: April 21, 2011

Manuscript Accepted: May 17, 2011

Published: July 12, 2011

**Citation**

Antonin Miks and Jiri Novak, "Estimation of accuracy of optical measuring systems with respect to object distance," Opt. Express **19**, 14300-14314 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14300

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### References

- N. Suga, Metrology Handbook: The Science of Measurement (Mitutoyo Ltd., 2007).
- T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).
- A. Miks, Applied Optics (Czech Technical University Press, 2009). [PubMed]
- H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge University Press, 1970).
- W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic Press, 1974).
- M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
- M. Herzberger, “Theory of image rrrors of the fifth order in rotationally symmetrical systems. I,” J. Opt. Soc. Am. 29(9), 395–406 (1939). [CrossRef]
- C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. 65B, 429–437 (1952).
- A. Walther, “Irreducible aberrations of a lens used for a range of magnifications,” J. Opt. Soc. Am. A 6(3), 415–422 (1989). [CrossRef]
- A. Mikš, J. Novák, and P. Novák, “Method of zoom lens design,” Appl. Opt. 47(32), 6088–6098 (2008). [CrossRef] [PubMed]
- J. B. Develis, “Comparison of methods for image evaluation,” J. Opt. Soc. Am. 55(2), 165–173 (1965). [CrossRef]
- http://www.schneiderkreuznach.com/pdf/div/optical_measurement_techniques_with_telecentric_lenses.pdf
- E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Inc., 1963).
- J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread function ” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51, pp. 349–468.
- W. B. King, “Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. 58(5), 655–661 (1968). [CrossRef]
- V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 72(9), 1258–1266 (1982). [CrossRef]
- V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73(6), 860–861 (1983). [CrossRef]
- V. N. Mahajan, “Zernike circle polynomials and optical aberrations of systems with circular pupils,” Appl. Opt. 33(34), 8121–8124 (1994). [CrossRef] [PubMed]
- V. N. Mahajan, “Zernike polynomials and optical aberrations,” Appl. Opt. 34(34), 8060–8062 (1995). [CrossRef] [PubMed]
- A. J. E. M. Janssen, S. van Haver, J. J. M. Braat, and P. Dirksen, “Strehl ratio and optimum focus for high-numerical-aperture beams,” J. Eur. Opt. Soc. Rapid Publ. 2, 07008 (2007). [CrossRef]

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