## Direct determination of *f*-number by using Ronchi test

Optics Express, Vol. 17, Issue 7, pp. 5107-5111 (2009)

http://dx.doi.org/10.1364/OE.17.005107

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### Abstract

A method to determine *f*-number for an imaging system is discussed. This method uses Ronchi test and is different in that the value of *f*-number of the system can be determined without depending on any particular parameters. Two different sizes of aperture for a common system were investigated and the corresponding *f*-numbers were compared with those calculated by a lens design software. In addition, the determined *f*-numbers are proved to be consistent with other values obtained in this study.

© 2009 Optical Society of America

## 1. Introduction

*f*-number by using a Ronchi test and with the

*f*-number to be determined, the analysis of Ronchi test can be completed by itself. The new method consists of several measurements of Ronchi images at various locations nearby Gaussian image plane for a Ronchi ruling. Even though it is not necessary to know the precise locations, it would be useful to set the difference in distance between the locations as a pre-determined value. Ronchi images at various locations will be converted into a set of Zernike polynomial coefficients and then the term

*a*

_{3}among 35 coefficients will be varied accordingly because the image plane where the ruling was located was changed. This variance of

*a*

_{3}enables us to determine

*f*-number directly. We investigated a thick lens system with two different sizes of aperture to confirm this methodology.

## 2. Theory to determine *f*-number

*f*-number of the system, the best fit of the polynomials coefficients can be determined by a simple matrix equation,

*A*is a row vector comprising the 35 coefficients, as

*E*is a row vector comprising the

*N*

_{x}+

*N*

_{y}experimental transverse ray aberration data collected from edge information for each pair of Ronchi images for vertical and horizontal ruling positions,

*Z*is the matrix, of 35 rows and

*N*

_{x}+

*N*

_{y}columns,

*a*

_{3,F/#=1}is the increment of the experimental coefficient

*a*

_{3}with a pre-assigned value of

*f*-number as 1. If the difference of ruling’s locations is kept constant, as it was done in this study, the increment should be a constant and Eq. (7) can be rewritten as

*δz*is the displacement of the image plane. Even though we do not know exact values of the displacement, because we vary the image plane with a certain increment

*Δz*, the defocus wavefront aberration must also vary accordingly as

*a*

_{3}, that is, 2

*a*

_{3}=

_{0}

*W*

_{20}, Eq. (10) can be rewritten as

## 3. Experiment and discussion

4. S. Lee and J. Sasian, “Ronchigram quantification via a non-complementary dark-space effect,” Opt. Express **17**, 1854–1858 (2009). [CrossRef] [PubMed]

4. S. Lee and J. Sasian, “Ronchigram quantification via a non-complementary dark-space effect,” Opt. Express **17**, 1854–1858 (2009). [CrossRef] [PubMed]

*a*

_{3}in black and

*a*

_{8}in red among them as a function of the locations of ruling. They were determined with a fixed value of F/#=1. The trend lines indicate the creditability of the whole experiment and the expectation.

*a*

_{3}is 7.78±0.41 and 11.76±0.57, respectively in unit of wavelengths. The

*f*-numbers are obtained by using Eq. (12) and are listed in Table 1 with those calculated by a lens design software, ZEMAX [6

6. ZEMAX Optical Design Program, ZEMAX Development Corporation, www.zemax.com.

*f*-numbers agree well each other within the uncertainties.

*f*-number are consistent with other values obtained during the analysis. Since the whole setup was not changed, the image distance from the lens must be the same for both and it can be determined as the

*f*-number multiplied by the diameter of aperture, and they are 127 and 126 mm, respectively. The two values agree well to each other as well as to the value calculated by ZEMAX, 131 mm within the uncertainty of thicknesses, 5 mm.

*a*

_{8}does not change for several different locations of ruling as shown in red symbols in Fig. 2, the “true” spherical aberration can be determined by finding an average of the experimental values in Fig. 2 and divided by the corresponding

*f*-number. They are 1.12±0.05 and 5.32±0.07, respectively. Since spherical wavefront aberration is governed by

*r*

^{4}, the one of the two values can be deduced from the other. Thus, since the radius of the aperture of 20 mm is 2/3 of that for 30 mm, the spherical aberration for the aperture of 20 mm can be deduced from that of 30 mm as 5.32 × (2/3)

^{4}and it is 1.05, which agrees well to the determined value of 1.12±0.05 within uncertainties. The values of the spherical aberration also agree well with those of calculated by ZEMAX, 5.27.

## 4. Summary

*f*-number for an optical system with Ronchi test. A single lens system was investigated with two different sizes of aperture and the

*f*-numbers were determined. Not only do they agree to the calculated values by ZEMAX, but also they are consistent with other numerical values, such as image distance from the lens and the spherical aberrations.

## Acknowledgments

## References and links

1. | R. Ditteon, |

2. | G. Smith and D.A. Atchison, |

3. | R.R. Shannon, |

4. | S. Lee and J. Sasian, “Ronchigram quantification via a non-complementary dark-space effect,” Opt. Express |

5. | There are many different versions, but we adopted one.
J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” in |

6. | ZEMAX Optical Design Program, ZEMAX Development Corporation, www.zemax.com. |

**OCIS Codes**

(120.4820) Instrumentation, measurement, and metrology : Optical systems

(080.2468) Geometric optics : First-order optics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: February 17, 2009

Revised Manuscript: March 12, 2009

Manuscript Accepted: March 12, 2009

Published: March 16, 2009

**Citation**

Sukmock Lee, "Direct determination of f-number by using Ronchi test," Opt. Express **17**, 5107-5111 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5107

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### References

- R. Ditteon, Modern Geometrical Optics, Wiley, New York (1998).
- G. Smith and D.A. Atchison, The Eye and Visual Optical Instruments, Cambridge University Press (1997). [CrossRef]
- R.R. Shannon, The Art and Science of Optical Design, Cambridge University Press (1997).
- S. Lee and J. Sasian, "Ronchigram quantification via a non-complementary dark-space effect," Opt. Express 17, 1854-1858 (2009). [CrossRef] [PubMed]
- There are many different versions, but we adopted one. J. C. Wyant and K. Creath, "Basic Wavefront Aberration Theory for Optical Metrology," in Applied Optics and Optical Engineering, XI, 1992, Academic Press, Inc.
- ZEMAX Optical Design Program, ZEMAX Development Corporation, www.zemax.com.

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