1. Introduction

An f-number is one of fundamental characteristics of an optical system and is defined by effective focal length divided by the exit pupil diameter [1, 2]. Although there are many ways to measure the focal length, the diameter of exit pupil cannot be measured, unless the aperture of the system is the last element. To our best knowledge, there is not a direct method for the f-number and it is desirable to develop a method to determine it directly because the direct measurement can be used as an independent check point for other parameters. An f-number relates not only to some geometrical and physical aspects, such as spot size and depth of focus [1, 2], but it is also used as a proportional constant between transverse ray aberration (TRA) and wavefront aberration [3]. Thus, for quantitative analysis in Ronchi test, an f-number is one of necessary conditions to know a priori. Recently, Lee has shown that Ronchi test can be used to quantify the spherical aberration of a lens system [4] with a concept of non-complementary dark-space effect to overcome the obstacles related to the diffraction. They however assume a value for the f-number of a thick-lens system to reconstruct wavefront aberration.

Therefore we present a method to determine the 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

When the transverse ray aberrations are described as a sum of Zernike polynomials [5],

where F/# is the image-side paraxial f-number of the system, the best fit of the polynomials coefficients can be determined by a simple matrix equation,

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

The entity 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,

and Z is the matrix, of 35 rows and N _x+N _y columns,

The solution, a set of Zernike polynomial coefficients, is then determined by,

For various locations of ruling, the increment of the corresponding coefficient is obtained from Eq. (6) as

where Δa _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

For an image plane displaced from Gaussian image plane, the defocus wavefront aberration is given by [3]

where 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

Since there is a factor of 2 between the defocus wavefront aberration and the Zernike polynomial a ₃, that is, 2a ₃ =₀ W ₂₀, Eq. (10) can be rewritten as

Lastly, by combining the two equations (8) and (11), f-number can be determined as

3. Experiment and discussion

The experimental setup with Ronchi ruling is shown in Fig. 1. We used a He-Ne laser as a light source (λ= 632.8 nm) whose spatial frequencies are filtered with a pinhole of 5 μm in diameter. We used a thick positive lens as a test optics and a Ronchi ruling (50% duty cycle) with a period of 200 μm placed near the Gaussian image plane produced by the lens. A circular aperture is located 37 cm to the right of the source and 2 cm to the left of the lens and the distances were measured with a precision of 0.5 cm. The light transmitted by the ruling was collected by a CMOS camera. An imaging lens is used to make the beam collimated over a finite region, where the camera should be situated. More experimental detail is described elsewhere [4].

 

Fig. 1. Ronchi test setup for measuring transverse ray aberrations in a beam.

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While the whole experimental setup is fixed, two different sizes of aperture were used and their diameters are 20 and 30 mm, respectively and Fig. 2 shows two typical Ronchi images at a common location of the horizontal ruling.

 

Fig. 2. Representing Ronchi images with two different apertures (a) 20 mm and (b) 30 mm.

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For each aperture, we followed the same procedure to determine the 35 Zernike polynomial coefficients, described in the reference [4], and Fig. 3 shows the two most important coefficients 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.

 

Fig. 3. Two important Zernike polynomial coefficients, a ₃ in black and a ₈ in red as a function of ruling’s location, determined by Ronchi test with two different apertures of (a) 20 mm and (b) 30 mm.

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The increment of the locations of ruling used in this work is fixed at 0.5 mm and the wavelength of the light used is 0.6328 μm. The average of the increment in 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]. Since the thicknesses used in the software were measured with uncertainty of about 5 mm, the f-numbers agree well each other within the uncertainties.

In addition, the values of 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.

Table 1. F-number measured with those calculated by ZEMAX for 2 different sizes of aperture.

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Another confirmation can be made through the spherical aberrations of the lens system measured in two different apertures. Since the term 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

We presented a method to determine 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.