Open Access
Add to BibSonomyAbstract
A method to determine the location of the paraxial image plane of an imaging system is discussed. This method uses a recently developed quantitative Ronchi test and is different in that the location of paraxial image plane of the system can be determined from the measured Ronchigrams alone. We validate the location determined by the method by modifying the optical setup and comparing the retrieved f-number of the system to the theoretical prediction.
©2010 Optical Society of America
1. Introduction
The paraxial image plane is important not only because it is the first-order characteristic of an optical system, but it is also the location at which the system will form a sharp image for a point source for aberration free systems [1, 2]. For an aberration-free system, it is simple to find the image plane because the smallest image for a point source is formed there. However, for an arbitrary optical system, for example with a few wavelengths of spherical aberration, the size of the image at paraxial image plane is no longer a minimum and the system should be defocused some amount depending on the spherical aberration to get a best image [3,4]. This implies that a series of direct images measured at various places for the image spot sizes can be used to get a reasonable estimate of the paraxial image plane location for an optical system with a known spherical aberration. However, a careful numerical simulation study by using Fourier transforms showed that the size of the circular image caused by the aberration at various image planes depends not only on the location of the plane but also on the saturation level of the detector used and that the minimum size locates at different locations for different saturation level. Thus, it is difficult to define the best focus in terms of the measured point spread function. Alternatively, the focal point can be obtained by combining two measurements of focal length, for example by Talbot interferometry [5], and of principal planes obtained using a Nodal slide. However, this is still a two-step process. Thus, it is desirable to develop a direct and quantitative method to measure the paraxial image plane for an arbitrary imaging system.
Recently, we showed that a Ronchi test could be used to quantify the spherical aberration for a single thick lens system in a self-consistent manner [6,7]. The quantification requires a priori an f-number of the system under test and the defocus aberrations determined by the quantitative Ronchi test, which depends on the location of the ruling, enabled to determine the system f-number. Since the f-number is defined by a ratio of the distance from the exit pupil of the system to the paraxial image plane to the pupil diameter, the determination of the paraxial image plane can be used to determine the distance from the real exit pupil. This allows computing the f-number alternatively and directly, which can be compared with the f-number determined by the quantitative Ronchi test. This additional determination enhances the validity and robustness of the quantitative Ronchi test in terms of the accuracy and the precision of the measurements.
2. Determination of paraxial image plane location by Ronchi test
One of many advantages of the quantitative Ronchi test is the ability of quantifying defocus wavefront aberrations, for example a 3 in Zernike polynomials [8], by varying the longitudinal position of the Ronchi ruling [6,7]. Because of this nature, the f-number of an optical system under test can be determined [6]. This same principle can be employed to determine the paraxial image plane location of the system.
For an arbitrary axial-symmetric optical system with an axial point source, the theoretical wavefront aberration is usually referred to the paraxial image point, and is composed of spherical aberration of different orders, with a zero defocus wavefront aberration. According to the third-order aberration theory, the defocus wavefront aberration 0 W 20 created by a defocus dz from paraxial image plane in air is [3,4]
where F/# is the f-number. Thus, the paraxial image plane can be identified by the plane where the left hand side of Eq. (1) is zero. The Ronchi test can be used to measure the defocus wavefront aberrations at several arbitrary locations near the paraxial image plane. The measured defocus wavefront aberrations as a function of the location can then be fitted as a straight line whose slope and intercept point can be used to determine the location for the zero defocus wavefront aberration.3. Experiments and discussion
A quantitative Ronchi test setup is shown in Fig. 1 . A diverging light (λ = 0.6328 μm) from a 5 μm diameter pinhole is incident on a bi-convex test lens (f = 100 mm). The lens is 50 mm diameter and a 25 mm diameter aperture stop located to the right of the lens limits the size of the converging beam. Ronchigrams are measured by a set of binary Ronchi ruling (period = 0.2 mm, 50% duty cycle), imaging lens (f = 25 mm), and CMOS camera detector. The Ronchi image is 640×512 pixels, with each pixel being 10.4 μm×10.4 μm, and the intensities stored in each image are in 256 gray scale. The global gain and the exposure time of the camera were adjusted so that the maximum intensity of the image is close to the maximum, 255.
Fig. 1 Ronchi test setup for measuring transverse ray aberrations in a beam. Notice that the aperture is located behind the test lens and that the location of the aperture was varied.
The experiment was performed with two location variables; the longitudinal locations of the aperture and the ruling. The aperture’s location is changed to vary the distance to the paraxial image plane (PIP) and the f-number, accordingly while the ruling’s location is varied to determine the f-number and the location of the PIP. The aperture was mounted on a translation stage having a precision of 10 μm, while the Ronchi ruling was on a XYZ translation stage of a precision of 2 μm. In order to distinguish the locations in this paper, we use two variables for the two locations, z AS for the aperture with respect to an arbitrary initial location (0 mm) and z RR for the ruling in terms of the scales of the translation stage moving the ruling.
In order to have a real exit pupil, we placed the aperture to the right of the test lens and measured the distance from the aperture to the ruling. Since the exact location of the PIP is not known a priori, we placed the ruling at the z RR=13.00 mm and the aperture at the initial location, z AS=0 mm. The distance between the aperture and the ruling was measured by a vernier caliper having a precision of 0.1 mm and it was 110.4±0.2 mm. The relatively large precision is due to the difficulties in aligning the two end points of the vernier parallel to the optic axis. If the location of the PIP is found in terms of the location of the ruling, the distance from the aperture to the PIP can be computed, and so can the f-number. In order to minimize the experimental error, the location of the aperture was varied with an increment of 2 mm from the initial location toward the PIP. This variation changes the characteristics of the system, such as the f-number and the primary spherical aberration, and the change of these parameters as a function of the location of the aperture will be used to validate the technique to determine the location of the PIP. For each location of the aperture, a set of 4 Ronchigrams, 2 images with the ruling in vertical orientation and 2 images in horizontal orientation, was measured. The 2 images are the 0 and π-phase Ronchigrams, respectively, and the latter was measured with the ruling shifted half a period laterally with respect to the former. Figure 2 shows a set of 4 typical Ronchigrams for the aperture located at the z AS=4.0 mm when the ruling is located at z RR=13.00 mm. The Ronchigrams were analyzed to determine a set of preliminary 35 Zernike polynomial coefficients with the f-number as 1. This analysis was repeated with 6 different locations of the ruling, separated by 0.5 mm, and the defocus aberration a 3 as a function of the ruling location was used to determine the f-number and a set of final 35 Zernike coefficients [6].
Fig. 2 A set of typical Ronchigrams. The ruling is in vertical orientation for (a) and (b) and horizontal orientation for (c) and (d). (A) and (c) are 0-phase and (b) and (d) are π-phase.
Figure 3 shows the two important Zernike polynomial coefficients in units of waves as a function of location of the ruling for the system with the aperture located at 4 mm to the right of the initial location. While the spherical aberration a 8 is constant, as expected, the defocus aberration a 3 is linearly proportional to the location. All other coefficients as well as other third-order coefficients came out as small as 0.1 waves or less in magnitude. For the system with the aperture at z AS=4.0 mm, the f-number was found as 4.272 [7].
Fig. 3 Defocus a3 (black squares) and spherical aberration a8 (red circles) for the aperture stop located at 4 mm are drawn as a function of ruling’s longitudinal position. The accompanied lines are the fitted lines.
The best fit equation of the line for the defocus aberration is a 3 = 5.4244zRR – 60.819, where z RR is the location of the ruling in units of mm. For zero defocus wavefront aberration 0 W 20, the two Zernike polynomial coefficients are related by a 3=3a 8 within the third-order aberration theory. Thus the location of PIP can be obtained by solving the equation, zRR = (3a 8 + 60.819)/5.4244, and the solution is found as 13.09 mm with the primary spherical aberration of 3.404 waves. The same procedure was repeated for the aperture stop located at various locations and the results are listed in Table 1 .
Table 1. The F/#, the primary spherical aberration a8, and the locations of the paraxial image plane in terms of Ronchi ruling locations, determined by the quantitative Ronchi test are listed for several locations of the aperture.
The averaged location of the PIP is then z RR=13.09±0.02 mm in terms of the ruling’s translation stage. This means that the first measurement (13.00 mm) in Fig. 3 was performed 0.09 mm to the left of the PIP, which the second was measured 0.41 mm to the right of the PIP, and so on. It is interesting to note that as the F/# decreases, the values of zPIP are monotonically increasing rather than randomly distributed around the mean value of 13.09 mm. We speculate that this could be due to the approximation for the relation a 3=3a 8 [4].
Because the distance between the aperture at the z AS=0.00 mm and the ruling at the z RR=13.00 mm was measured as 110.4 mm, the distance from the aperture to the PIP becomes 110.49 mm and the corresponding f-number can be computed as 4.4196 (=110.49/25). The change of the location of the aperture affects the f-number and the primary spherical aberration. Let us call the f-number for initial location of the aperture stop (F/#)0, which is 4.4196. Then the f-number can be written as
where D 0 is the diameter of the aperture, which is fixed in this work and z AS is the longitudinal distance of the aperture with respect to the initial location. Simultaneously, the primary spherical aberration can be written aswhere a 8,0 is the primary spherical aberration for the initial aperture location z AS=0 mm.Figure 4(a) shows the measured f-number as a function of the aperture location with the theoretical line, calculated by Eq. (2) with the fact that the PIP is located at z RR=13.09 mm in terms of the ruling’s translation stage. Because the uncertainty of the direct measurement of the distance from the aperture to the ruling is 0.2 mm, the numeric value 0.09 determined by the quantitative Ronchi test may not be needed. However, the f-number computed by Eq. (2) with the location of the PIP at 13.09 mm matches the best with the f-numbers determined by the Ronchi test. Simultaneously, the sum of differences between the primary spherical aberrations shown in Fig. 4(b), where the measured values are denoted in symbols and the theoretical values calculated by Eq. (3) are drawn in line, becomes the minimum with the a 8,0 being as 2.957 waves. Both data sets agree with the theoretical values within about 1% uncertainties.
Fig. 4 (a) The f-number and (b) the primary spherical aberration a 8 are drawn as a function of the aperture location, z AS. The black lines show the theoretical values with respect to those for the initial aperture location at 0 mm, respectively.
Even though we used a real exit pupil to determine the paraxial image plane, this method can be applied to any imaging systems including those having virtual exit pupils.
4. Summary
We demonstrated that a quantitative Ronchi test can be used to determine the paraxial image plane for an imaging system. In our experiments we used a real exit pupil, this enabled us to modify the system’s f-number in a known way, which can be compared with one determined by the Ronchi test. This confirmation validates the values of f-number and the aberrations of the system and suggests the quantitative Ronchi test as a true self-consistent wavefront sensing tool.
Acknowledgments
This work was supported by Inha University Research Grant. Authors want to thank Manuel Guizar-Sicairos in the Institute of Optics, University of Rochester for his proof reading and anonymous reviewers for the detailed reviewing comments, which surely made this paper more readable.
References and links
1. C. Donald, O’Shea, Elements of Modern Optical Design, John Wiley & Sons, New York (1985).
2. P. Mouroulis, and J. Macdonald, Geometrical Optics and Optical Design, Oxford University Press, New York (1997).
3. R. R. Shannon, The Art and Science of Optical Design, Cambridge University Press, Cambridge (1997).
4. V. N. Mahajan, “Zernike polynomials and aberration balancing,” Proc. of SPIE Vol. 5173, edited by P.Z. Mouroulis, W.J. Smith, and R.B. Johnson (SPIE, Bellingham, WA, 2003).
5. For example,Y. Nakano and K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24(19), 3162 (1985). [CrossRef] [PubMed]
6. S. Lee and J. Sasian, “Ronchigram quantification via a non-complementary dark-space effect,” Opt. Express 17(3), 1854–1858 (2009). [CrossRef] [PubMed]
7. S. Lee, “Direct determination of f-number by using Ronchi test,” Opt. Express 17(7), 5107–5111 (2009). [CrossRef] [PubMed]
8. There are many different conventions, but we followed the one in J. C. Wyant and K. Creath, “Basic Wavefront Aberration Theory for Optical Metrology,” in Applied Optics and Optical Engineering, XI, 1992, Academic Press, Inc.
