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Measurement of the Focal Lengths of Lenses
Source publication
A new and simple technique for the determination of the focal length of positive lenses and concave spherical surfaces by Talbot interferometry is described. Results of measurements are presented. We show that high measurement accuracy can be achieved with this technique.
Contexts in source publication
Context 1
... careful alignment of the traverse of the stage with the light beam is done before the measure- ments are taken. Table 1 shows the results of our measurements. There is significant agreement with the values obtained by a commercial focometer (with an accuracy of 0.3%). ...
Context 2
... A 1 and A 2 are the error contributions from the two settings made in the experiment on the f and 2f planes, respectively. The values of the standard deviations s 1 and S2 for a set of 10 readings as shown in Table 1 are quite small. For f = 500 mm s 1 and 82 are 0.04 and 0.05 mm, respectively, which give A 1 = 2.3s, and A 2 = 2.3S2 for 95% reliability. ...
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Citations
... Despite their importance, we decided to leave out the publications on X-rays, nonlinear optics, and quantum optics (Bachche et al., 2017;Bouffetier et al., 2020;Deng et al., 2020;Dennis et al., 2007;Morimoto et al., 2020;Nikkhah et al., 2018;Neuwirth et al., 2020;Hall et al., 2021;Gerlich et al., 2007;Wen et al., 2013). Here, we focus in the applications on the optical range, by noticing the usefulness of the self-imaging phenomena in optical interferometry (Lau, 1948;Yokozeki and Suzuki, 1971a;Yokozeki and Suzuki, 1971b;Lohmann and Silva, 1971;Lohmann and Silva, 1972;Bartelt and Jahns, 1979;Jahns and Lohmann, 1979;Bartelt and Li, 1983;Bolognini et al., 1985;Ojeda-Castañeda et al., 1988;Ibarra and Ojeda-Castañeda, 1993), for implementing novel optical sensors (Silva, 1971;Chavel and Strand, 1984;Nakano and Murata, 1985;Chang and Su, 1987;Bernardo and Soares, 1988;Su and Chang, 1990;Sriram et al., 1992;Gómez-Sarabia et al., 2019), and for setting theta demodulators (Andrés et al., 1986;Ojeda-Castañeda and Sicre, 1986;Chitralekha et al., 1989;Barreiro and Ojeda-Castañeda, 1993;Ojeda-Castañeda et al., 1998). The later type of optical setups leads to the implementation of a noncoherent version of the Abbe-Porter experiments (Ojeda-Castañeda et al., 1989a). ...
The Talbot effect and the Lau effect have been usefully applied in optical interferometry, and for designing novel X-ray devices, as well as for implementing useful instruments for matter waves. In temporal optics, the above phenomena play a significant role for reconstructing modulated, optical short pulses that travel along a dispersive medium. We note that the Talbot-Lau devices can be spatial frequency tuned if one employs varifocal lenses as a nonmechanical technique. Thus, we identify a pertinent link between the Talbot-Lau sensors and the development of artificial muscle materials, for generating tunable lenses. Our discussion unifies seemly unrelated topics, for providing a global scope on the applications of the Talbot-Lau effect.
... Many of these techniques, however, usually require rigorous alignment, which places a big impact on the measurement accuracy. There also exist other methods based on diffraction theory, such as Moiré deflectometry and Talbot interferometry [6], [7]. The measurement accuracy is mainly limited by fringe counting for Moiré deflectometry, and distance measurement between Moiré images for Talbot interferometry. ...
Intraocular lens (IOL) is widely used for cataract treatment. Its optical properties are crucial to obtain a good treatment efficacy and thus need to be evaluated and controlled. In this study, we propose a novel method based on optical coherence tomography (OCT) for noncontact and accurate in vitro measurement of thickness, refractive index and dioptric power of IOL implants. The OCT setup is specially designed to create two sampling optics, called dual-side view OCT (DSV-OCT), which allows for imaging IOL from the two opposite sides simultaneously in single OCT volume scanning. This can produce a three-dimensional surface contour without suffering image distortions due to the refraction of the curved surface. Then, the thickness and surface curvature can be easily computed from the surface contours. In addition, DSV-OCT is able to measure the refractive index. Three IOLs with different dioptric powers (5D, 20D and 28D) were chosen to evaluate this method. The results show that this method is capable to provide a comprehensive and accurate evaluation of IOLs from the aspects of thickness, refractive index and dioptric power, and hence can be potentially used as a quality assurance tool by IOL manufacturers.
... While DeBoo and Sasian [11,12] demonstrated a Fresnel hologram method for measuring a lens with a focal length of 9 m to a precision of 0.01%, this technique requires suitable holograms to be designed for different lenses, which is difficult for lenses with large curvature. Nakano and Murata [13][14][15][16] applied Talbot interferometry to measurement of long focal lengths, making use of the Talbot effect and Moiré pattern techniques to develop a flexible measurement platform that is insensitive to changes in the environment [17][18][19][20][21][22][23]. Chen et al. [24] analyzed the operational principle of measurement of long focal lengths performed with a double-grating interferometer. ...
In this paper, we propose a new calibration method for long-focal-length measurement using a double-grating interferometer. A numerical analysis of the relative error in measurement proves that the main factor affecting accuracy is the angle of inclination between grating grids. Hence, we present a least-squares calibration method using a series of lenses to correct this error. Experiments conducted to demonstrate high precision in wide-range focal length measurement (5–85 m) exhibited an error lower than 0.1% and a repeatability better than 0.03%.
... There exist various methods to measure the focal length such as the classical methods based on nodal slide and image magnification [1]. To improve the accuracy, some modern methods based on moire deflectometry [2,3] and Talbot interferometry [4,5] were proposed with measurement accuracy ranging from 0.19% to 0.8% for the lenses' focal length around 200mm. However, for the micro-lenses (4mm-50mm), the method based on moire deflectometry [6] yields a relatively low accuracy of 3%. ...
... As we want to fully utilize the resolution of the CCD sensor, the size of target is smaller than the CCD sensor due to magnification effect. Therefore, the maximum incident angle can be defined in Eq. (4). Once this angle is smaller than 15 degrees, the paraxial approximation is satisfied and the measurement method can be carried out. ...
In this paper, we realize an accurate and flexible measurement method for paraxial focal length with a simple apparatus. The proposed method utilizes the established measurement principle which is based on the differentiation of the Gaussian optical formula and geometrical relationship of camera-lens system to obtain the focal length of a lens. Some image processing methods are employed to experimentally realize this measurement principle with high accuracy and practicability. Image sharpness evaluation function is utilized to ensure the system conforms to the Gaussian optical formula. System calibration is adopted to compensate for the errors caused by the simple apparatus. In addition, aberration is considered to obtain a paraxial focal length of a lens. Experimental results indicate that repeatability error of measurement system is less than 0.11% for most lenses.
... Numerous methods have been developed to measure the focal length of imaging systems, such as the traditional methods, that is, those based on imaging properties of lens systems [1][2][3], and modern methods, based on diffraction and interferometry [4][5][6][7][8][9]. All of these methods have their own advantages and disadvantages, and most of them provide high-precision results only under restricted conditions with rather complicated equipment that are very sensitive to mechanical vibrations and optical noises. ...
An accurate and simple technique based on Fresnel diffraction of light from a phase wedge is described to measure the effective focal lengths (EFLs) of optical imaging systems. When the neighbourhood of a transparent wedge is illuminated by a monochromatic parallel beam of light, the Fresnel diffraction occurs as a result of sharp change in phase at the wedge boundary. The wedge with small apex angle imposes a linearly varying phase change that causes periodic diffraction fringes in the direction parallel to the wedge boundary. The EFL is determined by measuring the fringe spacing of the Fresnel diffraction both before and after installing the test lens in the setup. The method is easy to apply and it is less sensitive to vibration compared to interferometric methods. It covers a wide range of both positive and negative focal lengths. Experimental results for several different lenses are quite consistent with the values indicated by the lens manufacturer.
... Thus, the accuracy of this technique is relatively low. Moiré reflectometry [3] and Talbot interferometry [4][5][6] utilize a Moiré fringe and have achieved a measurement uncertainty of approximately 0.7% for a focal length of 210 mm. In these methods, the distance between two gratings is crucial and should be accurately known. ...
An approach for measuring the focal length of a lens using a diffractive beam sampler is presented. The high-precision measurement of the focal length is conducted while a specimen is continuously moved along the optical axis. Images of the fractional beam spots on a charge-coupled-device camera are tracked with respect to the position of the specimen to obtain the relation between the beam-spot distance and the lens–specimen distance. Accordingly, this relation can be used to infer the focal length via computation. The method is carried out by simulation, easy, and inexpensive. In addition, it is applicable to find the focal length of a thin lens as well as the effective focal length of a system of lenses because it does not require the movement or replacement of lenses. The results indicate a dramatically high precision.
... In this step, S is gradually moved away from the OL path and the change in the separation between beam spots on IS, which is improved to read directly the distance and show on the computer screen, with respect to OLeS distance, was tracked using 0.1 mm increments (Dv ¼ 0:1 mm). According to Eq. (8), the separation between the beam spots linearly decreases as the distance between S and OL increases. The linear ratio of the two distances was recorded to compute the focal length. ...
... In Fig. 5, the initial position of S is chosen randomly with the condition that the beam spots do not overlap. S is gradually shifted 20 times in increments of 0.1 mm (the distance between the initial point and the final point of specimen is 2 mm), and the beam spot separation on IS is recorded for each position of S. The slope of the linear plot in Fig. 5, theoretically given by Eq. (8), is represented by the parameter w, and can be computed by the method of least squares using the data in Fig. 5. The focal length is then calculated easily using Eq. ...
A cutting-edge precision technique for computation of focal length of a positive lens with double-hole mask is described. The technique is simple and versatile due to incorporation of the updated functions of image sensor device that supports reading the distance between beam spots instantaneously while the position of the specimen is being changed, as well as the reduction in several challenging measurement steps. Furthermore, this technique does not require prior knowledge of distances in the optical setup. High accuracy in focal-length measurement is obtained by precise beam spot distance analysis using image sensor integrated software. The acquired data exhibit considerably high precision and reproducibility.
... Several other bulk motion-based methods of focal length measurement have been reported in prior articles. Measurements using Talbot interferometry rely on recording moiré fringe tilt angles and displacements to estimate the focal length [13][14][15][16]. Similarly, measuring the Lau phase interferometric fringe intensity distribution for focal length measurements has also been reported [17,18]. ...
... As shown in Fig. 4, this results in deviations in focal length from the ideal design value, which leads to a minor shift in the rear principal plane of the lens and results in a offset in the distances by (ΔD 1 ) Manufac , (ΔD 2 ) Manufac , and (ΔD 3 ) Manufac . The total offset ΔD n Total between known and actual separation distance values is a sum of these three offsets, as expressed in Eq. (15), where the subscript n 1, 2, or 3 ΔD n Total ΔD n Manufac ΔD n Thick_Lens ΔD n Placement : ...
This paper presents a motion-free technique to characterize the focal length of any spherical convex or concave lens. The measurement test-bench uses a Gaussian laser beam, an electronically controlled variable focus lens (ECVFL), a digital micro-mirror device (DMD), and a standard photo-detector (PD). The method requires measuring beam spot sizes for different focal length settings of the ECVFL and using the measurement data to obtain a focal length estimate through an iterative least-squares-based curve-fitting algorithm. The method is also shown to overcome potential measurement errors that arise due to inaccurate placement of optical components on the test-bench as well as unknown principal plane locations of asymmetric lens samples such as plano-convex lenses. Contrary to the commercially deployed and other proposed methods of focal length characterization, this method does not involve any bulk mechanical motion of optical elements. This approach eliminates measurement errors due to gradual mechanical wear and tear and improves measurement repeatability by minimizing mechanical hysteresis. The compact and fully automated method delivers fast, repeatable, and reliable measurements, which we believe makes it ideal for deployment in industrial lens production units and characterizing lenses used in sensitive imaging systems and various other optical experiments and systems. Measured focal lengths are within the 1% manufacturer-provided tolerance values showing excellent agreement between theory and experiments. We also demonstrate measurement robustness by rectifying discrepancies between known and actual separation distances on the measurement test bench.
... Classical methods such as nodal slide and image magnification are based on imaging properties of a lens system under illumination of an incoherent light source that are suitable for short focal lengths [1]. The methods based on diffraction and interferometry, such as moiré deflectometry [2,3], Talbot interferometry [4][5][6], and other interferometric approaches, have been investigated. In moiré deflectometry and Talbot interferometry, the focal length of the lens is obtained from the tilt of a moiré fringe. ...
... 6(a)-6(e) represent the phase difference of the extrema from the first maximum on the right and left side of the optical axis versus x 2 for the lenses with nominal EFLs of 100 mm, 4 mm, 250 mm, 175 mm, and −100 mm by apertures with diameters of 38.00 mm, 3.36 mm, 38.00 mm, 22.10 mm, and 9.48 mm, respectively. The dashed curves are obtained by fitting Δϕ from Eq. (6). The corresponding EFLs with the associated uncertainties that are obtained by fitting on the experimental results are presented in Table 1. ...
A method based on the Fresnel diffraction of light from the phase step is introduced for measuring effective focal length (EFL) and back focal length (BFL) of optical imaging systems. It is shown that, as a transparent plane-parallel plate is illuminated at a boundary region by a monochromatic beam of light, Fresnel diffraction occurs because of the abrupt change in phase imposed by the finite change in refractive index at the plate boundary. Variation of the incident angle in a convergent (or divergent) beam of light causes the periodic intensity along the central fringe of the diffraction pattern. The measurement of the extrema position of the intensity distribution accurately provides the EFL and BFL. The technique is easy to apply and can measure a wide range of both positive and negative focal lengths. The measuring setup can be very compact with low mechanical and optical noises. As examples of this technique, the EFLs of five different lenses are experimentally obtained. The results are quite consistent with the values indicated by the lens manufacturer.
... Many approaches have been proposed to meet the rigorous measurement requirement. Nakano and co-workers proposed a Talbot interferometry method, which used the Talbot effect and moiré techniques to measure the focal length of the test lens, and the measurement repeatability was 0.03% for a test lens with a focal length of 500 mm [5][6][7][8]. Based on Talbot interferometry, Singh et al. improved the discriminant accuracy of the moiré fringe angle using the Fourier analysis to remove the image noise resulting from the grating lines, and its measurement error was less than 0.3% for a test focal length of 240 mm [9]. ...
We propose a new laser multi-reflection differential confocal focal-length measurement (LDCFM) method to meet the requirements of high-precision measurements of long focal lengths. An optical flat and a reflector are placed behind a test lens for reflecting the measuring beam repeatedly. Then, LDCFM uses the property that the null points of differential confocal response curves precisely correspond to the convergence points of the multi-reflected measuring beam to exactly determine the positions of the convergence points accurately. Subsequently, the position variation of the reflector is measured with different reflection times by using a distance-measuring instrument, and thereby the long focal length is measured precisely. Theoretical analyses and preliminary experimental results indicate that the LDCFM method has a relative expanded standard uncertainty ( k = 2 ) of 0.04% for the test lens with a focal length of 9.76 m. The LDCFM method can provide a novel approach for high-precision focal-length measurements.
