In this paper, we use the irradiance transport equation and the Fourier transform-based experimental solution given by Ichikawa–Lohmann–Takeda. We analyze experimental factors such as the digital filter, the introduced error for the rotation and period of the Ronchi ruling, and a new method is demonstrated for the measurement of 3D wavefront information.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Using previously derived relations between a wavefront and its irradiance distributions in different planes, Teague  derived the following so-called irradiance transport equation (ITE):
For the use of ITE [Eq. (1)] in the field of optical testing, an interesting, pioneering, and simple method was proposed by Ichikawa, Lohmann, and Takeda, called the ILT method . Figure 1 shows the experimental setup used in this paper.
An important aspect of the technique analyzed in this paper is the use of a Ronchi ruling for registering the irradiance. Consequently, the two planes for such irradiance measurements are very close, that is, less than a millimeter apart.
2. MATHEMATICAL SOLUTION
For solving the ITE equation, an equivalent expression of Eq. (1) is
The Fourier transform method, developed by Takeda et al.  is used by the ILT method to solve Eq. (2) using a Ronchi ruling. The main idea of this technique employs , where is the Ronchi ruling frequency. The final results derived for , in our case, are only for the first term on the right side of Eq. (2). The second term is eliminated because in the original paper by ILT, the results derived were difficult to understand . Afterward, that second term has been eliminated by several authors [8–10], considering if uniform illumination is used along the experiment; in our case, we verified such a condition of uniform illumination; from the experimental measurements of the irradiance in two close planes, the Fourier transform is used to obtain a spectrum for which the first-order harmonic is isolated with a digital filter. Figure 2 shows an example of the profile of the irradiance in one plane.
In Section 3, we explain in detail the effect of employing the Ronchi ruling with different spatial frequencies and the effect of rotation of the Ronchi grating with respect to the optical axis in the experimental setup. In Sections 4 and 5, we present experimental results for one-dimensional wavefronts and a new method for obtaining 3D wavefront information by rotating the Ronchi ruling around the optic axis and applying Takeda’s Fourier technique for each rotation. In Section 6, there is a comparison of different methods applied for testing diverse optical elements by means of ITE.
3. EXPERIMENTAL ANALYSIS
This section has two parts: (a) an analysis of the frequency of the Ronchi ruling used to locate the first-order harmonic in the Fourier plane and its ability to isolate that first order; and (b) a correction to small rotation of the Ronchi ruling around the optical axis as a possible source of error in the experiment.
The experimental setup used is shown in Fig. 1. The LED used by ILT was replaced by an He–Ne laser (LS) with power of 30 mW (); we used the laser because we obtain a better alignment of the experimental setup, and the irradiance measurements gave us better results; the point source (P) was produced with a microscope objective and numerical aperture of 0.65, with spatial filter with a diameter of 5 μm; the two lenses under test (L2) were focused close (2 mm) to the point source; the lenses have 200 and 250 mm focal length and a diameter of 50 mm. The Ronchi grating is located at the exit pupil of lens L2, and two intensity planes, separated by 0.7 mm, are close to the exit pupil, where intensity images are registered by a photographic camera. The camera is mounted on a mechanical mount with a micrometer for sliding it. The camera is a Canon EOS Rebel T5i, with a CCD with ; the pixel size is 4.29 μm.
A. Period of the Ronchi Ruling
To understand the effect of using Ronchi gratings with different periods, we performed a simulation using three rulings with periods of 20 line pairs/cm, 60 line pairs/cm and 120 line pairs/cm. The first column of Fig. 3 shows the Fourier spectrum results when the Ronchi ruling period is decreased, which shifts the first-order harmonic to the right. The second column shows how the circular filter is shifted to the position of the first-order harmonic of the spectrum, and the third column shows the registered experimental irradiance, which is used to find the wavefront .
It seems that a higher frequency in the Ronchi grating produces a more isolated first order in the Fourier plane; however, this effect causes notable problems in the experimental procedures because of the diffraction effects introduced during the experimental irradiance measurements. Hence, from our own experience, low-frequency gratings reduce problems with the irradiance images.
B. Rotation of the Ronchi Grating
An additional effect that has not yet been analyzed, until now, is when the grating has some rotation with respect to the optical axis of the lens in the experimental arrangement. This situation produces error in the period of the grating with respect to its nominal value. Therefore, a computer program was developed to correct for the ruling’s rotation and its effect on the computed ruling period.
Figure 4(a) shows a grating image with small rotation. To measure such rotation, in Fig. 4(b) five “dark” slits are isolated; the deviation of these dark slits from a vertical reference line is shown in Fig. 4(c). In Fig. 4(d), these differences are amplified. Using our developed computer program, such difference between the nominal and experimental grating periods was calculated and applied to improve the experimental results.
4. RESULTS FOR THE ONE-DIMENSIONAL WAVEFRONT
In this section, the experimental results for testing low-quality single lenses will be shown. The main purpose is to applied the method developed by ILT for such kinds of lenses, because when you applied interferometric methods to them, the interferograms become too complex to be analyzed easily. See Fig. 5 with information about the tested lenses with a ZYGO interferometer.
Based on the analysis in Section 3, Figs. 6 and 7 show the experimental results for the testing of two lenses with focal lengths of 250 and 200 mm, respectively; both have a diameter of 50 mm. Figures 6(a) and 6(b) show small defects (scratches) at the edge of the lens, in contrast to the lens shown in Figs. 7(a) and 7(b), where such defects are not observed; for the derived experimental wavefront of Fig. 6(c), some ripples can be seen, but they do not appear in Fig. 7(c).
5. RESULTS FOR THREE-DIMENSIONAL WAVEFRONTS
In this section, a novel method is presented to obtain 3D wavefronts from the difference of two irradiance measurements, using the method of ILT. In the original ILT method, a Ronchi ruling is used, and only a one-dimensional (1D) wavefront was derived in the lens testing explained in Section 4.
In order to obtain 3D wavefronts, the Ronchi ruling is rotated to different angles, as is shown in Fig. 8. For each ruling position, the Fourier transform method, developed by Takeda, is applied and four 2D wavefronts are derived, shown in Figs. 9(a) and 10(a). Using the Toolbox Curve Fit of MATLAB, Figs. 9(b) and 10(b) show the derived 3D wavefronts for the testing of the two lenses analyzed in Section 4; Figs. 9(c) and 10(c) show the wavefronts’ cutoff over the diameters of the lenses. The MATLAB program performs a polynomial fitting, up to the third order, to the four curves shown in Figs. 9(a) and 10(a).
6. COMPARISON OF SIMILAR METHODS TO ITE IN OPTICAL TESTING
From the comparison with the content of this paper with other similar methods using ITE and phase-retrieval techniques, we found two main different characteristics among them: a) the mathematical method to find the phase and wavefronts, and b) the location of the image planes to obtain the intensity measurements.
For the testing of astronomical telescopes, Fienup et al.  used laborious phase-retrieval algorithms for the analysis of the Hubble telescope defocused images in order to know, mainly, the value of the spherical aberration of the primary mirror. Roddier and Roddier , for recover wavefronts, used two separated intensity image planes located symmetrically from the focus plane of terrestrial telescopes; the mathematical method used is a closed-loop wavefront reconstruction Fourier transform. A comparison of their analysis was done with the results from a Shack–Hartmann sensor.
Quiroga et al.  tested a multifocal lens; the intensity image planes were located similarly to the previous proposal by ILT ; the mathematical procedure was by means of Fourier transform, the wavefronts derived are unidimensional, and the units used are arbitrary units. In a paper by Dorrer and Zuegel , the ITE method was applied for testing flat surfaces, polished with the magnetoreological technique. The solution to a simplified ITE was using the Fourier transform to the phase and intensity; however, the main aim in this work is to resolve the high-frequency surface modulation introduced by the polishing technique employed, and single intensity images were measured to obtain the phase.
For the analysis of an aspheric surface testing, Shomali et al.  used ITE, and only simulation results were provided. The solution to ITE was given by the iterative Fourier transform proposed by Roddier and Roddier ; analysis of close-intensity image planes with distance of 1 mm was given.
The analyses done by Soto  and Soto et al.  are interesting because an optimization of the intensity planes was done when ITE was used for phase retrieval, but the locations of the measured intensity planes are on both sides of the focal plane, according to the proposal by Roddier and Roddier .
From the previous analysis, we conclude that a clear and systematic comparison among the different approaches for the testing of optical components using ITE is hard to draw. This is difficult because diverse mathematical and experimental technologies were applied in each case, and different optical elements had been tested. On the other hand, it is important to mention that these methods, in which the ITE is used, must be considered as alternative techniques to the interferometric and screen ones regularly applied in the field of optical testing .
However, in this context, the method developed by ILT and analyzed in this paper improved results, and 3D wavefronts were obtained with two advantages: the distance between image intensity planes is less than 1 mm, and the well-known Takeda’s Fourier transform method is used .
In this work, we studied the effect of two factors in the method proposed by ILT : selection of the Ronchi ruling frequency to obtain better irradiance measurements and the separation of the first-order frequency in the Fourier spectrum, and solving a possible problem regarding the Ronchi ruling rotation with respect to the optical axis along the experiment. Therefore, we optimized the ILT technique results to find the wavefront of low-cost lenses with an accessible and simple experimental setup. Besides, a new method to obtain a 3D wavefront results was developed by rotating the Ronchi ruling to four different angles and fitting a polynomial to the four 2D curves. As can be seen from the comparison for the results for 1D and 3D wavefronts, improvement of the accuracy was reached for the fitting of the wavefront for several orientations of the ruling for the case of 3D; given the low quality of the lenses, the accuracy is close to . With these results, we plan to use a traditional nodal bench for measuring the wavefront coming from the lenses under test, according to the previous work developed by Magaña et al. . (A shorter version of this paper was presented at the 100th Anniversary Meeting of the Optical Society of America, in Rochester NY, 2016.)
Consejo Nacional de Ciencia y Tecnología (CONACYT).
The authors are grateful to Prof. Glenn Boreman, UCC, for his important and enlightening comments on the content of the paper, and also to the reviewers for their important comments. J. Arriaga Hernández was a graduate student with scholarship of CONACYT, México. The authors are grateful to Ana María Zárate Rivera and Fernando Romero for the editing of the paper, and to Elizabeth Percino Zacarías for the ZYGO interferograms.
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