INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
doi:10.1088/0957-0233/16/12/019
Meas. Sci. Technol. 16 (2005) 2534–2540
A spot pattern test chart technique for measurement of geometric aberrations caused by an intervening medium—a novel method A R Ganesan, P Arulmozhivarman and M Jesson Photonics Laboratory, Department of Physics, National Institute of Technology, Tiruchirappalli-620 015, India E-mail:
[email protected]
Received 1 July 2005, in final form 10 October 2005 Published 15 November 2005 Online at stacks.iop.org/MST/16/2534 Abstract Accurate surface metrology and transmission characteristics measurements have become vital to certify the manufacturing excellence in the field of glass visors, windshields, menu boards and transportation industries. We report a simple, cost-effective and novel technique for the measurement of geometric aberrations in transparent materials such as glass sheets, Perspex, etc. The technique makes use of an array of spot pattern, we call the spot pattern test chart technique, in the diffraction limited imaging position having large field of view. Performance features include variable angular dynamic range and angular sensitivity. Transparent sheets as the intervening medium introduced in the line of sight, causing aberrations, are estimated in real time using the Zernike reconstruction method. Quantitative comparative analysis between a Shack–Hartmann wavefront sensor and the proposed new method is presented and the results are discussed. Keywords: aberration measurement, Shack–Hartmann wavefront sensor, spot
pattern
1. Introduction An intervening medium always causes aberrations/distortions in the quality of the image ‘seen’ by an imaging system. For instance, seeing through a helmet visor, windscreen or windshield of a car or an aircraft degrades the image quality of scenery when compared to seeing with the naked eye. A similar problem arises in astronomical observations using ground-based telescopes because the system has to ‘see’ through the atmosphere. Thus, the measurement of aberration caused by the intervening medium becomes very essential in several ‘seeing’ applications [1]. Fundamentally, the intervening medium introduces phase variations in the optical wavefront chiefly due to the refractive index variations across the ‘seeing’ apertures. In the case of groundbased astronomical imaging, the phase variations are usually measured using a Shack–Hartmann wavefront sensor and compensated by deformable mirrors. This technology, 0957-0233/05/122534+07$30.00
termed adaptive optics, provides continuous measurement and compensation of the aberrations caused by the fluctuating intervening medium [2]. However, in the case of an intervening medium such as a helmet visor or windshield, which provides fixed aberrations, a single measurement across the ‘seeing’ aperture would normally be sufficient. By the ‘seeing’ aperture we mean the particular portion of the windscreen through which the driver sees the scenery. The ‘seeing’ aperture will be different when the driver looks in different directions. In other words, at a given instant of time the driver would not be seeing through the entire windshield but only through a portion of that. The measurement of the aberrations caused by such intervening media becomes very essential in the design of helmet visors, automobile screen, etc, especially so in the case of aircraft windshield. The measurement of geometric aberrations caused by such media can be done using interferometric methods or Shack– Hartmann wavefront sensors, but these methods are very
© 2005 IOP Publishing Ltd Printed in the UK
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A spot pattern test chart technique for measurement of geometric aberrations caused by an intervening medium
expensive and exclusive. The Shack–Hartmann (SH) method has been used for surface metrology of thin transparent optics such as glass substrates and photo masks in the semiconductor industry by analysing the reflected wavefronts in deep-UV wavelengths [3]. The measurement of surface shapes and refractive index inhomogeneity of an optical element has been carried out using a phase shifting interferometric technique [4, 5]. In another work, a comparative study of a Shack– Hartmann sensor and a phase shifting interferometer for largeoptics metrology has been done [6]. In this paper, we present a simple and cost-effective method analogous to Shack– Hartmann wavefront sensing by using a single lens and a spot pattern test chart instead of a microlens array. This new method, the spot pattern test chart technique, has lots of advantages over the conventional Shack–Hartmann wavefront sensor. It is simple, cost effective and ‘application adaptable’. This method is simple, as it needs only a lens, a spot pattern test chart and an imaging system. It has relaxed constrains of alignment and the need for a vibration isolation table is flexible and not rigid. It is cost effective, as it does not need a coherent monochromatic source, lenslet array, beam expander or density filters. No specific software is required for this method to calculate local slope, centroid and Zernike terms as that developed for the SH method could be adapted for this technique. Actually, this new method was inspired by the Shack–Hartmann method. The new method is application adaptable; for instance, the ‘seeing’ aperture in this method could be varied to a greater extent depending on the application requirements, especially in the large optics environment. The pitch of the spot pattern could also be varied according to the application requirement very easily. The source light could be an ordinary white light room condition. Experimental results obtained with our technique have been compared with those obtained by the conventional Shack–Hartmann technique.
2. The conventional Shack–Hartmann wavefront estimation technique The Shack–Hartmann wavefront sensor is the most popular method available so far to measure aberrations accurately in astronomical as well as in clinical optometry. The Shack– Hartmann sensor infers local near field wavefront gradient by measuring a corresponding focused spot position in the far field [7]. To achieve this, an array of lenslet is used which spatially samples the incoming wavefront and focuses each lenslet beam portion of the wavefront onto the CCD detector array. The average wavefront tilt across each lenslet aperture results in a shift of the respective focal spot. A planar wavefront produces a regular array of focal spots, while an aberrated wavefront produces a distorted spot pattern. Thus the centroid of the blur pattern does not lie at the spot in the focal plane at which the light was originally aimed. A comparison of these two produces a map of the wavefront slopes, and integration of the slopes allows reconstruction of the test wavefront [8]. Figure 1 shows a schematic representation of the aberration studies done using the Shack–Hartmann method. In this method, the light beam from a He–Ne laser source
Figure 1. Conventional Shack–Hartmann technique for intervening medium aberration studies. NDF—neutral density filter, SPF—spatial filter, CL—collimating lens and MLA—microlens array.
Figure 2. Schematic of the spot pattern test chart technique for intervening medium aberration studies.
is spatially filtered and collimated, after which the beam is spatially sampled through a microlens array unit, which produces an array of focus spots. At the focal point of the microlens array, a CCD camera detector is placed which is connected to a frame grabber card interfaced to a computer. The detector array of the CCD is divided into sub-arrays such that each sub-aperture focus spot is at its centre. The centroid coordinate is estimated for each focus spot and these values are taken as initial reference for a plane wavefront. Now the aberrating test object such as a glass or Perspex sheet is introduced into the path of the collimated beam. This would introduce phase variations across the wavefront, resulting in the shift of the centroids of the focus spots. By measuring the centroid shift of each spot, the local slopes are estimated from which the entire wavefront profile can be precisely calculated. The local slope data can also be fit into a Zernike polynomial, the coefficients of which give the various aberrations.
3. The proposed spot pattern test chart technique The proposed method uses a test chart consisting of an array of spots pattern as the object, which is imaged by a lens system on to the CCD camera detector. General room light conditions are enough to illuminate the spot pattern test chart, and it does not need a monochromatic or coherent light source. Figure 2 shows the schematic of the setup, while figure 3 shows a typical spot pattern test chart. The spot pattern test chart could be either just the printout of the spot pattern image from a Shack– Hartmann sensor or obtained through computer simulations by pre-defining the intensity distribution. The magnification is adjusted such that the image of the spot pattern test chart fills the CCD detector array. First, the image of the spot pattern is recorded by the frame grabber and stored as a reference, where the centroid of every spot is calculated. When an intervening medium under study, such as a glass or Perspex sheet, is introduced in 2535
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4. Experimental procedure
Figure 3. Spot pattern test chart (15×15 spots).
the line of sight between the imaging lens and the test chart, the positions of the image spots will be changed depending on the aberrations introduced by the medium. This image is again stored in the frame grabber and the new positions of the centroids are compared with respect to the reference image. The shift in the centroid position gives the local slope values. The centroid position formulae are expressed [9] as I J i=1 j =1 x(i, j ) s(i, j ) Xc (K) = I J i=1 j =1 s(i, j ) I J i=1 j =1 y(i, j ) s(i, j ) Yc (K) = I J i=1 j =1 s(i, j ) where x(i, j ) and y(i, j ) are the coordinate positions of the (i, j )th pixel in the Kth sub-aperture, s(i, j ) is the input wavefront signal of the (i, j )th pixel on the square sub-aperture of I × J pixels. With the formulae, the centroid position of the input wavefront at the Kth sub-aperture could be calculated as (Xc(K), Yc (K)). The slope values are fitted into the Zernike polynomials for the phase estimation and calculation of the various aberrations as well as the peak-to-valley (PV) and root mean square (RMS) values. The shifts in the centroids of the spots are produced only by the various aberrations contributed by the entire aperture of the intervening medium which is interrogated by the light beam, and hence the procedure of centroid estimation, local slope calculation and computation of various aberrations seem similar to the conventional Shack– Hartmann technique. In this method, a polychromatic and incoherent light source is used to measure the monochromatic aberrations of the intervening medium, unlike the Shack–Hartmann method where a monochromatic and coherent source is used to measure monochromatic aberration. In the case of the Shack– Hartmann method, use of an incoherent source is not feasible as one is aware that in the wavefront sensor setup, the lenslet is placed at its focal length to the CCD. If an incoherent source were used, it would add in off-axis aberrations such as coma and astigmatism in large quantity. Apart from this, each subaperture in the lenslet array area should be proportionately matched with those in the CCD chip imaging area such that each lenslet point should get focused within its pitch distance or sub-aperture on the CCD. So in the case of an incoherent source, the lenslet spot could well may fall outside its pitch or sub-aperture causing erratic calculations of local slope, centroid and, therefore, Zernike coefficients too. 2536
For the conventional Shack–Hartmann technique (figure1), we used a 16 × 16 microlens array of 417 µm pitch and 45 mm focal length. This lens array would be able to spatially sample a light beam of about 6.5 mm diameter. A Sony 2/3 monochrome CCD camera (XC-ST 70CE) and a Matrix-Vision PC-Image SG frame grabber were used for image capture. The positional coordinates of the spot centres were found using centroid estimation algorithms. The initial reference centroid values were stored for a reference plane wavefront. A sheet of glass or Perspex was introduced in the path of the beam before the microlens array as shown in figure 1 and the new positions of new centroids were estimated. From these data, the shift of the centroid for each spot was calculated, which gives the local slopes of the sub-aperture beams. The local slopes were used for the reconstruction of the wavefront phase profile by Zernike polynomial fitting (up to fourth order). The Zernike coefficients were then used for estimating the various aberrations [10, 11]. Experiments were carried out with various sheets of glass and Perspex of different thicknesses and the interrogation area on the sheets was marked. The aberration values through Zernike decomposition algorithms were studied in terms of PV and RMS error. For the proposed spot pattern technique, we took a printout of the image of the spot pattern obtained using the microlens array and used it as the test chart object (figure 3). It should be noted that this test chart pattern could also be a computer-simulated image. This test chart was imaged on the CCD array by proper imaging optics and the magnification is adjusted such that the spot pattern just fills the entire detector array. In our case, we took a spot pattern array test chart of size 125 mm × 125 mm containing 15 × 15 spots which was placed 1 m from the imaging lens of focal length 50 mm. The CCD array was placed at the focal plane of this lens, which amounts to a magnification of 0.05 and would match the microlens array pitch (417 µm) of the earlier case. With the pixel size of the CCD being 11.6 µm × 11.2 µm, the resultant Airy disc of each spot at 580 nm wavelength is about 10 pixels in the virtual lenslet array focal plane in the CCD detector. Such an arrangement would enable us to use the same algorithms and software developed for the earlier conventional Shack– Hartmann method. As before, the image of the spot pattern is recorded and stored as a reference in the frame grabber of the system where the centroid of every spot is calculated. When an intervening medium such as Perspex or glass sheet of varying thickness is introduced in the line of sight of the propagating medium (as shown in figure 2), the new centroid positions are estimated. The shift in the centroid position with respect to the reference image gives the local slope values, which are fitted into the Zernike polynomials for the phase estimation. Experiments were carried out with the same set of specimens and the same interrogation area as in the earlier case.
5. Results and discussion Aberration measurements were carried out on Perspex sheets of various thicknesses (3 mm, 4 mm, 6 mm) and glass sheets
A spot pattern test chart technique for measurement of geometric aberrations caused by an intervening medium
Figure 4. Phase map for reference condition when an intervening medium is not inserted.
(1.5 mm, 2 mm, 3.5 mm) for our studies. Each test object was marked with a circle of 6.5 mm diameter to match the area of the CCD chip and used with both the Shack–Hartmann technique and the spot pattern test chart method keeping the same seeing aperture. Conditions for experimenting with both techniques remain the same though a vibration isolation optical table was a requirement in the case of the Shack– Hartmann technique, while in the proposed technique this was a less stringent condition, a claimed advantage of the method. At the same time, it does not imply that the non-use of a vibration isolation optical table would increase the accuracy in the measurement of aberration in the new method.
(a)
The reason for using the Shack–Hartmann technique alongside the proposed technique is to comparatively (both qualitative and quantitative) verify the new method with the standard Shack–Hartmann method. Before the experimental analysis we verified the object material property of the test objects. Perspex sheets with excellent clarity and durability, easily sizable and having a refractive index of 1.49, and glass materials having a refractive index of 1.50 were taken for our aberration studies. The initial reference conditions were measured for the Shack–Hartmann technique as well as the spot pattern test chart method separately before introducing an intervening medium. Figure 4 shows the initial reference phase map for the spot pattern test chart technique corresponding to the absence of any intervening medium. Figures 5(a), (b), 6(a), (b), 7(a), (b) and 8(a), (b) show the phase maps of the Shack–Hartmann technique and spot pattern test chart method for 1.5 mm thick glass, 3 mm thick Perspex, 3.5 mm thick glass and 4 mm thick Perspex, respectively. The experimental values of root mean square (RMS) error, peak-to-valley (PV) and spherical aberrations obtained for various thickness sheets of Perspex and glass using the Shack–Hartmann and the proposed spot pattern test chart method are summarized in tables 1 and 2, respectively. A comparative analysis of the phase maps and the tabulated results show that both the techniques produce almost analogous results with a mean error difference of 0.000 046 µm in RMS and 0.029 µm in peak-to-valley value. In this case, for
(b)
Figure 5. (a) 1.5 mm glass measured with the Shack–Hartmann technique and (b) 1.5 mm glass measured with the spot pattern test chart technique.
(a)
(b)
Figure 6. (a) 3 mm Perspex measured with the Shack–Hartmann technique and (b) 3 mm Perspex measured with the spot pattern test chart technique.
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(a)
(b)
Figure 7. (a) 3.5 mm glass measured with the Shack–Hartmann technique and (b) 3.5 mm glass measured with the spot pattern test chart technique.
(a)
(b)
Figure 8. (a) 4 mm Perspex measured with the Shack–Hartmann technique and (b) 4 mm Perspex measured with the spot pattern test chart technique. Table 1. Root mean square, peak-to-valley and primary spherical aberration obtained using the Shack–Hartmann wavefront sensor technique. Materials Without intervening medium (reference condition) 1.5 mm glass 2 mm glass 3.5 mm glass 3 mm Perspex 4 mm Perspex 6 mm Perspex
RMS error (µm)
PV error (µm)
Spherical aberration
0.002
0.001
–
0.017 58 0.017 54 0.017 74 0.017 98 0.017 42 0.016 33
0.069 0.173 0.229 0.172 0.181 0.245
0.079 8 0.019 11 0.014 32 0.019 73 0.008 8 0.074 92
the square sub-aperture spatial sampling of Shack–Hartmann data, Cartesian Zernike polynomials are used. Figures 9 and 10 show the correlation graph between the two techniques in terms of RMS and PV values. It is interesting to observe that correlation exists for both Perspex and glass sheets and for all the thicknesses of materials. Figure 11 shows the primary spherical aberration for various thicknesses of glass and Perspex sheets, using both methods. It is evident from the comparative analysis that the proposed spot pattern technique experimentally goes well with the standard conventional Shack–Hartmann technique, with not so significant error percentage. As a further study, a spot pattern test chart technique was examined for varying ‘seeing’ apertures of 10 mm, 20 mm and 30 mm diameter and for a thickness of 3 mm, 4 mm and 2538
Table 2. Root mean square, peak-to-valley and primary spherical aberration obtained using the spot pattern test chart technique. Materials Without intervening medium (reference condition) 1.5 mm glass 2 mm glass 3.5 mm glass 3 mm Perspex 4 mm Perspex 6 mm Perspex
RMS error (µm)
PV error (µm)
Spherical aberration
0.00
0.00
–
0.017 63 0.017 54 0.017 77 0.017 90 0.017 48 0.016 26
0.051 0.183 0.214 0.230 0.150 0.287
0.077 2 0.024 98 0.010 44 0.022 98 0.005 8 0.094 37
6 mm in Perspex sheets placed at a distance of 70 mm, 95 mm and 195 mm from the imaging lens, respectively. The mean PV increases with increasing apertures and thickness almost linearly. The mean PV plotted against varying ‘seeing’ apertures is shown in figure 12. The maximum ‘seeing’ aperture size achievable is the object plane area, which is 125 mm × 125 mm in this case. This evidently proves that, as far as thin transparent optics such as Perspex, etc, are concerned, the proposed method would be a viable alternative in providing simplest implementation, easier determination of results and more, stretching its ability to measure the maximum aperture possible, especially with large optics. A simple rail system with a scaling device could be simply used for varying aperture condition measurement in realtime studies. This is not practically feasible with an interferometric technique where the system actually requires
A spot pattern test chart technique for measurement of geometric aberrations caused by an intervening medium
RMS Error in micrometre
RMS Error Correlation 0.02 0.018
SHWS 0.016
SPOT PATTERN
0.014 0.012 0.01 1.5mmG 2mmG 3.5mmG 3mmPP 5mmPP 6mmPP
Materials / Thickness
Figure 9. RMS error correlation studies for both techniques. G—glass, PP—Perspex.
G 3. 5m m G 3m m PP 5m m PP 6m m PP
2m
1.
5m
m
G
SHWS SPOT PATTERN
m
PV Error in micrometre
PV Error Correlation
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Materials/ Thickness
Figure 10. PV error correlation studies for both techniques. G—glass, PP—Perspex.
Zernike polynomial
Primary Spherical Aberration 0.1 0.08 0.06 0.04 0.02 0 1.5mmG 2mmG 3.5mmG
3mmP
4mmP
6mmP
Intervening medium Shack Hartmann method
Spot pattern method
Figure 11. Primary spherical aberration for various thicknesses of glass and Perspex.
high precision reference optics for large area measurement, which is prohibitively expensive for aberration studies of the intervening medium. Similarly, in the Shack–Hartmann technique again the entire beam has to be properly resized by using aberration-free optics for spatial sampling of the beam according to the size of the microlens array and CCD in the line of sight propagation direction for large aperture studies. With advanced software available, the aberrations for large transparent materials could be easily measured in real time. The industries manufacturing helmet visors, aircraft windshields, windscreens, etc, which depend on less reliable methods, could benefit by the proposed method for high precision quantitative estimation of aberrations. Some industries practice a method where they project the image of a grid through the test transparent medium onto a wall and
Peak to Valley (nm)
PV vs Varying aperture studies
500 400
3mm
300
4mm
200
6mm
100 0 10
20
30
Aperture Size (mm)
Figure 12. Plot of PV values obtained for different seeing aperture diameters with 3 mm, 4 mm and 6 mm thick Perspex sheets.
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measure the distortions manually. For a transparent medium such as an aircraft windshield the above method is not a scientifically viable practice. The design of such materials needs not only mathematical inculcations but also clinical comportment as it involves eye or vision (seeing through a medium), like depth of perception, vergence or visual disparity while seeing through the transparent medium. The proposed technique stands to fit in all these aspects to be a feasible alternate to the other methods in application to date.
6. Conclusion A simple method for the measurement of aberrations caused by an intervening medium in the line of sight has been proposed. The method, apart from being fast and cost effective, provides quantitative evaluation of the aberrations of seeing apertures of any required size. The method could be used for testing materials which are really useful in helmet visors, windshields or windscreens of a car or an aircraft, retail displays, glazing products, signage material, picture framing, museum quality applications, menu boards, architectural applications, artistic applications, transportation industries, flat screen display and sport facilities.
Acknowledgment This work was supported by grant no ERIP/ER/0204258/ M/01/683, DRDO, New Delhi, India.
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