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266ARTICLE IN PRESSM. Fernández Guasti et al. / Optik 116 (2005) 265–269record the diffraction patterns are also described in thissection. In Section 3, the experimental layout of one ormore pixels produced by a LCD mask is presented. InSection 4, the experimental results are exposed and theirmain features discussed. The shape of the LCD pixel isinferred from its diffraction pattern through comparisonwith the computer-generated apertures. The LCDaperture used in a PDI, rather than the ideal circularsymmetry, has the shape of a square with blunt corners.The conclusions are drawn in Section 5.2. Photographic masksIntermediate figures between a circle and a squaremay be described analytically through a single equationrestricted to the appropriate domain [6]. The aperturesproduced with this algorithm yield figures that resemblesquares with blunt corners. These silhouettes may beadequate for modeling the shape of pixels produced bydiverse digital imaging systems. This assertion will, infact, be supported by the results presented hereafter.The photographic diffracting apertures employed in thiswork were generated from the squircle equation, that is,the equation for a square/circular aperture. The polarequation for a squircle form is given bys 2 4 sin2 ð2yÞr 4 k 2 r 2 þ k 4 ¼ 0, (1)where ðr; yÞ are the polar coordinates at the apertureplane and ‘‘s’’ is the squareness parameter [6]; s ¼ 1represents a square with sides equal to2k; while s ¼ 0represents a circle of radius k. The equation for thesquare is perhaps more clearly visualized in Cartesiancoordinates with the substitution 4x 2 y 2 ¼ r 4 sin 2 ð2yÞ:The Cartesian equation for a squircle iss 2 x2 y 2 x 2k 2 k 2 k 2 þ y2k 2þ 1 ¼ 0. (2)This expression may be factored for s ¼ 1 into ð1x 2 =k 2 Þð1 y 2 =k 2 Þ¼0; which yields a square when therestriction x 2 ; y 2 pk 2 is imposed. The parameter ‘‘s’’ is agood measure of how sharp or blunt are the corners ofthe square. This parameter may be continuously variedto smoothly change the shape of the contour from acircle toa square. The change in the aperture shape ismore sensitive when ‘‘s’’ is close to one. A typicalcomputer-generated aperture plot for s ¼ 0:9 is shown inFig. 1. The area lying outside the curve was rendered inblack in order to produce the aperture mask. The plotsof the squircle equation for various squareness valueswere recorded in high-contrast photographic film reducingtheir size toabout 1 mm in diameter, that is k ¼0:5mm: The upper left-hand quadrant of the diffractingapertures was photographed with an optical microscopein order to monitor the actual shape of the apertures.Fig. 1. Diffracting aperture plot for squareness parameter s ¼0:9:The form of the aperture in the other three quadrants issymmetrical with respect tothe quadrant shown in themicrophotographs.The apertures were illuminated directly with a HeNelaser placed 2 m apart from the aperture plane. Thediffraction patterns were recorded on high-resolutioncolor film located 3 m apart from the diffractingapertures. This distance was appropriate in order toconsider the pattern tobe in the far field and alsotofillreasonably well the 35 mm film that was used tophotograph these patterns. This simple arrangementwas favored in order to avoid contributions fromintermediate optical elements.3. Experimental setupThe experimental setup used toobtain the images anddiffraction patterns of the LCD pixels is shown in Fig. 2.The display is a monochrome active matrix twistednematic crystal consisting of an array of 320 240pixels with maximum contrast ratio of 80 to 1 (Kopin,CyberDisplay 320). The pixel dimensions given in thespecifications is 15 mm 15 mm and the separationbetween pixels is 25 mm: This display was controlledwith a personal computer. The screen resolution wasadjusted so that individual pixels could be turned on(full transparency) or off (fully darkened). The displaywas mounted on an xyz mount and placed in the vicinityof the beam waist where it is commonly positioned in aPDI scheme.Images of the open pixels were taken using a travelingmicroscope and white light illumination. These imageswere used to confirm the position and the number ofactive pixels. The diffraction patterns of the pixels weretaken under HeNe laser illumination. The laser beamwas spatially filtered and collimated following the usuallayout of a PDI. The image was detected with a CCD
ARTICLE IN PRESSM. Fernández Guasti et al. / Optik 116 (2005) 265–269 267Fig. 2. Experimental setup. LCD is the liquid crystal displaydiffracting aperture. The diffraction pattern is observed via acharged-coupled device (CCD). The microscope and the whitelight source (WLS) are used in order to observe the pixelsdirectly but are removed when the diffraction pattern isrecorded.placed instead of the microscope 50 cm after the LCDplane. A second computer was used to acquire the CCDimage.4. ResultsThe computer-generated aperture shapes togetherwith their corresponding diffraction patterns are shownin Figs. 3a–e. The microscope photographs of theapertures reveal that their shape is not as smooth asthe original plots. This raggedness is likely to be due tothe grain of the photographic film. The larger the size ofthe apertures the smaller is the relative effect of the filmgrain; however, the diffraction patterns become correspondinglysmaller and a compromise has to beestablished.The diffraction patterns were photographed usingdifferent exposure times. Short times reveal the intensitydistribution of the central lobe but higher orders are notdetected whereas longer times saturate the central lobewhile higher orders are recorded. Diffraction patternsare usually difficult tophotograph due tothe large rangeof intensities that need to be recorded. A compromise isrequired in order to observe the desired features in theregion where the film exhibits a linear response.Digitizing the images is not of great help since it isdifficult tomaintain linear relationships between theneutral density of the film and the incident intensity.The sequence in Figs. 3a–e shows how the initialSincðxÞSincðyÞ diffraction pattern of a square is modifiedin various steps of the squareness parameter. Thesecond-order maxima in each figure illustrate mostclearly the evolution of the sequence. The four lobesaround the central maximum obtained for the squareaperture, merge into one another in the subsequentfigures until one has a squarish ring with uneven butfinite intensity for all angles as may be seen in Fig. 3c.Fig. 3. (a) s ¼ 1:0 – on the left a microphotograph of the upperleft-hand corner of the aperture is shown. The right-hand sideshows a photograph of the far-field diffraction patternproduced by the aperture; (b) s ¼ 0:995 – a small departureof the squareness parameter already reveals an appreciableroundness of the corner of the square; (c) s ¼ 0:9 – thesecondary maxima in the diffraction pattern begin to spreadout and commence to ‘‘touch’’ each other; (d) s ¼ 0:8 – thesecondary maximum is clearly a ring although it still exhibitslobes reminiscent of the square aperture and (e) s ¼ 0:001 – theAiry pattern corresponding to a circle is obtained. Thesecondary maximum is a ring with circular symmetry.The shape of the rings come closer to a circular contouras the aperture approaches the circular form. Simultaneously,the intensity distribution as a function of anglebecomes more smooth. Airy rings are obtained in Fig. 3ethat correspond to the diffraction pattern of a circle.These features are in qualitative accordance with thenumerical estimates obtained from the Fourier Besselintegrals [1]. Fig. 4 shows a representative contour plotand mesh that evaluate the Fourier Bessel transform ofthe wave amplitude in the far-field limit.It should be noted that higher-order rings show anunequal intensity as a function of the angle even for thecircular aperture. This irregular behavior is presumablydue tothe raggedness of the aperture. Higher spatial
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