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Fortin Vby pJohn Talbot *422530 Von  uHpFeb 9, 1993#  P7,P#у"0*0*0*4D"  Y0#Xw P7[hXP#  %8T#Xw P7[hXP# # P7zP#FRESNEL DIFFRACTION LAB#Xw P7[hXP#  X0 OBJECT  To make a quantitative study of fresnel diffraction at different obstacles and apertures.  x We have experimentally plotted the Iediffratio Ie diffraction Ie  Ien  pattern of the straight edge at various distances, and of wire obstacles of various widths and distances.  X 0 METHOD   xdThe beam from a 15 mW HeNe gas laser was Iefocusse Ie focused Ie  Ied  by a microscope objective, and spatially  xpfiltered by a pinhole. The undisturbed pattern several meters from the pinhole is TEM00 and almost exactly gaussian.  xVarious kinds of obstacles are placed between the detector and the pinhole, and a motorized  xscanning photodetector is swept across the detector plane. Its output is amplified and sent to a chart recorder.  xThe laser beam was aligned parallel to an optical track (Z-axis) and at the height of a travelling  x<photodetector. A microscope objective was placed near the output of the laser and was  xcollimated by using a small screen on the laser head to measure the weak reflection from the  xllens surfaces, by tilting and displacing the lens the 'diagnostic' spots could be Iecentere Ie centred Ie  Ied . A pinhole  x0was placed after the lens near the focal point, this produced a much brighter kind of spot on the  xLlaser head due to reflections from the metal substrate of the pinhole, the diameter of this spot is very sensitive to focal adjustments, the focal plane was precisely located by minimizing  xthe size of this spot. The pinhole itself was centred by adjusting the XY controls until the pattern  xat the detector was a bright bessel pattern. Finally, very small Ieadjustement Ie adjustments Ie  Ies  of the focal control  x`and the XY controls produced a correctly filtered gaussian beam. To reduce stray light the entire  xLoptical assembly is placed in an enclosure which has been painted flat black on most interior surfaces.  xA slit of width 0.2  0.1mm and height 2.2  0.2mm covers the photodetector, spatial detail  xHfiner than these dimensions are smoothed out. The detector scans along the X-axis and  xpositional accuracy is maintained by guiding the detector head along a machined screw using a  x synchronous motor. The output of the photodetector is amplified, then printed on a chart  x|recorder. To increase the long term temporal stability of the laser beam intensity, we allowed  xthe laser to warm up for at least 30 minutes. Even after this precaution, high frequency intensity  xoscillations of about 5 to 10 Hz where observed in the undisturbed laser beam, however the  x<output of the chart recorder does not suffer from this noise and probably filters out these higher  x,frequencies. The output was sensitive to vibrations caused by walking, so movement was restricted during the scans. "'0*0*0*'"  X0 OBSERVATIONS   x The beam is linearly polarized horizontally and temporally stable as expected from a brewster  x`window Melles Griot 05-LHP-171. The gaussian beam pattern is uniform and bright over a large  x field of view at the detector we found no need to set the source to obstacle distance greater than  xone meter as suggested by previous experiments using a narrower bessel beam. In fact a source  xto obstacle distance as little as 80 mm is possible without losing any detail in the pattern diffracted at larger angles.  xThe wavefront curvature near the pinhole changes greatly over small distances, the diffraction  x$pattern is highly sensitive to small obstacle displacements here. Combined with the uncertainty  xin the pinhole to obstacle distance, this can lead to some difficulty in matching the theoretical  xpwith the experimental pattern. In the fresnel approximation all objects are treated as purely 2  x0dimensional in the aperture plane, however near the pinhole the wire obstacle begins to display  x 3 dimensional effects such as specular reflection from the sides of the wire. This effect can seen  xdwhen visually inspecting the light reflected of the wire at very large angles such as 90 to the  xDoptic axis. The observer must remain in the XZ plane at Y=0 to see the effect since from a  x(geometrical optics standpoint this is the zone of ray reflection. There is also a faint, very low  x<spatial frequency sinusoidal pattern in this region, the pattern moves with the wire; as expected this effect is not predicted by theory.  X0 CALIBRATION   xThe chart recorder X-axis scale was calibrated by dividing the distance the photodetector head  xdtravels during a certain amount of time to obtain the scan rate, then dividing the chart rate by the scan rate.  s4$#d6X@8;z@# Distance = 290  0.5 mm Time = 14min 31.06sec  0.1 sec = 871.7  0.1 sec Error = SQR ( (0.5/290)^2 + (0.1/871.7)^2 ) = 0.001728 Scan Rate= Distance/Time = 0.3327 (1  0.0017) mm/sec = 0.3327  0.0006 mm/sec Chart Rate 1 = 100 mm/min= 1.667 mm/sec (As indicated on control knob) Chart Rate 2 = 250 mm/min= 4.167 mm/sec (As indicated on control knob) Chart Recorder Scale = Chart Rate/Scan Rate Chart Recorder Scale 1 = 5.0095 ( 1  0.0017 ) = 5.010  0.009 Chart Recorder Scale 2 = 12.5237 ( 1  0.0017 ) = 12.52  0.02  Y!0#Xw P7[hXP#  x\Depending on which chart speed used, positional units along the chart recorder X-axis are magnified by this factor.""0*0*0*$"  X0 GAUSSIAN OPTICS:  We used plot 1, at 100 mm/min to study the undisturbed laser pattern:  s4$#d6X@8;z@# Chart reading True Intensity Position Intensity Position Xc dXc Y dY X dX I dI mm mm mm Normalized Zero --- 15.0  1 ---- ------ Peak 2  4 148.0  1 0.0  5 1.000  0.017 Minimum -404  1 16.0  1 -81.0  5 0.008  0.001 Maximum +418  1 15.0  1 83.0  5 0.000  0.000 Gauss waist -265  1 33.0  1 -53.3  5 0.135  0.015 Gauss waist +257  1 33.0  1 50.9  5 0.135  0.015  Y 0#Xw P7[hXP#  s4 $#d6X@8;z@#X = ( Xc  dXc - 2  4 ) / ( 5.010  0.009 ) mm dX = X * SQR[ ((dXc+4)/Xc)^2 + (0.009/5.010)^2 ] mm I = ( Y  dY - 15.0  1) / ( 148.0  1 - 15.0  1) Normalized dI = I * SQR [ (2/133.0)^2 + ( (dY+1)/(Y-15) )^2 ] Normalized  Y0#Xw P7[hXP#  xWe measured at the detector the positions of the gaussian beam waists at 13.5% peak intensity  x and from an average of both positive and negative waists we can compute the beam waist at the focal point using the equations of gaussian optics: see ref.3  s4$#d6X@8;z@# W = (53.3  5mm + 50.9  5 mm) / 2 = 52.1  5/SQR(2) mm = 52.1  4.0 mm Average Beam waist at detector S = 337  5mm Position of Source on optical track scale P = 1809  1mm + 85  5mm Position of detector on optical track scale Z = P-S = 1557  11 mm Source to detector distance Wavelength = 632.82 nm Helium Neon laser W0 = Wavelength*Z/(PI*W) Beam waist at focal plane of lens W0 = 6.0198( 1  SQR [ (11/1557)^2 + (4/52.1)^2 ]) microns = 6.0  0.5 microns 3*W0 = 18  2 microns Focal diameter contain Ie in Ie g 99% of beam power P = 30  10 microns Measured pinhole width P = 25 microns Indicated by label on pinhole substrate  Y0#Xw P7[hXP#  xdThe laser spot at the focal point is smaller than the pinhole by a reasonable margin, leading to  x|the clean gaussian beam observed at the detector plane. If the pinhole were any smaller or not  xplaced in the focal plane, the truncated gaussian would produce a pattern similar to a bessel function outline. The Intensity profile of the gaussian beam is: I(R) = I0*EXP[-2*R^2/W^2] R^2 = X^2 + Y^2  x|If the detector is not at Y=0 when it scans the X-axis across the gaussian pattern, I(R) will not  xchange shape but just be multiplied by the constant EXP[-2*Y^2/W^2] as long as Y does not change. Thus Y-axis centering is not critical. "(0*0*0*,"  xWe sampled the undisturbed gaussian into a computer and compared it to a theoretical gaussian  xand the deviations were minimal. To compensate for the angular Iedependanc Ie dependence Ie  Iee  of the frequency  xdistribution of the undisturbed laser beam, we multiplied all theoretical computations by a gaussian outline of width 52.1  4.0 mm.  X0 DIFFRACTION PATTERNS  Circular Obstacles:  x|We mounted two ball bearings with superglue to Iemicrocop Ie microscope Ie  Iee  slides. The slides were placed in  xthe beam with the side with the ball facing away from the pinhole, a tilt of about 45$ to the optics axis was introduced to prevent internal reflections reaching the detector plane.  s4 $#d6X@8;z@# d = 3.0  0.5 mm Small spherical ball bearing d = 4.5  0.5 mm Large spherical ball bearing P = 34 - 26 = 8  1 cm Source to obstacle distance S = 181 + 30 + 26 - 34 = 203  1 cm Obstacle to screen distance  Yq0#Xw P7[hXP#  xXWidth of geometrical shadow of small ball bearing was 8cm as measured on the screen which agrees with the theoretical value of :  s4,$#d6X@8;z@# (d/P)*(S+P)= ( 3.0  0.5 / 80  10 ) * ( 203  1 + 8  1 ) = 7.9  0.2 cm.#Xw P7[hXP#  x The boundary of the geometrical shadow of both spheres resembles the pattern of a straight  xedge. When observing the small sphere, a poisson spot of less than 0.5mm was observed and  xtdisappeared for P less than 5 cm. When we replaced it by the large sphere the poisson spot  xLdisappeared for P less than 10 cm. This lower cutoff for P in both cases could indicate that at  x`that point the surface roughness of the spheres became comparable to the width of a fresnel zone  x<at the diameter of the sphere, thus destroying the symmetry. As we move further away from the pinhole the width of fresnel zones increases, and the poisson spot becomes visible. The theoretical width of the spot is R = SQR(2)*SQR [ Wavelength*S*(S+P)/2P] /(4*SQR(N)) N = Last fresnel zone intercepted by sphere -----------------------------------------------------------------------  X0 THEORY   xThe validity of the fresnel diffraction method is based on the expansion of the exponential in the  xFresnel Kirchhoff integral formula (ref7) which gives a quantitative criterion for wavefront curvature. let d1 = 0.080 m, d2 = 1.469 m, Width = 0.00011 m 0.5*( 1/d1 + 1/d2 )*Width^2 << Wavelength, for Franhofer approximation. 76nm << Wavelength,  xFor a wavelength of 632.82nm the wavefronts are almost plane waves but do have a certain  xamount of curvature, thus the Fresnel diffraction formula which is correct to second order should be quite accurate. "(0*0*0*0*"  X0 SIMULATION   xWe compared various methods of computing the fresnel integrals on a personal computer  xp(TESTFRES.BAS) and chose the method of ref.4 as the most accurate and numerically stable over all possible input parameters.  xIn the program FRESNEL.BAS we use this numerical me Ie thod to simulate diffraction patterns  xand print them to laser printer and to the screen. The programs has options to include the  xgaussian angular intensity modulation or to simulate the actual 2 dimensional visual laser pattern  xon the video screen using 64 increasing levels of luminosity, the similarities between the computed an the actual pattern are Ieremarquabl Ie remarkable Ie  Iee .  xThe program ANIMATE.BAS to compute time animations has not yet been fully converted to IBM format, however a sample of 5 animations was recorded on VHS tape.  xHWhen observing the computer simulated diffraction pattern of a slit, we increased the slit width  xparameter at fixed source to obstacle distance of 80mm, a pattern resembling the classical  x|franhoffer diffraction shrinks in extent and increases in intensity until the slit width is about  xd=0.05 mm (at P = 80 mm) then the pattern ceases to grow in intensity and begins to widen  xin synchronism with the geometrical shadow. When the width was increased beyond 1.4 mm the  xpattern began resembling the pattern of two straight edges at a similar source to obstacle distance.  xAs we move a straight edge further from the source or closer to the detector during a computer  xanimation, the spatial frequency at the detector increases, but the pattern remains almost identical in shape; theory also predicts this effect. Theoretical parameters used in the figures:  s47$#d6X@8;z@#Wavelength = 632.82 nm Width of gaussian at Screen = 52.1  4.0 mm Figure Obstacle Source to Obstacle to Width Comment number type obstacle Detector d mm mm mm 1 Wire 87.0 1469.0 0.110 Compare with Plot 1 2 Wire 87.0 1469.0 0.110 Compare with Plot 1 3 Edge 60.0 1469.0 0.000 Compare with Plot 2 4 Slit 88.0 1469.0 1.800 Visual comparison 5 Slit 88.0 1469.0 1.000 Visual comparison 6 Slit 88.0 1469.0 0.500 Visual comparison 7 Wire 88.0 1469.0 0.300 Visual comparison Video ( Theoretical parameters used in the video: ) Number 1 Wire 88.0 1469.0 0.10 to 0.90 2:00 min 2 Wire 88.0 1469.0 0.05 to 0.22 2:00 min 3 Wire 88.0 1469.0 0.80 to 1.70 2:00 Iemi Ie min Ie  Iem  4 Slit 88.0 1469.0 0.10 to 0.90 2:00 min 5 Slit 88.0 1469.0 0.80 to 1.70 2:00 min"C)0*0*0*,"  Y0 x0#Xw P7[hXP#On every printed figure the undisturbed laser intensity is dotted and a thick black bar indicates the boundaries of the geometrical shadow of the slit, wire or straight edge.  xHWhen we compare Fig 1 with Plot 1, the match is almost exact except for a minor intensity kink  xPin the node at position X = 17 mm. On Fig 2 we can clearly see that the pattern of nodes is qualitatively identical to plot 1, notice also how closely the horizontal scale matches Plot 1.  xFor position X > 10 mm, out of the geometrical shadow of the wire, the pattern resembles a  x4beat wave. Beat waves are normally encountered when two simple harmonic functions with  x8similar frequencies overlap producing a very low frequency amplitude modulation at the difference frequency.  xThe Babinet principle states that the optical amplitudes produced by the diffraction from two  xcomplementary apertures should sum to the Ieorgina Ie original Ie  Iel  undisturbed wavefront. From a visual  xcomparison of of the figures generated by the computer program it becomes obvious that for  xidentical parameters the diffraction patterns of complementary apertures (e.g. slit and wire) are  x$not as similar as we would expect from the Babinet principle. However once it realised that the  x$principle applies only to the vector sum of complex AMPLITUDES it becomes obvious that an  ximportant missing parameter in all the previous plots is the PHASE information, which is lost  xtwhen we take the square of the amplitude to obtain the irradiance. With the correct phase,  xconstructive and destructive interference between complementary patterns should compensate for the mismatch in visual intensities.  X0 CONCLUSION   x4 The theoretical patterns produced by the program quantitatively match the experimentally  xobtained data. The two Iedimensina Ie dimensional Ie  Iel  simulations of the diffraction pattern on video screen bear a remarkable similarity to the actual laser pattern observed visually.  XN0 APPENDIX : POLARIZATION   xT To observe the effect of polarization on the diffraction from a wire we rotate the plane of linear  xpolarization by 360$ at a rapid rate, and measure the differences at the photodetector using a chopper synchronized to the rotation rate.  x To Ieachiv Ie achieve Ie  Iee  this requires that the beam from a laser with a fixed sense of polarization be passed  xthrough a half-wave plate rotating at half the required angular frequency. However in practice  x$the unequal reflection coefficient of the half-wave plate for orthogonal polarizations introduces  xunwanted amplitude modulation at that angular frequency. To solve this problem we could in  x@a certain sense 'cut' the plate in half to produce two quarter-wave plates, the first of which is  x0stationary with its fast axis at 45$ to the incident plane of polarization. The light emerging from  x$the first quarter wave plate becomes circularly polarized, and is passed on to the second quarter  xwave plate which rotates at the required angular frequency. The reflection from the front surface  xXof the second plate is not amplitude modulated as the plate rotates since the light is circularly  xpolarized. The light emerging from the second plate becomes linearly polarized but rotating at"(0*0*0*0*"  xthe plate angular frequency. Internal reflections from the second surface of the second quarter  xHwave plate is not amplitude modulated either since the polarization has become linear Ieagiai Ie again Ie  Ien  and  xis fixed relative to the rotating frame of reference of the second plate. The net result of both  xquarter wave plates is to produce constant amplitude linear polarization rotating at the angular  x$frequency of the second plate. Quarter wave plates designed for 632.82 nm were not available,  xat present we attempting to substitute plates made for a shorter Iewavelengh Ie wavelength Ie  Iet  of 589 nm Computations on the quality of the output are currently in progress.  XH0 REFERENCES   Y 01.M.V.Klein, 'Optics' (1970) p.  Y 02.R.Boyer, E.Fortin, 'Intensity measurements in a fresnel diffraction  Y 0 ` ` Pattern', American Journal of Physics 40(1972)75  Y 03.'Introduction to Lasers and Masers' , Siegman.  Y 04.J.Boersma, 'Computation of Fresnel Integrals', Mathematics of  Y 0 ` ` Computation, 14(1960)380  Y05.M.Abramowitz and A.Stegun, 'Handbook of Mathematical Functions', p.321  Y06.W.H.Press et al, 'Numerical recipes in C' (1988) p.188  Yp07.G.R.Fowles, 'Introduction to Modern Optics', 1975. p.110