A slit of width 0.2 ± 0.1mm and height 2.2 ± 0.2mm covers the photodetector, spatial detail finer than these dimensions are smoothed out. The detector scans along the X-axis and positional accuracy is maintained by guiding the detector head along a machined screw using a synchronous motor. The output of the photodetector is amplified, then printed on a chart recorder. To increase the long term temporal stability of the laser beam intensity, we allowed the laser to warm up for at least 30 minutes. Even after this precaution, high frequency intensity oscillations of about 5 to 10 Hz where observed in the undisturbed laser beam, however the output of the chart recorder does not suffer from this noise and probably filters out these higher frequencies. The output was sensitive to vibrations caused by walking, so movement was restricted during the scans.
The wavefront curvature near the pinhole changes greatly over small distances, the diffraction pattern is highly sensitive to small obstacle displacements here. Combined with the uncertainty in the pinhole to obstacle distance, this can lead to some difficulty in matching the theoretical with the experimental pattern. In the fresnel approximation all objects are treated as purely 2 dimensional in the aperture plane, however near the pinhole the wire obstacle begins to display 3 dimensional effects such as specular reflection from the sides of the wire. This effect can seen when visually inspecting the light reflected of the wire at very large angles such as 90° to the optic axis. The observer must remain in the XZ plane at Y=0 to see the effect since from a geometrical optics standpoint this is the zone of ray reflection. There is also a faint, very low spatial frequency sinusoidal pattern in this region, the pattern moves with the wire; as expected this effect is not predicted by theory.
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
Depending on which chart speed used, positional units along the chart
recorder X-axis are magnified by this factor.
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
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
We measured at the detector the positions of the gaussian beam waists at
13.5 percent peak intensity 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
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 containg 99 percent of beam power
P = 30 ± 10 microns Measured pinhole width
P = 25 microns Indicated by label on pinhole substrate
The laser spot at the focal point is smaller than the pinhole by a
reasonable margin, leading to the clean gaussian beam observed at the
detector plane. If the pinhole were any smaller or not placed 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^2If the detector is not at Y=0 when it scans the X-axis across the gaussian pattern, I(R) will not change 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.
We sampled the undisturbed gaussian into a computer and compared it to a theoretical gaussian and the deviations were mimimal. To compensate for the angular dependance of the frequency distribution of the undisturbed laser beam, we multiplied all theoretical computations by a gaussian outline of width 52.1 ± 4.0 mm.
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 distanceWidth of geometrical shadow of small ball bearing was 8cm as measured on the screen which agrees with the theoretical value of : (d/P)*(S+P)= ( 3.0 ± 0.5 / 80 ± 10 ) * ( 203 ± 1 + 8 ± 1 ) = 7.9 ± 0.2 cm The boundary of the geometrical shadow of both spheres resembles the pattern of a straight edge. When observing the small sphere, a poisson spot of less than 0.5mm was observed and disappeared for P less than 5 cm. When we replaced it by the large sphere the poisson spot disappeared for P less than 10 cm. This lower cutoff for P in both cases could indicate that at that point the surface roughness of the spheres became comparable to the width of a fresnel zone 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
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,
For a wavelength of 632.82 nm the wavefronts are almost plane waves but
do have a certain amount of curvature, thus the Fresnel diffraction
formula which is correct to second order should be quite accurate.
In the program DIFFRACT.BAS we use this numerical me thod to simulate diffraction patterns and print them to laser printer and to the screen. The programs has options to include the gaussian angular intensity modulation or to simulate the actual 2 dimensional visual laser pattern on the video screen using 64 increasing levels of luminosity, the similarities between the computed an the actual pattern are remarquable.
The 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.
When observing the computer simulated diffraction pattern of a slit, we increased the slit width parameter at fixed source to obstacle distance of 80mm, a pattern resembling the classical franhoffer diffraction shrinks in extent and increases in intensity until the slit width is about d=0.05 mm (at P = 80 mm) then the pattern ceases to grow in intensity and begins to widen in synchronism with the geometrical shadow. When the width was increased beyond 1.4 mm the pattern began resembling the pattern of two straight edges at a similar source to obstacle distance.
As we move a straight edge further from the source or closer to the detector during a computer animation, the spatial frequency at the detector increases, but the pattern remains almost identical in shape; theory also predicts this effect.
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 mim
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
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.When we compare Fig 1 with Plot 1, the match is almost exact except for a minor intensity kink in 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.
For position X > 10 mm, out of the geometrical shadow of the wire, the pattern resembles a beat wave. Beat waves are normally encountered when two simple harmonic functions with similar frequencies overlap producing a very low frequency amplitude modulation at the difference frequency.
The Babinet principle states that the optical amplitudes produced by the diffraction from two complementary apertures should sum to the orginal undisturbed wavefront. From a visual comparison of of the figures generated by the computer program it becomes obvious that for identical parameters the diffraction patterns of complementary apertures (e.g. slit and wire) are not as similar as we would expect from the Babinet principle. However once it realised that the principle applies only to the vector sum of complex AMPLITUDES it becomes obvious that an important missing parameter in all the previous plots is the PHASE information, which is lost when we take the square of the amplitude to obtain the irradiance. With the correct phase, constructive and destructive interference between complementary patterns should compensate for the mismatch in visual intensities.