Inquisitive study of light has a pedigree of thousands of years and yet continues as a primary field of interest to this day. This seductive charm has called the curious down a path of thorough exploration that continues to interpret numerous areas of scientific and technical research. The readily observable phenomenon of diffraction offers a rich theoretical testing ground for insights into the nature of light.
Late in the 17th century, Francesco Grimaldi1 (1660) and Robert Hooke2 (1672) noticed colored fringes, or light bands, in the shadows cast by small apertures, or sharp straightedges, illuminated by a light whose source was near at hand. The remarkable variation of light and dark bands, and the surprising intrusion of light inside the geometrical shadow, demanded an explanation beyond 17th century understanding. Christiaan Huygens3(1690), in an attempt to bring theory into line with observation, proposed light to have properties of a speedy spherically expanding wave front made by the interference of a multitude of similarly expanding wavelets.4
From this conjecture, light is shown to exhibit wave-like properties through the existence of Grimaldi-Hooke fringe patterns cast by a razorblade of finite distance from the source. This special case is than generalized using the geometrical arguments of Augustin Fresnel5 (1788-1827). Rigorous mathematical derivation of scalar-wave diffraction theory is left for independent exploration.6 The Fresnel diffraction theory is then modeled by computer aided calculation, demonstrated by computer simulation, and experimentally validated by straightedge diffraction.
As we begin to explore Fresnel's work on the theoretical description of these fringes, it will prove useful to reproduce the phenomena that Grimaldi and Hooke observed so long ago with modern equipment and precision. In the production of Fresnel's near-field diffraction, we will require a monochromatic point source, a straightedge and a screen.
The experimental setup is quite simple and involves a small number of components. A 0.5-mW Helium-Neon laser7 serves as the monochromatic light source, a spatial filter intensifies the laser's coherent beam, a pinhole simulates the point source, a razorblade functions as straightedge and a CCD chip becomes the screen where results are recorded—see Figure 1.
The spatial filter serves as an added protection against out-of-phase light rays by focusing the beam on one small point, the pinhole, thus allowing only a small deviation in phase—see Figure 2. The beam is then diffracted by razorblade at a distance of a, 13.28-cm, from the pinhole point source and allowed to fall upon a CCD chip, acting as screen and photosensor, at a distance b, 17.90-cm, from the straightedge obstacle. Light intensity can now be measured by the one thousand twenty-four 25-µm × 2.5-mm elements of the photon-sensing diode array.8
The experimental measurement of fringe pattern intensity as a function of 1024 positions requires only the most basic of spectrometer functionality and software operation.9 Firstly, an unobstructed intensity reading was recorded in 1/20-sec and plotted verses position. Secondly, a portion of the beam is blocked by a razorblade and a second data set is gathered, recorded and plotted in likewise manner—see Figure 3 noticing the Gaussian distribution of intensity and the fringe pattern in Open. It is interesting to note that the intensity distribution results from beam dispersion and that the fringe pattern is caused by the diffraction of light through the pinhole aperture. Fortunately, we do not have to deal with these effects as we can normalize the razorblade data set by plotting the ratio of razorblade to open data10—see Figure 4. With these observations in mind, we now turn our attention to the development of a theory of diffraction.
Fresnel's treatment of Grimaldi-Hooke fringes, now called Fresnel or near-field diffraction, rests on Huygens' principle of secondary emission and interference. As mentioned in the introduction Huygens proposed that light rays propagate from a luminary source in all directions. He continued to suggest that any-given point on this spherical wave would form secondary wavelets, which would likewise spherically expand in all directions. The word wave implies periodicity in amplitude, therefore the concept of interference follows. On meeting, the mutual effect on two wave fronts would be to add their instantaneous amplitude; producing alternate areas of increased or decreased amplitude, according to the phase difference between the waves.
We turn attention to the derivation of an expression for the light wave
equations of position and motion by first looking at the wave’s basic properties
of wavelength, 
, position, amplitude, 
,
angular frequency, 
0, phase, 
, and propagation velocity, c. Wavelength, oscillation and velocity all
become, for our purposes, constants when a coherent monochromatic light
source is used. By energy conservation we know that the intensity, ||,
of any given point on expanding spherical wave fronts, drop off as the
inverse square of their distance from the source increases—even more if
we admit to energy lost to absorption. Phase difference arises when two
coherent waves travel different distances and change observed intensity
by the rule of wave interference.
To begin, we represent the light wave as a one-dimensional oscillating system with no tendency to disperse. In terms of a mechanical wave on a spring, as light was long believed to be, we can describe the restoring force as linear, or proportional, in
(1).
An equation of motion can quickly be derived using Newtonian mechanics as is shown as
(2).
The differential equation has a general solution of the form
(3)
and a particular solution of the form
(4).
Let us now consider the case when a spherically expanding wave front
encounters an opaque razorblade, AG, whose edge width is proportional to
monochromatic wavelength, , of source, C—see
Figure 5. The Grimaldi-Hooke fringes we are set out
to predict are seen on a large flat screen where an arbitrary point, P,
is defined to have an intensity, ||. By second order
approximation the difference between the shortest path, CP, and the diffracted
wave path, CA + AP, can be expressed as
(5)
where a and b are, as seen in Figure 5, CA and AB, and x is the distance from the geometrical shadow, BP. The path difference and phase difference are related by an inverse half wavelength so that
(6)
with v defined for simplicity as
(7).
Now we can write equation (4) as a differential representing the infinitesimally small intensity contribution from each point on a wave front as
(8).
As we are only interested in relative intensity, K’ can arbitrarily be set equal to 1. If we integrate over the entire wave front we will obtain an expression for the total amplitude at point any point P on the screen. Rewriting equation (7) by use of an Euler identity, we obtain
(9)
or, as an expression of intensity
(10)
with 
and 
being
Fresnel integrals defined as
(11)
and
(12).
These results are applicable to all near field diffraction situations and thus represent a comprehensive wave theory for diffraction.
The Fresnel integrals in equations (11) and (12) have no closed form and thus must be calculated in definite form. In the experimental process described in §II, this would require 1024 calculations for each integral. Hand calculation would take a large portion of one’s life; therefore, other methods were historically useful.
The integrals both converge11 to ±½ and can be plotted against each other to form an imaginary vs. real amplitude-phase plot—see Figure 6. This plot in the imaginary plane, called Cornu’s Spiral, proves to be less difficult to generate, as it only need be calculated once, therefore, tables were generated and used to create the plot. Amplitude could than be calculated by measuring the distances from infinity, (½,½), and fringe amplitude, as seen in Figure 7. The use of amplitude-phase diagrams could be applied to any other simple diffracting shape; although it was not always necessary, as in the case of opaque objects with circular symmetry.
Thanks to the emergence of desktop computers the complete intensity vs. position plot can be calculated in one step by raw computing power by use of equation (7) to calculate the wave front phase position and intensity equation (10). The results from this calculation were than plotted along side the experimental data set—see Figure 7.
By the manual variation of parameters, we were able to minimize the difference between theory and observation. Root mean square12 (RMS) was minimized, to 0.01398, by fitting the two horizontal and vertical parameters of geometrical shadow position, bin #600, and relative intensity variation. The division of RMS by the maximum amplitude, times 100, can give, because it is in percent form, the general feeling of the accuracy of the experiment and was calculated to be 1.029%.
The visual correlation in Figure 7 between Data and Theory is quite stunning, and provides observational confidence in the diffraction wave theory. This theoretical success gives computer programs the ability to predict and simulate Grimaldi-Hooke fringe patterns13—see Figure 8. The simulation process allows for the observation of diffraction patterns made by simple obstacles and is a good tool for the introduction to Fresnel diffraction.
We have now demonstrated that light seems to have wave-like properties, when an obstacle diffracts it, by the success of the theory’s ability to explain and accurately predict Grimaldi-Hooke fringe patterns. By the demonstration for Fresnel’s analytical derivation of the wave theory of diffraction, a general foundation has been set for rigorous mathematical and experimental exploration. The use of computer’s has greatly simplified the rather difficult evaluation of Fresnel’s integrals and has allowed for the simulated experimentation with various simple obstructions. Where once only the mathematically elite could go, curious people of all walks of life can now explore one of the most interesting areas of science.
Our appreciation goes to Dr. Kutzner for his supervision, guidance and help; to Dr. Lee and Duane Beardsley for their enhancement of the spatial filter/laser mounting adapter; and to the Andrews University Physics Department for giving us the opportunity to work with such high quality equipment.
