FRESNEL DIFFRACTION DERIVATION THAT est. the wave theory of light
Fresnel derives a diffraction intensity equation by summating the interfering light waves' amplitudes, at the diffraction screen, using a line integral (equ 1).
"Hence the intensity of the vibration at P resulting from all these small disturbances is
{ [ ʃ dz cos (π z2 (a + b) / abλ) ]2 + [ ʃ dz sin (π z2 (a + b) / abλ)]2 }1/2 "..................................1
(Fresnel, 53). Fresnel's derivation of the diffraction intensity equation uses a length integral where dz represents a segment of the wave AMI. Fresnel uses the length integral to summate the interfering light waves' amplitudes at the diffraction screen point P (fig 7) but the point P is not within the limits of Fresnel's length integral (equ 1). Also, the crests and nodes of Fresnel's light waves propagate in the forward direction. At the diffraction screen, the propagating light waves' amplitudes would result in the spherical waves' amplitudes to oscillate forming an average resultant amplitude of zero that would eliminate the diffraction pattern.
Fresnel is using spherical waves formed along the wave AMI to derive the diffraction intensity equation. The interfering spherical waves' amplitudes are dependent on the inverse of the distance that intensity formed by the spherical waves is dependent on I = (U)2 where U is the equation of the spherical waves U = A cos(kr)/r. Using the distance r1 = .1 mm where the spherical waves are formed near the wave AMI, and, the distance of r2 = 5 cm that represents the distance from the wave AMI to the diffraction screen. The average total light intensity formed by the spherical waves just after leaving the wave AMI is I = [A cos(kr)/r]2 = B/(.0001)2 = B(108), and the average total intensity at the diffraction screen is I = [A cos(kr)/r]2 = B/(.05)2 = B(2.5 x 104). The total light intensity just after leaving the wave AMI decreases by the factor of 4,000, at the diffraction screen, using Fresnel's spherical wave diffraction mechanism.
Using an analogy, the formation of the small rectangular aperture diffraction pattern (fig 8) is represented using wave interference but the destructive interference of the light waves' amplitudes (energy) used to form the dark fringes of the diffraction pattern represents the destruction of light intensity that would result in a reduction in the total light intensity of the diffraction pattern since the destroyed light waves' amplitudes do not contribute to the total light intensity of the diffraction pattern. Every point of the diffraction pattern is formed by the same number of interfering of light waves. More than 80% of the small rectangular aperture diffraction pattern is composed of dark areas which would result in more than a 60% reduction in the total light intensity of the diffraction pattern yet experimentally, the total light intensity that enters a small rectangular aperture (dt = 1s) is equal to the total light intensity of the diffraction pattern.
Does everyone understand the difficulties regarding the wave theory of light if the foundation derivation is flawed?
