Question

Youngs double slit experiment derivation by complex numbers

Answer 1

Let a diverging beam of light with complex amplitude distribution
is incident on the diffracting aperture. According to Fresnel-Kirchhoff theory, diffracted field at observation point P0 in terms of incident wave field and its first derivatives at an arbitrary closed surface surrounding P0 
Where ∂/∂n denotes differentiation along the outward normal, Q is a point situated in the diffracting aperture Σ and exp(iks/s) is Green’s free space function, r is distance between source of light and a point Q on diffracting aperture and s is distance between aperture point Q and observation point P 0. Maggi and Rubinowicz converted double integrals used in above formulation into a line integral using Stoke’s theorem, giving

represents Young’s boundary diffraction wave generated at the edge of the aperture by it's interaction with incident light. Recently a quantitative criterion has been developed for classifying whether diffraction pattern is of Fresnel or Fraunhofer type giving

Fraunhofer rigion γ ≤ 0.8

Fresnel rigion γ > 0.

Using parameters of our experimental setup the slit-width (1X) = 20 micron, wavelength of laser (λ) = 632.8nm, r = 2mm, and s = 30mm value of γ = 0.0019. Thus, we receive Fraunhofer diffraction pattern at the detector's surface.

Using the method of stationary phase and some approximations the expression of boundary diffraction wave becomes
Here dl is an infinitesimal element situated on illuminated edge Γ of the diffracting aperture, λ is wavelength of light used and ϕ is polar coordinate. In case of single slit having width lx the diffracted field can be written as sum of two boundary diffraction waves arising from each edge of the slit, giving
This equation represents amplitude distribution at observation screen resulting due to single slit diffraction. Here effect of geometrical beam UG, which is small due to small size of slit, has been neglected. Here two boundary diffraction waves staring at each edge of the slit interfere to generate the single slit diffraction pattern having different diffraction orders. Diffracted light propagates along the direction of incident beam with an additional effect of diverging out symmetrically with respect to initial direction of propagation as given by Keller’s geometrical theory of diffraction . If slit width is small than boundary diffraction waves originating from two edges of the slit interfere to generate a single fringe, which forms the beam of light passing through the slit. This is also evident from our earlier discussion in a previous paper where it was demonstrated by using a Lloyd mirror on boundary diffraction wave from a knife-edge . When incident beams at individual slits of the double-slit system enclose large angle (in our case when slits are placed in proximity of focus), light striking at individual slits generates their own single slit diffraction pattern which propagates in different directions (i.e. beams from individual slits are spatially separated) and thus individual slit diffraction pattern can be observed. When angle between beams incident on slits decreases (this can be experimentally realized by moving slits away from focus) separation of light striking on individual slit also decreases and consequently diffracted patterns come closer to each other.

Answer 2

Answer:

Young Double Slits Experiment Derivation

The two waves interfering at P have covered different distances. Here, a and b are amplitudes of the two waves resp. Φ is the constant phase angle by which the second wave leads the first wave. The wave equation (4) represents the harmonic wave of amplitude R.