Fraunhofer Diffraction through a Rectangular Aperture

CAPTION

This Demonstration illustrates Fraunhofer optical diffraction through a rectangular aperture. The length and breadth determine the dimensions of the aperture in cm (length is along the horizontal axis and breadth along the vertical axis). The distance control specifies the distance between the aperture and the screen. Color is related to the wavelength of light used to illuminate the aperture. Violet corresponds to 400nm, green is around 500nm and orange is at 600nm. You can change the parameters to observe the variation in the intensity profile on the screen.

DETAILS

The Fraunhofer diffraction pattern is the Fourier transform of the aperture function. It is obtained by evaluating the Fraunhofer diffraction integral.

Consider a screen in the x-y plane. The rectangular aperture is in a plane parallel to it (the ξ-η plane). The perpendicular distance between the screen and aperture is given by z. The dimensions of the aperture are b×a.

Consider a monochromatic plane wave normally incident on the aperture. The field distribution at a point P(x,y,z) on the screen is given by the Fraunhofer diffraction integral:

E(P)=(A/(iλz))(e^(ikz))exp[((ik)/(2z))((x^2)+(y^2))](∫_(-b/2)^(b/2))(e^(-iuξ))dξ(∫_(-a/2)^(a/2))(e^(-ivη))dη,

where

u=((2πx)/(λz)),

v=((2πy)/(λz)),

and A is the amplitude of the incident plane wave in the plane of the aperture. The value of the integral turns out to be

E(P)=((Aba)/(iλz))(e^(ikz)) exp[((ik)/(2z))((x^2)+(y^2))]((sin β)/β)((sin γ)/γ),

where

β=((πbx)/(λz))

and

γ=((πay)/(λz)).

The intensity of a light wave is proportional to the square of the amplitude, hence the intensity distribution on the screen is given by

I(P)=(Subscript[I, 0])(((sin^2)β)/(β^2))(((sin^2)γ)/(γ^2)),

where (Subscript[I, 0]) is a constant. In this Demonstration, the intensity distribution has been plotted schematically (for simplicity, (Subscript[I, 0]) has been taken to be unity).

Snapshot 1: pattern obtained from a square aperture (0.1cm x 0.1cm) and violet light at a distance of 1000cm

Snapshot 2: pattern obtained from a square aperture (0.1cm x 0.1cm) and green light at a distance of 1000cm

Snapshot 3: pattern obtained from a square aperture (0.1cm x 0.1cm) and orange light at a distance of 1000cm

As the wavelength of the light increases, the width (the spacing between bright or dark spots) of the pattern also increases. This is seen clearly in the first three Snapshots.

Snapshots 4 and 5: the single slit limit is obtained when one of the dimensions is much larger in comparison to the other

Snapshot 6: if the dimension of the aperture decreases (0.05cm x 0.05cm in this case), then the width of the pattern increases, and vice versa; compare with Snapshot 1

References

[1] A. Ghatak, Optics, 5th ed., India: Tata McGraw-Hill Education Pvt. Ltd, 2012.

[2] Wikipedia. "Visible Spectrum." (Mar 14, 2018) en.wikipedia.org/wiki/Visible_spectrum.

THIS NOTEBOOK IS THE SOURCE CODE FROM

Radhika Prasad

"Fraunhofer Diffraction through a Rectangular Aperture"
 http://demonstrations.wolfram.com/FraunhoferDiffractionThroughARectangularAperture/
 Wolfram Demonstrations Project
 Published: May 2, 2018

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