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Diffraction from a Rectangular Aperture

Module by: Paul Padley. E-mail the author

Summary: Derive the expression for diffraction from a rectangular aperture.

We consider diffraction from apertures other than a slit. For example consider a rectangular aperture as shown below. If ɛ A ɛ A is the source strength per unit area (assumed to be constant over the entire area in this example) and d S = y z d S = y z is an infinitesmal area at a point in the aperture then we have:

d E = ɛ A r e i ( k r ω t ) y z d E = ɛ A r e i ( k r ω t ) y z

We see from the figure that r = [ x 2 + ( Y y ) 2 + ( Z z ) 2 ] r = [ x 2 + ( Y y ) 2 + ( Z z ) 2 ] and that R 2 = x 2 + Y 2 + Z 2 . R 2 = x 2 + Y 2 + Z 2 . Thus we use x 2 = R 2 Y 2 Z 2 x 2 = R 2 Y 2 Z 2 to write r = [ R 2 Y 2 Z 2 + Y 2 + y 2 2 y Y + Z 2 + z 2 2 z Z ] 1 / 2 r = [ R 2 Y 2 Z 2 + Y 2 + y 2 2 y Y + Z 2 + z 2 2 z Z ] 1 / 2 or r = [ R 2 2 Y y 2 Z z + y 2 + z 2 ] 1 / 2 r = [ R 2 2 Y y 2 Z z + y 2 + z 2 ] 1 / 2 r = R [ 1 2 Y y / R 2 2 Z z / R 2 + ( y 2 + z 2 ) / R 2 ] 1 / 2 . r = R [ 1 2 Y y / R 2 2 Z z / R 2 + ( y 2 + z 2 ) / R 2 ] 1 / 2 . We are only considering Fraunhofer diffraction so R , Z , Y R , Z , Y are much larger than y y and z z and we can rewrite r R [ 1 2 ( Y y + Z z ) / R 2 ] 1 / 2 r R [ 1 2 ( Y y + Z z ) / R 2 ] 1 / 2 and then finally expanding using the binomial theorem and taking only the most significant terms r R [ 1 ( Y y + Z z ) / R 2 ] . r R [ 1 ( Y y + Z z ) / R 2 ] .

So we have d E = ɛ A r e i ( k r ω t ) y z d E = ɛ A r e i ( k r ω t ) y z or using the fact that R R is large d E = ɛ A R e i ( k r ω t ) y z d E = ɛ A R e i ( k r ω t ) y z which we integrate to get the field

E = b / 2 + b / 2 a / 2 + a / 2 ɛ A R e i ( k r ω t ) z y = ɛ A e i ( k R ω t ) R a / 2 + a / 2 e i k y Y / R y b / 2 + b / 2 e i k z Z / R z = ɛ A e i ( k R ω t ) R [ e i k y Y / R i k Y / R | a / 2 + a / 2 [ e i k z Z / R i k Z / R | b / 2 + b / 2 = ɛ A e i ( k R ω t ) R [ e i k a Y / 2 R e i k a Y / 2 R i k Y / R ] [ e i k b Z / 2 R e i k b Z / 2 R i k Z / R ] E = b / 2 + b / 2 a / 2 + a / 2 ɛ A R e i ( k r ω t ) z y = ɛ A e i ( k R ω t ) R a / 2 + a / 2 e i k y Y / R y b / 2 + b / 2 e i k z Z / R z = ɛ A e i ( k R ω t ) R [ e i k y Y / R i k Y / R | a / 2 + a / 2 [ e i k z Z / R i k Z / R | b / 2 + b / 2 = ɛ A e i ( k R ω t ) R [ e i k a Y / 2 R e i k a Y / 2 R i k Y / R ] [ e i k b Z / 2 R e i k b Z / 2 R i k Z / R ]

Now lets define α = k a Y / 2 R α = k a Y / 2 R β = k b Z / 2 R β = k b Z / 2 R and β = k b Z / 2 R β = k b Z / 2 R and we see that

E = ɛ A e i ( k R ω t ) R [ e i α e i k α i k Y / R ] [ e i β e i β i k Z / R ] E = ɛ A e i ( k R ω t ) R [ e i α e i k α i k Y / R ] [ e i β e i β i k Z / R ]

or rearranging

E = ɛ A e i ( k R ω t ) R [ e i α e i α 2 i α / a ] [ e i β e i β 2 i β / b ] E = ɛ A e i ( k R ω t ) R [ e i α e i α 2 i α / a ] [ e i β e i β 2 i β / b ]

E = ɛ A e i ( k R ω t ) a b R [ e i α e i α 2 i α ] [ e i β e i β 2 i β ] E = ɛ A e i ( k R ω t ) a b R [ e i α e i α 2 i α ] [ e i β e i β 2 i β ]

E = ɛ A e i ( k R ω t ) a b R [ sin α α ] [ sin β β ] E = ɛ A e i ( k R ω t ) a b R [ sin α α ] [ sin β β ]

So finally we can write the intensity as

I ( Y , Z ) = I ( 0 , 0 ) [ sin α α ] 2 [ sin β β ] 2 I ( Y , Z ) = I ( 0 , 0 ) [ sin α α ] 2 [ sin β β ] 2

Below is a plot of [ sin β β ] 2 [ sin α α ] 2 [ sin β β ] 2 [ sin α α ] 2

Below is a plot of [ sin β β ] [ sin α α ] [ sin β β ] [ sin α α ]

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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