Wave Interference and Diffracction

Written by Lic. Renan Cabrera Lafuente
Supported by "Academia Nacional de Ciencias de Bolivia"

renanbo@hotmail.com
rencabla@ceibo.entelnet.bo
La Paz Bolivia -2000

<<  Graphics`Graphics`

N point interference.
a = separation length.
λ = wave length

\(Clear[a]
\)\(InterferenceIntensity[θ_, {N_, a_, λ_}] = \ 

      \((Sin[\(\(N\ π\ a\)\/λ\) Sin[θ]]/
            \((Sin[\(\(π\ a\)\/λ\) Sin[θ]])\))\)\^2\)

\(Plot[

      Evaluate[	
        InterferenceIntensity[θ, {2, a, λ}] /. \ 
          \(a\/λ\) Sin[θ] -> η\ ]

      	, {η, \(-2\), 2}, Frame -> True, 
      PlotLabel -> "Interference Pattern N=2", 
	RotateLabel -> False, 
      
	FrameLabel
         -> {\*""I/\!\(I\_0\)"", \ 
          \*"" \!\(a\/λ\)Sin[θ]  ""}, 
	
      Background -> GrayLevel[0.95]]\)

\(Plot[

      Evaluate[	
        InterferenceIntensity[θ, {4, a, λ}] /. \ 
          \(a\/λ\) Sin[θ] -> η\ ]

      	, {η, \(-2\), 2}, Frame -> True, 
	PlotRange -> {0, 16}, 
	
      PlotLabel -> "Interference Pattern N=4", 
	RotateLabel -> False, 
      
	FrameLabel
         -> {\*""I/\!\(I\_0\)"", \ 
          \*"" \!\(a\/λ\)Sin[θ]  ""}, 
	
      Background -> GrayLevel[0.95]]\)

\(PolarPlot[
	
      InterferenceIntensity[θ, {4, λ\/2, λ}]
, {
        θ, 0, 2  Pi}, 
      PlotLabel -> \*""I/\!\(I\_0\) Angular Distribution, N=4 ""]\)

Fraunhofer Diffraction

Fraunhofer Diffraction for Thin Slits

For one slit.
L = slit length
λ = wave length

FrDifraction[θ_, {λ_, L_}] =  
      (((Sin[((2  π L)/(2 λ)) Sin[θ]]/
            (((2  π L)/(2  λ)) Sin[θ])))^2)

\(Plot[

      Evaluate[FrDifraction[θ, {λ, L}] /. \ 
          L\  -> 3  λ\ ]
, {θ, \(-π\)/2, π/2}, 
	
      PlotRange -> {0, 1}, 
	
      PlotLabel -> "One slit Difraction Pattern L = 3λ ", 
	
      Frame -> True, 
      FrameLabel -> {"θ", \*""I/\!\(I\_0\)""}, 	
      RotateLabel -> False]\)

N Slit Difraction Pattern.

N = Number of slits
λ = wave length
L = Slit Length
a = Slit separation

\(FrDifractionNSlit[θ, {N, λ, L, a}] = 

      FrDifraction[θ, {λ, L}] 
        InterferenceIntensity[θ, {N, a, λ}]\)

\(Plot[

	
      Evaluate[

        		{FrDifractionNSlit[θ, {N, λ, L, a}]
			, 
            FrDifraction[θ, {λ, L}]} /. 

          			\ {N -> 2, L -> 2  λ, a -> 14  λ}\ \ \ ]

      
, {θ, \(-π\)/2, π/2}, PlotRange -> {0, 4}, 
      Frame -> True	]\)

In the above plot we ploted the 2 Slit Difraction Pattern with the only one difraction pattern to show that the Difracction Pattern modulates the Interference pattern.

Fraunhofer Diffraction for Rectangular Apertures

For one rectangular aperture.
Lz = z side aperture length 
Ly = y side aperture length 

λ = wave length

R = Screen - APerture length

z  =  z  screen coordinate
y  =  y screen coordinate

\(FrDifractionRectangular[{y_, z_}, {Ly_, Lz_, R_, λ_}]\  = 
	
      \((Sin[\(\(2  π\ Ly\)\/λ\) y\/\(2  R\)]/
              \((\(\(2  π\ Ly\)\/λ\) y\/\(2  R\))\))\)\^2	
        \((Sin[\(\(2  π\ Lz\)\/λ\) z\/\(2  R\)]/
              \((\(\(2  π\ Lz\)\/λ\) z\/\(2  R\))\))\)\^2\)

\(DensityPlot[

      Evaluate[FrDifractionRectangular[{y, z}, {Ly, Lz, R, λ}] /. {
            Ly -> 40  λ, Lz -> 40  λ, R -> 100}]

      	, {y, \(-10\), 10}, {z, \(-10\), 10}, Mesh -> False, PlotPoints -> 50]
      \)

\(Plot3D[

	
      Evaluate[
		
        FrDifractionRectangular[{y, z}, {Ly, Lz, R, λ}] /. {
            Ly -> 40  λ, Lz -> 40  λ, R -> 100}]


      	, {y, \(-10\), 10}, {z, \(-10\), 10}, Mesh -> False, 
	
      PlotRange -> {0, 1}, PlotPoints -> 50, BoxRatios -> {1, 1, 1.5}]\)

Fraunhofer Diffraction for Circular Apertures

For one circular aperture
	
ρ = radius aperture 

λ = wave length

R = Screen - Aperture length

r = screen radius coordinate

\(\(FrDifractionCirculat[\ r_\ , {ρ_, R_\ , λ_}]\  = \ 

      \((2  BesselJ[1\ , \ \(2  π\)\/λ\ ρ\/R\ r]
                \/\(\(2  π\)\/λ\ ρ\/R\ r\)\ )\)^2; \)\)

\(DensityPlot[

      Evaluate[
	
        \(FrDifractionCirculat[\ r\ , {ρ, R\ , λ}] /. 
            \ {ρ -> λ, R -> 10}\)
	 /. {r -> Sqrt[x^2 + y^2]}

        	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
, {x, \(-10\), 10\ }, {y, \(-10\), 
        10}, Mesh -> False, PlotPoints -> 50]\)

\(Plot3D[

      Evaluate[
	
        \(FrDifractionCirculat[\ r\ , {ρ, R\ , λ}] /. 
            \ {ρ -> λ, R -> 10}\)
	 /. {r -> Sqrt[x^2 + y^2]}

        	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
, {x, \(-15\), 15\ }, {y, \(-15\), 
        15}, Mesh -> False, PlotPoints -> 50, PlotRange -> {0, 1}, 
	
      BoxRatios -> {1, 1, 2}]\)

Fresnel Diffraction

Fresnel Diffraction for Rectangular Aperture

We define the Cornu function

\(\(MyFresnelCos\  = \ 
      Interpolation[Table[{x, FresnelC[x]}, {x, \(-6\), 6, 0.04}]]; 


    MyFresnelSin\  = \ 
      Interpolation[Table[{x, FresnelS[x]}, {x, \(-6\), 6, 0.04}]]; 

\)\)

Cornu[x_] := {MyFresnelCos[x], MyFresnelSin[x]}

\(ParametricPlot[
	\ Cornu[x], 
	{x, \(-6\), 6}\ , 
	Frame -> True, 
      GridLines -> Automatic]\)

Now we define the difference cuadratic of Cornu function for a 2D aperure

\(\(FresnelE2D[{u1_, u0_}, {v1_, v0_}] := \ 
      \((\((MyFresnelCos[u1] - \ MyFresnelCos[u0])\)\^2 + 
            \((MyFresnelSin[u1] - \ MyFresnelSin[u0])\)\^2)\)*\ 
		
        \((\((MyFresnelCos[v1] - \ MyFresnelCos[v0])\)\^2 + 
              \((MyFresnelSin[v1] - \ MyFresnelSin[v0])\)\^2)\)/4; \)\)

The Difracction function is then:

R =  Sreen - Aperture length
ρ =  Sreen - Source length
λ = wave length

Lx1 - Lx0  = aperture length side x
Ly1 - Ly0  = aperture length side y

\(\(\(FresnelDifraction2D[{x1_, x0_}, {y1_, y0_}, 

        	{{Lx1_, Lx0_}, {Ly1_, Ly0_}}, {R_, ρ_, λ_}, n_] := 
	
      Block[{}, 
		
        a\  = \ \@\(\(2 \((ρ + R)\)\)\/\(λ\ ρ\ R\)\); 
		
        ax1\  = \ \(a*Lx1 // PowerExpand\) // N; 
		
        ax0\  = \ \(a*Lx0 // PowerExpand\) // N; 
\ \ \ \ 
        ay1\  = \ \(a*Ly1 // PowerExpand\) // N; 
\ \ 	
        ay0\  = \ \(a*Lx0 // PowerExpand\) // N; 	
	

        Table[
			\ 	\ 
          FresnelE2D[{ax1 - x, ax0 - x}, {ay1 - y, ay0 - y}]\ \ \ \ , {x, x0, 
            x1, \((x1 - x0)\)/n}, {y, y0, y1, \((y1 - y0)\)/n}\ ]
			

        		
		]; \)

\)\)

And we can calculate the  I/(Subscript[I, 0]) value at the axes center point

We can calculate the Difraction pattern changing the coordinate system by parametric offset.

Lets see a particular case

\(\(
\(MF2\  = 
      FresnelDifraction2D[{3, \(-3\)}, {3, \(-3\)}, 

          	{{Lx1, Lx0}, {Ly1, Ly0}}, {R, ρ, λ}, 41] /. 

        	{Lx1 -> 3  λ, Lx0 -> \(-3\) λ, Ly1 -> 3  λ, 
          Ly0 -> \(-3\) λ, R -> 5  λ, 
          ρ -> 30  λ}; \)\)\)

ListPlot3D[MF2, PlotRange -> {0, 3}]

Now we can increase the screen distance and verify its aproach to the Fraunhofer Difraction

\(\(
\(MF\  = 
      FresnelDifraction2D[{3, \(-3\)}, {3, \(-3\)}, 

          	{{Lx1, Lx0}, {Ly1, Ly0}}, {R, ρ, λ}, 31] /. 

        	{Lx1 -> 3  λ, Lx0 -> \(-3\) λ, Ly1 -> 3  λ, 
          Ly0 -> \(-3\) λ, R -> 30  λ, 
          ρ -> 30  λ}; \)\)\)

ListPlot3D[MF, PlotRange -> {0, 3}]

Finaly we are going to see the irradiance profile in one dimension for pass from ligth to darkness.

\(\(FresnelE1D[u1_] := \ 
      \((\((MyFresnelCos[u1] - \ 1/2)\)\^2 + 
            \((MyFresnelSin[u1] - \ 1/2)\)\^2)\)/2
	\)\)

\(Plot[FresnelE1D[u1], {u1, \(-6\), 6}, 
	Frame -> True, 
      PlotRange -> {0, 1.5}, GridLines -> Automatic]\)