All waves diffract if they pass  through or around obstacles and interfere, if  two or more waves arrive at the same place at the same time.  When a monochromatic plane wave passes through a single slit of the right width w, we observe a Fraunhofer single slit diffraction pattern a large distance L >> w away from the slit.  When the wave passes through multiple regularly-spaced slits with slit-spacing d, we observe a multiple-slit Fraunhofer interference pattern a large distance L >> d away from the slits.

In this lab you will use Excel to produce some plots of diffraction and interference patterns for electromagnetic waves.

Open a Microsoft Word document to keep a log of your  procedures.  This log will become your lab report.  Address the points highlighted in blue.  Answer all questions.

Single slit diffraction

Dark fringes in the diffraction pattern of a single slit are found at angles θ for which w sinθ = mλ,  where m is an integer, m = 1, 2, 3, ...  .

Details:

 Assume light from a distant source passes through a narrow slit as shown on the figure on the right.  Let the polarization be perpendicular to the plane of the figure.  What do we observe on a distant screen? According to the Huygens-Fresnel principle, the total field at a point y on the screen is the superposition of wave fields from an infinite number of point sources in the aperture region.  Each point s on the wave front inside the aperture (–a/2 ≤ s ≤ a/2) is the source of a spherical wave.  A distance r from the point s the electric field is due to this point sources is  dE = (Asds/r)cos(kr - ωt). If r0 is the distance from the point s = 0 on the optical axis to a point y on the screen, then the contribution dE to the total amplitude on the screen from the point at s = 0 is dE(y) = (Asds/r0)cos(kr0 - ωt).  Here As/r0 is the amplitude per unit width and ds is the infinitely small width of a point source.  For off-axis points for which s ≠ 0, the distance is longer or shorter than r0 by an amount Δ. The contribution dE(y) to the total amplitude on the screen from an off-axis point (s ≠ 0) is dE(y) = (Asds/(r0 + Δ(s))) cos(k(r0 + Δ(s)) - ωt). To find the total amplitude E(y) we have to add up the contributions from all points on the aperture.  Because there are an infinite number of points, the sum becomes an integral. E(y) = ∫-a/2+a/2(A/(r0 + Δ(s)))cos(k(r0 + Δ(s)) - ωt)ds. We define sinθ = Δ/s.  Since r0 >> Δ, we approximate 1/(r0+Δ) with 1/r0.  However we cannot drop the Δ inside the cosine function, since kΔ(s) is not necessarily much smaller than 2π.  We then have E(y) = (As/r0)∫-a/2+a/2cos[(ksinθ)s + (kr0 - ωt)]ds. Using ∫cos(ax + b)dx = (1/a)sin(ax +b) the integration yields E(y) = (As/r0)cos(kr0 - ωt)(sin(ka(sinθ)/2)/(ka(sinθ)/2) . or, inserting k = 2π/λ, E(y) = (As/r0)cos(kr0 - ωt)(sin(πa(sinθ)/λ)/(πa(sinθ)/λ).
The function sin(x)/x = sinc(x) is called the sinc function.
 In Excel, produce a plot of sin(x)/x vs x for -4π < x < 4π.   (Divide the region from -4π < x < 4π into ~ 200 data points.  You can use this spreadsheet to get started.)   Label the axes and past the plot into a Word document.
The intensity is proportional to the square of the field,  I(y) E2(y).  Since the square of a cosine function averages to ½, the time-averaged intensity is given by

<I(y)> = <I0>sin2(πa(sinθ)/λ)/(πa(sinθ)/λ)2,

where <I0> is the average intensity at the center.
The time-averaged intensity has a peak in the center with smaller fringes on the sides.  For small angles we may approximate sinθ ~ θ.  Then the first zeros on the sides of the central peak occur when πasinθ/λ ~ πaθ/λ = π, or θ = λ/a.

 In Excel, produce a plot of / = sin2(πa(sinθ)/λ)/(πa(sinθ)/λ)2 versus πa(sinθ)/λ,  for  -4π < πa(sinθ)/λ < 4π, versus πa(sinθ)/λ. If you set x =  πa(sinθ)/λ, the produce a plot of / = sin2(x)/x2 versus x,  for  -4π < x < 4π, versus x.  Label the x-axis with πa(sinθ)/λ instead of x.  Paste the plot into a Word document.Describe the main feature of the Fraunhofer diffraction pattern.  If λ = 500 nm, and you want to observe the diffraction patter you have plotted have a width of ~ ±2cm a distance L = 1 m away from the slits, what is a reasonable slit width?

Multiple slit interference

Calculate the intensity distribution of the interference pattern for up to four equally-spaced sources.  Assume light shines on a series of equally spaced slits.  The spacing between the slits is d.  The diffraction pattern is observed on a screen a distance L away from the slits, L >> d.

If we view the slits as sources of electromagnetic waves, then these sources are coherent,
the electric fields E(x,t) = Emaxcos(kx - ωt + φ)  of all the sources are in phase.
But if we observe the diffraction pattern on the screen a distance z away from the x-axis so that z/L = tanθ, then the electric field of source n is out of phase with the electric field of source 1 by (n - 1)δ,  where

δ = k d sinθ = (2πd/λ) sinθ.

The total electric field at z is the sum of the fields due to all of the sources.  The intensity at z is proportional to the square of the amplitude of the resultant field.  The resultant field at z is given by

,

where α is the phase of the electric field of source one at position z on the screen and N is the number of sources.

The amplitude of this field at z is given by

The intensity distribution as a function of δ is given by (Eres/Emax)2.

#### Procedure:

Let column A contain the phase shift δ, from -14 to +14 ins steps of 0.1, starting in row 3.
Let columns B, C, D, and E contain a*cos(0), b*cos(δ), c*cos(2δ), and d*cos(3δ), respectively, starting in row 3.
 The coefficients a, b, c, and d in row 2 are used to turn a source on or off.  If a coefficient is 1 the source is turned on and if it is zero a source is turned off.  Start with a = b = 1, c = d= 0.
Let column F contain the sum of columns B through E, starting in row 3.
Let columns G, H, I, and J contain a*sin(0), b*sin(δ), c*sin(2δ), and d*sin(3δ), respectively, starting in row 3.
Let column K contain the sum of columns G through J, starting in row 3.
Let column L contain the square of column F plus the square of column K, starting in row 3.  Column L contains the intensity distribution as a function of d, starting in row 3.
Construct a plot of the intensity distribution as a function of δ (column L versus column A).  Label the axes.

With a = b = 1 and c = d = 0 the plot shows the intensity distribution of two sources.

 Copy your plot to a Microsoft Word document.  For a single source the intensity is normalized to one.  What is the intensity of the central maximum for two sources?

Turn on another source by setting c =1.

 Copy your plot of the intensity distribution as a function of d for three sources to a Microsoft Word document.  What is the intensity of the central maximum for three sources?

Turn on another source by setting d = 1.

 Copy your plot of the intensity distribution as a function of d for four sources to a Microsoft Word document.  What is the intensity of the central maximum for four sources? In your own words, describe how the intensity distribution changes, as you add more equally-spaced sources.