Interference and Diffraction

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This is a topic from Higher Physics 1B

Contents

Interference

Interference refers to the superposition of individual waves when they cross paths.

  • Constructive interference is when the amplitude of the resultant wave is greater than the individual waves, while destructive interference is when the amplitude of the resultant wave is less than the individual waves
  • Conditions for observing interference:
    • The sources must be coherent, meaning that they must maintain a constant phase with respect to one another
    • The sources must be monochromatic meaning that they have a single wavelength rather than a spectrum (for example yellow light is acceptable, but not white light)
  • Constructive interference occurs when the path of the light from one slit is equal in distance to, or an integer multiple of wavelengths more or less than the path of light from the other slit
  • Destructive interference occurs when this path difference is an odd integer number of half wavelengths such that crests and troughs superimpose to cancel each other out

Diffraction

Diffraction refers to the phenomena exhibited by light when it interacts with barriers and obstacles. Consider the following 'double-slit' setup:

Screen Shot 2012-09-17 at 12.20.26 PM.png
  • From Huygen's principle we know that the waves spread out in a spherical fashion from the slits in the opaque barrier. This type of change in the path of light is an example of diffraction
  • The wavefronts of the diffracted light interfere with one another, leading to constructive and destructive interference at various points on the viewing screen that appears as a pattern of bright and dark parallel bands called fringes:
Screen Shot 2012-09-17 at 12.25.17 PM.png

Interference Equations for Diffraction Through Two Slits

Consider the above figure and the schematic diagram below:
Screen Shot 2012-09-17 at 1.10.11 PM.png


In deriving these equations the following assumptions are made:

    • L>>d (L is much greater than d)
    • d>>λ (d is much greater than λ)
    • These assumptions allow us to treat the paths r1 and r2 as parallel

Path Difference

  • As the paths are parallel, the yellow triangle can be treated as a right angled triangle
  • Thus with basic trigonometry we can find the path difference, δ
δ = r2 - r1 = dSinθ

Conditions for Constructive/Destructive Interference

  • Constructive interference will occur when the path difference is equal to an integer multiple of the wavelength [δ = mλ where m is zero or an integer] and so:
dSinθbright = mλ (for m = 0, ±1, ±2 etc.)
  • Destructive interference will occur when the path difference is equal to an odd integer multiple of half the wavelength [δ = (m + 1/2)λ where m is zero or an integer] so that the crests of one wave are superimposed upon the troughs of another, and so:
dSinθdark = (m + 1/2)λ (for m = 0, ±1, ±2 etc.)
  • The integer m is known as the order number

Linear Positions of Fringes

  • The previous equations provide the angles for fringe positions, and using the parallel assumption again, and the trigonometric result Tanθ = y/L we derive the expression for the vertical position of light and dark fringes:
ybright = LTanθbright
ydark = LTanθdark
  • For small angles Tanθ = Sinθ, and so the vertical distribution of fringes can be modelled as linear:
ybright = L(mλ/d) (for small angles only)

Intensity Distribution for Double Slit Interference

The bright fringes in the interference pattern for a double slit do not have clear cut edges, rather there is a variation in intensity, and the above equations provide the centre of this distribution.

  • In deriving equations for this distribution the following assumptions are made:
    • The two slits represent coherent sources of monochromatic light which is a sinusoidal wave
    • The waves from the slits have the same field intensity Eo, angular frequency ω, and constant phase difference φ
  • The phase difference is dependent upon the path difference:
φ = 2πδ/λ

Electric Field Intensity

  • The electric field intensity at point on the screen, P, is given as the sum of the intensities of the two waves:
EP = E1 + E2
= Eo[Sinωt + Sin(ωt + φ)]
= 2EoCos(φ/2)Sin(ωt + φ/2)

Light Intensity

  • Intensity is proportional to the square of the resultant electric field magnitude [2EoCos(φ/2)] and so introducing the constant Imax, and thus removing other constants Eo and 2 gives:
I = ImaxCos2(φ/2)
= ImaxCos2[(2πδ)/(2λ)]
= ImaxCos2[πdSinθ / λ)]
I = ImaxCos2[πdy / λL]

This light intensity equation is a specialised case for light intensity (which is a time average of the Poynting vector, derived as I = EmaxBmax / 2μo.)

Phasors

A phasor or phase vector is a method of representing sinusoidal relationships which is used in the addition of many waves upon one another. The angle the phasor makes against its source is equal to the phase difference given in radian form (by multiplying the difference in length by 2π/λ). The phase vectors are added head to tail such that the vector sum of the individual phasors gives the resultant value:

Screen Shot 2012-09-17 at 2.07.37 PM.png

Phase diagrams can be used to calculate the distribution of minima and maxima for diffraction through many slits, such as the following diagrams showing the constructive and destructive interference patterns for three slits:

Screen Shot 2012-09-17 at 2.11.35 PM.png

Interference Equations for Diffraction Through One Slit

Interference still occurs with one slit, because by Huygen's principle all points of the wavefront within the slit can be considered to be point sources of wavelets which can in turn interfere with one another. The typical pattern is seen below: Screen Shot 2012-09-17 at 4.24.23 PM.png


Consider the following figure:
Screen Shot 2012-09-24 at 10.55.58 AM.png

  • Slit width a can be thought of as being split into zones of width Δy with equal contribution to the intensity of the light and to the electric field ΔE
  • Rays of light from different zones will arrive at a given point P at different times due to the different distance travelled (ΔySinθ), which results in a phase difference between the rays
  • The incremental fields between adjacent zones are out of phase by an amoount Δβ given by:
Δβ = (2π/λ) Δy Sinθ
  • An in-depth analysis of these phase differences leads to the following equations:

Intensity:
Screen Shot 2012-09-24 at 11.43.33 AM.png

where Imax is the intensity at θ = 0

Minima (dark bands):

Sinθdark = mλ/a
where m is a natural number

Interference Equations for Diffraction Through Three or More Slits

As the number of slits increases the primary maxima increase in intensity and become narrower, while the secondary maxima decrease in intensity relative to the primary maxima. If the number of slits is N, there are (N-2) secondary maxima.
Ipat.jpeg

Interference in Thin Films

In thin films such as soap bubbles interference is observed due to the reflection and refraction of the film's surfaces, leading to optical effects such as the rainbow coloration of the surface.

  • As previously noted, light undergoes a phase change of 180 degrees when moving from a less dense to a more dense medium.
Screen Shot 2012-09-17 at 4.04.01 PM.png
  • In the figure above, ray 1 undergoes a phase change, while ray 2 does not
  • Ray 2 also travels 2t further than ray 1
  • Combining these two affects leads to the following equations which are in fact quite the opposite to those derived for diffraction interference:
  • Constructive interference: 2nt = (m + 1/2)λ
  • Destructive interference: 2nt = mλ where n is the refractive index and m is the order number
  • These equations apply to cases such as the given one, whereby the the medium below the bottom surface is the same as that on the top, or alternatively to situations in which the medium below the bottom surface as a refractive index of less than n
  • If instead the film is between a surface of higher refractive index and a surface of lower refractive index then there are two phase changes which equate to no phase change, and so the conditions for constructive and destructive interference are reversed

Newton's Rings

Newton's rings refer to a phenomenon observed when a plano-convex lens is placed on top of a flat glass surface such that the air film between the glass surfaces varies in thickness

Screen Shot 2012-09-17 at 4.21.55 PM.png

It can be shown that:

rn= √(Rλ(N-1/2))

Where rn is the radius of the Nth bright ring

N is the ring number
R is the radius of curvature of the lens
λ is the wavelength of the light

Resolution

Resolution is a phenomenon relating to the ability to distinguish between two sources of light. Anything that you see can be considered to be a source of light, in that seeing something necessitates that there are rays of light travelling between the object and your eye.


Consider the following figure: Screen Shot 2012-09-24 at 12.20.33 PM.png

  • The sources, or images, S1 and S2 are far apart enough that when light rays travel from the sources through the slit the maxima in the diffraction pattern do not overlap
  • Instead of noting that they are far apart enough, it can be more useful to just consider whether the angle θ is large enough
  • Because the maxima do not overlap both sources are visible and distinguishable from one another, and so the images are resolved
  • Alternatively if the angle is too small the maxima overlap, the images are not distinguishable and the images are not resolved

Rayleigh's Criterion

The limiting condition of resolution is known as Rayleigh's Criterion.

  • The limit of resolution is when the central maximum of one image falls on the first minimum of another image
  • Therefore minimum angle θmin is given by the equation for the first minimum in one slit diffraction:
Sinθmin = λ/a
  • The angle is typically very small, and so the approximation Sinθmin ≈ θmin gives:
θmin = λ/a

Circular Apertures

For a circular aperture the limiting angle of resolution is given by:

θmin = 1.22λ/D
where D is the diameter of the aperture
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