Interference and Diffraction
This is a topic from Higher Physics 1B
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 'doubleslit' setup:

 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:
Interference Equations for Diffraction Through Two Slits
Consider the above figure and the schematic diagram below:
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 r_{1} and r_{2} 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 = r_{2}  r_{1} = 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:
 dSinθ_{bright} = 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:
 dSinθ_{dark} = (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:
 y_{bright} = LTanθ_{bright}
 y_{dark} = LTanθ_{dark}
 For small angles Tanθ = Sinθ, and so the vertical distribution of fringes can be modelled as linear:
 y_{bright} = 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 E_{o}, angular frequency ω, and constant phase difference φ
 The phase difference is dependent upon the path difference:
 φ = 2πδ/λ
 In deriving equations for this distribution the following assumptions are made:
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:
 E_{P} = E_{1} + E_{2}
 = E_{o}[Sinωt + Sin(ωt + φ)]
 = 2E_{o}Cos(φ/2)Sin(ωt + φ/2)
 E_{P} = E_{1} + E_{2}
Light Intensity
 Intensity is proportional to the square of the resultant electric field magnitude [2E_{o}Cos(φ/2)] and so introducing the constant I_{max}, and thus removing other constants E_{o} and 2 gives:
 I = I_{max}Cos^{2}(φ/2)
 = I_{max}Cos^{2}[(2πδ)/(2λ)]
 = I_{max}Cos^{2}[πdSinθ / λ)]
 I = I_{max}Cos^{2}[πdy / λL]
 I = I_{max}Cos^{2}(φ/2)
This light intensity equation is a specialised case for light intensity (which is a time average of the Poynting vector, derived as I = E_{max}B_{max} / 2μ_{o}.)
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:
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:
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:
Consider the following figure:
 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 indepth analysis of these phase differences leads to the following equations:
 where I_{max} is the intensity at θ = 0
Minima (dark bands):
 Sinθ_{dark} = 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 (N2) secondary maxima.
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.
 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 planoconvex lens is placed on top of a flat glass surface such that the air film between the glass surfaces varies in thickness
It can be shown that:
 r_{n}= √(Rλ(N1/2))
Where r_{n} is the radius of the N_{th} 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:
 The sources, or images, S_{1} and S_{2} 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 θ_{min} is given by the equation for the first minimum in one slit diffraction:
 Sinθ_{min} = λ/a
 The angle is typically very small, and so the approximation Sinθ_{min} ≈ θ_{min} gives:
 θ_{min} = λ/a
Circular Apertures
For a circular aperture the limiting angle of resolution is given by:
 θ_{min} = 1.22λ/D
 where D is the diameter of the aperture