Diffraction 1 1 1

“Any deviation of light rays from rectilinear path which is neither Diffraction “Any deviation of light rays from rectilinear path which is neither reflection nor refraction is known as diffraction.” (Sommerfeld) Types or kinds of diffraction: 1. Fraunhofer (1787-1826) 2. Fresnel (1788-1827)

In addition to interference, waves also exhibit another property – diffraction. It is the bending of the waves as they pass by some objects or through an aperture. The phenomenon of diffraction can be understood using Huygen’s principle 3

Diffraction of ocean water waves Ocean waves passing through slits in Tel Aviv, Israel Diffraction occurs for all waves, whatever the phenomenon. 4

Huygens Principle : when a part of the wave front is cut off by an obstacle, and the rest admitted through apertures, the wave on the other side is just the result of superposition of the Huygens wavelets emanating from each point of the aperture, ignoring the portions obscured by the opaque regions.

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Secondary wavelets from apertures

μv > μR Refraction Deviation for blue is larger than that for red

Diffraction Deviation for red is larger than that for blue

Fraunhofer diffraction

Single slit diffraction Principal maximum

First minimum

Second minimum

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Young’s Two Slit Experiment and Spatial Coherence If the spatial coherence length is less than the slit separation, then the relative phase of the light transmitted through each slit will vary randomly, washing out the fine-scale fringes, and a one-slit pattern will be observed. Fraunhofer diffraction patterns Good spatial coherence Poor spatial coherence 18

Diffraction from one- and two-slits Fraunhofer diffraction patterns One slit Two slits 19

Diffraction from small and large circular apertures Far-field intensity pattern from a small aperture Far-field intensity pattern from a large aperture 20

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Diffraction from multiple slits Slit Diffraction Pattern Pattern 22

Superposition of large number of phasors of equal amplitude a and equal successive phase difference δ. We have to find the resultant phasor. 23

GP series

δ 8δ 7δ 6δ 5δ 4δ 3δ 2δ δ/2 25 25

2 26

f Problem: Obtain intensity formula by integration Integrate: a exp(iδ) x sin θ is the path difference at a point x where the slit element dx is placed. 27

Δ = b Sin θ β = nδ/2 = πb sin θ/λ For single slit path difference between the two ends of the slit Δ = b Sin θ Phase difference = 2π Δ/λ = nδ β = nδ/2 = πb sin θ/λ 28

β = π b sin θ / λ tan β = β β = + m π m = 1, 2, 3… β Intensity for single slit 2 β = π b sin θ / λ Maxima at I Minima at tan β = β β = + m π _ m = 1, 2, 3… β 29

2 For Principal maxima β = π b Sin θ / λ

For extrema of I(q) tan β = β

Maxima tan β = β y= β y=tan β β 32

For subsidiary maxima

= λ / b Diffraction envelope size Δ x0 = f λ / b Δ x0 Full width of central maximum (First minima at sin  ~  = /b) = λ / b Diffraction envelope size Δ x0 = f λ / b Δ x0 Point source f Smearing effect of diffraction 34

Double slit 35

2 Phase=(2π/) * x sin 36

Double slit intensity pattern for d=5b 37

Single slit diffraction pattern X double slit interference pattern 38

b sin θ = m λ m = 0 / d sin θ = n λ Condition for Missing orders Diffraction Minima at b sin θ = m λ m = 0 / d sin θ = n λ Interference Maxima at When the above two equations are satisfied at the same point in the pattern (same θ), dividing one equation by the other gives the condition for missing orders.

Missing orders 5, 10, 15, 20…. d/b = 5 40

When we use the double-source equation to find locations of bright spots, we find that there are some places where we expect to see bright spots, but we see no light. This is known as a missing order, and it happens because at that location there's a zero in the single slit pattern.

Remember! If the zero in the single slit pattern, and a zero in the double slit pattern coincides, it is not called a missing order, as there is no order to be missing! Also, if there is a local peak in the single slit pattern, and a zero in the double source pattern, there will still be a zero (remember, we multiply the functions!) - this is also not a missing order.

N slit grating

Normal incidence Transmission grating

= Number of slits We know that the amplitude due to each single slit: = δ /2

Intensity pattern Diffraction Interference for small b

Behavior of sin2(x) sin2(5*x)

sin2(5*x) / sin2(x) sin2(10*x) / sin2(x)

When  = 0, or mπ (m = 0, ±1, ±2,…) (m = 0, ±1, ±2,…) The intensity is maximum when this condition is satisfied. These are called the Primary maxima. m gives the order of the maximum. The intensity drops away from the primary maxima. The intensity becomes zero (N-1) times between two successive primary maxima. There are (N-2) secondary maxima in between. As N increases the number of secondary maxima increases and the primary maxima becomes sharper.

Intensity I γ

The intensity becomes zero (N-1) times between two successive primary maxima. There are (N-2) secondary maxima in between. N=15

For two wavelengths

d/b = 4 1st Order 5th Order 6th Order 7th Order

Principal maxima Minima Because at minima we should have, sin   0 When  = 0, or mπ (m = 0, ±1, ±2,…) Principal maxima (m = 0, ±1, ±2,…) Minima Because at minima we should have, sin   0 If maximum is at m and Δw is the width, then the intensity should be zero at (m + Δw )

I π + π/N π - π/N γ

m Oblique incidence Maxima

Let us estimate the width of the mth order Principal maximum sin (A+B) = sin A cos B + cos A sin B d sin θm = mλ

Width of principal maxima When does the principal maxima get sharper? Dispersive power of grating is defined as Differentiate:

Chromatic resolving power of a grating = m N

Chromatic resolving power of a prism -

Transmission grating Reflection grating

Bragg’s law X-ray diffraction from crystals: 2d Sin θ = n λ

X-ray diffraction from (95% Cu 5% Co) crystals:

Visibility of the fringes (V) Maximum and adjacent minimum of the fringe system Reference: Eugene Hecht, Optics, 4th Ed. Chapter 12 69