Part8 - Diffractionsmartynk/Resources/Phys-130/Dr... · 2011. 12. 12. · (Chapter 36) Phys130, A04 Dr. Robert MacDonald Diffraction vs Geometry Geometric optics says shadows should

8: Diffraction(Chapter 36)

Phys130, A04Dr. Robert MacDonald

Diffraction vs GeometryGeometric optics says shadows should be sharp (if the light is from a point source).

That’s not what actually happens! In fact, all shadows have some blurring.

• They’re not even smooth shadows. You get a series of lines (fringes!).

This is an interference effect, the same idea as we’ve been discussing.

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Fig. 36.2

Shadow of a razor blade in monochromatic (single-colour) light.

Single-slit diffractrion

Fig. 36-1

Wikipedia

~85% of the lightfalls in the central band

Back to Huygens

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Consider a very long, very narrow slit. (Extends out of the page.)

Every point in the slit opening is a source of new wavelets, according to Huygens’s principle.

The resulting wave is the combination of all the wavelets.

Let’s divide the slit up into even smaller strips and look at wavelets coming from each strip.

We’ll be assuming the screen is far away (Fraunhofer diffraction). Wikipedia (Arne Nordmann)

Diffraction vs slit width

width ≈ λ (or less)! resembles single source! spreads in all directions

width >> λ! interference!! less spread

Approximation: far screenUsually we’re looking at the diffraction pattern a long way from the slit.

• Farther screen means larger pattern, easier to see.

We’ll be assuming that D >> a, so that all rays we draw from the slit are approximately parallel.

• This is very close to true.(Work it out to see how.)

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Pattern: Dark fringesWe’ll study the single-slit diffraction pattern using Huygens’s Principle.

This means we consider the slit as many, many tiny sources of light.

At any given angle, the rays from some of these sources can interfere destructively (cancel), while others interfere constructively, or something in between.

• Very hard to find the bright fringes!At a few specific angles, every ray is canceled by another ray.! Dark fringes at these angles.

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Note: At θ = 0, all path lengths are (roughly) equal.! All constructive interference. Bright fringe!

First dark fringeConsider a ray r1 from the top of the slit, and another ray r2 from the centre.

When these rays meet at the screen, their phase difference ϕ is just due to the path difference (r2 traveled farther).

• If D ≫ a, the rays are close to parallel. ! path difference is (a/2)sinθ.

Any pair of rays separated by a/2 has the same phase difference.

Path diff = λ/2 gives phase diff. = π.

• Each ray from the top half is cancelled by a ray from the bottom half.Dark fringe when sinθ = λ/a.

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Other dark fringes

Same argument holds for rays separated by a/4.! ! Destructive int. when (a/4)sinθ = λ/2 ! sinθ = 2λ/a.Repeat for a/6, a/8, ... Dark fringes occur when

asinθ = mλ, m = ±1, ±2, ...This formula does not hold for m=0! !

m = 1 m = 2 m = 3 m = 4m = –1m = –2m = –3m = –4

Note: At θ = 0, all path differences are 0.! ! All rays arrive in phase. Bright fringe!

Dark fringes at asinθ = mλ.• This says the central bright fringe is twice as wide as

any other fringe.

Behaviour:

• No dark fringes if a < λ; just one broad central fringe. (Get mλ/a > 1!)

• More fringes & narrower pattern for larger a or smaller λ.

Single-slit diffraction: intensity

The intensity I of light on the screen at angle θ is given by:

where Im = max intensity (at θ=0), and

Notice that this gives I(θ) = 0 when asinθ = mλ, as it should.

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a = λ

a = 5λ

a = 10λ

e.g. Intensities of maximaWhat are the intensities of the first two secondary maxima (bright fringes), relative to the central max (Im)?

Dark fringes located at asinθ = mλ, or α = mπ.! Bright fringes ~ halfway between: asinθ = (m + 1/2)λ, or α = (m + 1/2)π, for m = ±1, ±2, ±3, ...

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Circular AperturesCircular apertures are particularly interesting because so many devices — telescopes, cameras, eyes! — have circular apertures.

The idea’s the same, but shape is quite different.

For a circular aperture with diameter D:

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Location of first dark fringe:Some other fringes:

Fig. 36.10

Resolution

How close can two objects (e.g. stars) be in angle such that we can still tell them apart? (Call the min. angle θR.)Useful rule of thumb: Rayleigh’s criterion:

• The angle separating the two objects must be at least as wide as the diffraction pattern’s first minimum.

These are small angles, so sinθR ≈ θR. Then:14

Fig. 36.11

Dominion Radio Astrophysical ObservatoryPenticton, BC

W.H. Keck 10m optical telescopesMauna Kea, Hawai’i

Eg: Milky Way Black HoleA super-massive black hole is thought to sit at the centre of our galaxy (and almost every other galaxy, too), with a mass of four million suns (!).

The black hole is 26,000 light-years away (2.5x1017"km), and is 2.4x107"km across.

What is the angular size of the black hole?

If you’re using a radio telescope array with 0.13"mm light waves, how big does your telescope have to be to resolve the black hole?

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Two “wide” slitsWhen we studied two-slit interference, we assumed perfect circular waves from each slit.

• Only possible if the slits have zero width!In reality, the amount of light reaching the screen from each slit follows a single-slit diffraction pattern.

• That is the light that interferes to form the two-slit interference pattern “on top”.

To find the real interference pattern, multiply the single-slit and double-slit intensity formulas.

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Fig. 36-16

a = slit width

Single slit diffraction intensity

d = slit separation

xDouble slit interference intensity

Interference intensity for two “wide” slits:

Two-slit: Missing maximaIn the two-slit interference pattern, a bright fringe is formed wherever dsinθ = m2λ.

But, because of single-slit diffraction, no light reaches the screen wherever asinθ = m1λ.

• If this happens where a two-slit bright fringe is “supposed” to be, that fringe is “missing”.

If d = na (n = 1, 2, 3, ...) then this happens exactly, and removes every nth bright fringe.

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Here the m1 = 1 dark fringe kills the m2 = 5 bright fringe.

d = separation

a = width

More SlitsWhen you have 3 or more slits instead of just 2, you get constructive interference whenever the light from all of the slits arrives in phase.

This happens whenever the path difference for two adjacent slits is a whole number of cycles: ! d sinθ = mλ(Same as for 2 slits!)

But for other angles, it’s more likely that some of the rays will cancel each other (destructive interference) with more slits.

• Result: with many slits, the interference pattern is a series of sharp lines. 20

Fig.36.20

Fig.36.19

Diffraction GratingA diffraction grating is a device with a huge number of lines.

Each wavelength produces a line at a different angle. Used to study spectra!

Each wavelength produces more than one line, so the spectrum repeats. Each copy is called an “order”.

Fig.36-25: Emission spectrum of cadmium

Fig.36-25: First, second, and fourth orders of the emission spectrum of hydrogen.Third order not shown (it overlaps the second and fourth).

e.g. Sodium emission linesIn a sodium arc lamp (street lights, pickles, etc), the main orange colour is due to two lines, at 589.00 nm and 589.59 nm.

If your diffraction grating has 800 lines/mm, where are these two lines found in the second-order spectrum?

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