lecture9 diffraction(1)

Physics 2C: Fluids, Waves, Thermodynamics and Optics

Lecture 9: Diffraction

No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage, and there is no specific, important physical difference between them. -Richard Feynman

Diffraction (conceptual)Diffraction (conceptual)

Diffraction is a wave phenomena that occurs when a wave encounters an obstacle.

When light encounters a localized object it bends around it.

When light encounters a small opening the waves bend and spread out after passing through it.

Diffraction is a fundamental characteristic of waves.

Diffraction is most pronounced when the wavelength is proportional to the scale of the object causing the diffraction.

It is distinct from reflection or refraction.

Diffraction examplesDiffraction examples

Diffraction: Huygens perspectiveDiffraction: Huygens perspective

Huygens Principle is essential to understanding diffraction. Consider a plane wave incident on a narrow slit.

Finite opening restricts the spherical wave point sources to just that region when applying Huygens principle.

The curvature of the wavefront is more pronounced (stronger diffraction) when the opening is similar to the wavelength.

When

Diffraction double slitDiffraction double slit

Consider a perfectly coherent plane wave of wavelength lambda incident on two infinitesimally small slits space a distance d apart.

Question: What is the interference pattern produced (I(y)) on a screen a distance L from the slits?

Each slit will diffract the light such that it's emitted as a spherical wave. This will best be understood using the ray model formalism.

We must determine how light along paths P1 and P2 interfere at some point, y, along the screen.

Variables:

From the geometry we should be able to calculate the amount of interference from the path length difference:

In order to simplify the geometry we make a useful approximation small angle approximation

This has two important results, one being that P1 and P2 can be considered parallel:

Thus the path length difference is given by:


Constructive interference:

Destructive interference:

NOTE: P1 and P2 are NOT actually parallel, but if L is much much greater than d

In all real world cases d is very small.

Thus the bright fringes (constructive) are located at locations along the screen given by:

m is an integer

The second result of the small angle approximation:


Diffraction, wavelength comparisonDiffraction, wavelength comparison

Ignore the big broad fluctuations for now. It's the small equally spaced bands that are caused by double slit diffraction.

Note the wavelength dependence!

Diffraction - single slitDiffraction - single slit

So it seems clear why we get diffraction from two separate slits, but what about a single slit? In fact we do!

A single slit placed between a distance light source and a screen products a diffraction pattern as seen here.

It will have a broad, intense central band.

The central band will be flanked by a series of narrower, less intense secondary bands (secondary maxima)

Conditions: We can't consider the slit to be infinitesimally small compared to the wavelength anymore! The slit is of finite size.

Heuristically: We consider different parts of the single slit to be interfering with each other.

We are considering the finite width of the slit!

To use a similar geometrical argument as the double slit we must make the same small angle assumption:


Interference of a/2 rays: Interference of a/4 rays:Consider destructive interference:

In general destructive interference occurs when:Etc...thus since we can continue

to divide the slit into smaller and smaller spacings.

Sum over all pairs of points a/m apart.

m, non-zero integer!!


So what are the locations of the minima ym (destructive interference) on the screen?

Small angle approximation:

Width of central maxima:

Minima locations:


Clicker QuestionClicker QuestionYou're friend is playing hide-and-seek with you at night. Looking out the front door into the dark; you shine your flashlight through the doorway, onto the porch and yell your friends name.

You're friend, who is hiding just around the corner outside hears you call their name, but the light from your flashlight doesn't illuminate them. Why?

A.) Sound waves have a different law of diffraction than light waves do.

B.) Reflection of sound is the only reason your friend would hear your call.

C.) The light source is incoherent so diffraction doesn't apply.

D.) The wavelength of sound is much larger than the wavelength of visible light, therefore it diffracts more through the doorway.

So we've located the minima for the single slit diffraction. But we can actually calculate the entire diffraction pattern.

The intensity of the diffraction pattern depends on the angle you are at with respect to the central line, theta, the slit width, a, and the wavelength of light used.

Where:

As the size of the slit width increases (relative to the wavelength of light used), then the central peak gets narrower.


Single AND Double slit...Single AND Double slit...

Central single slit peak.

Double slit peaks.

Diffraction GratingsDiffraction Gratings

Consider a grating consisting of N number of slits separated by a distance d.

We now have N different light rays to consider.

If every wave coming from a slit has a path length difference of relative to the next, we still have constructive interference.

Same path length difference!

So the condition for constructive interference is the same as that for a double slit.

Diffraction grating patternsDiffraction grating patterns

So then how is this different than double slit?

The destructive interference is more complicated many different waves to interfere (kind of like the single slit)!

The difference arises when we consider the locations of the minima (destructive).

Consider the first minimum. This will occur when the top and bottom slits are a full wavelength out of phase akin to the first minimum of the single slit diffraction. Thus:

Half-width of central max:

Width of central max depends on number of rulings (N)!

In general:Take away: Peaks get sharper for larger N.

Clicker QuestionClicker QuestionWhich best characterizes the three diffraction patterns below (1,2,3)?

A.) Double slits w/ large slit widthDouble slit w/ small slit widthSingle slit w/ large slit width

B.) Single w/ small slitDouble w/ large slitsSingle w/ large slit

C.) Double w/ large slits Single w/ large slitDouble w/ small slits

D.) WTF?!

Circular apertures and Circular apertures and resolvability resolvability

We've only looked at diffraction from slits which only have 1D patterns, BUT diffraction in 2D and 3D is possible!

Diffraction of a laser through a pinhole 2D analog of single slit

This second one is more complex:

Diffraction through a hexagonally shaped hole

Fringes are circularly symmetric.

Circular apertures and Resolution Circular apertures and Resolution

Rayleigh's Criterion: When the central maximum of one image falls on the first minimum of another image, the images are said to be just resolved.

Resolution is the ability of an optical system to distinguish between closely spaced objects. This is limited due to the wave nature of light.

Consider two pinholes (incoherent with one another), if they are separated so that their central maxima do not overlap then their images are said to be resolved.

The limiting condition for resolution:

The images are just resolved if their angular separation satisfies Rayleigh's criterion

Here are two objects that are barely resolvable.

a.) The angular separation is too small. They appear as one object.

b.) They have moved apart and you can marginally distinguish one from another.

c.) Once separated further they are easily resolvable.

Part (b) satisfies Rayleigh's criterion.


I'm not discussing it in class, but read the section on dispersion and resolving power in your book (sec. 36-9)!!

For a slit of width, a, and applying Rayleigh's criterion, the limiting angle of resolution is:

For the images to be resolved, the angle subtended by the two sources at the slit must be greater than

For a circular opening of diameter D, applying Rayleigh's criterion the angle of resolution is:

Small angle approx:


Resolution example Resolution example

An apt analogy for resolution is the style of art called pointillism Here you see a blown up portion of a pointillism painting.

Length from observer to point (L) is small can resolve separate points (D).

But if we zoom out (L large) then the image appears less granular can no longer resolve as small of D values!!

Clicker QuestionClicker Question

What is the smallest object a 3 meter NSA spy telescope mounted on a satellite can observe in a city 100 kilometers below it? It is observing visible light with an average wavelength of 600 nm. This is a rough calculation, so approximate!

A.) Your DNA (~100nm)!!

B.) Your credit card you're showing to the cashier (~5 cm)

C.) The newspaper you're reading while sitting on a park bench (~1m)

D.) The trees in the park (~10m)

E.) The NSA doesn't have spy satellites...don't be silly!

Bragg Diffraction, crystallographyBragg Diffraction, crystallography

Atomic scale crystalline planes give rise to macro-scale crystal structure.

Bragg diffraction is a constructive reflection off the equally spaced lattice planes in a crystal. At some reflection angle when the path length difference is equal to a integer wavelength the reflection will be constructive.

It all sounds so familiar! That's because it is! Kind of like thin-film interference crossed with slit diffraction.

Bragg's Law gives constructive interference condition:

What to do before next class?

Reading: Principles of Physics, Sections 14.1 14.7

Homework 1II: Due next Monday (8/25)

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lecture9 diffraction(1)

Documents

double slit diffraction

diffraction pattern

small slits

small compa

wavelength comparisonignore

wavelength dependence

wave phenomena

interference pattern