diffraction

PHY 1214

General Physics II

PHY 1214

General Physics II

Weldon J. Wilson

Professor of Physics and Engineering

Howell Hall 221H

[email protected]

Lecture 30

Diffraction Gratings

July 25, 2005

Lecture 30

Diffraction Gratings

July 25, 2005

Lecture Schedule (Weeks 7-8)

Lecture Schedule (Weeks 7-8)

July 25, 2005 PHY 1214 - Lecture 30 3

Diffraction

Diffraction

• Huygen’s principle

requires that the waves spread out after they pass through slits

• This spreading out of light from its initial line of

travel is called diffraction

– In general, diffraction occurs when wave pass through small openings, around obstacles or by sharp edges

Diffraction, 2

Diffraction, 2

• A single slit placed between a distant light

source and a screen produces a diffraction

pattern

– It will have a broad, intense central band

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

• Called secondary maxima

– The central band will also be flanked by a series of dark bands

July 25, 2005 PHY 1214 - Lecture 30 5

Diffraction Grating

Diffraction Grating

• The diffracting grating consists of many

equally spaced parallel slits

– A typical grating contains several thousand

lines per centimeter

• The intensity of the pattern on the screen

is the result of the combined effects of

d

Path length difference

d

θ

Young’s Double Slit Review

Young’s Double Slit Review

θ θ

L

= d sinθθθθ

2) 1)

d sin θ = (m + 1 2)λ d sin θ = mλ

where m = 0, or 1, or 2, ... Which condition gives

July 25, 2005 PHY 1214 - Lecture 30 7

d

Path length difference 1-2

Multiple Slits

(Diffraction Grating – N slits with spacing d)

Multiple Slits

(Diffraction Grating – N slits with spacing d)

θ θ

L

= d sinθθθθ

d sin

θ =

m

λ

Constructive interference for all paths when

d

Path length difference 1-3 = 2d sinθθθθ

d

1

2 3

Path length difference 1-4 = 3d sinθθθθ

=λ =λ =λ =λ =2λ =2λ =2λ =2λ =3λ =3λ =3λ =3λ 4

July 25, 2005 PHY 1214 - Lecture 30 8

N slits with spacing d

Constructive Interference Maxima are at:

sin

θ =

m

λ

d

* screen VERY far away

θθθθ

Diffraction Grating

Diffraction Grating

Same as for Young’s Double Slit !

July 25, 2005 PHY 1214 - Lecture 30 9

d

Question

Question

L

d

1

2 3

All 3 rays are interfering constructively at the point shown. If the intensity from ray 1 is I₀ , what is the combined intensity of all 3 rays? 1) I₀ 2) 3 I₀ 3) 9 I₀

Each slit contributes amplitude E_o at screen. E_tot = 3 E_o. But I α E2_{. I}

d

Question

Question

θ θ

L

d

1

2 3

When rays 1 and 2 are interfering destructively, is the intensity from the three rays a minimum? 1) Yes 2) No

Rays 1 and 2 completely cancel, but ray 3 is still there. Expect intensity I = 1/9 I_max

d sin θ = λ2

this one is still there! these add to zero

July 25, 2005 PHY 1214 - Lecture 30 11 0 3 6 9 λ

3 λ

2λ 3 λ 2 ₁₉ 0 3 6 9 λ 3 2λ 3 λ 2

d sin θ =

Three slit interference

I₀ 9I₀

For many slits, maxima are still at

sin

θ =

m

λ

d

Region between maxima gets suppressed more and more as no. of slits increases – bright fringes become narrower and brighter.

10 slits (N=10)

d sin θ

in te n s it y

λλλλ _2λ2λ2λ2λ

0

2 slits (N=2)

d sin θ

in te n s it y

λλλλ 2λ2λ2λ2λ

0

Multiple Slit Interference

(Diffraction Grating)

Multiple Slit Interference

(Diffraction Grating)

Peak location depends on wavelength!

July 25, 2005 PHY 1214 - Lecture 30 13

Constructive interference:

2

dsin

θ =

m

λ

d

≈

0

.

5

nm

For λ=.017nm

X-ray

d

1st _{maximum will be at 10}0

X-Ray Diffraction

X-Ray Diffraction

Crystal solid such as sodium

θ

dsin

θ

Grating spectrometer

Grating spectrometer

• The light to be analyzed

passes through a slit and

is formed into a parallel

beam by a lens. The

diffracted light leaves the

grating at angles that

July 25, 2005 PHY 1214 - Lecture 30 15

Diffraction Grating

Diffraction Grating

Spectroscopy

Spectroscopy

Look at the first spectrum maximas. Can accurately

measure the wavelengths from the size of the angle using dsin

θ

λ

We know different substances have different spectra, so we can identify by these spectra.

July 25, 2005 PHY 1214 - Lecture 30 17

A Grating Spectroscope

A Grating Spectroscope

White Light

Light with λλλλ₁₁₁₁=400 nm and λλλλ₂=700 nm.

Light can be dispersed by wavelength using a diffraction grating. A

grating spectroscope provides a way of making precise measurements of wavelength by noting the angular positions at which bright fringes occur. If photographic plates or CCDs are used to make a more permanent

record, the device is called a grating spectrograph.

Atoms in a low pressure electrical discharge produce light with characteristic wavelengths that can be identified with a spectrograph.

Example: Measuring

Wavelengths Emitted by Sodium

Example: Measuring

Wavelengths Emitted by Sodium

Light from a sodium lamp passes through a

diffraction grating having 1000 slits per millimeter. The interference pattern is viewed on a screen

1.000 m behind the grating. Two bright yellow fringes are visible at distances of 72.88 cm and 73.00 cm from the central maximum.

Assuming that m=1, what are the wavelengths of these two fringes?

tan

m m

y = L θ

( )

1 1 tan 1/

36.08 for 72.88 cm 36.13 for 73.00 cm

m m

y L y y

θ ₌ −

= ° =

= ° =

1

sin

589.0 nm for 72.88 cm 589.6 nm for 73.00 cm

m m

d

y y

λ = θ

= =

= =

3 6

1.0 10 m /1000 1.0 10 m d = × − = × −

sin

July 25, 2005 PHY 1214 - Lecture 30 19

Single Slit Diffraction

Single Slit Diffraction

• According to Huygen’s

principle, each portion of the

slit acts as a source of

waves

• The light from one portion of

the slit can interfere with

light from another portion

• The resultant intensity on

the screen depends on the

direction

Single Slit Diffraction, 2

Single Slit Diffraction, 2

• All the waves that originate at the slit are in phase

• Wave 1 travels farther than wave 3 by an amount

equal to the path difference (a/2)sin

• If this path difference is exactly half of a

wavelength, the two waves cancel each other and

destructive interference results

• In general,

destructive interference

occurs for a

single slit of width a when sin

=m /a

July 25, 2005 PHY 1214 - Lecture 30 21

Single Slit Diffraction, 3

Single Slit Diffraction, 3

• The general features of the intensity distribution are

shown

• A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes

• The points of constructive interference lie

approximately halfway between the dark fringes

July 25, 2005 PHY 1214 - Lecture 30 22

• •

Diffraction/ Huygens

Every point on a wave front acts as a source of tiny wavelets that move forward.

We will see maxima and minima on the wall.

Light waves originating at different points within

opening travel different distances to wall, and can interfere!

July 25, 2005 PHY 1214 - Lecture 30 23

W

w

2 sin θ

θ θ θ W 2 1 1′

Rays 2 and 2′ also start W/2 apart and have the same path length difference.

2

2′

1st _{minimum at sin} θθθθ ₌ λλλλ_/w

When rays 1 and 1′

interfere destructively.

w

2 sin θ = λ2

Under this condition, every ray originating in top half of slit interferes destructively with the corresponding ray originating in bottom half.

Single Slit Diffraction

Single Slit Diffraction

July 25, 2005 PHY 1214 - Lecture 30 24

w

Rays 2 and 2′ also start w/4 apart and have the same path length difference.

2nd _{minimum at sin} θθθθ _{= 2}λλλλ_/w

Under this condition, every ray originating in top quarter of slit interferes destructively with the corresponding ray originating in second quarter.

Single Slit Diffraction

Single Slit Diffraction

4 w θ 1 1′ ) sin( 4 w θ 2 2′

When rays 1 and 1′

will interfere destructively.

2 ) sin( 4 w λ θ =

July 25, 2005 PHY 1214 - Lecture 30 25

Condition for quarters of slit to destructively interfere

sin

θ =

m

λ

w

(m=1, 2, 3, …)

Single Slit Diffraction

Summary

Single Slit Diffraction

Summary

Condition for halves of slit to

destructively interfere

_w

λ

θ

)

=

sin(

w

λ

θ

)

2

sin(

=

Condition for sixths of slit to

destructively interfere

_w

λ

θ

)

3

sin(

=

THIS FORMULA LOCATES MINIMA!! Narrower slit => broader pattern All together…

Maxima and minima will be a series of bright and dark

rings on screen

Central maximum

sin

θ =

1.22

λ

D

First diffraction minimum is at

Diameter D

light

Diffraction from Circular

Aperture

Diffraction from Circular

Aperture

1st _{diffraction minimum}

July 25, 2005 PHY 1214 - Lecture 30

θ

≈

θ

sin

D

λ

22 1.

I

Intensity from Circular

Aperture

Intensity from Circular

Aperture

These objects are

just resolved

Two objects are just resolved when the maximum of one is at the minimum of the other.

July 25, 2005 PHY 1214 - Lecture 30 29

Resolving Power

Resolving Power

To see two objects distinctly, need

θθθθ

>

θθθθ

θθθθ_min

θθθθ_objects

Improve resolution by increasing θθθθ_objects or decreasing θθθθ_min

θθθθ_objects is angle between objects and aperture:

θθθθ_objects ≈ tan (d/y)

sin θθθθ_{min ≈} θθθθ_min = 1.22 λ/D

θθθθ_min is minimum angular separation that aperture can resolve:

D

d

Question

Question

Astronaut Joe is standing on a distant planet with binary

suns. He wants to see them but knows it’s dangerous to look straight at them. So he decides to build a pinhole camera by poking a hole in a card. Light from both suns shines through the hole onto a second card.

But when the camera is built, Astronaut Joe can only see one spot on the second card! To see the two suns clearly, should he make the pinhole larger or smaller?

Decrease θθθθ_min = 1.22λ / D Want θθθθ_objects > θθθθ_min

July 25, 2005 PHY 1214 - Lecture 30 31

sin

θ

≈ θ

=

1.22

λ

D

Resolving Power

Resolving Power

How does the maximum resolving power of your eye change when the brightness of the room is decreased.

1) Increases 2) Constant 3) Decreases

When the light is low, your pupil dilates (D can increase by factor of 10!) But actual limitation is due to density of rods and cones, so you don’t notice an effect!

Recap.

Recap.

• Interference:

– Full wavelength difference = Constructive – ½ wavelength difference = Destructive

• Multiple Slits

– Constructive d sin(θ) = m λ (m=1,2,3…) – Destructive d sin(θ) = (m + 1/2) λ 2 slit only – More slits = brighter max, darker mins

• Huygens’ Principle:

• Single Slit:

– Destructive: w sin(θ) = m λ (m=1,2,3…) – Resolution: Max from 1 at Min from 2

op po sit e!

End of Lecture 30

End of Lecture 30