Diffraction

Goals for lecture

To see how a sharp edge or an aperture affect light

To analyze single-slit diffraction and calculate the intensity

of the light

To investigate the effect on light of many closely spaced

slits

To learn how scientists use diffraction gratings

To see what x-ray diffraction tells us about crystals

To learn how diffraction places limits on the resolution of a

telescope

Introduction

How can we use coherent light to visually see the

difference in pit density on CDs, DVDs and Blu-Ray disks?

Why does light from a point source form light and dark

fringes ?

We will continue our exploration of the wave nature of light

with diffraction.And we will see how to form three-

dimensional images using a hologram.

Diffraction

According to geometric optics, a light source shining on an

object in front of a screen should cast a sharp shadow.

Surprisingly, this does not occur because of diffraction.

Diffraction and Huygen’s Principle

Huygens’s principle can be used to analyze diffraction.

Fresnel diffraction: Source, screen, and obstacle are close

together.

Fraunhofer diffraction: Source, screen, and obstacle are far

apart

Diffraction from a single slit

In Figure below, the prediction of geometric optics in (a)

does not occur. Instead, a diffraction pattern is produced,

as in (b).

The narrower the slit, the broader the diffraction pattern.

Fresnel and Fraunhofer diffraction by a single slit

Figure below shows Fresnel (near-field) and Frauenhofer

(far-field) diffraction for a single slit.

Locating the dark fringes

• Follow the single-slit diffraction discussion in the text.

Figure below shows the geometry for Fraunhofer diffraction

An example of single-slit diffraction

Figure

36.6 (bottom left) is a photograph of a Fraunhofer pattern

of a single horizontal slit.

Example 36.1: You pass 633-nm light through a narrow slit

and observe the diffraction pattern on a screen 6.0 m away.

The distance at the screen between the center and the

first minima on either side is 32 mm long. How wide is the

slit?

(6000 mm)(0.000633 mm)0.24 mm

32 mmm

xmy a

a

Intensity in the single-slit pattern

Follow the text discussion of the intensity in the single-slit

pattern using the phasor diagrams in Figure below.

Quantitative Intensity in the single-slit pattern

Follow the text discussion of the intensity in the single-slit

pattern using the phasor diagrams in Figure below.The angle

b is the phase angle of the ray from the top of the slit,

while the phase angle from the bottom of the slit is 0. The

vectors lie along a circle whose center is at C, so Ep is a

chord of the circle. The arc length E0 is subtended by this

same angle β, so the radius of the circle is E0/β.

From the diagram

since

• We have

(sinc function)

00

sin / 22 sin

2 / 2p

EE E

2sina

2

0

sin sin /

sin /

aI I

a

Intensity maxima in a single-slit pattern

Figure at the right shows the intensity versus angle in a

single-slit diffraction pattern.

The minima occur when βis a multiple of 2π, i.e. at

Width of the single-slit pattern

The single-slit diffraction pattern depends on the ratio of

the slit width a to the wavelength

Example : (a) The intensity at the center of a single-slit

diffraction pattern is I0. What is the intensity at a point in

the pattern where there is a 66-radian phase difference

between wavelets from the two edges of the slit? (b) If this

point is 7 degrees from the central maximum, how many

wavelengths across is the slit?

a)

b)

sin( 1, 2, 3, ...)

am m

2 2

4

0 0 0

sin sin / sin 33 rad9.2 10

sin / 33 rad

aI I I I

a

sin 33 rad33 rad 86

sin 7

aa

Two slits of finite width

When we discussed two-slit interference, we ignored the

width of each slit. When we demonstrated it, however, we

saw clearly the effect of the slit widths.

The overall pattern of two finite-width slits is the product

of the two patterns, i.e.

2 2

0

sinsinc cos sinc

2 2

xI I x

x

Several slits

In Figure below, a lens is used to give a Fraunhofer pattern on

a nearby screen. It’s function is to allow the pattern to be

seen nearby, without having the screen really distant.

The phasor diagrams show the electric vectors from each slit

at different screen locations.

Interference pattern of several slits

The figure below shows the interference pattern for 2, 8,

and 16 equally spaced narrow slits.

By making the slits really close together, the maxima

become more separated. If the light falling on the slits

contains more than one wavelength (color), there will be

more than one pattern, separated more or less according to

wavelength, although all colors have a maximum at m = 0.This

means that the different orders make rainbows—separating

wavelengths into a spectrum, with the separation being

greater for greater order m.

The diffraction grating

A diffraction grating is an array of a large number of slits

having the same width and equal spacing. The intensity

maxima occur at

Example 36.4: The wavelengths of the visible spectrum are

approximately 380 nm (violet) to 750 nm (red). (a) Find the

angular limits of the first-order visible spectrum produced by

a plane grating with 600 slits per millimeter when white light

falls normally on the grating. (b) Do the first order and second

order spectra overlap? What about the 2nd and 3rd orders?

(a)distance between slits is

Violet light for 1st order occurs at

Red light for 1st

order occurs at

(b) recalculate for m = 2 and m = 3

The 2nd-order spectrum extends from 27.1-63.9° while the 3rd

order is from 43-90.

sind m

61 mm1.67 10 m

600 slitsd

7 6arcsin / arcsin 3.8 10 /1.67 10 13.2d

7 6arcsin / arcsin 7.5 10 /1.67 10 26.7d

Grating spectrographs

A diffraction grating can be used to disperse light into a

spectrum. The greater the number of slits, the better the

resolution.

Figure (a) below shows our sun in visible light, and in (b)

dispersed into a spectrum by a diffraction grating.

Diagram of a grating spectrograph

Figure below shows a diagram of a diffraction-grating

spectrograph for use in astronomy.

X-ray diffraction

When x rays pass through a crystal, the crystal behaves like

a diffraction grating, causing x-ray diffraction. Figure below

illustrates this phenomenon.

A simple model of x-ray diffraction

Follow the text analysis using Figure below.

The Bragg condition for constructive interference

2d sin = m.

Circular apertures

An aperture of any shape forms a diffraction pattern

Figures below illustrate diffraction by a circular aperture.

The airy disk is the central bright spot.

Figures below illustrate diffraction by a circular aperture.

The airy disk is the central bright spot.

Diffraction limits the resolution of optical equipment, such

as telescopes. The larger the aperture, the better the

resolution. Figure (right) illustrates this effect.

Bigger telescope, better resolution

Because of diffraction, large-diameter telescopes, such as the

VLA radio telescope below, give sharper images than small ones

What is holography?

By using a beam splitter and mirrors, coherent laser light

illuminates an object from different perspectives.

Interference effects provide the depth that makes a

three-dimensional image from two-dimensional views. Figure

below illustrates this process.