Chapter 4
Diffraction of Light Waves
Diffraction
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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 waves
pass through small
openings, around
obstacles or by sharp
edges
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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
 Called minima
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The results of the
single slit cannot be
explained by
geometric optics
 Geometric optics
would say that
light rays traveling
in straight lines
should cast a
sharp image of the
slit on the screen
Fraunhofer Diffraction
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Fraunhofer Diffraction occurs
when the rays leave the
diffracting object in parallel
directions
 Screen very far from the slit
 Converging lens (shown)
A bright fringe is seen along
the axis (θ = 0) with alternating
bright and dark fringes on
each side
Single Slit Diffraction
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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 θ
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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 θ
If this path difference is exactly half of a
wavelength, the two waves cancel each
other and destructive interference results
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In general, destructive interference occurs
for a single slit of width a when
sin θdark = mλ / a
 m = 1, 2, 3, …
Doesn’t give any information about the
variations in intensity along the screen
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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
Resolution of Single-Slit
and Circular Apertures
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The resolution is the ability of optical
systems to distinguish between closely
spaced objects, which are limited because
of the wave nature of light
If no diffraction occurred, two distinct bright
spots would be observed on the viewing
screen. However, because of diffraction,
each source is imaged as a bright central
region flanked by weaker bright and dark
bands.
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If the two sources are separated enough to keep their
central maxima from overlapping, their images can be
distinguished and are said to be resolved.
If the sources are close together, however, the two
central maxima overlap and the images are not resolved.
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Rayleigh's criterion
To decide when two images are resolved,
the following criterion is used:
When the central maximum of one image
falls on the first minimum another image, the
images are said to be just resolved.
This limiting condition of resolution is known
as Rayleigh's criterion.
The diffraction patterns of two point sources (solid curves)
and the resultant pattern (dashed curves) for various
angular separations of the sources
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From Rayleigh's criterion, we can determine
the minimum angular separation, θmin ,
subtended by the sources at the slit so that
their images are just resolved.
the first minimum in a single-slit diffraction
pattern occurs at the angle for which
sin θ = λ / a
where a is the width of the slit. According to
Rayleigh's criterion, this expression gives
the smallest angular separation for which
the two images are resolved.
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Because λ « a in most situations,
sin θ is small and we can use the
approximation
sin θ ≈ θ . Therefore, the limiting angle
of resolution for a slit of width a is
θmin = λ / a
where θmin is expressed in radians.
Hence, the angle subtended by the two
sources at the slit must be greater than
λ / a if the images are to be resolved.
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The diffraction pattern of a circular
aperture consists of a central circular
bright disk surrounded by progressively
fainter rings. The limiting angle of
resolution of the circular aperture is:
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Where D is the diameter of the aperture.
Diffraction Grating
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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 interference
and diffraction
Diffraction Grating
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The condition for
maxima is
 d sin θbright = m λ
 m = 0, 1, 2, …
The integer m is the
order number of the
diffraction pattern
If the incident radiation
contains several
wavelengths, each
wavelength deviates
through a specific angle
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All the wavelengths are
focused at m = 0
 This is called the zeroth
order maximum
The first order maximum
corresponds to m = 1
Note the sharpness of the
principle maxima and the
broad range of the dark
area
 This is in contrast to the
broad, bright fringes
characteristic of the twoslit interference pattern
diffraction grating spectrometer.
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The collimated
beam incident
on the grating is
spread into its
various wavelength
components with
constructive interference for a particular wavelength
occurring at the angles that satisfy the equation
Resolving power of the diffraction grating
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The diffraction grating is useful for
measuring wavelengths accurately.
Like the prism, the diffraction grating can be
used to disperse a spectrum into its
components.
Of the two devices, the grating may be more
precise if one wants to distinguish between
two closely spaced wavelengths.
If λ1 and λ2 are the two nearly equal
wavelengths between which the
spectrometer can barely distinguish, the
resolving power R is defined as
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where λ = ( λ1 + λ2 ) / 2 , and
Δ λ = λ2 - λ1
a grating that has a high resolving power
can distinguish small differences in
wavelength.
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if N lines of the grating are illuminated, it
can be shown that the resolving power in
the mth order diffraction equals the product
N m:
R = N m
Thus, resolving power increases with
increasing order number.
R is large for a grating that has a large
number of illuminated slits.
Consider the second-order diffraction pattern
(m = 2) of a grating that has 5000 rulings
illuminated by the light source.
 The resolving power of such a grating in
second order is: R = 5000 x 2 = 10,000.
 The minimum wavelength separation
between two spectral lines that can be just
resolved, assuming a mean wavelength of
600 nm, is
 Δλ = λ / R = 6.00 X 10-2 nm.
 For the third-order principal maximum,
R = 15 000 and Δλ = 4.00 x 2 nm, and so on.
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