Diffraction of Light Waves

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Chapter 4

• Diffraction of Light Waves

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Diffraction

• Huygens 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

4

• 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

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Fraunhofer Diffraction

• 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 (? 0)
  with alternating bright and dark fringes on each
  side

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Single Slit Diffraction

• According to Huygens 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 ?

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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 ?
• 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 ?dark m? / a
• m ?1, ?2, ?3,
• Doesnt 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

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Resolution of Single-Slitand Circular Apertures

• 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.

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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, ?min , 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 ? ? / 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 ? a in most situations,
• sin ? is small and we can use the approximation
• sin ? ? . Therefore, the limiting angle of
  resolution for a slit of width a is
• ?min ? / a
  
• where ?min is expressed in radians. Hence, the
  angle subtended by the two sources at the slit
  must be greater than ? / 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
• Where D is the diameter of the aperture.

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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 interference
  and diffraction

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Diffraction Grating

• The condition for maxima is
• d sin ?bright m ?
• 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 two-slit interference
  pattern

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diffraction grating spectrometer.

• 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

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Resolving power of the diffraction grating

• 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.

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• If ?1 and ?2 are the two nearly equal
  wavelengths between which the spectrometer can
  barely distinguish, the resolving power R is
  defined as
• where ? ( ?1 ?2 ) / 2 , and
• ? ? ?2 - ?1
• 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.

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• 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
• ?? ? / R 6.00 X 10-2 nm.
• For the third-order principal maximum,
• R 15 000 and ?? 4.00 x 2 nm, and so on.