38.2 Diffraction Pattern from Narrow Slits, cont

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38.2 Diffraction Pattern from Narrow Slits, cont
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Intensity of Two-Slit Diffraction Patterns

• When more than one slit is present, consideration
  must be made of
  
• The diffraction patterns due to individual slits
• The interference due to the wave coming from
  different slits
  
• The single-slit diffraction pattern will act as
  an envelope for a two-slit interference pattern

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Intensity of Two-Slit Diffraction Patterns,
Equation

• To determine the maximum intensity
• (38.6)
• The factor in the square brackets represents the
  single-slit diffraction pattern
  
• This acts as the envelope
• The two-slit interference term is the cos2 term

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Intensity of Two-Slit Diffraction Patterns, Graph
of Pattern

• The broken blue line is the diffraction pattern
• The red-brown curve shows the cos2 term
• This term, by itself, would result in peaks with
  all the same heights
  
• The uneven heights result from the diffraction
  term (square brackets in the equation)

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Two-Slit Diffraction Patterns, Maxima and Minima

• To find which interference maximum coincides with
  the first diffraction minimum
  
• (38.7)
• The conditions for the first interference
  maximum
  
• d sin ? m?
• The conditions for the first diffraction minimum
• a sin ? ?

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Example 38.3 Two-Slit Diffraction (Quiz 38.3)

• Using the previous figure, make a sketch of the
  combined diffraction and interference pattern for
  650-nm light waves striking two 3.0-?m slits
  located 9.0-?m apart.
• First case d 18.0-?m, a 3.0-?m, so m 6
• In our case m 9/3 3.
• The 3rd interference maximum coincides with the
  1st diffraction minimum

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Active Figure 38.11
(SLIDESHOW MODE ONLY)
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38.3 Resolution

• The ability of optical systems to distinguish
  between closely spaced objects is limited because
  of the wave nature of light
  
• If two sources are far enough apart to keep their
  central maxima from overlapping, their images can
  be distinguished
  
• The images are said to be resolved
• If the two sources are close together, the two
  central maxima overlap and the images are not
  resolved

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Resolved Images, Example

• The images are resolved
• The images are far enough apart to keep their
  central maxima from overlapping
  
• The angle subtended by the sources at the slit is
  large enough for the diffraction patterns to be
  distinguishable

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Images Not Resolved, Example

• The images are not resolved
• The sources are so close together that their
  central maxima do overlap
  
• The angle subtended by the sources is so small
  that their diffraction patterns overlap

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Rayleighs Criterion

• When the central maximum of one image falls on
  the first minimum of another image, the images
  are said to be just resolved
  
• This limiting condition of resolution is called
  Rayleighs criterion
  

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Rayleighs Criterion, Equation

• The angle of separation, ?min, is the angle
  subtended by the sources for which the images are
  just resolved
  
• Since ? small and sin ? ?
  
• Therefore, the limiting angle (radians) of
  resolution for a slit of width a is
  
• (38.8)
• To be resolved, the angle subtended by the two
  sources must be greater than ?min

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Circular Apertures

• Many optical systems use circular apertures
  rather than slits
  
• The diffraction pattern of a circular aperture
  consists of a central bright disk surrounded by
  progressively fainter bright and dark rings
  
• The limiting angle of resolution of the circular
  aperture is
  
• (38.9)
• D is the diameter of the aperture

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Circular Apertures, Well Resolved

• The sources are far apart
• The images are well resolved
• The solid curves are the individual diffraction
  patterns
  
• The dashed lines are the resultant pattern

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Circular Apertures, Just Resolved

• The sources are separated by an angle that
  satisfies Rayleighs criterion
  
• The images are just resolved
• The solid curves are the individual diffraction
  patterns
  
• The dashed lines are the resultant pattern

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Circular Apertures, Not Resolved

• The sources are close together
• The images are unresolved
• The solid curves are the individual diffraction
  patterns
  
• The dashed lines are the resultant pattern

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Resolution, Example

• Pluto and its moon, Charon
• Left Earth-based telescope is blurred
• Right Hubble Space Telescope clearly resolves
  the two objects

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Example 38.4 Eye and Telescope Resolution

• Estimate ?min for the human eye. Using a light
  of 500 nm and the daytime average diameter for a
  human pupil of 2.0 mm
  
• The Keck telescope at Mauna Kea, Hawaii, has an
  affective diameter of 10 m. Find the resolution
  ?min for 600-nm light

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Material for the Final Exam

• Examples to Read!!!
• Example 38.3 (Page 1216)
• Homework to be solved in Class!!!
• Problems 12, 21