wug.physics.uiuc.educcIAStatePhys222falllectu

							  Lecture 41
Single and double slit
     diffraction.
                    No sharp edges

According to geometrical optics, the shadow of a razor blade
should have a sharp boundary.




This is instead what is seen
(with monochromatic light)
                         Single-slit diffraction

    Slit of width a
        • consider it made of a large number of point sources
        • interference by N slits

                                            N slits   N phasors
                               P
                          θ
a                                                EP

                R >> a                                       β
                                            E0

                                                             2
Angle between first and last phasors:         ka sin          a sin 
                                                             
                                   Length of the arc is NE0 = Emax
                 r
                                   (amplitude when β = 0)
          β/2
    β/2                                         E max      (β in radians)
                                           
r               EP                               r
                                                                          EP
                                   From the triangle:      sin         
                                                                   2       2r
                           β
                                                       E max     
                                     EP  2r sin     2       sin
    E0                                             2             2
                                                    Emax
                                                r 
                                                     

                                                               2
                                                            
                         sin                          sin     
           E P  E max         2                           2
                                          IP  I max          
                                                            
                          2                                   
                                                      2       
 Minima:
  2m m  1, 2...
a sin  m m  1, 2...                                 2
                                                        
                                                 sin     
                                         I            2
                                                        
                                        Imax            
  Maxima:                                                
                                                 2       
       (
  b » 2m + 1 p )
  (and b = 0)

                                                              θ




                             This is the             This is light
    DEMO:
 Single-slit                “normal” part          “bending around
diffraction                                          the corner”!
                     In-class example: Single slit

    A single slit is cut in a dark film and placed R = 4.00 m away from a screen. A
    laser with λ = 700 nm shines on the slit. The first dark fringe is y = 1.00 cm above
    the center of the most intense (central) bright fringe. What is a, the width of
    the slit?
                        Destructive interference: a sin   m m  1, 2...
     A. 350 nm
                                                      y
    B. 700 nm            From the triangle: tan  
                                                      R
    C. 28 μm
                         For small angles,   sin  tan
    D. 70 μm
    E. 0.28 mm                                             y
                                                         a 
                                                           R
                                 y
                                           R  700  10 m  4.00 m 
                                                           9
                     θ
a                                       a                             2.8  10   4
                                                                                         m
                                            y      1.00  10 m 2



                 R
                                ACT: Shadow

       Visible light shines on a slit 1 cm by 5 cm. Qualitatively, which is the
       approximate picture of light on the screen?

   A                                           B




   a ~ 10-2 m        First maximum after center:   2
   λ ~ 10-9 m              2
                              a sin   2  107 sin   2         sin   107
                            
This is geometrical optics (light traveling only in a straight line)… because a >>λ
            Diffraction and wavelength

Diffraction effects (= light bending around the corner) are
most important when a ~λ




Diffraction pattern
of a square slit (with
both sides a ~λ)
            Light bends around the corner

    Huygens’ model: each point in the wavefront emits a wavelet.




                        obstacle
• Block the lower sources of induced fields with an obstacle
• Induced fields from upper sources/wavelets are still spherical
   – reach beyond blockage
   – waves effectively “bend” around corners                DEMO:
                                                          Arago (or
                                                     Poisson’s) spot
                     AM/FM radio

Examples:

  AM signal with f = 1000 kHz        λ = 300 m
  FM signal with f = 100 MHz         λ=3m

In mountainous areas, AM radio station reception is much
better because AM waves are diffracted “around” the
mountain tops and into the valleys

FM waves propagate “in a straight line” and cannot reach
the valleys (unless an antenna on top of the mountain is
used)
        Your roommate’s (awful) music

Like interference, diffraction applies to ALL waves (not just
light)

 Human ear perceives f ~ 20 Hz – 20 kHz         λ ~ 2 cm – 2 m

A door left ajar (opening ~ 1-5 cm) protects you from most of
your roomie’s awful music, but not from the bass track.



Also: That’s why all home theater speakers must be “in line of
sight”, but subwoofers can be behind the sofa.
                   Realistic double-slit

We need to take diffraction into account!
Superposition of interference and diffraction effects:

                                                   2
    Interference: IP  Imax cos2                          d sin 
                                  2                 
                                         2
                                       
                                sin     
                                     2                2
    Diffraction:    IP  I max                            a sin 
                                
                                        
                                         
                                                      
                                2       

                                                           2
                                                        
                                                 sin     
                                         2         2
  The whole nine yards:     IP  Imax cos              
                                            2  
                                                         
                                                          
                                                 2       
                                    d

 Two slit interference
                              m
 MAXIMUM when sin 
                              d




   Each slit diffraction                  a
                              m
   MINIMUM when sin 
                               a




Interference pattern modulated
by diffraction pattern
(with d >>a )

                   DEMO:
                Double-slit
   Example: Two-slit interference-diffraction


Light of λ = 550 nm illuminates two slits of width 0.03 mm and
separation 0.15 mm. How many interference maxima fall within
the full width of the central diffraction maximum?

First diffraction minimum: a sin 1  
                         550  10 9 m
                sin 1                    1  1.05
                        a 0.03  10 3 m
                                              d sin 1 d
Interference maxima: d sin  m           m          5
                                                      a
Central maxima + 5 on each side = 11

                    (But those at   1.05 are not visible
                    because of the diffraction minimum)
                   0


              –1       1




             –2            2




        –3                     3




       –4                          4
(–5)                                   (5)