Interference

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					Interference and Diffraction
Certain phenomena require the light
  (the electromagnetic radiation)
       to be treated as waves.

    Two relevant examples are:
    interference and diffraction
             Plane Electromagnetic Waves


               Ey                                     ˆ
                             E(x, t) = EP sin (kx-t) j
                                                      ˆ
                             B(x, t) = BP sin (kx-t) z
       Bz


Waves are in phase,                                c
but fields oriented at 900
Speed of wave is c                                     x
c  1 / 00  3 108 m / s
At all times E = c B
          Plane Electromagnetic Waves


                                   E and B are perpendicular
                                   to each other, and to the
                                   direction of propagation
                                   of the wave.



The direction of propagation
is given by the right hand            Electromagnetic waves
rule: Curl the fingers from E         propagate in vacuum
to B, then the thumb points           with speed c, the speed
in the direction of propagation.      of light.
                 Combination of Waves

In general, when we combine two waves to form a composite wave,
the composite wave is the algebraic sum of the two original waves,
point by point in space [Superposition Principle].
When we add the two waves we need to take into account their:
•   Direction
•   Amplitude
•   Phase




                 +              =
                Combination of Waves
The combining of two waves to form a composite wave is called:
                        Interference



            +              =

                               Constructive interference
(Waves almost in phase)

              The interference is constructive
             if the waves reinforce each other.
                 Combination of Waves
The combining of two waves to form a composite wave is called:
                        Interference

                                 (Waves almost cancel.)
             +               =
                                 Destructive interference
 (Close to p out of phase)


                The interference is destructive
           if the waves tend to cancel each other.
       Interference of Waves




       +                =

                            Constructive interference
  (In phase)




       +            =
                                (Waves cancel)

( p out of phase)           Destructive interference
                  Interference of Waves
           When light waves travel different paths,
           and are then recombined, they interfere.
     1                           Each wave has an electric field
*   2
                                  whose amplitude goes like:
                                     E(s,t) = E0 sin(ks-t) î

         Mirror                  Here s measures the distance
                                 traveled along each wave’s path.



              +              =


Constructive interference results when light paths differ
  by an integer multiple of the wavelength: s = m 
                 Interference of Waves
          When light waves travel different paths,
          and are then recombined, they interfere.
    1                           Each wave has an electric field
*   2
                                 whose amplitude goes like:
                                    E(s,t) = E0 sin(ks-t) î

        Mirror                  Here s measures the distance
                                traveled along each wave’s path.


                 +          =


 Destructive interference results when light paths differ
by an odd multiple of a half wavelength: s = (2m+1) /2
              Interference of Waves

Coherence: Most light will only have interference for small
optical path differences (a few wavelengths), because the
phase is not well defined over a long distance. That’s
because most light comes in many short bursts strung
together.
                           Incoherent light: (light bulb)



               random phase “jumps”
              Interference of Waves

Coherence: Most light will only have interference for small
optical path differences (a few wavelengths), because the
phase is not well defined over a long distance. That’s
because most light comes in many short bursts strung
together.
                           Incoherent light: (light bulb)



                random phase “jumps”

Laser light is an exception:     Coherent Light: (laser)
        Thin Film Interference
We have all seen the effect of colored reflections
from thin oil films, or from soap bubbles.




                 Film; e.g. oil on water
        Thin Film Interference
We have all seen the effect of colored reflections
from thin oil films, or from soap bubbles.

                           Rays reflected off the lower
                           surface travel a longer
                           optical path than rays
                           reflected off upper surface.
                 Film; e.g. oil on water
        Thin Film Interference
We have all seen the effect of colored reflections
from thin oil films, or from soap bubbles.

                           Rays reflected off the lower
                           surface travel a longer
                           optical path than rays
                           reflected off upper surface.
                 Film; e.g. oil on water

                           If the optical paths differ by
                           a multiple of , the reflected
                           waves add.
                           If the paths cause a phase
                           difference p, reflected waves
                           cancel out.
                    Thin Film Interference
Ray 1 has a phase change of phase of p upon reflection
Ray 2 travels an extra distance 2t (normal incidence approximation)


         1
                2      n=1
                               oil on water
 t                     n>1     optical film on glass
                               soap bubble



Constructive interference: rays 1 and 2 are in phase
 2 t = m n + ½ n  2 n t = (m + ½)  [n = /n]
Destructive interference: rays 1 and 2 are p out of phase
 2 t = m n  2 n t = m 
                  Thin Film Interference

 When ray 2 is in phase with ray 1, they add up constructively
 and we see a bright region.
Different wavelengths will tend to add constructively at
different angles, and we see bands of different colors.

        1
              2      n=1
                              oil on water
t                    n>1      optical film on glass
                              soap bubble
                      Thin films work with even low
                      coherence light, as paths are short

When ray 2 is p out of phase, the rays interfere destructively.
This is how anti-reflection coatings work.
               Diffraction
What happens when a planar wavefront
of light interacts with an aperture?

                     If the aperture is large
                     compared to the wavelength
                     this would be expected....




                       …Light propagating
                       in a straight path.
              Diffraction

If the aperture is small
compared to the wavelength,
would this be expected?




                              Not really…
                        Diffraction

If the aperture is small compared to the wavelength,
would the same straight propagation be expected? … Not really

                                        In fact, what
                                        happens is that:
                                        a spherical wave
                                        propagates out
                                        from the aperture.

                                          All waves behave
                                               this way.

  This phenomenon of light spreading in a broad pattern,
  instead of following a straight path, is called: DIFFRACTION
                    Diffraction
Slit width = a, wavelength =                  I




                                         q



                   Angular Spread: q ~   /a
                Huygen’s Principle

Huygen first explained this in 1678 by proposing that all
planar wavefronts are made up of lots of spherical
wavefronts..
               Huygen’s Principle

Huygen first explained this in 1678 by proposing that all
planar wavefronts are made up of lots of spherical
wavefronts..

                        That is, you see how light propagates
                        by breaking a wavefront into little
                        bits
               Huygen’s Principle

Huygen first explained this in 1678 by proposing that all
planar wavefronts are made up of lots of spherical
wavefronts..

                        That is, you see how light propagates
                        by breaking a wavefront into little
                        bits, and then draw a spherical wave
                        emanating outward from each little
                        bit.
                Huygen’s Principle

Huygen first explained this in 1678 by proposing that all
planar wavefronts are made up of lots of spherical
wavefronts..

                        That is, you see how light propagates
                        by breaking a wavefront into little
                        bits, and then draw a spherical wave
                        emanating outward from each little
                        bit. You then can find the leading edge
                        a little later simply by summing all
                        these little “wavelets”
                  Huygen’s Principle
      Huygen first explained this in 1678 by proposing
      that all planar wavefronts are made up of lots of
      spherical wavefronts..

                         That is, you see how light propagates
                         by breaking a wavefront into little
                         bits, and then draw a spherical wave
                         emanating outward from each little
                         bit. You then can find the leading edge
                         a little later simply by summing all
                         these little “wavelets”



It is possible to explain reflection and refraction this way too.
Diffraction at Edges




              what happens to the
              shape of the field at this
              point?
               Diffraction at Edges

Light gets diffracted at the edge of an opaque barrier
 there is light in the region obstructed by the barrier

                                                   I
Double-Slit Interference



                  Because they
                  spread, these
                  waves will
                  eventually
                  interfere with
                  one another and
                  produce
                  interference
                  fringes
Double-Slit Interference
Double-Slit Interference
Double-Slit Interference
Double-Slit Interference




                           Bright
                           fringes
Double-Slit Interference

                                     screen




                           Bright
                           fringes
     Double-Slit Interference

                                                  screen




                                       Bright
                                       fringes




   Thomas Young (1802) used double-slit
interference to prove the wave nature of light.
                    Double-Slit Interference
Light from the two slits travels different distances to the screen.
The difference r1 - r2 is very nearly d sinq. When the path
difference is a multiple of the wavelength these add constructively,
and when it’s a half-multiple they cancel.

                          P


              r1
                            y
               r2

  d
          q
               L
                    Double-Slit Interference
Light from the two slits travels different distances to the screen.
The difference r1 - r2 is very nearly d sin q. When the path
difference is a multiple of the wavelength these add constructively,
and when it’s a half-multiple they cancel.

                                 d sin q = m   bright fringes
                           P
                                d sin q = ( m+1/2)   dark fringes
              r1
                                  Now use y = L tan q; and for
               r2           y     small y  sin q  tan q = y / L

  d                                   y bright = mL/d
          q
               L                      y dark = (m+ 1/2)L/d
               Multiple Slit Interference

With more than two slits, things get a little more complicated


                              P


                               y



  d

                 L
               Multiple Slit Interference

With more than two slits, things get a little more complicated
                                    Now to get a bright
                                    fringe, many paths
                              P
                                    must all be in phase.

                               y    The brightest fringes
                                    become narrower but
                                    brighter;
  d
                                    and extra lines show up
                 L
                                    between them.
Such an array of slits is called a “Diffraction Grating”
             Multiple Slit Interference

   The most intense diffraction lines appear when:
                      d sin(q) = m 
      Note that each wavelength  is diffracted
                at a different angle q

  S


                                                         q
All of the lines (more intense and less intense) show up at the
set of angles given by: d sinq = (n/N)  (N = number of slits).
                   Single Slit Diffraction
Each point in the slit acts as a source of spherical wavelets

                                Slit width a



                                                       q



For a particular direction q, wavelets will                               I
                                                            Intensity
interfere, either constructively or destructively,         distribution
resulting in the intensity distribution shown.
                   Angular Spread: q ~  / a
               The Diffraction Limit
           Diffraction imposes a fundamental limit
           on the resolution of optical systems:
        Suppose we want to image 2 distant points,
        S1 and S2, through an aperture of width a:

Two points are resolved when
  the maximum of one is at
 the minimum of the second                         L
 The minima occurs for           S1
     sin q =  / a                                q      D
     a = slit width              S2

Using sin q  q  qmin =  / a        Dmin / L   / a
       Example: Double Slit Interference
Light of wavelength  = 500 nm is incident on a double
slit spaced by d = 50 m. What is the fringe spacing on
the screen, 50 cm away?


               d
                      50 cm
       Example: Double Slit Interference
Light of wavelength  = 500 nm is incident on a double
slit spaced by d = 50 m. What is the fringe spacing on
the screen, 50 cm away?


               d
                        50 cm

   y  L / d
                  
       50  10 m 500  10 m 50  10 m
                   2            9              6


       5mm
         Example: Single Slit Diffraction
Light of wavelength  =
500 nm is incident on a
slit a=50 m wide. How
                            a
wide is the intensity
distribution on the
screen, 50 cm away?                50 cm
          Example: Single Slit Diffraction
Light of wavelength  =
500 nm is incident on a
slit a=50 m wide. How
                                  a
wide is the intensity
distribution on the
screen, 50 cm away?                          50 cm
 q   / a
                       
 y  L q  50  10 m 500  10 m 50  10 m  5mm
                     2            9                6




What happens if the slit width is doubled?
The spread gets cut in half.
Beams of green ( = 520 nm) and red ( = 650 nm) light
impinge normally on a grating with 10000 lines per cm.
What is the separation y of the first order diffracted beams
on a screen, parallel to the grating, and located at a distance
L = 50 cm away from it?

           Grating equation  d sin q = m 

   sin q  y/L for small angles, and m = 1 for first order
                10000 lines/cm  d = 0.0001
                 d y / L =   y =  L /d

            Y green = 520x10-7x50 / 0.0001 = 26 cm
            Y red = 650x10-7x50 / 0.0001 = 32.5 cm
                          y = 6.5 cm

				
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