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					Reflection & Refraction
             The Phase Difference
  1  kx1  t  10                     2  kx2  t  20
     1  2  (kx1  t  10 )  ( kx2  t  20 )
                                                      x
         k ( x1  x2 )  (10  20 )  2                 0
                                                       

            Path-length difference                              Inherent phase difference

If the waves are initially in-phase

                                     2          2
                                     x         n x  k0 
                                                0

 n x  nx1  x2       The optical path difference (OPD)


If 0 is constant waves are said to be coherent.
             The Phase Difference
The condition of being in phase, where crests are
aligned with crests and troughs with troughs, is that
   = 0, 2, 4, or any integer multiple of 2.

        For constructive interference:
                    x                     x 0
            2         0  2m or           m
                                         2    2

 For identical sources, 0 = 0 rad , maximum constructive
 interference occurs when x = m ,

 Two identical sources produce maximum constructive
 interference when the path-length difference is an
 integer number of wavelengths.
             The Phase Difference
The condition of being out of phase, where crests are
aligned with troughs of other, that is,
   =, 3, 5 or any odd multiple of .
               For destructive interference:
                           x
                   2         0  2(m  1 ) or
                                             2

                  x 0
                          m 1
                 2    2      2


 For identical sources, 0 = 0 rad , maximum constructive
 interference occurs when x = (m+ ½ ) ,

 Two identical sources produce perfect destructive
 interference when the path-length difference is
 half-integer number of wavelengths.
We’ll check the interference one
direction at a time, usually far away.
This way we can approximate spherical waves by plane waves in
that direction, vastly simplifying the math.




                                                       Far away,
                                                       spherical wave-
                                                       fronts are almost
                                                       flat…



Usually, coherent constructive interference will occur in one direction,
and destructive interference will occur in all others.
If incoherent interference occurs, it is usually omni-directional.
To understand scattering in a given
situation, we compute phase delays.

                                     Wave-fronts
 Because the phase is
 constant along a
                                               L1
 wave-front, we
 compute the phase                              L2
 delay from one wave-
                                                    L3      Potential
 front to another                                           wave-front
 potential wave-front.                               L4
                                                          i  k Li
                                  Scatterer

If the phase delay for all scattered waves is the same (modulo 2),
then the scattering is constructive and coherent. If it varies
continuously from 0 to 2, then it’s destructive and coherent.
If it’s random (perhaps due to random motion), then it’s incoherent.
Scattered spherical
waves often combine
to form plane waves.


  A plane wave impinging on a
  surface (that is, lots of very small
  closely spaced scatterers!) will
  produce a reflected plane wave
  because all the spherical
  wavelets interfere constructively
  along a flat surface.
Coherent constructive scattering:
Reflection from a smooth surface when angle
of incidence equals angle of reflection
    A beam can only remain a plane wave if there’s a direction for which
                            coherent constructive interference occurs.

  The wave-fronts are
  perpendicular to
  the k-vectors.                qi qr


                                                    Consider the
                                                   different phase
                                                      delays for
                                                   different paths.


Coherent constructive interference occurs for a reflected beam if the
angle of incidence = the angle of reflection: qi = qr.
Coherent destructive scattering:
Reflection from a smooth surface when the
angle of incidence is not the angle of reflection
                       Imagine that the reflection angle is too big.
    The symmetry is now gone, and the phases are now all different.


       = ka sin(qtoo big)   qi qtoo big        = ka sin(qi)




                                                  Potential
                                                  wave front
                               a


Coherent destructive interference occurs for a reflected beam direction
if the angle of incidence ≠ the angle of reflection: qi ≠ qr.
Coherent scattering usually occurs in one (or
a few) directions, with coherent destructive
scattering occurring in all others.

A smooth surface scatters light coherently and constructively only in
the direction whose angle of reflection equals the angle of incidence.




Looking from any other direction, you’ll see no light at all due to
coherent destructive interference.
Incoherent scattering: reflection from a
rough surface


                                    No matter which direction we
                                    look at it, each scattered wave
                                    from a rough surface has a
                                    different phase. So scattering is
                                    incoherent, and we’ll see weak
                                    light in all directions.




Coherent scattering typically occurs in only one or a few directions;
incoherent scattering occurs in all directions.
Why can’t we see a light beam?

Unless the light beam is propagating right into your eye or is scattered
into it, you won’t see it. This is true for laser light and flashlights.

This is due to the facts that air is very sparse (N is relatively small), air
is also not a strong scatterer, and the scattering is incoherent.


                                          This eye sees almost no light.



                                                  This eye is blinded
                                                  (don’t try this at home…)


To photograph light beams in laser labs, you need to blow some
smoke into the beam…
What about light that scatters on
transmission through a surface?
Again, a beam can remain a plane wave if there is a direction for
which constructive interference occurs.

                                       Constructive interference
                                       will occur for a transmitted
                                       beam if Snell's Law is
                                       obeyed.
ni sin q i  nt sin q t   Snell’s Law
On-axis vs. off-axis light scattering
 Forward (on-axis) light                 Off-axis light scattering: scattered
 scattering: scattered                      wavelets have random relative
 wavelets have nonrandom                  phases in the direction of interest
 (equal!) relative phases in                 due to the often random place-
 the forward direction.                       ment of molecular scatterers.




   Forward scattering is coherent—            Off-axis scattering is incoherent
   even if the scatterers are randomly        when the scatterers are randomly
   arranged in space.                         arranged in space.

       Path lengths are equal.                   Path lengths are random.
Scattering from a crystal vs. scattering
from amorphous material (e.g., glass)
A perfect crystal has perfectly regularly spaced scatterers in space.


                                             So the scattering from
                                             inside the crystal cancels
                                             out perfectly in all directions
                                             (except for the forward and
                                             perhaps a few other
                                             preferred directions).


Of course, no crystal is perfect, so there is still some scattering, but
usually less than in a material with random structure, like glass.

There will still be scattering from the surfaces because the air nearby
is different and breaks the symmetry!
Scattering from large particles
For large particles, we must first consider the fine-scale scattering
from the surface microstructure and then integrate over the larger
scale structure.
If the surface isn’t smooth, the scattering is incoherent.
If the surfaces are smooth, then we use Snell’s Law and angle-of-
incidence-equals-angle-of-reflection.




Then we add up all the
waves resulting from all
the input waves, taking into account their coherence, too.
Diffraction Gratings
If light impinges on a periodic
array of grooves, scattering ideas
tell us what happens. There will
be constructive interference if the
delay between adjacent beamlets
is an integral number of
wavelengths.

 a sin(q m )  sin(qi )  m
where m is any integer.                  Path
                                      difference
A grating can have solutions for      AB – CD =    AB = a sin(qm)
                                          m       CD = a sin(qi)
zero, one, or many values of m, or
“orders.”

Remember that m and the dif-
fracted angle can be negative, too.
  Diffraction orders

                                                            Because the
                 Diffraction angle, qm                      diffraction
                                                            angle depends
                                             First order    on , different
                                                            wavelengths
                                             Zeroth order   are separated
                                                            in the +1 (and
                                             Minus          -1) orders.
                                             first order    No wavelength
                                                            dependence in
                                                            zero order.


The longer the wavelength, the larger its diffraction angle in nonzero orders.
Diffraction-grating dispersion
It’s helpful to know the variation of the diffracted angle vs. wavelength.

Differentiating the grating equation,   a sin(qm )  sin(qi )  m
with respect to wavelength:

                                    dq m
                        a cos(q m )      m           [qi is a constant]
                                    d
Rearranging:

                       dq m    m                  Gratings typically have an
                                                 order of magnitude more
                       d  a cos(q m )            dispersion than prisms.



Thus, to separate different colors maximally, make a small, work in high
order (make m large), and use a diffraction angle near 90 degrees.
Wavelength-dependent
incoherent molecular scat-
tering: Why the sky is blue

Light from the sun                                         Air molecules
                                                           scatter light,
                                                           and the
                           Air                             scattering is
                                                           proportional to
                                                           4.




 Shorter-wavelength light is scattered out of the beam, leaving longer-
 wavelength light behind, so the sun appears yellow.
 In space, the sun is white, and the sky is black.
Sunsets involve longer path lengths
and hence more scattering.
                                                  Note the cool sunset.
       Noon ray
                      Sunset ray

         Earth

                      Atmosphere

  As you know, the sun and
  clouds can appear red.


     Edvard Munch’s “The Scream”
     was also affected by the eruption
     of Krakatoa, which poured ash
     into the sky worldwide.                 Munch Museum/Munch
                                         Ellingsen Group/VBK, Vienna