Interference and Diffraction
of EM waves
Maxwell Equations in General Form
Differential form Integral Form
Gauss’s Law for E
field.
Gauss’s Law for H
field. Nonexistence
of monopole
Faraday’s Law
Ampere’s Circuit
Law
vD
0 B
t
BE
t
DJH
v
v
s
dvdSD
0s
dSB
sL
dSBt
dlE
sL
dSt
DJdlH
Terms
• E = electric field intensity [V/m]
• D = electric field density
• H = magnetic field intensity, [A/m]
• B = magnetic field density, [Teslas]
Maxwell’s Equations
• Additionally the equation of continuity
• Maxwell added the term to Ampere’s Law so
that it not only works for static conditions but
also for time-varying situations.
• This added term is called the displacement
current density, while J is the conduction current.
tJ v
t
D
Energy (intensity) of electromagnetic waves
• The frequency of light is very high,
• There is no such detector to measure the electric
field changes,
• We are able only to measure the mean value of the square
root of the electric field,
• our eyes can detect only intensity of light, not phase.
Energy (intensity) of electromagnetic
waves
Poynting vector
Intensity of EM wave
• Light intensity I is a mean velue of square root of the
electric field intensity and is defined in W/m²
• Taking into account the spectral characteristic of human eye,
light intensity is defined in candelas, lumens or lux
Two waves interfering with each other
If two monochromatic waves described as:
will overlap in some plane x=const, then:
Responsible for interference
Two waves interfering with each other
Responsible for interference
For : > 0 constructive interference
= 0
destructive interference< 0
The same phases The oposite phases
Constructive interference Destructive interference
The key principle: Huygen’s Principle
Christian Huygens
1629-1695
All points in a wavefront serve as point
sources of spherical secondary waves.
After a time t, the new wavefront will be
the tangent to all the resulting spherical
waves.
Huygen’s Principle
For plane waves entering a single slit, the waves emerging from the slit start spreading out, diffracting
Young’s Double Slit ExperimentFor waves entering two slits, the emerging waves interfere and form an interference (diffraction) pattern
Young experiment in 1801: light is wave phenomenon
First plane wave through a small slit yields coherent spherical wave
Then interposed two slits: interference of two spherical waves on a screen
Interference• Phase difference between two waves can change for paths of different lengths
• Each point on the screen is determined by the path length difference DL of the rays reaching that point
Path Length Difference: sinL d D
If sin integer bright fringeL d D
Maxima-bright fringes:
sin for 0,1,2,d m m
Minima-dark fringes: 12
sin for 0,1,2,d m m
1 1.51 dark fringe at: sinm
d
1 22 bright fringe at: sinm
d
Interference
Two sources can produce an interference that is stable over
time, if their light has a phase relationship that does not
change with time: E(t)=E0cos(wt+f).
Coherent sources: Phase f must be well defined and constant
Sunlight is coherent over a short length and time range
Since laser light is produced by cooperative behavior of atoms,
it is coherent of long length and time ranges
Incoherent sources: f jitters randomly in time, no stable
interference occurs
When the interference is possible
Red laser light (=633nm) goes through two slits d=1cm apart, and produces a
diffraction pattern on a screen L = 55cm away. How far apart are the fringes near
the center?
For the spacing to be 1mm, we need d~ L/1mm=0.35mm
If the fringes are near the center, we can use
sin ~ , and then
m=dsin~d => =m/d is the angle for each
maximum (in radians)
D= /d =is the “angular separation”.
The distance between the fringes is then
Dx=LD=L/d=55cm 633nm/1cm=35 mm
Example of interference
Example 2
In a double slit experiment, we can measure the wavelength of the light if we
know the distances between the slits and the angular separation of the fringes.
If the separation between the slits is 0.5mm and the first order maximum of the
interference pattern is at an angle of 0.059o from the center of the pattern, what
is the wavelength and color of the light used?
d sin=m => =0.5mm sin(0.059o)= 5.15 x 10-7m=515nm ~ green
1 0 2 0sin and sinE E t E E tw w f
E1
E2
2 10 2
4 cosI I f2
sind
f
1 1 12 2 2
Minima when: sin for 0,1,2, (minima)m d m mf
12
2Maxima when: for 0,1,2, 2 sin
sin for 0,1,2, (maxima)
dm m m
d m m
f f
avg 02I I
Intensity in Double-Slit Interference
ExampleA double slit experiment has a screen 120cm away from the slits, which are 0.25cm apart. The slits are illuminated with coherent 600nm light. At what distance above the central maximum is the average intensity on the screen 75% of the maximum?
I/I0=4cos2f/2 ; I/Imax=cos2f/2 =0.75 => f=2cos–1 (0.75)1/2=60o=/3 radf=(2d/)sin => = sin-1(/2d)f0.0022o40 mrad (small!)y=L48mm
Interferometers
Michelson’s
Mach-Zehnder’s
Ring
Temporal coherence
White light
LED
SLED
LD
Gas laser
He-Ne
D
Measuring the distance with Michelson’s interferometer
Optical coherence tomography
Diffraction
Huygen’s Principle
Christian Huygens
1629-1695
All points in a wavefront serve as point
sources of spherical secondary waves.
After a time t, the new wavefront will be
the tangent to all the resulting spherical
waves.
Huygen’s Principle
For plane waves entering a single slit, the waves emerging from the slit start spreading out, diffracting
Young’s Double Slit ExperimentFor waves entering two slits, the emerging waves interfere and form an interference (diffraction) pattern
Young experiment in 1801: light is wave phenomenon
First plane wave through a small slit yields coherent spherical wave
Then interposed two slits: interference of two spherical waves on a screen
• Path length difference between rays
r1 and r2 is /2
• Two rays out of phase at P1 resulting in
destructive interference
• Path length difference is distance from
starting point of r2 at center of the slit to
point b
• For D>>a, the path length difference
between rays r1 and r2 is (a/2) sin
Diffraction by a Single Slit:
Locating the Minima
Repeat previous analysis for pairs of rays, each separated by a
vertical distance of a/2 at the slit.
Setting path length difference to /2 for each pair of rays, we obtain
the first dark fringes at:
(first minimum)sin sin2 2
aa
For second minimum, divide slit into 4 zones of equal widths a/4
(separation between pairs of rays). Destructive interference occurs
when the path length difference for each pair is /2.
(second minimum)sin sin 24 2
aa
Dividing the slit into increasingly larger even numbers of zones, we
can find higher order minima:
(minima-dark fringes)sin , for 1,2,3a m m
Diffraction pattern from a single narrow slit.
Diffraction and the Wave Theory of Light
Central
maximum
Side or secondary
maxima
Light
Fresnel Bright Spot.
Bright
spot
Light
These patterns cannot be explained
using geometrical optics!
Single Slit Diffraction
When light goes through a narrow slit, it spreads out to form a diffraction pattern.
Here we will show that the intensity at the screen due to a single slit
is:
Intensity in Single-Slit Diffraction,
Quantitatively
2
sin (36-5)mI I
1where sin (36-6)
2
a f
, for 1,2,3m m
In Eq. 1 , minima occur when:
sin , for 1,2,3
or sin , for 1,2,3
(minima-dark fringes)
am m
a m m
If we put this into Eq. 2 we find:
(1)
(2)
Diffraction of a laser through a slit
(example)
Light from a helium-neon laser ( = 633 nm) passes through a narrow slit and is seen on a screen 2.0 m behind the slit. The first minimum of the diffraction pattern is observed to be located 1.2 cm from the central maximum.
How wide is the slit?
11
(0.012 m)0.0060 rad
(2.00 m)
y
L
74
3
1 1
(6.33 10 m)1.06 10 m 0.106 mm
sin (6.00 10 rad)a
1.2 cm
Width of a Single-SlitDiffraction Pattern
; 1,2,3, (positions of dark fringes)p
p Ly p
a
2(width of diffraction peak from min to min)
Lw
a
w
-y1 y1 y2 y30
X-band: =10cm
You are doing 137 mph on I-10 and you pass a little old lady doing 55mph when a cop, Located 1km away fires his radar gun, which has a 10 cm opening. Can he tell you from the L.O.L. if the gun Is X-band? What about Laser?
1m
1m
10 m1000m
w 2L
a2 0.1m 1000m
0.1m 2000m w
2L
a2 0.000001m 1000m
0.1m 0.02m
Laser-band: =1mm
Angles of the Secondary Maxima
The diffraction minima
are precisely at the angles
where
sin q = p l/a and a = pp (so
that sin a=0).
However, the
diffraction maxima are not
quite at the angles where
sin q = (p+½) l/a
and a = (p+½)p
(so that |sin a|=1).
p (p+½) /a Max
1 0.00475 0.00453
2 0.00791 0.00778
3 0.01108 0.01099
4 0.01424 0.01417
5 0.01741 0.01735
1
2
34 5
l = 633 nm
a = 0.2 mm
q (radians)
2
max
sinI I
A device with N slits (rulings) can be used to manipulate light, such as separate
different wavelengths of light that are contained in a single beam. How does a
diffraction grating affect monochromatic light?
Diffraction Gratings
Fig. 36-17 Fig. 36-18
sin for 0,1,2 (maxima-lines)d m m
(36-11)
Circular Apertures
When light passes through a circular aperture instead of a vertical slit, the diffraction pattern is modified by the 2D geometry. The minima occur at about 1.22/D instead of /a. Otherwise the behavior is the same, including the spread of the diffraction pattern with decreasing aperture.
Single slit of aperture aHole of diameter D
The Rayleigh Criterion
The Rayleigh Resolution Criterion says that the minimum separation to separate two objects is to have the diffraction peak of one at the diffraction minimum of the other, i.e., D 1.22
/D.
Example: The Hubble Space Telescope has a mirror diameter of 4 m, leading to excellent resolution of close-lying objects. For light with wavelength of 500 nm, the angular resolution of the Hubble is D = 1.53 x 10-7 radians.
Example
A spy satellite in a 200km low-Earth orbit is imaging the Earth in the visible wavelength of 500nm.
How big a diameter telescope does it need to read a newspaper over your shoulder from Outer Space?
D 1.22 /D (The smaller the wavelength or the bigger the telescope opening — the better the angular resolution.)
Letters on a newspaper are about Dx = 10mm apart.Orbit altitude R = 200km & D is telescope diameter.
Formula:Dx = RD = R(1.22/D)
D = R(1.22/Dx)
= (200x103m)(1.22x500x10–9m)/(10X10–3m)
= 12.2m
Example Solution
R
Dx
D
Holography
Brief history of holography
• Invented in 1948 by Dennis Gabor – to improve
the resolution in electron microscopy, before the
invention of the laser (this time light sources
were not coherent)
• Leith and Upatnieks (1962) applied laser light to
holography and introduced an important off-axis
technique (the first holographic picture, laser
was necessary to see the picture)
• The pioneer of holography in Poland –
prof. Mieczysław Wolfke (professor from Faculty
of Physics WUT),
Holos - whole, grapho – drawing
•Holography is a method of producing a three-dimensional (3-D) image of an object. (The three dimensions are height, width, and depth.)
•Later the object can be reconstructed.
•The hologram is actually a recording of the difference between two
beams of coherent light
Can be used as optical store disk, in information processing,
Conventional vs. Holographic picture
• Conventional:
– 2-d version of a 3-d scene
– Photograph lacks depth perception or parallax
– Film sensitive only to radiant energy
– Phase relation (i.e. interference) are lost
Conventional photography
Light
Object
Reflected
wave
Photographic film:
The intensity is
recorded
Conventional photography
Light
• Hologram:– Freezes the intricate wavefront of light that carries all
the visual information of the scene
– To view a hologram, the wavefront is reconstructed
– View what we would have seen if present at the original scene through the window defined by the hologram
– Provides depth perception and parallax
Conventional vs. Holographic picture
Conventional vs. Holographic picture
– Converts phase information into amplitude information (in-phase - maximum amplitude, out-of-phase – minimum amplitude)
– Interfere wavefront of light from a scene with a reference wave
– The hologram is a complex interference pattern of microscopically spaced fringes
How hologram is made?
• Need a laser, lenses, mirror, photographic film,
and an object
• The laser light is separated into two beams,
reference beam and object beam
• Reference beam enlarged and aimed at a piece
of holographic film
How hologram is made?
• Object beam directed at subject to be recorded
and expanded to illuminate subject,
• Object beam reflects off of object and meets
reference beam at film,
• Produces interference pattern which is recorded,
Reference wave
Photographic film.
Interference of reference and
reflected waves is recorded
How hologram is made?
How hologram is made ?
• Film is developed,
• Hologram illuminated at same angle as
reference beam during original exposure
to reveal holographic image,
Types of holograms
Transmission hologram:
reference and object waves
traverse the film from the
same side
Reflection hologram:
reference and object waves
traverse the emulsion from
opposite sidesInvented by Benton,
Can be reconstructed
in normal light
Applications of holography
• Credit cards carry monetary value,
• Supermarket scanners,
• Optical Computers,
• Used in aircraft “heads-up display”,
• Art,
• Archival Recording of fragile museum artifacts,
Holography in the future
• Medical Purposes
• Gaming Systems
• Personal Defense
• Computers
• Artwork
• Amusement Park Rides
• Movie production
– Holodeck from Star Trek Holodeck Clip
– Star Wars Chess Game
Summary
• Interference only for coherent light, i.e., with a phase relationship that is time independent
• Intensity in double-slit interference:
• Use Huygens’ Principle to find positions of
diffraction minima of a single slit by subdividing
the aperture
(minima-dark fringes)sin , for 1,2,3a m m
2 10 2
4 cosI I f2
sind
f
Summary
• Diffraction of light occurs at openings of the order of the wave length of the light
• Double slit experiment:
• Intensity in double-slit interference:
Maxima-bright fringes:
sin for 0,1,2,d m m
Minima-dark fringes: 12
sin for 0,1,2,d m m
2 10 2
4 cosI I f2
sind
f
Summary
• To predict the interference pattern of a multi-slit
system, we must combine interference and
diffraction effects.
• Rayleigh’s Criterion for the separability of two
points
• Intensity in single-slit diffraction:
• Double-slit diffraction:
2
sin (36-5)mI I
1where sin (36-6)
2
a f
2
2 sincos (double slit)mI I
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