Download Fresnel and Fraunhofer diffraction

Diffraction
• Interference with more than 2 beams
– 3, 4, 5 beams
– Large number of beams
• Diffraction gratings
– Equation
– Uses
• Diffraction by an aperture
–
–
–
–
–
–
Huygen’s principle again, Fresnel zones, Arago’s spot
Qualitative effects, changes with propagation distance
Fresnel number again
Imaging with an optical system, near and far field
Fraunhofer diffraction of slits and circular apertures
Resolution of optical systems
• Diffraction of a laser beam
LASERS 51
April 03
Interference from multiple apertures
L
Bright fringes when OPD=nλ
x
d
source
OPD
nLλ
x=
d
Intensity
40
two slits
position on screen
screen Complete destructive interference halfway between
OPD 1
OPD 2
OPD 1=nλ, OPD 2=2nλ
40 interfere constructively
all three waves
Intensity
d
source
three
equally spaced
slits
screen
position on screen
OPD 2=nλ, n odd
outer slits constructively interfere
middle slit gives secondary maxima
LASERS 51
April 03
Diffraction from multiple apertures
• Fringes not sinusoidal for
more than two slits
• Main peak gets narrower
2 slits
– Center location obeys same
equation
3 slits
• Secondary maxima appear
between main peaks
4 slits
– The more slits, the more
secondary maxima
– The more slits, the weaker the
secondary maxima become
5 slits
• Diffraction grating – many slits, very narrow spacing
– Main peaks become narrow and widely spaced
– Secondary peaks are too small to observe
LASERS 51
April 03
Reflection and transmission gratings
• Transmission grating – many closely spaced slits
• Reflection grating – many closely spaced reflecting regions
Input
wave
screen
opaque
Huygens
wavelets
transmitting
opening
path length to
observation point
Input
wave
wavelets
path length to
observation point
screen
Transmission grating
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absorbing
reflecting
Reflection grating
April 03
Grating equation – transmission
grating with normal incidence
input
•
•
•
•
Θd
Diffracted
light
Θd is angle of diffracted ray
λ is wavelength
l is spacing between slits
p is order of diffraction
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pλ
sin θ d =
l
Except for not making a
small angle approximation,
this is identical to formula
for location of maxima in
multiple slit problem earlier
April 03
Diffraction gratings – general
incidence angle
• Grating equation
pλ
sin θ d − sin θ i =
l
l=distance between grooves (grating spacing)
Θi=incidence angle (measured from normal) Θd
Θi
Θd=diffraction angle (measured from normal)
p=integer (order of diffraction)
• Same formula whether it’s a transmission or reflection
grating
– n=0 gives straight line propagation (for transmission grating) or
law of reflection (for reflection grating)
LASERS 51
April 03
Intensities of orders – allowed orders
• Diffraction angle can be found only for
certain values of p
strong diffracted
– If sin(Θd) is not
between –1 and 1,
there is no allowed Θd
order
weak diffracted
order
input
beam
• Intensity of other orders
are different depending
on wavelength, incidence angle,
and construction of grating
• Grating may be blazed to make
a particular order more intense than
others
Blazed grating
– angles of orders unaffected by blazing
LASERS 51
April 03
Grating constant (groove density) vs.
distance between grooves
• Usually the spacing between grooves for a grating
is not given
– Density of grooves (lines/mm) is given instead
1
– g=
l
– Grating equation can be written in terms of grating
constant
sin(Θ d ) − sin (Θ i ) = pgλ
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April 03
2nd
order
Diffraction grating - applications
1st
order
• Spectroscopy
grating
– Separate colors, similar to
prism
• Laser tuning
– narrow band mirror
– Select a single line of
multiline laser
– Select frequency in a
tunable laser
negative
orders
Littrow mounting – input
and output angles identical
Θ
2 sin (Θ ) =
λ
d
grating
• Pulse stretching and
compression
– Different colors travel
different path lengths
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two identical
gratings
April 03
Fabry-Perot Interferometer
Input
transmitted through
Beam is partially reflected and
first mirror
partially transmitted at each
mirror
Transmitted All transmitted beams interfere
Reflected
field
field
Partially
reflecting
mirrors
with each other
All reflected beams interfere with
each other
OPD depends on mirror
separation
• Multiple beam interference – division of amplitude
– As in the diffraction grating, the lines become narrow as
more beams interfere
LASERS 51
April 03
Fabry-Perot Interferometer
transmission
1
free
spectral
range,
fsr
0
Linewidth=
fsr*finesse
frequency or wavelength
• Transmission changes with frequency
– Can be very narrow range where transmission is high
• Width characterized by finesse
• Finesse is larger for higher reflectivity mirrors
– Transmission peaks are evenly spaced
• Spacing called “Free spectral range”
• Controlled by distance between mirrors, fsr=c/(2L)
• Applications
– Measurement of laser linewidth or other spectra
– Narrowing laser line
LASERS 51
April 03
Diffraction at an aperture—observations
Aperture
Light
through
aperture on
screen
downstream
• A careful observation of the light transmitted by an
aperture reveals a fringe structure not predicted by
geometrical optics
• Light is observed in what should be the shadow region
LASERS 51
April 03
Pattern on screen at various distances
2.5mm
Near Field
Immediately
behind screen
Intermediate field
25 mm from screen,
bright fringes just
inside edges
250 mm
light penetrates
into shadow
region
2500 mm
pattern doesn’t
closely resemble
mase
Far field – at a large enough distance
shape of pattern no longer changes but
it gets bigger with larger distance.
Symmetry of original mask still is
evident.
LASERS 51
April 03
Huygens-Fresnel diffraction
screen
with
aperture
Point
source
observing
screen
Wavelets
generated in
hole
• Each wavelet illuminates the observing screen
• The amplitudes produced by the various waves at the
observing screen can add with different phases
• Final result obtained by taking square of all amplitudes
added up
– Zero in shadow area
– Non-zero in illuminated area
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April 03
Fresnel zones
• Incident wave propagating to right
• What is the field at an observation point a
distance of b away?
• Start by drawing a sphere with radius
b+λ/2
• Region of wave cut out by this sphere is
the first Fresnel zone
• All the Huygens wavelets in this first
Fresnel zone arrive at the observation
point approximately in phase
• Call field amplitude at observation point
due to wavelets in first Fresnel zone, A1
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b + λ/2
First Fresnel
zone
observation
point
b
incident
wavefront
April 03
Fresnel’s zones – continued
• Divide incident wave into
additional Fresnel zones by
drawing circles with radii,
b+2λ/2, b+3λ/2, etc.
• Wavelets from any one zone
are approximately in phase
at observation point
b +λ/2 b +λ
observation
point
b
– out of phase with wavelets from a
neighboring zone
incident
wavefront
• Each zone has nearly same area
• Field at observation point due to second Fresnel zone
is A2, etc.
• All zones must add up to the uniform field that we must
have at the observation point
LASERS 51
April 03
Adding up contributions from Fresnel
zones
• A1, the amplitude due to the first zone and A2, the amplitude
from the second zone, are out of phase (destructive
interference)
– A2 is slightly smaller than A1 due to area and distance
• The total amplitude if found by adding contributions of all
Fresnel zones
A=A1-A2+A3-A4+…
minus signs because the amplitudes are out of phase
amplitudes slowly decrease
So far this is a complex way
of showing an obvious fact.
LASERS 51
April 03
Diffraction from circular apertures
• What happens if an aperture the diameter of the
first Fresnel zone is inserted in the beam?
• Amplitude is twice as high
as before inserting aperture!!
– Intensity four times as large
• This only applies to
intensity on axis
b +λ/2 b +λ
observation
point
b
incident
wavefront
Blocking two Fresnel zones gives almost zero
intensity on axis!!
LASERS 51
April 03
Fresnel diffraction by a circular aperture
• Suppose aperture size and observation distance chosen so
that aperture allows just light from first Fresnel zone to pass
– Only the term A1 will contribute
– Amplitude will be twice as large as case with no aperture!
• If distance or aperture size changed so two Fresnel zones are
passed, then there is a dark central spot
– alternate dark and
light spots along
axis
– circular fringes
off the axis
LASERS 51
April 03
Fresnel diffraction by circular obstacle—
Arago’s spot
• Construct Fresnel zones just as
before except start with first zone
beginning at edge of aperture
• Carrying out the same reasoning
as before, we find that the
intensity on axis (in the
geometrical shadow) is just what
it would be in the absence of the
obstacle
• Predicted by Poisson from
Fresnel’s work, observed by
Arago (1818)
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b
observation
point
b+λ/2
incident
wavefront
April 03
Character of diffraction for different
locations of observation screen
• Close to diffracting screen (near field)
– Intensity pattern closely resembles shape of aperture, just like
you would expect from geometrical optics
– Close examination of edges reveals some fringes
• Farther from screen (intermediate)
– Fringes more pronounced, extend into center of bright region
– General shape of bright region still roughly resembles
geometrical shadow, but edges very fuzzy
• Large distance from diffracting screen (far field)
–
–
–
–
Fringe pattern gets larger
bears little resemblance to shape of aperture (except symmetries)
Small features in hole lead to larger features in diffraction pattern
Shape of pattern doesn’t change with further increase in distance,
LASERSbut
51 it continues to get larger
April 03
How far is the far field?
z = distance from aperture to observing screen
A = area of aperture
Fresnel number
λ = wavelength
characterizes importance
A
of diffraction in any
Fresnel number, F =
λz
situation
• A reasonable rule: F<0.01, the screen is in the far
field
– Depends to some extent on the situation
• F>>1 corresponds to geometrical optics
• Small features in the aperture can be in the far
field even if the entire aperture is not
• Illumination of aperture affects pattern also
LASERS 51
April 03
screen
with
aperture
Imaging and diffraction
Lens
Image of aperture
observing
screen at
image of
plane P
Diffraction pattern
at some plane, P
• Image on screen is image of diffraction pattern at P
– Same pattern as diffraction from a real aperture at image location
except:
• Distance from image to screen modified due to imaging equation
• Magnification of aperture is different from magnification of diffraction
pattern
• Important: for screen exactly at the image plane there is no
diffraction (except for effects introduced by lens aperture)
LASERS 51
April 03
Imaging and far-field diffraction
screen
with
aperture
observing
screen
Lens
f
• Looking from the aperture, the observing screen
appears to be located at infinity. Therefore, the
far-field pattern appears on the screen even though
the distance is quite finite.
LASERS 51
April 03
Fresnel and Fraunhofer diffraction
• Fraunhofer diffraction = infinite observation distance
– In practice often at focal point of a lens
– If a lens is not used the observation distance must be large
– (Fresnel number small, <0.01)
• Fresnel diffraction must be used in all other cases
• The Fresnel and Fraunhofer regions are used as synonyms
for near field and far field, respectively
– In Fresnel region, geometric optics can be used for the most part;
wave optics is manifest primarily near edges, see first viewgraph
– In Fraunhofer region, light distribution bears no similarity to
geometric optics (except for symmetry!)
– Math in Fresnel region slightly more complicated
• mathematical treatment in either region is beyond the scope of this course
LASERS 51
April 03
Fraunhofer diffraction
at
a
slit
Observation
small
• Traditional (pre laser)
setup
Light
source
– source is nearly
monochromatic
• Condenser lens collects
light
Condenser
lens
screen
source slit
Collimating Diffracting
lens
slit
f2
f1
• Source slit creates point source
Focusing
lens
– produces spatial coherence at the second slit
• Collimating lens images source back to infinity
– laser, a monochromatic, spatially coherent source, replaces all this
• second slit is diffracting aperture whose pattern we want
• Focusing lens images Fraunhofer pattern (at infinity) onto
screen
LASERS
51
April 03
Fraunhofer diffraction by slit—zeros
• Wavelets radiate in all
directions
– Point D in focal plane is at
angle Θ from slit, D=Θf
– Light from each wavelet
radiated in direction Θ arrives
at D
field radiated by
wavelets at angle Θ
D= λ
f
d
Θ
λ/2
λ
• Distance travelled is different for
Slit
each wavelet
width = d
• Interference between the light
from all the wavelets gives the
diffraction patter
f
– Zeros can be determined easily
• If Θ=λ/d, each wavelet pairs with one exactly out of phase
– Complete destructive interference
– additional zeros for other multiples of λ, evenly spaced zeros
LASERS 51
April 03
Fraunhofer diffraction by slit—complete
pattern
slit
Diffraction pattern,
short exposure time
Diffraction pattern,
longer exposure time
• Evenly spaced zeros
• Central maximum brightest, twice as wide as
others
LASERS 51
April 03
Multiple slit diffraction
• In multiple slit patterns discussed earlier, each slit
produces a diffraction pattern
• Result: Multiple slit interference pattern is
superimposed over single slit diffraction pattern
Intensity
Three-slit interference
pattern with single-slit
diffraction included
position on screen
LASERS 51
April 03
Fraunhofer diffraction by other apertures
• Rectangular aperture
– Diffraction in each direction is
just like that of a slit
corresponding to width in that
direction
– Narrow direction gives widest
fringes
• Circular aperture
–
–
–
–
circular rings
central maximum brightest
zeros are not equally spaced
diameter of first zero=2.44λf2/d
where d= diameter of aperture
– Note: this is 2.44λf/#
– angle=1.22λ/d
LASERS 51
April 03
Resolution of optical systems
• Same optical system
as shown previously
without diffracting slit
– produces image of
source slit on
observing screen
– magnification f2/f1
Light
source
small
source slit
Collimating
lens
f2
f1
Condenser
lens
Observation
screen
Focusing
lens
• We’ve assumed before that the source slit is very small,
let’s not assume that any more
– each point on source slit gives a point of light on screen
– if we put the diffracting aperture back in, each point gives rise to
its own diffraction pattern, of the diffracting slit
– ideal point image is therefore smeared
LASERS 51
April 03
Resolution of optical systems (cont.)
• With two source
slits we can ask the
question, will we see
two images on the
observation screen
or just a diffraction
pattern?
Main lobe of
pattern due to
one slit
Light
source
Observation
screen with
screen
two source slits
Collimating
lens
Diffracting
slit
f2
f1
Condenser
lens
Focusing
lens
Rayleigh criterion-images are just
resolved if minimum of one
coincides with peak of neighbor
• Answer: If the spacing between the images is larger
than the diffraction pattern, then we see images of two
slits, i.e. they are resolved. Otherwise they are not
distinguishable
and we only see a diffraction pattern April 03
LASERS
51
Resolution of optical systems (cont.)
• Limiting aperture is usually a round aperture stop, so
Rayleigh criterion is found using diffraction pattern of a
round aperture
1.22λf
minimum resolvable distance = R =
= 1.22λf /#
D
f= focal length
D=diameter of aperture stop
R= distance spots which are just resolved
Diffraction Limited System: Resolution of an optical system
may be worse than this due to aberrations, ie not all rays
from source point fall on image point. An optical system for
which aberrations are low enough to be negligible
compared to diffraction is a diffraction limited system.
If geometrical spot size is 2 times size of diffraction spot,
LASERS 51 then system is 2x diffraction limited, or 2 XDL
April 03
Resolution of spots and Rayleigh limit
A
Well resolved
A
Rayleigh limit
A
Slightly closer, are you
sure it’s really two spots?
• At the Rayleigh limit, two spots can be
unambiguously identified, but spots only slightly
closer merge into a blur
LASERS 51
April 03
Diffraction of laser beams
• Till now, disscussion has been of uniformly illuminated
apertures
– mathematical diffraction theory can treat non-uniform
illumination and even non-plane waves
• A TEM00 laser beam has a Gaussian rather than uniform
intensity pattern
– no edge to measure from so we use 1/e2 radius, w
– wo is radius where beam is smallest (waist size)
– relatively simple formulae for diffraction apply both in near field
(Fresnel) and far field (Fraunhofer) zones
– only far field result will be presented here
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λ
far field divergence half angle,θ =
πw0
λz
far field beam radius, w =
πw0
April 03
Diffraction losses in laser resonators
2a
L
•
•
•
•
Light bounces back and forth between mirrors
Spreads due to diffraction as it propagates
Some diffracted light misses mirror and is not fed back
Resonator Fresnel Number measures diffraction losses
πa
F=
λL
2
LASERS 51
If index of refraction in
laser resonator is not 1,
multiply by n
April 03