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Fraunhofer Diffraction. Last Lecture Dichroic Materials Polarization by Scattering Polarization by Reflection from Dielectric Surfaces Birefringent Materials Double Refraction The Pockel’s Cell. This Lecture Fraunhofer versus Fresnel Diffraction - PowerPoint PPT Presentation
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Fraunhofer DiffractionFraunhofer Diffraction
Last Lecture• Dichroic Materials• Polarization by Scattering• Polarization by Reflection from Dielectric Surfaces• Birefringent Materials• Double Refraction• The Pockel’s Cell
This Lecture• Fraunhofer versus Fresnel Diffraction• Diffraction from a Single Slit• Beam Spreading• Rectangular and Circular Apertures
Optical Diffraction Optical Diffraction
• Diffraction is any deviation from geometric optics that results from the obstruction of a light wave, such as sending a laser beam through an aperture to reduce the beam size. Diffraction results from the interaction of light waves with the edges of objects.
• The edges of optical images are blurred by diffraction, and this represents a fundamental limitation on the resolution of an optical imaging system.
• There is no physical difference between the phenomena of interference and diffraction, both result from the superposition of light waves. Diffraction results from the superposition of many light waves, interference results from the interference of a few light waves.
Optical Diffraction Optical Diffraction
Hecht, Optics, Chapter 10
Fraunhofer versus Fresnel Diffraction Fraunhofer versus Fresnel Diffraction
• The passage of light through an aperture or slit and the resulting diffraction patterns can be analyzed using either Fraunhofer or Fresnel diffraction theory. In Fraunhofer (far-field) diffraction theory the source is far enough from the aperture that the wavefronts are planar at the aperture, and the image plane is far enough from the aperture that the wavefronts are planar at the image plane.
• If the curvature of the optical waves must be taken into account at the aperture or image plane, then we must use Fresnel (near-field) diffraction theory.
• The Huygens-Fresnel principle is used in diffraction theory, in that every point of a given wavefront of light can be considered as a source of secondary wavelets. To analyze two-slit interference, we assumed that the slits were point sources. To analyze diffraction, we need to consider the generation of wavelets at different spatial positions within the slit.
Fraunhofer versus Fresnel Diffraction Fraunhofer versus Fresnel Diffraction
• We can move to the Fraunhofer regime by placing lenses on each side of the aperture. Lens 1 is place a focal length away from the point source so that the wavefronts are planar at the aperture. The observation screen is in the focal plane of lens 2 so that the diffraction pattern is imaged at infinity.
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
• Consider the geometry shown below. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction.
7
Fraunhofer vs. Fresnel diffractionFraunhofer vs. Fresnel diffraction
• In Fraunhofer diffraction, both incident and diffracted waves may be considered to be plane (i.e. both S and P are a large distance away)
• If either S or P are close enough that wavefront curvature is not negligible, then we have Fresnel diffraction
P
S
Hecht 10.2Hecht 10.2 Hecht 10.3Hecht 10.3
8
Fraunhofer Vs. Fresnel DiffractionFraunhofer Vs. Fresnel Diffraction
2
2222
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11
2
1
'
'
2
11
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2
11'
2
11
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2
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'''
ddd
h
d
h
d
hd
d
hd
d
hd
d
hd
hdhdhdhd
Now calculate variation in (r+r’) in going from one side of aperture to the other. Call it
9
Fraunhofer diffraction limitFraunhofer diffraction limit
• If aperture is a square - X • The same relation holds in azimuthal plane
and 2 ~ measure of the area of the aperture
• Then we have the Fraunhofer diffraction if,
apertureofaread
or
d
,
2
Fraunhofer or far field limit
10
Fraunhofer, Fresnel limitsFraunhofer, Fresnel limits
• The near field, or Fresnel, limit is
2
d
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
0
0
0
0
exp
0.
P
P
The contribution to the electric field amplitude
at point P due to the wavelet emanating from
the element ds in the slit is given by
dEdE i kr t
r
Let r r for the source element ds at s
Then for any element
dEdE
r
0
0
exp
, .
, , .
sin
L L
i k r t
We can neglect the path difference in the amplitude term but not in the phase term
We let dE E ds where E is the electric field amplitude assumed uniform over the width of the slit
The path difference s
/ 2
0 0 / 20 0
/ 2
00 / 2
.
exp sin exp exp sin
exp sinexp
sin
bL L
P P b
b
LP
b
Substituting we obtain
E ds EdE i k r s t E i kr t i k s ds
r r
i k sEIntegrating we obtain E i kr t
r i k
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
00
00
00
exp expexp
sin
1sin
2
exp exp exp2
exp2
LP
LP
L
Evaluating with the integral limits we obtain
i iEE i kr t
r i k
where
k b
Rearranging we obtain
E bE i kr t i i
r i
E bi kr t
r i
00
2 2 2* 2
0 0 0 02 20
sin2 sin exp
1 1 sin sinsinc
2 2
L
LP P
E bi i kr t
r
The irradiance at point P is given by
E bI = c E E c I I
r
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
2 2 2* 2
0 0 0 02 20
0 0
1 1 sin sinsinc
2 2
sinsinc 1 0, lim sinc lim 1
1sin 0, sin 1, 2,
2
LP P
The irradiance at point P is given by
E bI = c E E c I I
r
The function is for
The zeroes of irradiance occur when or when k b m m
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
,
sin , 2 / ,
1 2
2
In terms of the length y on the observation screen
y f and in terms of wavelength k
we can write
y b yb
f f
Zeroes in the irradiance pattern will occur when
b y m fm y
f b
The maximum in the irradiance pattern is at β = 0
2 2
sin cos sin cos sin0
sintan
cos
.
Secondary maxima are found from
d
d
Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit
Note: x- and y-axes switched in book, Figs. 16-5a (here) and Fig. 16-1 do not match.
Beam SpreadingBeam Spreading
sin
min
1 2
The angular width of the central
maximum is defined as the angular
separation Δθ between the first minima on
either side of the central maximum,
y
f
The first ima in the irradiance pattern
will occur when
fm fy Δθ
b b b
The
2
width W of the diffraction pattern thus
increases linearly with distance from the slit,
in the regions far from the slit where Fraunhofer
diffraction applies
LW = L Δθ
b
Rectangular AperturesRectangular Apertures
When the length a and width b of the
rectangular aperture are comparable,
a diffraction pattern is observed in
both the x - and y - dimensions, governed
in each dimension by the formula we
have already developed. The irradiance
pat
2 20 sinc sinc
1sin
2
tern is
I I
where
k a
Zeroes in the irradiance pattern are observed
when
m f m fy or x
b a
x
y
Square AperturesSquare Apertures
Fraunhofer Diffraction from General Apertures
Fraunhofer Diffraction from General Apertures
1/ 22 22
1/ 22 2 2
expAP
For the general aperture
EdE i t kr dA
r
where
dA dy dz
r X Y y Z z
R X Y Z
Fraunhofer Diffraction from General Apertures
Fraunhofer Diffraction from General Apertures
1/ 22 2
2 2
1/ 2
2
21
,
21 1
2 2
2
We can combine the relations for r and R to obtain
y z Yy Zzr R
R R
In the far field R is very large compared to the aperture dimensions, and
y + zwecan neglect the term and wecan write
R
Yy Zz Yr R R
R
2
exp exp
exp exp
AP
aperture
A
aperture
y Zz Yy ZzR
R R
Therefore, the total electric field at point P is given by
EE i t kR ik Yy Zz dA
R
Ei t kR ik Yy Zz dy dz
R
Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures
.
cos sin
cos sin
sin sinexp exp
P
AP
Now we specialize to a circular aperture of radius a At this point we switch to cylindrical coordinates
z y
Z q Y q
dA d d
Substituting into our expression for E we obtain
ik qEE i t kR
R
2
0 0
2
0 0
cos cos
exp exp cos
a
aA
qd d
R
E i k qi t kR d d
R R
Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures
2
0 0
2
0
0
.
exp exp cos
exp cos .
1( )
2
aA
P
Because of symmetry our solution will be the same for any angle Choosing Φ = 0, we obtain
E i k qE i t kR d d
R R
k qBut i d is in the form of a Bessel function
R
J u
2
0
00
2
0
exp cos .
exp 2
,
( ) exp cos2
v
v
aA
P
m v
m v
i u v dv is a Bessel function of the order zero
We can write
E k qE i t kR J d
R R
In general a Bessel function of the order m is given by
iJ u i mv u v dv
Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures
1
1
00
,
exp 2
m mm m
u
00
AP
A useful recurrence relation for Bessel functions is
du J u u J u
du
When m = 1, we can integrate the expression to find
J u du u J u
k q RDefine w then d dw gives us
R kq
E k qE i t kR J d
R R
2/
00
21
22* 2
0 1
exp 2
exp 2
1
2
a w kaq RA
AP
AP P
E Ri t kR J w wdw
R kq
From the recurrence relation we obtain
E R kaqE i t kR a J
R kaq R
The irradiance is given by
E R kaqI c E E A J
R kaq R
2
Fraunhofer Diffraction from Circular AperturesFraunhofer Diffraction from Circular Apertures
2 2
1 1
1
0
sin / ,
2 2 sin0 0
sin
1lim
2u
Assuming that R is essentially constant over the observation screen and recognizing that
q R we can write the irradiance as
J kaq R J k aI I I
kaq R k a
where we have used the
J urelation that
u
0
0.
recognizing that u when
q
Fraunhofer Diffraction from Circular Apertures: Bessel Functions
Fraunhofer Diffraction from Circular Apertures: Bessel Functions
Fraunhofer Diffraction from
Circular Apertures: The Airy Pattern
Fraunhofer Diffraction from
Circular Apertures: The Airy Pattern
min
min
2sin 3.83
2
1.22
2 .
First minimum in the Airy pattern is at
Dk a k a
D
where D a
Circular AperturesCircular Apertures