10'3 Fresnel diffraction

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February 27, March 2 Fresnel zones

• 10.3 Fresnel diffraction
• 10.3.1 Free propagation of a spherical wave
• Fresnel diffraction For any R1 and R2.
• Fraunhofer diffraction is a special case of
  Fresnel diffraction.
• The integration for Fresnel diffraction is
  usually complicated. Fresnel zones will be
  introduced to estimate the diffraction pattern.

Directionality of secondary emitters Inclination
factor (obliquity) (To be proved later. )
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Free propagation of a spherical monochromatic
wave Primary spherical wave Question What is
the field at P which is r0 away from the sphere?
Contribution from the sources inside a slice ring
dS
S
P
x
r
r0
O
r
r
dS
The area of the slice ring is
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Contribution from the l th zone to the field at P
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Sum of disturbance at P from all zones
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Note Huygens-Fresnel diffraction theory is an
approximation of the more accurate
Fresnel-Kirchhoff formula.
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Read Ch10 3 Homework Ch10 42,43 Due March 6
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March 4, 6 Circular apertures 10.3.2 The
vibration curve (phasor representation) A graphic
method for qualitatively analyzing diffraction
problems with circular symmetry.

• For the first zone
• Divide the zone into N subzones.
• Each subzone has a phase shift of p/N.
• The phasor chain deviates slightly from a circle
  due to the inclination factor.
• When N? 8, the phasor train composes a smooth
  spiral called a vibration curve.

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2) P out-of axis As P moves outward, portions of
the zones (defined by P, S and O) will be
uncovered and covered, resulting in a series of
relative maxima and minima. (The integration will
be very complicated.)
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II. Plane waves The radius of the mth zone
The area of one zone
Example
On-axis field
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Read Ch10 3 Homework Ch10 52,53 Due March 13
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March 23 Fresnel zone plate
10.3.4 Circular obstacles Poissons spot Bright
spot always appears at the center of the shadow
of a circular obstacle. Poisson intended to use
this unusual conclusion to deny Fresnels wave
description of light, but this prediction was
soon verified to be true. The spot is ironically
called Poissons spot. May have been observed by
ancient people.
The spot is everywhere along the axis except
immediately behind the obstacle. The irradiance
is not very different from that of the
unobstructed wave.
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10.3.5 Fresnel zone plate Zone plate A device
that modifies light by using Fresnel zones.
Modification can be either in amplitude or in
phase. Example Transparent only for odd (or
even) zones. The first 10 odd (even) zones will
result in an intensity of 400 times larger
compared to the unobstructed light.
Radii of the zones
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For plane waves
Primary focal length
Third-order focal length because
Fabrication of zone plates Photographically
reduce large drawings. Newtons rings serves as
good pictures for this purpose. Modification
light in phase by zone plates is difficult in
practice.
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Read Ch10 3 No homework
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March 25 Introduction to Mathematica

• Some basic rules for beginners
• 0. Never guess when not sure. You pay for
  guessing.
• Build-in functions begin with capital letters.
  Arguments are in square brackets.
• Some letters and words are reserved I, D, Pi, E,
  N, Sin,
• Multiplication is or space.
• It remembers assignments. Use Clear.
• Exit when finish because we have limited number
  of license.
• 6. F1 keyHit it as often as you can.
• Let us learn
• Numerical calculations 3.168, N, integers and
  decimals
• Derivatives Df,x, Df,x,n
• Integrals Integratef,x, Integratef,x,xmin,xma
  x , BasicMathInput Palette
• Power series expansion Series
• Solving equations Solvelhsrhs,var
• Numerical equations NSolvelhsrhs,var
• Find numerical root FindRootf,x,x0

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• Differential equations DSolveeqn,y,x
• Functions fx_3 x5
• Special Functions Gamma, Erf, FresnelC,
  FresnelS, LegendreP, BesselJ,
• Lists a,b,c, Table, Array
• Plot Plotf,x,xmin,xmax
• Options AspectRatio, AxesLabel, PlotPoints,
  PlotRange, PlotStyle? Red,Thickness
• Plot3D
• ContourPlot
• ParametricPlot

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Homework
All these problems should be neatly done by
Mathematica. If you do not have a color printer,
send me your solutions by emailing me your .nb
file. You must have your name in the file. 1.
Numerical calculations (give approximate
numerical values) 1) 1 ii, where
isqrt(-1). 2) cosh(4) sin4. 3) J0 (p/2) ,
Bessel function.
2. 3.
4. Plot y sinx, sin2x, sin3x, sin4x, sin5x,
0ltxltp, in one graph with different line
colors. 5. Three dimensional plot. z
(cos4xsin4y) exp-(x2y2)/5, -5lt
xlt5,-5ltylt5. 6. Contour plot. z(sin2x cos2y)(x2
y2), -10lt xlt10,-10ltylt10.
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March 27 Rectangular apertures
10.3.6 Fresnel integrals and the rectangular
apertures Fresnel diffraction with no circular
symmetry. The zone idea does not work.
The contribution to field at P from sources in dS

• K(q) 1 if the aperture is small (ltltr0, r0).
• In the amplitude r r0, r r0.
• In the phase

Half of the unobstructed field Eu/2
Fresnel integrals
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Fresnel integrals

• Ep and Ip can be evaluated using a look-up
  table.
• Off-axis P points can be estimated by
  equivalently shifting the aperture and changing
  the limits (u1, u2, v1, v2) in the integrals
  according to the new values of (y1, y2, z1, z2).

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Plane wave incidence
Fresnel diffraction of a plane wave incidence on
a rectangular aperture Example Aperture 2 mm2
mm, l500 nm. (a2/l 8 m) For any point P (X/m,
Y/mm, Z/mm)
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• Read Ch10 3
• Homework
• Using any software available (e.g., Mathematica),
  draw the following diffraction patterns (contour
  plots) from a plane wave incidence on a
  rectangular aperture.
• Aperture 2 mm2 mm l 500 nm Screen 0.4 m,
  4 m and 40 m away.
• Note
• Describe the procedures of how you calculate the
  intensity distribution.
• In Mathematica the Fresnel integral functions
  are FresnelC and FresnelS. You may need to
  study ListContourPlot or ListPlot3D.
• For each distance adjust the screen area you
  plot so that you can see the main features of the
  pattern.
• Use logarithmic scale for the intensity
  distribution. Let each picture span the same
  orders of magnitude of intensity down from its
  maximum.
• Discuss the evolution of the diffraction
  patterns for the above three distances.
• Due April 3

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March 30 Cornu spiral 10.3.7 Cornu spiral Cornu
spiral (clothoid) The curve generated by a
parametric plot of Fresnel integrals C(w) against
S(w).
Arc length parameter w
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Diffraction from a rectangular aperture

• Practice
• Off-axis point Slide arc (string) of constant
  length along the spiral.
• Expanding the aperture size Extend endpoints
  of arc along the spiral.
• Very large aperture

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10.3.8 Fresnel diffraction by a slit From
rectangular aperture to slit

• Practice
• Off-axis point Slide arc of constant length Dv
  along the spiral.
• Relative extrema occur. Small Dv has broad
  central maximum.
• Expanding the aperture size Extend ends of arc
  along the spiral.
• Relative extrema occur.

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Fresnel diffraction by a slit
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• Read Ch10 3
• Homework
• 1. Draw a Cornu spiral using Mathematica.
• 2. The diffraction pattern from a single slit
  with plane wave incidence is
• a) Draw the 3D picture of B122/2 as a
  function of , for the
• range of
• b) Discuss the evolution of the diffraction
  patterns when you change (v2-v1)/2.
• c) For a given slit width a, where and how
  much is the highest diffraction intensity?
• Due April 10

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April 1 Kirchhoffs diffraction theory 10.3.9
Semi-infinite opaque screen
From slit to semi-infinite edge
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10.3.10 Diffraction by a narrow obstacle

• There is always an illuminated region along the
• central axis
• For off-axis points, slide the obstacle
  along the spiral.

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10.3.11 Babinets principle
Babinets principle The fields from two
complementary diffraction apertures satisfy
Example Thin slit and narrow obstacle. Special
cases This happens in Fraunhofer diffraction
when P is beyond the Airy disk.
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10.4 Kirchhoffs scaler diffraction theory
Greens theorem
Helmholtz equation
Kirchhoff integral theorem
Apply to unobstructed spherical wave from a point
source
Fresnel-Kirchhoff diffraction formula
Obliquity factor
Differential wave equation ? Huygens-Fresnel
principle
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Read Ch10 3-4 Homework Ch10 46,48,54,55 Due
April 10