Chap 4 Fresnel and Fraunhofer Diffraction

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Chap 4 Fresnel and Fraunhofer Diffraction
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Content

• 4.1 Background
• 4.2 The Fresnel approximation
• 4.3 The Fraunhofer approximation
• 4.4 Examples of Fraunhofer diffraction patterns

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4.1 Background

• These approximations, which are commonly made in
  many fields that deal with wave propagation, will
  be referred to as Fresnel and Fraunhofer
  approximations.
• In accordance with our view of the wave
  propagation phenomenon as a system, we shall
  attempt to find approximations that are valid for
  a wide class of input field distributions.

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• 4.1.1 The intensity of a wave field
• Poyntings thm.

When calculation a diffraction pattern, we will
general regard the intensity of the pattern as
the quantity we are seeking.
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• 4.1.2 The Huygens-Fresnel principle in
  rectangular coordinates
• Before we introducing a series of approximations
  to the Huygens-Fresnel principle, it will be
  helping to first state the principle in more
  explicit from for the case of rectangular
  coordinates.
• As shown in Fig. 4.1, the diffracting aperture is
  assumed to lie in the plane, and is illuminated
  in the positive z direction.
• According to Eq. (3-41), the Huygens-Fresnel
  principle can be stated as

(1)
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Fig. 4.1 Diffraction geometry
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and therefore the Huygens-Fresnel principle can
be rewritten
(2)

(3)

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• There have been only two approximations in
  reaching this expression.
• One is the approximation inherent in the scalar
  theory

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4.2 Fresnel Diffraction

• Recall, the mathematical formulation of the
  Huygens-Fresnel , the first Rayleigh- Sommerfeld
  sol.
• The Fresnel diffraction means the Fresnel
  approximation to diffraction between two parallel
  planes. We can obtain the approximated result.

(1)
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The quadratic-phase exponential with positive
phase i.e, ,for zgt0
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• Note The distance from the observation point to
  an aperture point
• Using the binominal expansion, we
  obtain the approximation to

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• as the term
• is sufficiently small.
• The first Rayleigh Sommerfeld sol for
  diffraction between two parallel planes is then
  approximated by

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• ( ) ,
  the r01 in denominator of the integrand is
  supposed to be well approximated by the first
  term only in the binomial expansion, i.e,
  
• In addition, the aperture points and the
  observation points are confined to the ( ,
  ) plane and the (x,y) plane ,respectively. )
• Thus, we see

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• Furthermore, Eq(1) can be rewritten as
• 
  (2a)
• where the convolution kernel is
• 
  
  (2b)
• Obviously, we may regard the phenomenon of wave
  propagation as the behavior of a linear system.

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• Another form of Eq.(1) is found if the term
• is factored outside the integral signs, it
  yields

(3)
which we recognize (aside from the multiplicative
factors) to be the Fourier transform of the
complex field just to the right of the
aperture and a quadratic phase exponential.
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• We refer to both forms of the result Eqs.
  (1) and (3), as the Fresnel diffraction integral
  . When this approximation is valid, the observer
  is said to be in the region of Fresnel
  diffraction or equivalently in the near field of
  the aperture.

• Note
• In Eq(1),the quadratic phase exponential in the
  integrand

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• do not always have positive phase for zgt0
  .Its sign depends on the direction of wave
  propagation. (e.g, diverging of converging
  spherical waves)
• In the next subsection ,we deal with the
  problem of positive or negative phase for the
  quadratic phase exponent.

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• 4.2.1 Positive vs. Negative Phases
• Since we treat wave propagation as the behavior
  of a linear system as described in chap.3 of
  Goodman), it is important to descries the
  direction of wave propagation.
• As a example of description of wave propagation
  direction, if we move in space in such a way as
  to intercept portions of a wavefield (of
  wavefronts ) that were emitted earlier in time.

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In the above two illustrations, we assume the
wave speed vzc/tc where zc and tc are both
fixed real numbers.
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• In the case of spherical waves,

Diverging spherical wave
Converging spherical wave
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Consider the wave func.
,where
and r gt0 and
If
,then
(?Positive phase)
implies a diverging spherical wave.
Or if
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(?Negative phase)
implies a converging spherical wave.

Note For spherical wave ,we say they are
diverging or converging ones instead or saying
that they are emitted earlier in time or
later in time.
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Specifically, for a time interval tc gt0, we see
the following relations,
The term standing for the time
dependence of a traveling wave implies
that we have chosen our phasors to rotate in the
clockwise direction.
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• Therefore, we have the following seasonings
• Earlier in time Positive phase
• (e.g., diverging spherical waves)
• Later in time Negative phase
• (e.g., converging spherical waves)

Note Earlier in time means the general
statement that if we move in space in such a way
as to intercept wavefronts (or portions of a
wave-field ) that were emitted earlier in time.
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To describe the direction of wave propagation for
plane waves, we cannot use the term diverging or
converging .Instead .we employ the general
statement ,for the following situations.
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, (where
gt0)
The phasor of a plane wave,
multiplied by the time dependence gives
, where
We may say that ,if we move in the positive y
direction , the argument of the exponential
increases in a positive sense, and thus we are
moving to a portion of the wave that was emitted
earlier in time.
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In a similar fashion , we may deal with the
situation for
Propagation direction
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Note Show that the Huygens-Fresnel principle
can be expressed by
ltpfgt Recall the wave field at observation point
P0
(1)
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For the first Rayleigh Sommerfeld solution ,the
Green func.
(2)
Note we put the subscript -, i.e, G- to signify
this kind of Green func. Substituting Eq(2)
into Eq.(1) gives
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(3)
or
(4)

where the Green func. proposed by Kirchhoff
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The term in the integrand of Eq.(4)

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as
or
(5)
Finally, substituting Eq.(5) into Eq.(4) yields

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• 4.2.2 Accuracy of Fresnel Approximation
• Recall Fresnel diffraction integral

Parabolic wavelet
(4.14)
Aperture point (varying withS)
observation point (fixed)
We compare it with the exact formula
Spherical wavelet
where
(or
)
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since the binomial expansion
where
The max.approx.error (i.e.,(
)max)
and the corresponding error of the exponential
is maximized at the phase (or approximately 1
radian)
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A sufficient condition for accuracy would be
ltlt1
For example
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or
This sufficient condition implies that the
distance z must be relatively much larger than
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Since the binomial expansion
(high order term)
where
we can see that the sufficient condition leads to
a sufficient small value of b
However, this condition is not necessary. In the
following, we will give the next comment that
accuracy can be expected for much smaller values
of z (i.e., the observation point (x , y) can be
located at a relatively much shorter distance to
an arbitrary aperture point on the (?,?) plane)
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We basically malcr use of the argument that for
the convolution integral of Eq.(4-14), if the
major contribution to the integral comes from
points (?,?) for which ??x and ??y, then the
values of the HOTs of the expansion become
sufficiently small.(That is, as (?,?) is close to
(x , y)
gives a relatively small value
Consequently, can be well
approximated by . )
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In addition it is found that the convolution
integral of Eq.(4-14),
or
where and ,
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can be governed by the convolution integral of
the function with a second
function (i.e., U(?,?)) that is smooth and slowly
varying for the rang 2 lt X lt 2 and 2 lt Y lt 2.
Obviously, outside this range, the convolution
integral does not yield a significant addition.
( Note For one dimensional case
is governed by
we can see that
is well approximated by
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Finally, it appears that the majority of the
contribution to the convolution integral for the
range -8 lt X lt 8 and -8 lt Y lt 8 or the aperture
area S comes from that for a square in the (?,?)
plane with width and centered on the
point ? x,? y (i.e., the range 2 lt lt2
and 2lt lt2 or lt
and lt )
As a result within the square area, the expansion
as well approximated, since
is small enough.
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From another point of view, since the Fresnel
diffraction integral
Corresponding square area
yields a good approximation to the exact formula
where
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we may say that for the Fresnel approximation
(for the aperture area S or the corresponding
square area) to give accurate results, it is not
necessary that the HOTs of the expansion be
small, only that they do not change the value of
the Fresnel diffraction integral significantly.
Note From Goodmans treatment (P.69 70), we
see that
can well approximate
or
Where the width of the diffracting aperture is
larger than the length of the region 2 lt X lt 2
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For the scaled quadratic-phase exponential of
Eqs.(4-14) and Eq.(4-16), the corresponding
conclusion is that the majority of the
contribution to the convolution integral comes
from a square in the (?,?) plane, with width
and centered on the point (? x ,? y)

• In effect,
• When this square lie entirely within the open
  portion of the aperture, the field observed at
  distance z is, to a good approximation, what it
  would be if the aperture were not present. (This
  is corresponding to the light region)

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• When the square lies entirely behind the
  obstruction of the aperture, then the observation
  point lies in a region that is, to a good
  approximation, dark due to the shadow of the
  aperture.
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• When the square bridges the open and obstructed
  parts of the aperture, then the observed field is
  in the transition (or gray) region between light
  and dark.
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• For the case of a one-dimensional
  rectangular slit, boundaries among the regions
  mentioned above can be shown to be parabolas, as
  illustrated in the following figure.
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The light region
W x ? , x ? 0 W x ? , x lt
0
Thus, the upper (or lower) boundary between the
transition (or gray) region and the light region
can be expressed by
(or )
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• 4.2.3 The Fresnel approximation and the Angular
  Spectrum
• In this subsection, we will see that the Fourier
  transform of the Fresnel diffraction impression
  response identical to the transfer func. of the
  wave propagation phenomenon in the angular
  spectrum method of analysis, under the condition
  of small angles.
• From Eqs.(4-15)and (4-16), We have

Where the convolution kernel (or impulse
response) is
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• The FT of the Fresnel diffraction impulse
  response becomes

The integral term
can be rewritten a
where
and
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• (because the exponents

P
where

q
where

as a result,
1
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• so

On the other hand, the transfer function of the
wave propagation phenomenon in the angular
spectrum method of analysis is expressed by
under the condition of small angles (as noted
below the term)
can be approximated by
(because
)
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• (Note because

For Fresnel approximation, the sufficient
condition ma be
The obliquity factor
then approaches 1
That is,
is small angle
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• Which is the transfer function of the wave
  propagation phenomenon in the angular spectrum
  method of analysis under the condition of small
  angles.

Therefore, we have shown that the FT of the
Fresnel diffraction impulse response
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4.2.4 Fresnel Diffraction between Confocal
Spherical surfaces.
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as
are all very close to zero, (i.e, the paraxial
condition)
Recall the Rayleigh Sommerfeld sol, (for the
paraxial condition
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• as a result, for the paraxial region,

(including the paraxial representation of
spherical phase)
This Fresnel diffraction eq. expresses the field
observed on the right hand spherical cap as the
FT of the filed U(x,y) on the left-hand
spherical cap.
Comparison of the result with Eq(4-17),the
Fresnel diffraction integral (including
Fourier-transform-like operation)
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quadratic phase parabolic phase
Note Recall
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• The two quadratic phase factors in Eq(4-17)are in
  fact simply paraxial representations of spherical
  phase surfaces, (since the Rayleigh Sommerfeld
  sol. can be applied only to the planar screens),
  and it is therefore reasonable that moving to the
  spheres has eliminated them.
• For the diffraction between two spherical caps,
  it is not really valid to use the
  Rayleigh-Sommerfeld result as the basis for the
  calculation (only for the diffraction between two
  parallel planes).
• However, the Kirchhoff analysis remains valid,
  and its predictions are the same as those of the
  Rayleigh-Sommerfeld approach provided paraxial
  conditions hold.

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4.3 The Fraunhofer approximation

• From Eq(4-17), We see

(4-17)
If the exponent
We have
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• The observed filed strength U(x,y) can be found
  directly from a FT of the aperture function
  itself (because )

That is, Eq.(4-17)with the Fraunhofer
approximation becomes
(4-25)

(Aside from the multiplicative phase factors,
this expression is simply the FT of the aperture
distribution)
where
(4-26)
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• Note
• Recall the different forms of Fresnel
  diffraction integral

where the Fresnel diffraction impulse response
(4-16)
and that of Eq(4-17)
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• Comparison of Eqs(4-15)and (4-16) with
  Eqs.(4-25)and (4-26) tell us that there is no
  transfer function for the Fraunhofer (or
  far-field) diffraction since Eqs(4-25) and (4-26)
  do not include impulse response.
• Nonetheless, since Fraunhofer diffraction is
  only a special case of Fresnel diffraction, the
  transfer function Eq(4-21) remains valid
  throughout both the Fresnel and the Fraunhofer
  regimes. That is, it is always possible to
  calculate diffracted field in the Fraunhofer
  region by retaining the full accuracy of the
  Fresnel approximation.

Treating the wave propagation phenomenon as a
linear system
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4.4 Examples of Fraunhofer diffraction patterns

• 4.4.1 Rectangular Aperture
• If the aperture is illuminated by a
  unit-amplitude, normally incident, monochromatic
  plane wave, then the field distribution across
  the aperture is equal to the transmittance
  function .Thus using Eq.(4-25), the Fraunhofer
  diffraction pattern is seen to be

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• 4.4.2 Circular Aperture

Suggests that the Fourier transform of Eq.(4-25)
be rewritten as a Fourier-Bessel transform. Thus
if
is the radius coordinate in the
observation plane, we have
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• 4.4.3 Thin Sinusoidal Amplitude Grating
• In practice, diffracting objects can be far more
  complex. In accord with our earlier definition
  (3-68),the amplitude transmittance of a screen is
  defined as the ratio of the complex field
  amplitude immediately behind the screen to the
  complex amplitude incident on the screen . Until
  now ,our examples have involved only
  transmittance functions of the form

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• Spatial patterns of phase shift can be introduced
  by means of transparent plates of varying
  thickness, thus extending the realizable values
  of tA to all points within or on the unit circle
  in the complex plane.
• As an example of this more general type of
  diffracting screen, consider a thin sinusoidal
  amplitude grating defined by the amplitude
  transmittance function

(4-33)
where for simplicity we have assumed that the
grating structure is bounded by a square
aperture of width 2w. The parameter m represents
the peak-to-peak change of amplitude
transmittance across the screen, and f0 is the
spatial frequency of the grating.
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• 4.4.4 Thin sinusoidal phase grating

Binary phase grating