Moller K.D. Optics. Learning by Computing, with Examples Using Mathcad, Matlab, Mathematica, and Maple
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3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION 147
FileFig 3.5 (D5RECTS)
A 3-D chart of the diffraction pattern of a rectangular aperture. By changing
the lower limits of x and y and enlarging N, one may get a more densely lined
pattern. Changing d and a to larger values will result in a narrower pattern. If
d is equal to a, we have a square pattern.
D5FARECTS
Diffraction Pattern of a Rectangular Aperture
The width in the x-direction is d,inthey-direction, a. One may look at the plot
from different angles, change colors, and make a contour plot.
i : 0 ...N j : 0 ...N
x
i
: (−6) + .20001 · iy
j
: 6 + .20001 · j
λ ≡ 4
M
i,j
: f (x
i
,y
j
)
f (x, y):
sin
2 ·π · d ·
x
2·λ
2 ·π · d ·
x
2·λ
2
·
sin
2 ·π · a ·
y
2·λ
2 ·π · a ·
y
2·λ
2
N ≡ 60 d ≡ 3 a ≡ 2.
Application 3.5.
1. Using a 3-D contour and a 3-D surface plot and making changes in d and a,
one may study the intensity of the diffraction of the rectangular aperture for
a. A square aperture;
148
3. DIFFRACTION
b. A long strip aperture which should have the pattern of a slit on one side.
2. One may change the ratio of d/λ and a/λ and observe a wider or more
narrow diffraction pattern.
3.4.4 Circular Aperture
Diffraction on a circular aperture is present on all optical devices and instruments
with circular symmetry. Although diffraction seems to be a minor effect, the size
of astronomical telescope mirrors is large in order to reduce the limitations of
image quality by diffraction. Even the Mount Palomar telescope mirror with a
diameter of about 5 m reduces the image quality of a star by diffraction.
For the calculation of the diffraction pattern of a circular aperture we look at
the integral for the rectangular aperture (Eq. (3.41)) and integrate over a circular
opening
u(Y, Z)
circular opening
(e
−ik(zZ+yY)/X
) dz dy. (3.44)
Since the problem is circular symmetric, one changes the coordinate system
for the mathematical treatment to the coordinate system shown in Figure 3.14.
z r cos φZ R cos ψ (3.45)
y r sin φY R sin ψ, (3.46)
where
0 ≤ r ≤ a, 0 ≤ φ ≤ 2π, 0 ≤ ψ ≤ 2π, (3.47)
FIGURE 3.14 Coordinates for diffraction on a circular aperture.
3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION
149
and k 2π/λ. For the integral we obtain
u(r, φ) u
o
+π
−π
a
0
e
−i2π (rR/λX) cos(ψ−φ)
rdrdφ. (3.48)
This integral can be expressed with the Bessel function of zero order
J
o
(q) 1/2π
+π
−π
e
iq cos(ψ−φ)
dφ, (3.49)
where q 2π (rR/λX) and therefore r (λX/2πR)q, and we get for
Eq. (3.48),
u u
o
(λX/2πR)
2
2π
q2π (aR/λX)
q0
J
o
(q)qdq. (3.50)
Using the relation
q
0
J
o
(q)qdq q
J
1
(q
) (3.51)
one has for the intensity
I I
0
{J
1
(2π(aR/λX))/(2π (aR/λX))}
2
, (3.52)
where I
0
is the normalization constant.
The three graphs in FileFig 3.6 show that we have a narrow diffraction pattern
for large diameters and vice versa as we found for the diffraction on a slit. The
FIGURE 3.15 Diffraction pattern of a round aperture (from M. Cagnet, M. Francon, and J.C.
Thrierr, Atlas of Optical Phenomena, Springer-Verlag, Heidelberg, 1962).
150 3. DIFFRACTION
angle from the center of the circular aperture to the first minimum of the diffracted
intensity (Eq. (3.52)), is the diffraction angle, equal to 1.22 λ/2a where 2a is the
diameter of the opening. Comparing to a slit, we obtain λ/d, where d is the width
of the slit. One has for the slit the factor 1, instead of 1.22 for the round aperture.
The determination of the factor 1.22 is done in Application FF8. The factor 1.22
appears in catalogues of optical devices to specify limitations by diffraction. A
photograph of the diffraction pattern of a round aperture (Airy disc) is shown in
Fig. 3.15 and a 3-D graph in FileFig 3.7.
FileFig 3.6 (D6FARONS)
A graph of the intensity of the diffraction pattern of a round aperture. By changing
the radius a of the aperture, we see that the width of the diffraction pattern is
inversely proportional to the diameter of the round opening. This is a general
property one observes for the diffraction pattern, as well as the corresponding
Fourier transformation.The size of the pattern is proportional to the wavelength.
D6FARON is only on the CD.
Application 3.6.
1. Do the normalization of Eq. (3.52) by dividing I(R) by I (R 0).
2. The dependence of the width of the diffraction pattern on values of (and a
may be studied by changing λ to λ/2 and 2λ and 2a to 2a/2 and 2 times 2a
and considering the ratio of 2a/λ. (The diameter of the round aperture 2a is
used here for comparison with the slit width d.)
FileFig 3.7 (D7FARON3DS)
A 3-D graph of the round aperture.
D7FARON3DS
3-D Diffraction Pattern of a Round Aperture as a Circular Symmetric Plot Using Two
Coordinates
Radius of aperture is a. The coordinate on the observation screen in R. Wave-
length λ, distance from aperture to screen, is X. One may look at the plot from
different angles, change colors, and make a contour plot.
i : 0 ...N j 0 ...N
x
i
: (−40) + 2.0001 · iy
j
:−40 + 2.0001 · jλ≡ .0005
R(x, y):
(x)
2
+ (y)
2
3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION 151
N ≡ 60 X : 4000 a ≡ .05
g(x, y):
J 1
2 ·π · a ·
R(x,y)
X·λ
2 ·π · a ·
R(x,y)
X·λ
2
M
i,j
: g(x
i
,y
j
).
Application 3.7. Change the radius of the aperture and wavelength and make
both twice as large and reduce both to
1
2
. Consider the ratio of 2a/λ and change
it to twice the value and to
1
2
.
FileFig 3.8 (D8RONEXS)
Graph of diffraction pattern of the round aperture for R 3 to 10, X 1000,
and λ 0.01 for the determination of the diffraction angle of a round aperture.
The first minimum is at 1.22λ/2a.
D8RONEX is only on the CD.
Application 3.8.
1. Modify FileFig 3.8 and plot the Bessel function J 1(q) depending on q for 0
to 20. Normalize the Bessel function and determine the first zero at around
q 3.9 to five digits.
2. Make a graph of the diffraction pattern as described in FileFig 3.8 and
determine the first minimum (R/X (λ/a)(q/2π) 0.003).
3. We saw that for a slit the diffraction angle was Y/X (λ/d) or
(Y/X)/(λ/d) 1. We want to calculate the diffraction angle for the round
152 3. DIFFRACTION
aperture. Calculate the first zero of I (R) J 1(2π aR/λX)/(2πR/λX) for
λ 0.01 mm, X 1000 mm, and a 1.5 mm, and call it RR. Insert it into
R/X (RR/2π)(λ/a), but leave (λ/a) symbolically. Use the diameter of
the opening d 2a and calculate (R/X)/(λ/d) and get the value 1.22.
4. For comparison of the diffraction angle for a slit and a round aperture, plot
on the same graph for the same choice of parameters the diffraction pattern
of a slit and the diffraction pattern of a round aperture. Take the same value
for the width of the slit and the diameter of the aperture; that is, 2a d.
Compare the values of the first minima and observe that the overall width of
the diffraction pattern of the circular aperture is larger than the one for the
slit. The height for the first minimum of the round aperture is smaller than
that for the slit. Note that the plot for the slit is a linear plot whereas the plot
for the round aperture is a radial plot.
3.4.5 Gratings
Gratings are used in spectrometers from the near-infrared to the X-ray region.
They are usually reflection gratings with a zig-zag profile and called echelette
gratings. Simple transmission gratings may be produced as plastic films as repli-
cas of reflection gratings. We discuss the amplitude transmission grating with
larger or smaller transmitting areas at normal incidence or under an angle. We
also discuss a transmission echelette grating with the incident light at a specific
angle. In the appendix we discuss the step grating because of its potential for use
in Fourier transform spectroscopy.
3.4.5.1 Amplitude Grating
Amplitude gratings are periodic structures with alternating transmissive and
opaque strips. Similar to what we did for single apertures, we now have to
integrate over the open areas of the periodic set of slits (Figure 3.16). The
integration over one slit
u(θ)
y
2
d/2
y
1
−d/2
e
−ik(y sin θ)
dy (3.53)
is now extended to many slits; that is, we have a summation as
U(θ)
d/2
−d/2
e
−ik(y
1
sin θ)
dy
1
+
a+d/2
a−d/2
e
−ik(y
2
sin θ)
dy
2
+
2a+d/2
2a−d/2
...dy
3
+,.... (3.54)
We substitute into the second integral y
1
y
2
−a, similarly into the third integral
y
1
y
3
− 2a, and so on. Changing the integration variable to y in all integrals
3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION 153
FIGURE 3.16 Coordinates for the diffraction on an amplitude grating.
we get
{1 + e
−ika sin θ
+ e
−ik2a sin θ
+ e
−ik3a sin θ
+···e
−ik(N−1)a sin θ
}
·
d/2
−d/2
e
−ik(y sin θ)
dy. (3.55)
The bracket contains the amplitude array factor or interference factor (Ia)
(Chapter 2.4) multiplied by the amplitude diffraction factor (Da), equal to the
diffraction on a single slit. Symbolically we have
Diffraction amplitude Pa Interference factor Ia
∗
Diffraction factor Da (3.56)
or
Pa(θ ) Ia(θ )Da(θ ). (3.57)
154 3. DIFFRACTION
We sum up the interference factor
Ia(θ)
N−1
m0
e
−ikma sin θ
(1 − e
−ikNa sin θ
)/(1 − e
−ika sin θ
)
and have with Eq. (2.116) for the normalized intensity
{sin(πNasin θ/λ)/(N sin(πa sin θ/λ))}. (3.58)
The intensity is obtained as P (θ) Pa(θ)Pa(θ)
∗
,
P (θ) {[sin(πd sin θ/λ)]/[πd sin θ/λ)]}
2
·{sin(πNa sin θ/λ)/(N sin(πasin θ/λ))}
2
, (3.59)
and in the small angle approximation
P (Y ) {[sin(πdY/Xλ)]/[(πdY/Xλ)]}
2
·{sin(πNaY/Xλ)/(N sin(πaY/Xλ))}
2
. (3.60)
We have normalized the resulting pattern by division with 1/N
2
. The intensity
P (θ) of interference and diffraction of the amplitude grating is shown in the
graphs of FileFig 3.9 where we used values such as .001 instead of.000 in the
numerical calculations when approaching 0/0 at θ 0.
The first graph shows separately the numerator y(θ ) of the interference factor
and on the same graph the intensity I (θ ). The numerator y(θ ) is zero for the main
maxima and all side minima of I(θ). At the main maximum both the numerator
and the denominator are zero (i.e., one has 0/0), which results in 1, as discussed
for the interference factor in Chapter 2. The second graph of FileFig 3.9 shows the
zeroth order (main maximum) of the interference factor, at θ 0. There are N-1
side minima and N-2 side maxima between main maxima. Two side maxima
do not appear. The second graph P (θ ) shows the interference and diffraction
factors separately. The interference factor describes the maxima and minima,
and the envelope of the diffraction factor limits the intensity of the peaks of the
interference factor. The graph in FileFig 3.10 shows the corresponding pattern
for a slit opening 10 times smaller than the periodicity constant a.
For integer ratios of a/d, some maxima of the pattern can be seen while others
are suppressed. Taking as an example a/d 2, the zeroth and first orders of the
pattern are seen, and the second order is suppressed. If a/d has the integral value
nth, more orders may be observed and the resulting pattern becomes wider and
the nth order is suppressed.
A photograph of the diffraction pattern for the case of N 2, 3, 4, and 5 is
shown in Fig. 3.17. One observes one side maximum for N 3, two for N 4,
and three for N 5.
3.4. FAR FIELD AND FRAUNHOFER DIFFRACTION 155
FIGURE 3.17 Diffraction pattern of an amplitude grating: (a) N 2; (b) N 3; (c) n
4; (d) N 5 (from M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena,
Springer-Verlag, Heidelberg, 1962).
FileFig 3.9 (D9FAGRAMPS)
A graph for the amplitude grating is plotted using λ 0.0005, d 0.001, a
0.002, and N 6.We have plotted the intensity of the interferenceand diffraction
factor separately as well as the product.The numerator of the interference pattern
is shown and, one has 0/0 for all main maxima. The diffraction factor corresponds
to a slit pattern. The X scale has its origin at zero and there we have the zeroth
order (main maxima) of the interference factor. We have normalized the resulting
pattern by division with 1/N
2
and fixed the 0/0 problem at X 0 by using for
θ values such as 0.001 instead of 0.000. There are N -1 side minima and N-2
side maxima between the two main maxima.
D9FAGRAMPS
Diffraction on an Amplitude Grating at Normal Incidence
Width of openings d, center-to-center distance of strips a, wavelength λ, dis-
tance from grating to screen X, and coordinate on screen Y . All distances and
wavelengths are in mm; number of lines N. All parameters are globally defined
above the graph. D(A) is the diffraction factor. I (A) is the interference factor
156 3. DIFFRACTION
normalized to 1. The numerator is plotted separately to show where the main
maxima are located (0, 0). P (A) is the product of interference and diffraction
factor.
θ :−.5001, −.4999 ....5
D(θ):
sin
π ·
d
λ
· (sin(θ ))
!
π ·
d
λ
· (sin(θ ))
!
2
I (θ ):
sin
π ·
a
λ
· (sin(θ )) ·N
!
N · sin
π ·
a
λ
· (sin(θ ))
!
2
P (θ): D(θ) · I (θ) y(θ): sin
π ·
a
λ
· (sin(θ )) ·N
!
2
d ≡ .001 λ ≡ .0005 a ≡ .002 N ≡ 6.
Application 3.9.
1. Change FileFig 3.9 to small angle approximation. Make a graph and initially
use (all in mm) d 0.01, λ 0.0005, a 0.02, N 20, X 4000, and
Y −200 to 200. Compare the interference factor I (Y ) with the intensity
P (Y ) and observe;
a. that P (Y ) is 1 for I (Y ) at 0;
b. that the number of the side maxima is N-2 and that the maxima of I (θ)
close to the main maxima do not appear. Show that this is true for other
values of N;
c. that there are N − 1 side minima.