This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.
One of the most spectacular successes for phasor analysis arises in the
study of light diffraction by a narrow slit. The experiment is to shine laser
light through a slit in an otherwise opaque sheet and observe the pattern of
light that falls on a distant screen. When the slit is very narrow compared
with the wavelength of the light, then the light that falls on the screen is nearly
uniform in intensity. However, when the width of the slit is comparable to the
wavelength of the light, the pattern of light that falls on the screen is scalloped
in intensity, showing alternating light and dark bands. The experiment, and
the observed results, are illustrated in Figure 1.
Why should this experiment produce this result? Phasor analysis illuminates the question and produces an elegant mathematical description of a profoundly important optical experiment.
Huygens's Principle. We will assume, as Christiaan Huygens did, that the light incident on the slit sets up a light field in the slit that may be modeled by
N
N discrete sources, each of which radiates a “spherical wave of light.” This model is illustrated in Figure 2. The distance between sources is
d
d, and Nd=LNd=L is the width of the slit. Each source is indexed by
n
n, and
n
n runs from 0 to N-1N-1. The 0th0th source is located at the origin of our coordinate system.
The spherical wave radiated by the nthnth source is described by the equation
E(rn,t)= Re {ANej[ωt-(2π/λ)rn]}.E(rn,t)= Re {ANej[ωt-(2π/λ)rn]}.
(1)
The function E(rn,t)E(rn,t) describes the “electric field” at time t and distance rn
from the nthnth source. The field is constant as long as the variable ωt-(2π/λ)rnωt-(2π/λ)rn
is constant. Therefore, if we freeze time at t=t0t=t0, the field will be constant
on a sphere of radius
r
n
r
n. This is illustrated in Figure 2.
Fix
r
n
r
n
in
E(rn,t)E(rn,t) and show that E(rn,t)E(rn,t) varies cosinusoidally
with time
t
t. Sketch the function and interpret it. What is its period?
Fix
t
t in E(rn,t)E(rn,t) and show that E(rn,t)E(rn,t) varies cosinusoidally
with radius
r
n
r
n
. Sketch the function and interpret it. Call the “period in
r
n
r
n
"
the “wavelength.” Show that the wavelength is λλ.
The “crest of the wave E(rn,t)E(rn,t)" occurs when ωt-(2π/λ)rn=0ωt-(2π/λ)rn=0.
Show that the crest moves through space at velocity v=ωλ/2πv=ωλ/2π.
Geometry. If we now pick a point
P
P on a distant screen, that point
will be at distance
r
0
r
0
from source 0,...,rn0,...,rn from source n,...n,..., and so on. If we
isolate the sources 0 and
n
n, then we have the geometric picture of Figure 3.
The angle
φ
φ is the angle that point
P
P makes with the horizontal axis. The
Pythagorean theorem says that the connection between distances
r
0
r
0
and
r
n
r
n
is
(rn-nd sin φ)2+(nd
cos φ)2=r02.(rn-nd sin φ)2+(nd cos φ)2=r02.(2)
Let's try the solution
rn=r0+ndsinφ.rn=r0+ndsinφ.(3)
This solution produces the approximate identity
r02+(nd cos φ)2≅r02r02+(nd cos φ)2≅r02(4)
r02+(nd cos φ)2≅r02
1
+
(
n
d
r
0
c
o
s
φ
)
2
≅
1
.
r02+(nd cos φ)2≅r02
1
+
(
n
d
r
0
c
o
s
φ
)
2
≅
1
.
(5)
This will be close for ndr0<<1ndr0<<1. We will assume that the slit width
LL is small
compared to the distance to any point on the screen. Then Ndr0=Lr0<<1Ndr0=Lr0<<1,
in which case the approximate solution for
r
n
r
n
is valid for all
n
n. This means
that, for any point
P
P on the distant screen, the light contributed by the nthnth
source is approximately
E
n
(
φ
,
t
)
=
Re
{
A
N
e
j
[
ω
t
-
(
2
π
/
λ
)
(
r
0
+
n
d
sin
φ
)
]
}
=
Re
{
A
N
e
-
j
(
2
π
/
λ
)
r
0
e
-
j
(
2
π
/
λ
)
n
d
sin
φ
e
j
ω
t
}
.
E
n
(
φ
,
t
)
=
Re
{
A
N
e
j
[
ω
t
-
(
2
π
/
λ
)
(
r
0
+
n
d
sin
φ
)
]
}
=
Re
{
A
N
e
-
j
(
2
π
/
λ
)
r
0
e
-
j
(
2
π
/
λ
)
n
d
sin
φ
e
j
ω
t
}
.
(6)
The phasor representation for this function is just
E
n
(
φ
)
=
A
N
e
-
j
(
2
π
/
λ
)
r
0
e
-
j
(
2
π
/
λ
)
n
d
sin
φ
.
E
n
(
φ
)
=
A
N
e
-
j
(
2
π
/
λ
)
r
0
e
-
j
(
2
π
/
λ
)
n
d
sin
φ
.
(7)
Note that E0(φ)E0(φ), the phasor associated with the 0th0th source, is ANe-j(2π/λ)r0ANe-j(2π/λ)r0.
Therefore we may write the phasor representation for the light contributed by the nthnth source to be
En(φ)=E0(φ)e-j(2π/λ)ndsinφ.En(φ)=E0(φ)e-j(2π/λ)ndsinφ.
(8)
This result is very important because it shows the light arriving at point P
from different sources to be “out of phase” by an amount that depends on the
ratio ndsinφλndsinφλ
Phasors and Interference. The phasor representation for the field observed at point
P
P on the screen is the sum of the phasors contributed by each source:
E
(
φ
)
=
∑
n
=
0
N
-
1
E
n
(
φ
)
=
E
0
(
φ
)
∑
n
=
0
N
-
1
e
-
j
(
2
π
/
λ
)
n
d
sin
φ
.
E
(
φ
)
=
∑
n
=
0
N
-
1
E
n
(
φ
)
=
E
0
(
φ
)
∑
n
=
0
N
-
1
e
-
j
(
2
π
/
λ
)
n
d
sin
φ
.
(9)
This is a sum of the form
E(φ)=E0(φ)∑n=0N-1ejnθE(φ)=E0(φ)∑n=0N-1ejnθ
(10)
where the angle
θ
θ is (2π/λ)dsinφ
(2π/λ)dsinφ. This sum is illustrated in Figure 4 for
several representative values of
θ
θ. Note that for small θθ, meaning small φφ,
the sum has large magnitude, whereas for θθ on the order of 2π/N2π/N, the sum
is small. This simple geometric interpretation shows that for some values of
φφ, corresponding to some points PP on the screen, there will be constructive
interference between the phasors, while for other values of
φφ there will be
destructive interference. Constructive interference produces bright light, and
destructive interference produces darkness.
The geometry of Figure 4 is illuminating. However, we already know
from our study of complex numbers and geometric sums that the phasor sum
of Equation 3.85 may be written as
E(φ)=E0(φ)1-e-j(2π/λ)Ndsinφ1-e-j(2π/λ)dsinφ.E(φ)=E0(φ)1-e-j(2π/λ)Ndsinφ1-e-j(2π/λ)dsinφ.
(11)
This result may be manipulated to produce the form
E
(
φ
)
=
A
N
e
-
j
(
2
π
/
λ
)
r
0
e
-
j
(
π
/
λ
)
(
N
-
1
)
d
sin
φ
sin
(
N
π
d
λ
sin
φ
)
sin
(
π
d
λ
sin
φ
)
E
(
φ
)
=
A
N
e
-
j
(
2
π
/
λ
)
r
0
e
-
j
(
π
/
λ
)
(
N
-
1
)
d
sin
φ
sin
(
N
π
d
λ
sin
φ
)
sin
(
π
d
λ
sin
φ
)
(12)
The magnitude is the intensity of the light at angle φ from horizontal:
|
E
(
φ
)
|
=
|
A
N
sin
(
N
π
d
λ
sin
φ
)
sin
(
π
d
λ
sin
φ
)
|
.
|
E
(
φ
)
|
=
|
A
N
sin
(
N
π
d
λ
sin
φ
)
sin
(
π
d
λ
sin
φ
)
|
.
(13)
Limiting Form. Huygens's model is exact when dd shrinks to 0 and
NN increases to infinity in such a way that Nd→LNd→L, the slit width. Then
|
E
(
φ
)
|
→
|
A
sin
(
π
L
λ
sin
φ
)
π
L
λ
sin
φ
|
.
|
E
(
φ
)
|
→
|
A
sin
(
π
L
λ
sin
φ
)
π
L
λ
sin
φ
|
.
(14)
This function is plotted in Figure 5 for two values of LλLλ the width of the
slit measured in wavelengths. When Lλ<<1Lλ<<1 (i.e., λL>>1λL>>1), then the light
is uniformly distributed on the screen. However, when Lλ>1(λL<1)Lλ>1(λL<1), then
the function has many zeros for || sin φ|<1φ|<1, as illustrated in the figure. These
zeros correspond to dark spots on the screen where the fields radiated from
the infinity of points within the slit interfere destructively.
(MATLAB) Plot the discrete approximation
A
N
sin
(
N
π
d
λ
sin
φ
)
sin
(
π
d
λ
sin
φ
)
A
N
sin
(
N
π
d
λ
sin
φ
)
sin
(
π
d
λ
sin
φ
)
(15)
versus sin
φ
φ for Lλ=Ndλ=10Lλ=Ndλ=10 and N=2,4,8,16,32N=2,4,8,16,32. Compare with the
continuous, limiting form
A
sin
(
π
L
λ
sin
φ
)
π
L
λ
sin
φ
A
sin
(
π
L
λ
sin
φ
)
π
L
λ
sin
φ
(16)
(What does "Endorsed by" mean?)
"Reviewer's Comments: 'I recommend this book as a "required primary textbook." This text attempts to lower the barriers for students that take courses such as Principles of Electrical Engineering, […]"