NUMERICAL ANALYSIS OF OPTCIAL AND ELECTROMAGNETIC PHENOMENA
PROPAGATION OF ELECTROMAGNETIC WAVES IN FREE SPACE
THE WAVE EQUATION IN FREE SPACE
Electromagnetism is the fundamental theory that underlies most of optics associated with wave
phenomena. Maxwell’s equations provide the basic information needed to describe electromagnetic
phenomena. Equation 1 gives a summary of Maxwell’s equation for free space (vacuum).
(1A)
Gauss’s Law
(1B)
(1C)
Faraday
(1D)
Ampere’s Law
electric field [V.m-1]
magnetic
field [T]
= 8.85x10-12
F.m-1
=4
x10-7
electric permittivity of free space
N.A-2
magnetic permeability of free space
We can use Maxwell’s equation to derive the wave equation using the identify for the vector
(2)
The speed of light in free space is
(3)
The speed of light in free space is defined to be
So, the wave equations for the electric and magnetic fields are (4A)
(4B)
PLANE HARMONIC (MONOCHROMATIC) WAVES
The vectors for the electric field
Cartesian components and each
component of
and
and magnetic field
can be resolved into their X, Y and Z
satisfies the scalar wave equation
(5)
where
is called the wavefunction representing any one of the field components
.
It is often convenient to assume that the wavefunction
varies
harmonically (sinusoidal variation with time), so the wavefunction
can be expressed the product of a spatial wavefunction
and a harmonic time-varying function
(6)
By substitution of equation 6 into equation 5, we can derive the
Helmholtz equation
(7)
The Maxwell equations 1C and 1D, for harmonically varying fields, reduce to
(8C)
(8D)
We will consider the special case of a plane harmonic electromagnetic wave propagating in the +Z
direction, where
(9)
Amplitude of the wave [V.m-1]
Phase of the wave [rad]
Propagation constant or wave number [m-1 or rad.m-1]
Angular frequency [s-1 or rad.s-1]
By direct substitution of equation 9 into equation 5, it is to verify that the electric field given by equation 9
is a solution to the wave equation
(5) provided that the ratio of the constants
and
is
(10)
We can then substitute equation 9 into the Maxwell equation 1C to find the magnetic field component of
our plane harmonic electromagnetic wave
(11)
The ratio
is called the impedance of free space (wave impedance of a plane wave)
Our plane harmonic electromagnetic wave is a transverse wave propagating in the +Z direction with the
electric field varying in the +X direction and the magnetic field varying in the +Y direction. The vibrations of
the electric field and the magnetic field are in phase at the same frequency at all times. This type of
wave, in which the electric field vector is always parallel or antiparallel to a fixed direction is called a
plane-polarized wave. For the solution given by equation 9, the plane of polarization is the XY plane (figure
1).
Fig. 1.
A plane electromagnetic wave propagating in the +Z direction. The direction of
propagation and the directions of the oscillations for the electric field and magnetic field are
mutually orthogonal. The plane of polarization is the XY plane.
Fig. 2. Wavefronts of a plane wave travelling in [3D] space. The direction of propagation is
perpendicular to the planes of constant phase.
The wave advances such that the phase in the plane perpendicular to the direction of propagation remains
constant.
We can differentiate this expression for the phase with respect to time t
But
is the velocity
at which the phase advances. Therefore, the plane wave given propagates
in the +Z direction with a phase
velocity
where (10)
When the phase of the wave increases by
unchanged
rad (one complete cycle), the value of the electric field is
For one cycle, where the wave advances a distance
the same
and the electric field values at
and
are
(12)
is the wavelength and corresponds to the distance measured along the direction of propagation such
that the electric field or magnetic field goes through one complete cycle. Hence, the propagation
constant k is the spatial frequency and is the number of cycles in the distance .
For one cycle, where the time advances by T and the electric field values at times
and
are the
same
(13)
T is the period and is the time for one complete oscillation. The reciprocal of the period T is the frequency
f and is the number of cycles per unit time.
T
period [s]
f
frequency [Hz]
Hence, the wave travels a distance
in the time interval T. The phase velocity
of the wave is defined as
(10)
For a plane wave given by
the phase velocity is
, hence the wave travels in the
Z direction.
The wavefunctions are expressed as complex function. However, it is only the real part of these functions
that represent actual physical quantities.
ENERGY FLOW AND THE POYNTING VECTOR
The rate at which energy is transferred by a plane wave is given by the Poynting vector S [W.m-2] which is
defined as the cross produce of the electric and magnetic fields
(15)
For our harmonic wave (9)
(11)
Taking the real part for the actual value of the Poynting vector
The average value of the cosine squared is equal to ½. Hence, the time average value of the Poynting
vector is
The flow of energy is in the same direction in which the wave propagates.
The magnitude of the average Poynting vector is called the intensity
or irradiance, so
(16)
Thus, the rate of energy flow is proportional to the square of the amplitude of the electric field.
The temporal and spatial variations of the electromagnetic field for a 1.0 mW plane monochromatic wave
propagating in the +Z direction with its plane of polarization directed long the X-axis are displayed in
figure 3, 4 and 5 for green light
. A wavelength in the range from 380 nm to 780 nm can
entered in the INPUT section of the script. The wavelength is assigned a color to match its spectral color
by calling the function colorCode.m.
Fig. 3.
A plot of the electric field as a function of position at three time steps. Note: the
wave advances to the right a distance of one wavelength in a time interval of one period.
op_001.m
Fig. 4. The oscillations of the electric field and magnetic field are in phase and
perpendicular to each other and perpendicular to the direction of propagation as shown in
figure 1. op_001.m
Fig. 5. Animation of the vibration of the electric field as the wave propagates in the +Z
direction. From the plot, you can
verify the relationship
. How far does the wave advance in one period?
Fig. 6. The electromagnetic fields for a plane wave with
.
It is easy to create an animation of a plane wave propagating in two- dimensions. The [2D] version of a
plane wave shows how the wavefronts are straight lines (lines of constant phase) that move in the
direction of propagation. In [3D], you can think of the straight lines of constant phase as planes
extending out of the screen and moving in the direction of propagation