Download Light 1 Mathematical representation of light (EM waves)

Light
(Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 1,2, 3, 8)
Light is an Electromagnetic (EM) field arising from the non-uniform motion of charged particles. It is
also a form of EM energy that originates from the motion of charged particles between various energy states.
The primary characteristics of light are its velocity denoted by c, frequency ν, wavelength λ, Energy E and
momentum p. These quantities are related as follows:
• velocity of light: c(in m/s) = νλ; ν is in Hz (s−1 ); and λ is in m
• Energy of Photon: E(in J) = hν where h is Planck’s constant in Js
• Also, since ν = λc , the frequency and wavelength are closely related
Light exhibits wave-like or particle-like behavior depending on conditions. For instance, in the photoelectric
effect light behaves like particles (referred to as photons) while in interference conditions (e.g. Young’s double slit experiment) it manifests wave-like properties. Wave-particle duality can be expressed conveniently
as a relation between wavelength (a wave characteristic) and momentum ( a particle characteristic) using de
Broglie’s relation:
p=
1
h
λ
Mathematical representation of light (EM waves)
Light is a transverse EM oscillation travelling at speed c. A monochromatic (single frequency) light wave
can be expressed as:
# r, t) = E
# o (#r, t)e−i("k"r−ωt+φ)
E(#
where,
# is the electric field vector with peak value | Eo | and has units of volts/m
• E
• #r is position vector identifying a location in space in Cartesian coordinates
−1
• #k is the wave-vector with magnitude | k |= 2π
λ and has units of m
• t is time in s−1
• ω is the period of oscillation in radians and has magnitude ω = 2πν
1
(1)
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Light
• φ is a starting phase angle in radians
Alternately, light can also be expressed in terms of an oscillating magnetic field as:
# r, t) = B
# o (#r, t)e−i("k"r−ωt+φ)
B(#
# r, t) is the space and time-dependent magnetic field in T esla = N/amp − m with peak value
where B(#
of | Bo |
1.1
Wave and phase velocity
Eq. 1 is a general representation of a plane wave propagating in the positive x-direction including both the
real and imaginary components of the wave. Often we are interested in the real part of the wave, the part
that actually carries the energy. A much simplified form of the wave can be obtained by first making use of
Euler’s complex notation through which the plane wave of eq. 1 can also be expressed as:
# r, t) = E
# o (#r, t)[cos(#k#r − wt + φ) + isin(#k#r − wt + φ)]
E(#
and as will be often see, the real part of the wave can simply be denoted as:
# r, t) = E
# o (#r, t)cos(#k#r − wt + φ)
E(#
(2)
Using this form we can express some important features of wave motion:
1. The phase of the wave, θ: The argument of the cosine term is the phase of the travelling wave, i.e:
θ = kr − wt + φ
(3)
2. Rate of change of phase with time or distance gives us the frequency and wavenumber respectively as
follows:
∂θ
∂t |= ω
| ∂θ
∂r |= k
(a) |
(b)
In the above partial derivatives it was assumed that the initial phase angle φ is independent of time
and/or position. This need not always be the situation
From the rate of change of the phase with time and position and important quantity known as the phase
velocity can be extracted as:
v=
|
∂r
=
∂t
|
∂θ
∂t
∂θ
∂r
|
|
=
ω
2πν
= 2π = νλ = c
k
λ
So we see that the phase velocity is equivalent to the wave velocity, i.e. the speed of light.
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1.2
Light
Relation between E and B
Faraday’s law which states that a changing magnetic field generates an electric current (i.e. an electric field)
relates E and B through the relation:
# =
where, "
∂#
∂x i
+
∂#
∂x j
+
∂ #
∂x k.
#
# ×E
# = − ∂B
"
∂t
The above relation gives: | B |=
|E|
c
# E,
# i.e.
and the condition that B⊥
the magnetic and electric fields are always perpendicular to each other.
1.3
Plane monochromatic linearly polarized light wave
Referring to Fig. 1 a monochromatic light wave travelling in a single direction (i.e. x−direction) that has
an electric (or magnetic) field oscillating in a single direction (linearly polarized) say y can be expressed as:
#
E(x,
t) = Eo (x, t)e−i(kx x−ωt+φ)#j
and from Faraday’s law, the corresponding magnetic field can be expressed as:
Bo (x, t) −i(kx x−ωt+φ) # #
#
e
(i × j)
B(x,
t) =
c
where #i is the unit vector in the direction of propagation of the wave and #j is the polarization vector
denoting the direction of oscillation of the electric field.
Figure 1: Plane monochromatic linearly polarized wave travelling along the x-direction with the electric
field oscillating in the y-direction.
1.4
Energy and Pressure in light
EM waves carry energy simultaneously through their electric and magnetic components with the instantaneous energy density, i.e. energy per unit volume U in J/m3 expressed for each part as:
)
UE = E 2
2
UB =
1 2
B
2µ
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where ) and µ are the permittivity and permeability of the medium in which light is travelling. For the case
of vacuum or air, ) = )o = 8.85 × 10−12 C 2 − s2 m−3 kg −1 and µ = µo = 4π × 10−7 m − Kg − C −2 .
The total energy density carried by light is then:
1 2
)
B
UT otal = UE + UB = E 2 +
2
2µ
further, since E = cB and c =
√1
(µ
the above can be expressed as:
UT otal = )E 2 =
1 2
B
µ
• Pressure (P): Since the unit of pressure is N/m2 = J/m3 which is the same as U so the energy
density carried by the EM field is also the pressure of the EM field, i.e. P = )E 2
• Rate of energy flow S: The flow of energy in EM fields is often expressed by estimating the instan-
taneous energy flowing across a unit are per unit time. Since the energy is being transported in the
direction of the light beam, it is useful to express the flow as a vector quantity known as the Poynting
vector S which has units if W/m2 given by:
# = 1E
# ×B
# = c2 )E
# ×B
#
S
µ
(4)
• Intensity of light I: Often it is useful to evaluate the time averaged light energy incident on a unit area
per unit time (as compared to the instantaneous energy transported by the Poynting vector). Using eq.
4 the time averaged intensity can be evaluated as:
# ×B
# |
I =< S >t = c2 ) | E
Using the fact that the intensity is a real quantity the above equation can be evaluated using the real
components of E and B to give:
1
I = c2 ) | Eo × Bo |< Cos2 (kr − ωt + φ) >= c2 )E 2
2
2
Polarization
In the previous section, linearly polarized light was introduced by stating that the electric (or magnetic)
field oscillated in a single direction. In general a beam of light can have the E and B fields oscillating in
any arbitrary direction with the only constraint that the oscillation is perpendicular to the wave propagation
direction. The polarization state of a beam is a function of two quantities: (i) the oscillation direction of the
E field and (ii) the behaviour of this oscillation with time. Based on this, and referring to Fig. various types
of polarization states may be defined:
1. Random polarization: here the oscillation of the E field is equally likely to be in any direction at a
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given instant of time (but always perpendicular to the direction of propagation). In other words, if
snapshots of the E-field vector were to be captured, the vectors would lie in random directions.
2. Circular polarization: here the E-field is observed to rotate about the direction of propagation maintaining equal magnitude as a function of time.
3. Elliptical polarization: the electric field again rotates about the direction of propagation nut its magnitude varies with position.
The various polarization types may be visualized in Fig. 2. To visualize the different polarizations one may
make use of an oscilloscope and combine two sinusoidal signals in perpendicular directions to create Lissajous figures. This web-link allows to to do this through java applets: http://www.surendranath.org/Applets.html
Figure 2: Different polarization states viewed along the direction of propagation. The direction of the
electric field represents a time average over sufficiently long times.
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Huygens-Fresnel principle
While a ray of light travels in a straight line, a collection of rays originating from a light source varies
spatially according to the nature of the source and other obstacles in the path of light. Huygens came up
with an approach to determine the distribution of light intensity for an arbitrary situation given its value
at some earlier instant. To understand the Huygens principle, we must first understand the concept of a
wavefront. The wavefront is the surface or locus of all point on the travelling wave having identical phase.
From eq.’s 2 and 3 the mathematical condition describing a wavefront is that all points have the same θ
value. Fig. 3(a) denotes a light source and its wavefront. Huygens’ theorem is especially important because
a simple geometrical construction can be used to determine the wavefront at any later time. The theorem
goes as follows:
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Every point of a wavefront may be considered a source of small secondary wavelets, which
spread out in all directions from their centers with a velocity equal to the velocity of the propagating wave. Tye new wavefront it then found by constructing a surface tangent to the secondary
wavelets, thus giving the locus of all points with the same phase.
Such a construction is demonstrated in Fig. 3(b). The original wavefront, S-S, is traveling as indicated by
the arrows. The shape after a time t can be obtained by constructing a number of circles centred on S-S with
radius r = vt, where v is the propagation velocity. The common tangent drawn between these circles will
give the new wavefront. One of the problems with the Huygens’ construction is that according to the form,
a wave must be travelling in the backward direction - but this is not the case. This problem was solved by
Fresnel when he modified the construction to account the magnitude and phase of the local electric field, i.e.
the math that gives us the interference phenomenon (to be discussed later). Now the modified theorem is
called the Huygens-Fresnel principle.
Figure 3: Example of an expanding wavefront and the construction of the Huygens wavefronts.
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Geometrical forms of Light sources
The spatial variation of light intensity differentiates some common types of propagating waves. These
propagating waves can also be distinguished on the basis of the shape of their wavefronts and with reference
to Fig. 4 the three common forms are:
1. Plane waves: Here the wavefront is a planar surface perpendicular to the propagation direction. The
plane EM wave can be expressed as:
#
E(x,
t) = Eo ei(kx−ωt+φ)#i
# E
# ∗ ∝ Eo2 and is independent
The intensity of a plane wave as a function of space and time is I ∝ E.
of r i.e. the distance from the source.
2. Spherical waves: Here the wavefronts are concentric spheres about the source of the waves and the
electric field can be expressed as:
E(r, t) =
Eo i(kr−ωt+φ)
e
p#
r
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where p# is the polarization direction. The intensity of a spherical wave as a function of its distance
from the source will be I ∝ E 2 ∝
Eo2
,
r2
which falls of as the square of the distance from the source.
3. Cylindrical waves: Here the wavefronts are concentric cylinders about the source of the waves and
the electric field can be expressed as:
Eo
E(r, t) = √ ei(kr−ωt+φ) p#
r
2
from which the intensity will change as I ∝ Ero
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(a)
Light
(b)
(c)
Figure 4: Types of wavefronts and light sources. (a) Plane wave, (b) spherical waves; (c) cylindrical waves.
Source of figures: Hecht, Ch 2.
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Light sources and the EM spectrum
The most important natural light source is the sun, which as Newton discovered, consists of white light.
Unlike monochromatic light, white light is composed of numerous frequencies (or wavelengths). The reason
the sun emits white light is primarily because it is a very dense and hot body of gases comprising various
neutral and charged species, including ions and electrons. The random collision of the charged particles
leading ti acceleration/deceleration, i.e. non-uniform motion, results in various frequencies being emitted.
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In comparison to the sun, artificial light sources can be made to emit in specific frequencies. These light
sources are known as spectral lamps and typically make use of the well defined energy levels in atoms to
emit light of well defined frequencies, as indicated by the relation between the energy and frequency of light.
The principle of a spectral lamp is as follows: A lamp consists of gas of a specific element or compound,
such as Na, Hg, Ne, etc. An electric discharge is generated in the lamp by using a high-voltage source to
produce excitation of the electrons in the gas atoms to higher energy levels. When these electrons fall back
to their original energy levels, they radiate light of frequency proportional to the difference in energy levels.
Consider a Na atom with equilibrium electron configuration given by 1s2 2s2 2p6 3s1 . By the high-voltage,
lets say the 3s1 electron is excited to the 3p level. On falling back to the ground state the frequency of light
emitted will be:
ν=
E3p − E3s
h
The web-link http://webexhibits.org/causesofcolor/3B.html introduces you to various sources of light
and color. The study involving the observation of frequencies emitted/absorbed by various objects is called
spectroscopy while the measurement of the emitted/absorbed frequencies is called spectrometry. The range
of frequencies commonly encountered by us has been categorized into a spectrum (ref. p 74, Hecht) based
on wavelength, energy, color etc. Some examples are given in table 1.
name
frequency ν in
Hz
wavelength λ
Energy E in eV
Uses
Radio
frequency
waves (RF)
1kHz-1GHz
km’s to m’s
≤ 10−6
Radio communication; power
lines
Microwaves
109 − 1011
30 cm ∼ 1 mm
10−6 ≤ E ≤ 10−3
Infrared
(IR)
3 × 1011 − 4 ×
1014
1 mm ∼ 780 nm
10−3 ≤ E ≤ 1
Visible light
4 × 1014 −
7.7 × 1014
760 nm ∼ 380 nm
1≤E≤3
1014
ultraviolet
(UV)
8×
−
3.4 × 1016
10 nm ∼ 400 nm
3.2 ≤ E ≤ 100
x-rays
2.4 × 1016 −
5 × 1019
0.01 Å ∼ 100 Å
100 ≤ E ≤ 2 × 105
Gamma
rays
> 5 × 1019
< 0.01 Å
> 2 × 105
microwaves resonate with
vibration frequencies in water
and is used in a microwave over
Most solids absorb and emit IR
and so these are most efficient at
delivering energy to solids
visible colors in this range
Rays harmful to the skin. Also
absorbed by the cornea of the
eye, which is the primary cause
of snow blindness.
Wavelength are
comparable/smaller then
inter-atomic distances in solids
and so x-ray diffraction is a
powerful materials science tool
Table 1: EM spectrum. Source: Hecht, Ch 3
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