Light Waves

Pablo A. Costanzo Caso

 The Nature of Light
• There are three theories explain the nature of light:
– Quantum Theory – Light consists of small particles (photons)
– better explains light detection and generation
– Electromagnetic Theory – Light is modeled as two coupled
vector waves, E and H. Polarization explain devices as
beamsplitters.
– Wave Theory – Light travels as a scalar wave function. It
explain interference and diffraction, which involves phase.
– Ray Theory – Light travels along a straight line and obeys
laws of geometrical optics – useful tool when the objects are
much larger than the wavelength of light. Explain focusing
and imaging.
The Nature of Light
Wavelength Ranges
 Wave Theory of Light

• Electromagnetic light signal has electric and magnetic

fields orthogonal to each other.
• The frequency of this EM wave is in the order of THz.
Therefore, it is convenient to measure it in terms of
wavelength.
c  
• where, c - speed of light 3 X 108 m/s in air, ν - frequency
and λ- wavelength
 Wave Theory of Light

Frequency and waveength

• Ex. Bandwidth of 2 channels of 100GHz at 700nm and 1550nm.

 Quantum Theory of Light
• Light consists of discrete units called photons. The
energy in a photon
Eg = hν
h= 6.6256 x10(-34) J.s is the Planck’s constant and
ν is the frequency.

• Ex1: Find the energy of a photon travelling with 200 THz frequency
• Ex2: Show
1.24
Eg  eV
 ( m)
Maxwell's Equations in Dielectric Media

In free space, P=M=0

ME in Free Space

0  1/(36)10-9 F/m ; 0  410-7 H/m

D: electric flux density P: polarization density

B: magnetic flux density M: magnetization density

Equations relating P and E, as well as M and H, are established once the medium is specified.
Depend on the electric and magnetic character of the medium, respectively.

The Wave Equation

A necessary condition for E and H to satisfy Maxwell's equations is that
each of their components satisfy the wave equation
𝑐0 𝜀𝜇
𝑛= = 𝜀0 𝜇0
𝑐

The Helmholtz Equation Substituting the monochromatic electric and magnetic fields in the wave
equation. (Linear, Nondispersive, Homogeneous, and Isotropic Media)

𝑘 = 𝑛𝑘0 = 𝜔 𝜀𝜇 𝑘0 = 𝜔/𝑐0

U represents the complex amplitude of any of the three components of E or H

ELECTROMAGNETIC WAVES IN DIELECTRIC MEDIA

 The character of the medium is determined by the relation between the polarization
and magnetization densities, P and M, and the electric and magnetic fields, E and H.
These are known as the constitutive relation.
 In most media, the constitutive relation separates into a pair of constitutive relations,
one between P and E, that describes the dielectric properties, and another between M
and H, that describes its magnetic properties.
 It is useful to regard the P-E constitutive relation as arising from a system in which E is
the input and P is the output or response. Note that E=E(r,t) and P=P(r,t) are functions of
both position and time.
Definitions for Dielectric Media

Linear: P(r,t) is linearly related to E(r,t). (superposition applies)

Nondispersive: The response is instantaneous. P at time t is determined by E at the
same time t and not by prior values of E. (Nondispersiveness is an idealization).
Homogeneous: The relation between P and E is independent of the position r.
Isotropic: The relation between the vectors P and E is independent of the direction
of the vector E, so that the medium exhibits the same behavior from all directions. The
vectors P and E must then be parallel. (Anisotropic then X is tensor 3x3)
Spatially nondispersive: Relation between P and E is local. P at each position r is
influenced only by E at the same position r.
Linear, Nondispersive, Homogeneous, and Isotropic Media

X is called the electric susceptibility

electric
  permittivity
of the
medium
relative permittivity

Dispersive Media
The relation between the vectors P and E in a
dispersive dielectric medium is dynamic rather
than instantaneous. (Convolution, X is the
impulse response)

Nonlinear Media

A nonlinear dielectric medium is defined as one in

which the relation between P and E is nonlinear
 Plane Waves
• Most light waves are plane waves
• For plane waves the transverse condition requires
that the electric and magnetic field be
perpendicular to the direction of propagation and
to each other.
• The electric field vector of a plane wave may be
arbitrarily divided into two perpendicular
components labeled x and y (with z indicating the
direction of travel).
Field distributions in plane E&M waves

E and H fields are orthogonal to

each other and to the direction
of propagation Z (or k)
 Basics about Plane Waves
U (r )  A exp(  jkz)
u (r, t )  A cos[2t  kz  arg( A)] Plane wave travelling at z direction

 A cos[2 (t  z / c)  arg( A)]

c0 0 2
c  k  nk0  remains unchanged
n n 
 Phase Velocity

vp

 c0 k  nk0
Phase velocity: v p  c   co: Speed of light in air
k n 0  n n: Refractive index
PULSE PROPAGATION IN DISPERSIVE MEDIA

Carrier and Envelope

vp vg
PULSE PROPAGATION IN DISPERSIVE MEDIA

Group of Waves
Consider two plane waves Ex1 and Ex2 propagating in the +ve z direction

𝐸𝑥1 = 𝐸0 cos 𝜔0 + ∆𝜔 𝑡 − 𝑘0 + ∆𝑘 𝑧

𝐸𝑥2 = 𝐸0 cos 𝜔0 − ∆𝜔 𝑡 − 𝑘0 − ∆𝑘 𝑧

The superposition of the waves (wavepacket) is given by

𝐸 = 𝐸𝑥1 + 𝐸𝑥2 = 2𝐸0 cos ∆𝜔𝑡 − ∆𝑘𝑧 cos 𝜔0 𝑡 − 𝑘0 𝑧

Slowly varying envelope Rapid oscillation (carrier frequency)

𝑘0 𝑧
𝐸 = 2𝐸0 cos ∆𝜔 𝑡 − ∆𝑘𝑧 ∆𝜔 cos 𝜔0 𝑡 − 𝜔0

= 2𝐸0 cos ∆𝜔 𝑡 − 𝑧 𝑇 cos 𝜔0 𝑡 − 𝑧 𝑇

𝑔 𝑝

= 2𝐸0 cos ∆𝜔 𝑡 − 𝑇𝑔 cos 𝜔0 𝑡 − 𝑇𝑝

PULSE PROPAGATION IN DISPERSIVE MEDIA

Group of Waves
𝑘0 𝑧
𝐸 = 2𝐸0 cos ∆𝜔 𝑡 − ∆𝑘𝑧 ∆𝜔 cos 𝜔0 𝑡 − 𝜔0

= 2𝐸0 cos ∆𝜔 𝑡 − 𝑧 𝑇 cos 𝜔0 𝑡 − 𝑧 𝑇

𝑔 𝑝

= 2𝐸0 cos ∆𝜔 𝑡 − 𝑇𝑔 cos 𝜔0 𝑡 − 𝑇𝑝

We define phase and group delay and velocity
𝑧 ∆𝑘𝑧
𝑧 𝑘0 𝑧 𝑇𝑔 = =
𝑇𝑝 = = 𝑣𝑔 ∆𝜔
𝑣𝑝 𝜔0
𝜔0 ∆𝜔 𝑑𝜔 1
𝑣𝑝 = = 𝑐0 𝑣𝑔 = =
𝑘0 ∆𝑘 𝑑𝑘 𝑑𝑘
𝑑𝜔
In a medium with refraction index n  k=nk0 =n/c, and n depend on the frequency n()

1 1 1 𝑐0
𝑣𝑔 = = = =
𝑑𝑘 (1/𝑐0 )𝑑 𝜔𝑛(𝜔) (1/𝑐) 𝑛 + 𝜔𝑑𝑛 𝑑𝜔 𝑛𝑔
𝑑𝜔 𝑑𝜔
PULSE PROPAGATION IN DISPERSIVE MEDIA

Group Velocity

vg 
1

c0  n 
k ng
m/s ng   n   
   
• The function k(ω) is known as the dispersion relationship.
• If k is proportional to ω, then the group velocity is exactly
equal to the phase velocity.
• Usually it is not the case; the envelope becomes distorted
as it propagates.
• This is called the "group velocity dispersion“.
• The GVD is important single mode optical fibers.
 Refractive Index Dependency of
Light Velocity
• Light travels faster in air than water: apparent and
true depth
• Light in the fiber core travels slower than light in the
fiber cladding ‘waveguide dispersion’
ABSORPTION AND DISPERSION

A. Absorption
The dielectric media considered thus far have been assumed to be fully transparent,
i.e., not to absorb light.
We adopt a phenomenological approach to the absorption of light in linear media.
Consider a complex electric susceptibility

For monochromatic light, the Helmholtz equation for the complex amplitude U(r)
remains valid, but the wavenumber k becomes complex valued:

ko=w/c is the wave

number in free space.
Writing k in terms of real and imaginary parts
ABSORPTION AND DISPERSION
As a result of the imaginary part of k, a plane wave U traveling through such a medium in the
z-direction undergoes a change in magnitude (as well as the usual change in the phase)

For >0 which corresponds to absorption in the medium, the envelope A of the original
plane wave is attenuated by a factor exp(- z/2) and the intensity by exp(- z)

 is recognized as the absorption coefficient (or attenuation coefficient or extinction

coefficient) of the medium.

 is the rate at which the phase changes with z, it represents the propagation constant of
the wave. The medium therefore has an effective refractive index n defined by (for
absorbent media)

Substituting for k it is relates the refractive index n and the absorption coefficient  to the
real and imaginary parts of the susceptibility X' and X":
The impedance associated with
the complex susceptibility

In general, therefore, X, k, , and  are complex quantities while , , and n are real.
ABSORPTION AND DISPERSION

B. Dispersion

Dispersive media are characterized by a frequency dependent (and wavelength dependent)

susceptibility X(v) and electric permittivity (v), refractive index n(v) and speed c0/n(v).

Since the angle of refraction in Snell’s law depends on refractive index, which is wavelength
dependent, optical components fabricated with dispersive materials (lenses, prisms) bend
light of different wavelengths by different angles.

Due to the frequency-dependent speed of light in a dispersive medium, each of the

frequency components comprising a short pulse of light experiences a different time delay.
PULSE PROPAGATION IN DISPERSIVE MEDIA
A dispersive medium is characterized by a frequency- dependent refractive index and
absorption coefficient, so that monochromatic waves of different frequencies travel
through the medium with different velocities and undergo different attenuations.
Since a pulse of light comprises a sum of many monochromatic waves, each of which is
modified differently, the pulse is delayed and broadened (dispersed in time); in general its
shape is also altered.

Group Velocity
Consider a pulsed plane wave traveling in the z direction through a lossless dispersive
medium with refractive index n(w).

z = 0  U(0, t) = A(t) exp(jwot), A(t) is the complex envelope

z = z  U(z, t) = A(t - z/v) exp[jwo(t – z/c)]

c = co/n(wo) speed of light in the

medium, v is the group velocity
(velocity at which the envelope
travels)
PULSE PROPAGATION IN DISPERSIVE MEDIA

Group Velocity for absorbent media (w) = k0n(w)

Frequency-dependent propagation constant

Group velocity at w=w0

Phase velocity

In an ideal (nondispersive) medium, (w) = w/c whereupon v = c and the group and
phase velocities are identical.
PULSE PROPAGATION IN DISPERSIVE MEDIA

Group Velocity
Since the index of refraction of most materials is typically measured and tabulated as a
function of optical wavelength rather than frequency, it is convenient to express the
group velocity v in terms of n().

From the chain rule

v Group Velocity
N Group Index
Group Velocity Dispersion (GVD)

Since the group velocity v =1/(d/dw) is often frequency dependent

 different frequency components of the pulse undergo different delays Td = z/v
 group velocity dispersion (GVD).

To estimate the spread associated with GVD we consider a pulse with two frequency
components: v and v + v, which suffers a differential group delay (for a distance o
propagation z)

Dispersion Coefficient (s/m-Hz)

Group Velocity Dispersion (GVD)

If the pulse has an initial spectral width a v(Hz), its temporal spread is

Pulse Spread
Group Velocity Dispersion (GVD)

If the refractive index is specified in terms of the wavelength, n(o), from

Dispersion Coefficient
(s/m-nm)

It is also common to define a dispersion coefficient in terms of the wavelength D

instead of the frequency. Using

Dispersion Coefficient Pulse Spread

(s/m-nm)

In fiber-optics applications D  is usually specified in units of ps/ km-nm: the pulse

broadening is measured in ps, the length of the medium in km, and the source
spectral width in nm.
Normal and Anomalous Dispersion

The sign of the dispersion coefficient Dv (or D  ) does not affect the pulse broadening rate.
It affects the phase of the complex envelope of the optical pulse and plays an important role
in pulse propagation through media.
If D  < 0 (Dv > 0), the medium is said to exhibit normal dispersion (travel time for higher-
frequency components is greater)
If D  > 0 (Dv < 0), the medium is said to exhibit anomalous dispersion, (higher-frequency
components travel faster and arrive earlier)
 Changing Refractive Index
• In general, the refractive index n is some
function of the wavelength of light, n = n(λ).
• The wavelength dependence of a material's
refractive index is usually quantified by an
empirical formula, the Cauchy or Sellmeier
equations

red   yellow  blue

 Sellmeier Equation

• Refractive Index n is a nonlinear function of

wavelength
• The relationship is governed by ‘Sellmeier
equation’
EXAMPLE 5.6-1. Dispersion in a Multi-Resonance Medium: Fused Silica.
In the region between 0.21 and 3.71 um, the wavelength dependence of the refractive index n for
fused silica at room temperature is well characterized by the Sellmeier equation

Expressions for the group index N and the dispersion

coefficient D are readily derived from this equation as

The refractive index n decrease monotonically with

increasing wavelength, exhibit a point of inflection at
o = 1.276um. At this wavelength, the group index N
is minimum so that the group velocity v = co/N is
maximum.

Since the dispersion coefficient D is proportional to the second derivative d2n/do2, it vanishes at this
wavelength. Zero dispersion coefficient signifies minimal pulse broadening.
At s < 1.276 um, D < 0 (medium exhibits normal dispersion). At s > 1.276 um, D > 0 and the
dispersion is anomalous.
 Polarization
• Polarization of a plane wave is the
orientation of the oscillations of the E field;
perpendicular to the direction of propagation
• For a simple harmonic wave, the electric
vector in orthogonal directions may have:
– Different amplitude
– Different phase
• The resulting wave is
– Linearly, elliptically or circularly polarized
 Polarization states
• If the resulting electric field is oscillating
along a straight line, it is called a linearly
polarized (LP) or plane polarized wave
• If the E field rotates in a circle (constant
magnitude) it is called circularly polarized
• If the E field rotates in an ellipse then, it is
called elliptically polarized wave
• Natural light has random polarization – The
orientation of E filed keep changing randomly
 Polarization

0x 0y

0x 0y

0x 0y 0x 0y

Ax  E0 x exp( j x )
Complex Envelopes
Ay  E0 y exp( j y )
Adding two linearly polarized waves with a
phase shift will produce an elliptically
polarized light
Elliptical Polarization
Adding two linearly
polarized waves with zero
phase shift will generate
another linearly polarized
wave
Linear Polarization

0y

0x
Adding two linearly
polarized waves with
equal amplitude and
90o phase shift results
in circular polarized
wave
Circular Polarization
0x 0y
Poincare Sphere and Stokes Parameters
SOP can be described by two real parameters: E0x=ax y E0y=ay
r=ay/ax=E0y/E0x and = y-x. (or  and )
It is defined complex polarization ratio r.exp(j)

The Poincare sphere

SOP represented by a point on the surface of a sphere with coordinates (r=1 ;  = 90 - 2 ;
 = 2)

Points on the equator (X = 0°) represent states of LP

North and south poles (2X = + /- 90°) represent RCP and LCP, respectively
Other points: elliptical polarization.
(r, ) , or (,) describe the SOP but contain no information about the intensity of the wave.
Another representation that does contain such information is the Stokes vector.

Stokes vector and Stokes parameters

Stokes parameters: S0, S1, S2, S3

S0 proportional to the optical intensity

S1, S2, S3 proportional to Cartesian coordinates on the poincare sphere (by S0)

then only three of the four components of the Stokes vector are independent
Matrix Representation - The Jones Vector

monochromatic plane wave of frequency v traveling in the z direction is completely characterized by

the complex envelopes

Written as the
Jones vector:

Given J, we can determine intensity

the ratio
the orientation
 the Poincare sphere and the Stokes parameters

There are four equivalent representations for describing the SOP of an optical field:
(1) the polarization ellipse
(2) the Poincare sphere
(3) the Stokes vector
(4) the Jones vector
The Jones Vector

the complex envelopes

Special polarization states for normalizaed intensity and

Orthogonal Polarizations: inner product between J1 and J2 is zero

Expansion of Arbitrary Polarization as a Superposition of Two Orthogonal Polarizations

An arbitrary Jones vector J can always be analyzed as a weighted superposition of
two orthogonal Jones vectors

normalized

coefficients

EXAMPLE 6.1-1. Expansions of J in LP and CP basis

LP Basis LP Basis CP Basis

J is LP at 

1 and 2 in CP basis are

EXERCISE 6. 1-1
Measurement of the Stokes Parameters of an
arbitrary wave from intensities.

Matrix Representation of Polarization Devices

plane wave

The matrix T, called the Jones matrix, describes the optical system, whereas the vectors J1
and J2 describe the input and output waves.
The structure of the Jones matrix T of a given optical system determines its effect on the
polarization state and intensity of the wave.
Linear polarizers.

Linear polarizer at x direction,

eliminates the y component

Wave retarders (modify polarization)

Polarization rotators (not modify polarization)

Cascaded Polarization Devices

LP LP
Use matrix multiplication. T=Tn....T2.T1
Coordinate Transformation
Jones vectors J and Jones matrices T are dependent on the coordinate system.

If J is in the x-y coordinate system, then in a new coordinate system

x’-y', with the x' direction making an angle  with the x direction,
the Jones vector J' is

The Jones matrix T, which represents an optical system, is similarly transformed into T',
in accordance with the matrix relations

Demonstration:
EXERCISE 6. 1-3. Demonstrate that the
Jones Matrix of a Linear Polarizer with
transmission axis angle 
 Faraday Effect
• When a magnetic field is applied to linearly
polarized light, the plane of polarization
rotates.
• The rotation is proportional to the intensity
of the applied magnetic field in the
direction of the beam of light

This effect is used in

Optical Isolators
 Optical Isolator
output polarizer (45 degrees)
• Forward traveling light
becomes polarized vertically.
The Faraday rotator will
rotate by 45 degrees. Output
polarizer passes the light.
• Light traveling backward
Faraday rotator
direction will get horizontal
Input (vertical) Polarizer polarization and will be
extinguished.
Polarization Control

Polarisers are used to

block reflected light in
optical fibres –
Optical Isolators
Polarized Sunglasses
• These provide
better view by
blacking
reflected
(glare) light
that has
horizontal
polarization
• What happen in fiber optic?

• How polarization can impact in the propagation?

End