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Wave Optics

Content

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Introduction What are waves? Wave equation Wave optics Gaussian Beam Diffraction Interference Coherence

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Introduction

Light is part of the family of electromagnetics radiation

“Electromagnetics is the study of electric and magnetic phenomena and their engineering applications”

Light, i.e. a family of Electromagnetic radiation or wave, constitutes a time varying electric and magnetic fields propagating in certain direction

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Electromagnetic radiation propagating in z direction

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Electromagnetic (EM) waves includes many types of signal: visible light, radio waves, infrared waves, gamma rays, x rays, …

All EM waves share the following properties– Phase velocity in vacuum is c = 3 x

108m/s– In vacuum, for any EM wave = c/ f

Each EM waves is distinguished by its own wavelength, or equivalently by its own oscillation frequency f.

The Electromagnetic Spectrum

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What are waves?

Waves are a natural consequence of many physical processes:

- waves and ripples on oceans and lakes,- mechanical waves on stretched strings,- sound wave that travel trough air,- electromagnetic waves (light,..)- earthquake waves,…

Common properties of EM waves

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Moving waves carry energy Waves have velocity

It takes time for a wave to travel from one point to another

Light wave travel at 3 x 108 m/sSound wave travel at 330 m/s

Some waves exhibit a property called linearityWaves that do not affect the passage of other

waves are called linear. The total of two linear waves is the sum of the

two

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Wave equation

The basis for understanding the wave theory of transmission is through 4 sets of equations known as Maxwell's equations.

Maxwell’s EquationsModern electromagnetism is based on these set of four fundamental relations given by:

•D = v, E = -dB/dt •B = 0, H = J + dD/dt

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Symbols

E and D are electric field quantities interrelated by D = E, with being the electrical permittivity of the material;

B and H are magnetic field quantities interrelated by B = H, with being the magnetic permeability of the material;

v is the electric charge density per unit volume; and J is the current density per unit area.

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Wave equation

The wave equation can be derived from the Maxwell’s Equations by assuming the conductivity and volume charge density are both zero.

2

2

2

22

tc

n

The wave equation

Where may represent a component of the E or H field and c is the velocity given by c=1/(oo)0.5 in the dielectric medium.From here the general solution is given by

ztj exp0

0 – amplitude - angular frequencyt – time - propagation constant Complex representation

Wave equationWave in a lossless Medium

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A medium is said to be lossless if it does not attenuate the amplitude of the wave traveling within it or on its surface.

y(x,t) = A cos (t-kx)

where A amplitude

f (Hz) Frequency

T (s) Period

(rad/s) = 2/T = 2f Angular frequency

(m) Wavelength

k (rad/m) = 2/ Phase constant / wavenumber

up = f = / phase velocity

Phase & Phase Velocity

Phase was defined as the argument of the sine function.

j = kx wt

At t = x = 0,

which is a special case.

The sine function can be rewritten as

whereis the initial phase.

00,0,00

txtx

)sin(, tkxAtx

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The initial phase angle is just the constant contribution to the phase arising from the generator.

It is independent of how far or how long the wave has travelled.

The phase (kx – t) and (t – kx) were used to describe the sine wave functions earlier. Both describe waves moving in the positive x–direction that are

otherwise identical except for a relative phase difference of p. The initial phase is of no particular significance; thus literatures

abound with both expressions.

The phase of a disturbance is a function of x and t.

The partial derivative of wrt t or the rate-of-change of phase with time,

It is given as the angular frequency of the wave at any fixed location.

It indicates the rate at which a point oscillates up and down.

For each cycle, changes by 2.

tkxtx,

xt

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The partial derivative of wrt x or the rate-of-change of phase with distance,

These two expressions combined yields

The term on the LHS represents the speed of propagation of the condition of constant phase.

It gives the speed at which the profile moves and is known commonly as the phase velocity of the wave.

It is positive when the wave moves in positive x direction; negative when in direction of decreasing x.

v

kx

t

t

x

t

x

kx t

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In a lossy medium, the amplitude of the wave will decrease as an exponential decaying factor e-x

Thus

y(,x, t)= A e-x cos (t-x+0)

e-x : attenuation factor and: attenuation constant of the medium (Neper

per meter, Np/m)

Notice the wave amplitude is now A e-x.

Wave in a Lossy Medium

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Wave Optics

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In describing the propagation of light as a wave we need to understand:

wavefronts: The surfaces joining all points of equal phase are known as wavefronts or a surface passing through points of a wave that have the same phase and amplitude.

rays: a ray describes the direction of wave propagation. A ray is a vector perpendicular to the wavefront.

Wavefronts

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We can chose to associate the wavefronts with the instantaneous surfaces where the wave is at its maximum.

Wavefronts travel outward from the source at the speed of light: c.

Wavefronts propagate perpendicular to the local wavefront surface.

Light Rays

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The propagation of the wavefronts can be described by light rays.

In free space, the light rays travel in straight lines, perpendicular to the wavefronts.

Huygens’ Principle:

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Huygens’ principle (Christaan Huygens, 1629-1695, published about 1690) describes how a wavefront moves in space.

According to this principle, we imagine that each point on the wavefront acts as a point source that emits spherical wavelets.

These wavelets travel with the velocity of light in the medium.

At any later time, the total wavefront is the envelope that encloses all of these wavelets.

That is, the tangent line that joins the front surface of each one of them.

Huygens’ Principle:

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All the points on a wavefront can be considered as point source for the production of spherical secondary wavelets. At the later time, the new position of the wavefront will be the surface of tangency to these secondary wavelets.

Superposition Principle of Waves

The 1-D differential wave equation reveals an intriguing property of waves; its solutions are according to the Superposition Principle.

This can be easily proven to be true. Consider two different wave functions y1 and y2; both are separate solutions to the differential wave equation. Thus,

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2

222

2

21

2

221

2 1and

1

tvxtvx

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Adding those two equations yield

It is established that (1 + 2) is indeed a solution itself.

Physically it translate that when two separate waves arrive at the same place in space wherein they overlap, these waves simply add (or subtract from) one another without permanently destroying of disrupting either wave.

212

2

2212

2

22

2

221

2

222

2

21

2

1

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tvx

tvtvxx

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The resulting disturbance at each point in the region of overlap is the algebraic sum of the individual constituent waves at that location.

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Superposition of two equal-wavelength sinusoids

-2

-1

0

1

2

-1 0 1 2 3 4

kx (rad)

(x, 0)

1

2

)(1 ox

)(2 ox

)(1 ox

)()( 21 oo xx

At every point, the resultant wave is the summation of the individual waves

rad0.1sin9.0sin0.1

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kxkx

The two constituent waves are in-phase (phase-angle difference is zero).

The composite wave of substantial amplitude is sinusoidal with the same frequency and wavelength as the component waves.

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-2

-1

0

1

2

-1 0 1 2 3 4 5 6 7

kx

1

2

The resultant amplitudes diminishes as the phase-angle difference increases.

It is expected to vanish when the phase difference equals . At that point, the waves are said to be out-of-phase.

The fact that at this point the resultant wave amplitude is zero gives rise to the name interference.

-2

-1

0

1

2

-1 0 1 2 3 4 5 6 7kx

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-2

-1

0

1

2

-1 0 1 2 3 4 5 6 7

kx

-2

-1

0

1

2

-1 0 1 2 3 4 5 6 7kx

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Before proceeding with other wave phenomena, it should be obvious that analysis using sine and cosine functions involving trigonometric manipulations are rather involved and somewhat tedious.

Consider the following solution for the algebraic addition of two (or more) overlapping waves that have the same frequency and wavelength.

Algebraic Method

A solution of the differential wave equation can be written in the form

in which Eo is the amplitude of the harmonic disturbance propagating along the positive x-axis.

To separate the space and time part of the phase, let

so that

)(sin),( kxtEtxE o

),(sin),( xtEtxE o

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kxx,

Suppose then that there are two such waves

and

each with the same frequency and speed, coexisting in space.

The resultant disturbance is the linear superposition of these waves:

E = E1 + E2

111 sin),( tEtxE o

222 sin),( tEtxE o

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222

111

sincoscossin

sincoscossin

ttE

ttEE

o

o

Separate the time-dependent terms,

Let

Take the summation of the square of the two equations above:

which the sought-after expression for the amplitude (Eo) of the resultant wave.

2211

2211

sinsinsin

coscoscos

ooo

ooo

EEE

EEE

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tEE

tEEE

oo

oo

cossinsin

sincoscos

2211

2211

122122

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2 cos2 ooooo EEEEE

The phase of the resultant wave is obtained from

The total disturbance then becomes

or

The composite wave is harmonic and of the same frequency as the constituents, although its amplitude and phase are different.

tEtEE oo cossinsincos

tEE o sin

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2211

2211

coscos

sinsintan

oo

oo

EE

EE

When Eo1 >> Eo2, 1 and when Eo2 >> Eo1 , 2 . The resultant wave is in-phase with the dominant component wave.

The flux density of a light wave is proportional to its amplitude squared.

The resultant flux density is not simply the sum of the component flux densities; there is an additional contribution, known as the interference term.

2211

2211

coscos

sinsintan

oo

oo

EE

EE

122122

21

2 cos2 ooooo EEEEE

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The crucial factor is the difference in phase between the two interfering waves E1 and E2,

2 1

When = 0, 2, 4,… the resultant amplitude is maximum, whereas

= , 3,… the resultant amplitude is minimum at any point in space.

In the former case, the waves are said to be in-phase; while in the latter case, the waves are 180o out-of-phase.

-2

-1

0

1

2

-1 0 1 2 3 4 5 6 7kx

35-2

-1

0

1

2

-1 0 1 2 3 4 5 6 7

kx

1

2

Gaussian Beams

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Light can take the form of beams that come as close as possible to spatially localised & non-diverging waves – paraxial approximation of the wave equation.

The beam power is concentrated within a small cylinder surrounding the beam axis and intensity distribution in the transverse plane follows Gaussian distribution.

The wavefronts are approximately planar near the beam waist but they gradually curve and become approximately spherical far from the waist.

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Gaussian Beam Math

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The expression for a light beam's electric field can then be written as:

where: w(z) is the spot size (1/e2 width) vs. distance from the waist,R(z) is the beam radius of curvature, and(z) is a phase shift.This equation is the solution to the wave equation when we require that the beam be well localized at some point (i.e., its waist).

E x ,y ,z exp ikz i z

w z exp

x2y2

w2 z i

x2y2

R z

The paraxial approximation ensures that the plane wave is modulated by a complex amplitude that is a slowly varying function of position

Gaussian Beam Spot, Radius, and Phase

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The expressions for the spot size,

radius of curvature, and phase shift:

where zR is the Rayleigh Range (the distance over which the

beam remains about the same diameter), and it's given by:

w z w0 1 z / zR 2

R z z zR

2 / z

z arctan z / zR

zR w0

2 /

Gaussian Beam Collimation

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Twice the Rayleigh range is the distance over which the beamremains about the same size,that is, remains “collimated.”

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.225 cm 0.003 km 0.045 km 2.25 cm 0.3 km 5 km

22.5 cm 30 km 500 km_____________________________________________ Tightly focused laser beams expand quickly. Weakly focused beams expand less quickly, but still expand.

Collimation CollimationWaist spot Distance Distance size w0 = 10.6 µm = 0.633 µm

Longer wavelengths expand faster than shorter ones.

202 2 /Rz w

Gaussian Beam Divergence

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Far away from the waist, the spot size of a Gaussian beam will be:

The beam 1/e2 divergence half angle is then w(z) / z as z :

or…

The Rayleigh range zR is the ratio of the beam waist radius to the half angle divergence.

If we substitute the expression: into the one above, we obtain:

So the product of the waist and the divergence depends only on the wavelength. For a specific laser wavelength,the smaller the waist and the larger the divergence angle.

w z w0 1 z / zR 2 w0 z / zR 2 w0z / zR

tan1/e2

w z z

w0zzR z

w0

zR

zR w0

zR w0

2 /

w0 /

Focusing a Gaussian Beam

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A lens will focus a collimated Gaussian beam to a new spot size. It turns out that there is a relationship (derived later) between the

input beam size and the new beam waist:

w0 f / w,

where w is the beam radius at the lens. So the smaller the desired focus, the BIGGER the input beam must be!

It should be noted that the beam radius must be less than 2/3 of the lens diameter D. Beyond this the beam will lose its Gaussian character and diffraction rings will begin to emerge.