Optics+lesson+8+The+Superposition+of+Waves+for+upload

Optics Course

(Phys 311)

Wave Optics

The Superposition of Waves (1 of 2)

Lecturer: Dr Zeina Hashim

Phys

311

1. The Principle of Superposition.

2. The Addition of Waves of the Same Frequency Along the Same Direction:

a. algebraic method:

- The resultant wave equation, its new (amplitude, phase, and flux density).

- The Phase difference between two superimposed waves.

- The resultant wave equation in the case where two identical waves are following each

other by 𝚫𝒙 .

- The special cases of constructive and destructive interferences.

- The superposition of many waves. b. complex method. c. phasors method.

Objectives covered in this lesson :

Lesson 1 of 2

Slide 1 Wave Optics: The Superposition of Waves

Why study this process? Because it underlies

the three phenomena we will soon study:

Interference Diffraction Polarization

The Principle:

Phys

311

The Principle of Superposition :

Lesson 1 of 2


The resultant disturbance at any point in a medium is

the algebraic sum of the separate constituent waves

𝜓 𝑓𝑖𝑛𝑎𝑙= 𝑐1𝜓1 + 𝑐2𝜓2 +⋯

How did we know this?

From the 3D differential equation:

All solutions of this equation are linear (i.e. to the first power).

This implies that any linear combination of these solutions is also a solution.

Therefore:

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The Principle of Superposition :

Lesson 1 of 2


Are all EM waves linear ??

No ! We have “non-linear waves”, which we will not study.

They are solutions to non-linear partial differential wave equations.

These waves do not satisfy the Principle of Superposition.

A nonlinear wave is caused by a VERY LARGE force.

An example is: A focused beam of a high-intensity laser (Electric field = 1010 V/cm).

So, our differential equation should be called:

“linear partial differential wave equation”

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Lesson 1 of 2


Phys

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Lesson 1 of 2

Slide 5

Addition of Waves (travelling along the x-axis, with the same 𝜔):

Algebraic Method Complex Method Graphical Method (Phasors)

Wave Optics: The Superposition of Waves

Phys

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Lesson 1 of 2

Slide 6

Algebraic Method

𝐸 = 𝐸𝑜1 sin 𝜔𝑡 + 𝛼1 + 𝐸𝑜1 sin(𝜔𝑡 + 𝛼2)

sin 𝑥 + 𝑦 = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦

Let:


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Lesson 1 of 2

Slide 7

Algebraic Method

Using: sin 𝑥 + 𝑦 = sin 𝑥 cos 𝑦 + cos 𝑥 sin 𝑦

This is the new wave equation which resulted from the superposition of the two waves.

It: is harmonic

and has: the same frequency as its constituents,

but it has: a new 𝑬𝒐 and a new 𝜶.

𝐸 = 𝐸𝑜 sin(𝜔𝑡 + 𝛼)


Phys

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Lesson 1 of 2

Slide 8

What is its new amplitude (𝑬𝒐)?

Square each equation, and add them together

𝐸𝑜2 sin2 𝛼 = 𝐸𝑜1

2 sin2 𝛼1 + 𝐸𝑜22 sin2 𝛼2 + 2𝐸𝑜1𝐸𝑜2 sin 𝛼1 sin 𝛼2

,

𝐸𝑜2 cos2 𝛼 = 𝐸𝑜1

2 cos2 𝛼1 + 𝐸𝑜22 cos2 𝛼2 + 2𝐸𝑜1𝐸𝑜2 cos 𝛼1 cos 𝛼2

sin2 𝜃 + sin2 𝜃 = 1

𝐸𝑜2 = 𝐸𝑜1

2 + 𝐸𝑜22 + 2𝐸𝑜1𝐸𝑜2 (cos 𝛼1 cos 𝛼2 + sin𝛼1 sin 𝛼2)

Algebraic Method

𝑬𝒐𝟐 = 𝑬𝒐𝟏

𝟐 + 𝑬𝒐𝟐𝟐 + 𝟐𝑬𝒐𝟏𝑬𝒐𝟐 𝐜𝐨𝐬(𝜶𝟏 − 𝜶𝟐) its new amplitude.

أو العكس ؟ 2ناقص ألفا 1هل تفرق إذا حطينا ألفا

The resultant’s amplitude depends on what? 𝜶𝟐 − 𝜶𝟏 is called “the phase difference 𝜹”


𝐜𝐨𝐬(𝜶𝟏 − 𝜶𝟐)

When the two wave equations are:

In-phase 𝜹 = 𝟎 , ±𝟐𝝅 ,±𝟒𝝅 ,…

Out-of-phase 𝜹 = ±𝝅 ,±𝟑𝝅 ,…

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Lesson 1 of 2

Slide 9

Homework: Q1:

Show that when the two wave equations

𝐸1 = 𝐸𝑜1 sin(𝜔𝑡 + 𝛼1) and

𝐸2 = 𝐸𝑜2 sin(𝜔𝑡 + 𝛼2) are in-phase, the resulting

amplitude squared is a maximum (and equals

(𝐸𝑜1 + 𝐸𝑜2)2), and when they are out-of-phase it is

a minimum (and equals (𝐸𝑜1 − 𝐸𝑜2)2).

Algebraic Method The Phase Difference:


𝜹 = 𝜶𝟐 − 𝜶𝟏 but

So, 𝜹 = 𝒌𝒙𝟏 + 𝜺𝟏 − 𝒌𝒙𝟐 + 𝜺𝟐

𝜹 = 𝒌( 𝒙𝟏 + 𝜺𝟏 − 𝒙𝟐 + 𝜺𝟐 )

𝜹 =𝟐𝝅

𝝀 𝒙𝟏 − 𝒙𝟐 + 𝜺𝟏 − 𝜺𝟐

𝜹 =𝟐𝝅

𝝀𝒐𝒏 𝒙𝟏 − 𝒙𝟐 + 𝜺𝟏 − 𝜺𝟐

Phys

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Lesson 1 of 2

Slide 10

Distances

from

sources

Initial

phases

𝜆 =𝜆𝑜𝑛

If 𝜀1 − 𝜀2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 the waves are coherent



If the two waves were initially in phase (example: they came from the same source):

𝜺𝟏 = 𝜺𝟐 𝜹 =𝟐𝝅

𝝀𝒐 𝒏 𝒙𝟏 − 𝒙𝟐

𝜹 = 𝒌𝒐𝚲

Q: How can two waves from the same source have different distances from the

source?

Phys

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Lesson 1 of 2

Slide 11

Optical Path:

∆= 𝑛𝑑

d here is x

Optical path

difference (𝚲)



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Lesson 1 of 2

Slide 12

Algebraic Method Now, what is its new phase (𝜶) ?

Divide the second equation by the first equation:

,

its new phase.

Q: When does 𝜶 ≈ 𝜶𝟏 and when does 𝜶 ≈ 𝜶𝟐 ?


Now… if I gave you the amplitudes and phases of two waves, can you write down

the wave equation of their resultant superposition ?

Phys

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Lesson 1 of 2

Slide 13

Homework: Q2:

Determine the resultant of the superposition of the parallel waves:

𝐸1 = 𝐸𝑜1 sin(𝜔𝑡 + 𝜀1) and 𝐸2 = 𝐸𝑜2 sin(𝜔𝑡 + 𝜀2)

when 𝜔 = 120𝜋 , 𝐸𝑜1 = 6 , 𝐸𝑜2 = 8 , 𝜀1 = 0 , and 𝜀2 =𝜋

2 .

Algebraic Method


What is the flux density of the resultant wave?

The flux density has an extra term.

Phys

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Lesson 1 of 2

Slide 14

∴ 𝐼 =𝑐𝜖𝑜2(𝑬𝒐𝟏

𝟐 + 𝑬𝒐𝟐𝟐 + 𝟐𝑬𝒐𝟏𝑬𝒐𝟐 𝐜𝐨𝐬 𝜶𝟏 − 𝜶𝟐 )

= 𝑐𝜖𝑜

2𝑬𝒐𝟏𝟐 +𝑐𝜖𝑜

2𝑬𝒐𝟐𝟐 +𝑐𝜖𝑜

2𝟐𝑬𝒐𝟏𝑬𝒐𝟐 𝐜𝐨𝐬 𝜶𝟏 − 𝜶𝟐

= 𝐼1 + 𝐼2 +𝑐𝜖𝑜2𝟐𝑬𝒐𝟏𝑬𝒐𝟐 𝐜𝐨𝐬 𝜶𝟏 − 𝜶𝟐

𝟐𝑬𝒐𝟏𝑬𝒐𝟐 𝐜𝐨𝐬 𝜶𝟏 − 𝜶𝟐 is called: “Interference Term”

Algebraic Method


If we have two waves which have the same frequency & same amplitude & same initial

phase, but one is following the other by

Then,

And the resultant wave equation can be written as:

Phys

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Lesson 1 of 2

Slide 15

Algebraic Method


The special case of

Constructive Interference Destructive Interference

Phys

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Lesson 1 of 2

Slide 16

Algebraic Method


The special case of

Constructive Interference Destructive Interference

Phys

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Lesson 1 of 2

Slide 17

Algebraic Method 𝑬𝒐𝟐 = 𝑬𝒐𝟏

𝟐 + 𝑬𝒐𝟐𝟐 + 𝟐𝑬𝒐𝟏𝑬𝒐𝟐 𝐜𝐨𝐬(𝜶𝟏 − 𝜶𝟐) the new amplitude.

The peaks occur at the same time.

The resultant’s amplitude becomes

(𝐸𝑜 = 𝐸𝑜1 + 𝐸𝑜2). This happens when 𝛼1 − 𝛼2 = 0 , 2𝜋 , …

Two waves with the same frequencies,

amplitudes, initial phases, and follow

each other by Δ𝑥 will interfere

constructively if

The peak and trough occur at the same time.

The resultant’s amplitude becomes

(𝐸𝑜 = 𝐸𝑜1 − 𝐸𝑜2). This happens when 𝛼1 − 𝛼2 = 𝜋 , 3𝜋 , …

Two waves with the same frequencies,

amplitudes, initial phases, and follow each

other by Δ𝑥 will interfere destructively if

𝐸𝑜 = 2𝐸𝑜1 𝐸𝑜 = 0


Phys

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Lesson 1 of 2

Slide 18

Algebraic Method


The Superposition of Many waves

The superposition of N number of:

a. coherent

b. harmonic waves

c. having a given frequency (i.e. the same 𝜔)

d. travelling in the same direction

Its wave equation will be:

with a new amplitude: and a new phase:

Phys

311

Lesson 1 of 2

Slide 19

Algebraic Method

Leads to a harmonic wave of

that same frequency.

= 𝐸𝑜𝑖 cos(𝛼𝑖 ± 𝜔𝑡)

𝑁

𝑖=1


Phys

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Lesson 1 of 2

Slide 20




If we have N waves with the same frequency, travelling in the positive x-direction,

each having a wave equation of:

𝐸𝑗 = 𝐸𝑜𝑗𝑒𝑖(𝛼1+𝜔𝑡)

The resultant wave of their superposition will be:

with a new complex amplitude:

and a new phase: Not given here

Phys

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Lesson 1 of 2

Slide 21

Complex Method

𝐸 = 𝐸𝑜𝑒𝑖(α+𝜔𝑡)

𝐸𝑜2𝑒𝑖𝛼 = 𝐸𝑜𝑗𝑒

𝑖𝛼𝑗

𝑁

𝑗=1


Phys

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Lesson 1 of 2

Slide 22




Phys

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Lesson 1 of 2

Slide 23

Link to: Phasors and waves Graphical Method (Phasors)


http://www.animations.physics.unsw.edu.au/jw/phasor-addition.html

It is a graphical method to obtain the new amplitude and new phase.

It is useful when we have more than two waves which we need to combine.

Each wave is described by a vector: its length = amplitude of wave.

its direction from the positive x-axis = its 𝜶.

Steps: a. Draw each vector.

b. Shift them so that they are head-to-tail, head-to tail.

c. Draw the resultant wave vector (from tail of first wave to head of last wave.

d. Resultant vector length = its amplitude. Its angle from + x-direction = its phase.

Phys

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Lesson 1 of 2

Slide 24

Graphical Method (Phasors)

Phasors can be represented by:


Example:

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Lesson 1 of 2

Slide 25



The resultant wave leads or lags the constituent waves ?

wave 1 leads wave 2 means: peak of 1 occurs at an earlier location than peak of 2.

wave 1 lags wave 2 means: peak of 1 occurs at a later location than peak of 2.

If it leads its phase is positive

(counter-clockwise from x-axis)

If it lags its phase is negative (clockwise)

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Lesson 1 of 2

Slide 26



Q3:

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Lesson 1 of 2

Slide 27

Homework :


Q4:

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Lesson 1 of 2

Slide 28

Homework :


1. The Principle of Superposition.

2. The Addition of Waves of the Same Frequency Along the Same Direction:

a. algebraic method:

- The resultant wave equation, its new (amplitude, phase, and flux density).

- The Phase Difference between two superimposed waves.

- The resultant wave equation in the case where two identical waves are following each other

by 𝜟𝒙 .

- The special cases of constructive and destructive interferences.

- The superposition of many waves. b. complex method. c. phasors method.

Phys

311

Lesson 1 of 2

Slide 29 (last)

Summary: Any Questions? Next lesson will cover:

Superposition of waves (2)


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