Optics Course (Phys 311)

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Optics Course (Phys 311)

Wave Optics Wave Motion

Lecturer: Dr Zeina Hashim

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Phys 311

1. Introduction to waves.

2. One dimensional waves.

3. The differential wave equation in 1D and 3D.

4. Harmonic waves.

5. Complex Representation of waves.

6. Plane waves.

7. Spherical waves.

8. Cylindrical waves.

Wave Optics: Wave Motion

Objectives covered in this lesson :

Lesson 1 of 1 Slide 1

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A classical travelling wave:

is a self-sustaining disturbance of a medium, which moves through space transporting energy and momentum.

What is energy?

What is momentum?

The disturbance can be:

1. a displacement (like in a rope).

2. a change in pressure (example: air pressure  sound waves)

3. a change in an electric field (example: traveling EM waves), etc.

Phys 311

Introduction to Waves :

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In a longitudinal wave:

the medium is displaced in the direction of motion of the wave.

Example: Sound waves.

In a transverse wave:

the medium is displaced in a direction perpendicular to that of the motion of the wave.

Example: waves on a string, EM waves.

In all cases; the disturbance advances, not the material medium (the individual participating atoms remain in the vicinity of their equilibrium positions).

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Let us give this disturbance a mathematical character; a wave equation 𝜓 , which for now, is for a 1D-wave moving in the positive x-direction with a constant speed.

Because the disturbance is moving, it must be a function of both position and time and can be written as:

𝝍 𝒙, 𝒕 = 𝒇(𝒙, 𝒕)

where f(x,t) corresponds to some specific function or wave shape.

At t=0:

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One Dimensional Waves :

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Example: where a is constant.

Drawing that function gives a Gaussian function:

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Because the material is not moving but is just vibrating around its equilibrium:

The origin in the graph represents the equilibrium state of that part of the material.

When the other part vibrates, it has a separate position of its origin.

We can therefore consider a moving reference frame, moving with the wave, from one equilibrium state to the next.

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If the shape of the wave does not change as it travels:

after a time t : f(x) will be f(x’), where now:

Remember, we assumed that the wave is traveling with a constant speed v.

From the point of view of the first reference frame (S), the moving wave has the mathematical form:

𝜓 𝑥, 𝑡 = 𝑓 𝑥^′ = 𝑓(𝑥 − 𝑣𝑡)

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Phys 311

If, instead of moving in the positive x-direction, the wave was moving in the negative x-direction, then the sign of v must be reversed, and the wave function will be:

𝜓 𝑥, 𝑡 = 𝑓 𝑥^′′ = 𝑓(𝑥 + 𝑣𝑡)

Therefore, the general mathematical form of a travelling one dimensional wave is:

a one-dimensional wavefunction travelling with constant speed 𝝍 𝒙, 𝒕 = 𝒇(𝒙 ± 𝒗𝒕)

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Phys 311

Q: If 𝜓 𝑥, 0 = 𝑒^−𝑎𝑥² has a bell-shape (a Gaussian function), what is the shape of 𝜓 𝑥, 𝑡 = 𝑒^{−𝑎(𝑥−𝑣𝑡)}² ?

Individual Activity

One Dimensional Waves:

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Phys 311

Q: why is the wavefunction called “one dimensional” if we can see that the material is moving in two dimensions? (Example: a wave in a rope, as in figure).

Individual Activity

One Dimensional Waves:

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There are lots of kinds of waves, each has its our wavefunction (depending on its shape).

The good news is that all those wavefunctions are solutions for a one “partial differential wave equation”:

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The Differential Wave Equation :

One-dimensional differential wave equation

Homework: Q1: Go back to Hecht (p. 13) and study how that equation was derived.

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In three dimensions: that equation has the form:

But we know that the Laplacian Operator has the form:

Then, Phys 311

Three-dimensional differential wave equation

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k = a constant called: the propagation number (unit = 1/(unit of x)).

A = amplitude.

The argument of the sine function, i.e. the angle of the sine = the phase 𝝋 (unit = radians). Here, the phase 𝜑 is kx.

Phys 311

Harmonic Waves :

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sine (𝜑) = -1 to 1  So, maximum value of 𝜓 𝑥 = A.

Therefore, we can define the amplitude as: the maximum disturbance of the wave.

For a progressive wave, traveling with speed v : Replace x with (x - vt):

The wave is periodic in both space and time:

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Wavelength (𝝀): is the special period. It is the number of units of length per wave. It is the distance between two maxima (wave crests) or two minima (wave troughs). (unit of wavelength = nanometer (nm) (recent), micron (µm) (in some references), and angstrom (Å) (in older references)).

An increase or decrease in x by λ leaves the wavefunction unaltered, that is:

The phase does not change with ±2𝜋, therefore:

From this, we derive that:

One wavelength corresponds to a change in the phase 𝜑 of 2𝜋 rad.

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The temporal period (or simply period) (𝝉):

is the time period. It is the amount of time it takes for one complete wave to pass a stationary observer. It is the number of units of time per wave.

A change in the time by 𝜏 leaves the wavefunction unaltered, that is:

The phase does not change with ±2𝜋, therefore:

From this, we derive that Phys

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The temporal frequency (or simply frequency): is the inverse of the temporal period.

It is the number of waves per unit of time. (unit = cycles per second = vibrations per second = Hertz (Hz)).

The speed of the wave therefore is:

The angular temporal frequency (unit = radians per second) is:

The spatial frequency (or wave number) is the number of waves per unit of length (unit = inverse meters), and is given by:

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All of these quantities apply equally well to waves that are not harmonic, as long as each such wave is made up of a single

Regularly repeated profile-element.

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Using the above definitions, a number of equivalent expressions can be written for the traveling harmonic wave:

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Each such wave has a single constant frequency

 monochromatic (also called monoenergetic)

Real waves are never monochromatic.

All waves comprise of a band of frequencies,

and when that band is narrow, the wave is said to be quasimonochromatic.

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The simple harmonic wavefunction can be generalized to include an initial phase (𝜺):

The initial phase is also called an Epoch Angle.

when x=0 and t=0  𝜓 𝑥, 𝑡 = 𝐴 sin 𝜀 Phys

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Homework:

Q2:

Prove the above statement.

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For the wave velocity and phase velocity:

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𝝋 = 𝒌(𝒙 ± 𝒗𝒕)

When x and t change together so that the phase is constant, the disturbance:

𝜓 = 𝐴 sin 𝜑 is also constant.

The condition of constant phase definitely describes the motion of a fixed point on the waveform (phase velocity), which moves with the velocity of the wave (wave velocity).

Mathematically: 𝒅𝝋 = 𝟎 = 𝒌(𝒅𝒙 ± 𝒗𝒅𝒕) 𝒅𝒙

𝒅𝒕 = ±𝒗 Wave velocity Phase velocity

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Complex Representation of Waves :

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Therefore,

This wavefunction has two parts: a real part (𝜓 𝑥, 𝑡 = 𝐴 cos(𝜔𝑡 − 𝑘𝑥 + 𝜀)) and an imaginary part (𝜓 𝑥, 𝑡 = 𝐴 sin(𝜔𝑡 − 𝑘𝑥 + 𝜀))

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In three dimensions (and in the Cartesian coordinates):

and

Therefore, the wavefunction in 3D (at t=0) will have the forms:

Or:

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Plane Waves :

Wave equation of a plane wave

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The surfaces joining all points of equal phase are known as wavefronts:

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Any three-dimentional wave can be expressed as a combination of plane waves, each having a distinct amplitude and propagation direction.

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Spherical Waves :

or Spherical coordinates

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Cylindrical Waves :

Cylindrical coordinates

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Cylindrical Waves :

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Q3:

Q4:

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Homework :

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Q5: (optional):

Q6: (optional):

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Homework :

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1. Introduction to waves.

2. One dimensional waves.

3. The differential wave equation in 1D and 3D.

4. Harmonic waves.

5. Complex Representation of waves.

6. Plane waves.

7. Spherical waves.

8. Cylindrical waves.

Phys 311

Lesson 1 of 1 Slide 34 (last)

Summary:

Any Questions?

Next lesson will cover:

EM waves Wave Optics: Wave Motion