[Ch5] Fundamentals of photonics Summarization

Q1. Is Magnetic Displacement existing term?

there is actually no term called "Magnetic Displacement" in this text. You likely guessed that name because of the pattern with electricity: • Electricity: Electric Field ($\mathcal{E}$) and Electric Displacement ($\mathcal{D}$)1. • Magnetism: Magnetic Field ($\mathcal{H}$) and Magnetic Flux Density ($\mathcal{B}$)2. In magnetism, the "partner" to Electric Displacement ($\mathcal{D}$) is called Magnetic Flux Density ($\mathcal{B}$). Here is the easiest way to understand the difference between the Magnetic Field ($\mathcal{H}$) and Magnetic Flux Density ($\mathcal{B}$):

Q.2 Since longer wavelengths diffract at larger angles?

This is summarization of Chapter 5: ELECTROMAGNETIC OPTICS of ‘Fundamentals of photonics’

Word dictionary in this chapter:

Letter	Name	Meaning with analogy
	Electric Field	The Push. The external force that pushes or pulls electric charges. (Analogy: Wind blowing against trees.)
	Magnetic Field	The Driver. The external force that tries to align magnetic poles. (Analogy: A large magnet held near a pile of compasses.)
	Electric Permittivity (of vacuum)	The Baseline Stiffness. How difficult it is to create an electric field in empty space. (Analogy: The tension of a drum skin in a vacuum.)
	Magnetic Permeability (of vacuum)	The Baseline Flow. How easily a magnetic field can form in empty space. (Analogy: How easily water flows through an empty pipe.)
	Polarization Density	The Stretch. How much the positive and negative charges in the material have separated. (Analogy: Trees leaning over because of the wind.)
	Magnetization Density	The Alignment. How many atomic "magnets" inside the material have rotated to face the same way. (Analogy: The compass needles turning to point North.)
	Electric Flux Density	The Total Count. The sum of the external push () plus the material's reaction (). (Analogy: The total movement of air plus the leaves.)
	Magnetic Flux Density	The Actual Field. The true magnetic strength you would feel at a specific point (External drive + Material alignment ).
	Electric Susceptibility	The Sensitivity. A number indicating how easily the material "stretches" () when pushed by a field (). (Analogy: How flexible a tree is. High = loose rubber; Low = stiff wood.)
	Electric Permittivity (of material)	The Storage Capacity. The material's total ability to store electrical energy in its bonds. (Analogy: A combination of the drum skin's tension and the rubber coating on top of it.)
	Poynting Vector	The Flow of Energy. The direction and amount of light power traveling per second. (Analogy: Water blasting out of a fire hose. It has direction and power.)

5.1 ELECTROMAGNETIC THEORY OF LIGHT

Maxwell’s Equations in Free Space

Ampère-Maxwell Law: A swirling magnetic field is created by an electric field that changes over time.

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(Curl): Represents a swirling or circulating field. Imagine water swirling down a drain.

(Divergence): Represents a field spreading out from a point (like light from a bulb) or converging into a point.

: electric permittivity

Faraday’s Law of Induction: A swirling electric field () is created by a magnetic field () that changes over time.

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negative sign (): → Lenz’s Law: the created Electric Field spins in a direction that tries to cancel out the change in the Magnetic Field.

: magnetic permeability

In (5.1-1) and (5.1-2), the Electric and Magnetic fields constantly "recreate" each other. This traveling chain reaction is what we call Light (an Electromagnetic Wave).

Gauss’s Law for Electricity: The Electric field lines cannot start (born) or stop (die) in this space. They must pass all the way through, or form closed loops.

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Gauss’s Law for Magnetism: Magnetic field lines never begin or end at a single point. They always form continuous, unbroken loops. Even inside a magnet, the lines connect South back to North.

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Maxwell’s Equations in a (source free) Medium

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: Electric flux density (or electric displacement).

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: Magnetic flux density.

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The flux densities depend on the properties of the medium via the polarization density and magnetization density .

In free space, , so that and

Boundary Conditions

Tangential Components: and are continuous across the boundary. Normal Components: and are continuous across the boundary. Perfect Conductor: At a boundary with a perfect conductor (like a mirror), the tangential electric field must be zero. This implies that the reflected wave has a phase shift of to cancel the incident wave at the surface29.

Poynting vector

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Represents the Power. It tells you where the energy is going and how much energy is flowing per second.

If you point your thumb up () and your fingers right (), your palm pushes forward (). This "push" is the direction the light travels.

: Optical intensity is time average of poynting vector.

Unit:

Poynting theorem

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Left Side: Energy flowing out of a region

Right Side 1st and 2nd terms: Energy densities per unit volume stored in electric and magnetic fields.

Right Side 3rd and 4th terms: Power densities associated with the material’s electric and magnetic dipoles.

The power flow escaping from the surface of an incremental volume equals the time rate of change of the energy stored inside the volume.

Momentum

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Electromagnetic Momentum Density.

In vacuum/non-magnetic media, . Substitute into (5.1-13):

Speed of light formula:

Substitute that into the equation. Then we get (5.1-15)

5.2 ELECTROMAGNETIC WAVES IN DIELECTRIC MEDIA

Type	Main Concern	Property	Example
Isotropic Media	Direction	Properties are the same in all directions.	Glass, Plastic, Metal
Anisotropic Media	Direction	Properties change depending on the direction.	Wood, Graphite, Crystal
Dispersive Media	Frequency	Speed changes depending on wave frequency/wavelength.	Prism

A. Linear, Nondispersive, Homogeneous, and Isotropic Media

Electric Susceptibility ():

How easily the material is polarized.

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Electric Permittivity ():

The "total" electrical capability of the material.

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Refractive Index

Speed of Light (in a Medium):

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Refractive index :

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For nonmagnetic media (),

B. Nonlinear, Dispersive, Inhomogeneous, or Anisotropic Media

Anisotropic Media

In anisotropic media, and are not necessarily parallel; the response depends on the direction of the field.

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represent coordinates (). The susceptibility becomes a tensor rather than a scalar.

Dispersive Media

The response is not instantaneous; the polarization depends on the history of the field (memory).

Time Domain (Convolution):

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Frequency Domain:

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5.4 ELEMENTARY ELECTROMAGNETIC WAVES

The Transverse Electromagnetic (TEM) Plane Wave

Fields

For monochromatic waves, the fields are represented by complex amplitudes and .

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and are constant complex vectors.

is the wavevector.

Orthogonality Relations:

Substituting these into Maxwell's equations yields:

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(Travel): The direction light moves (e.g., Forward).

(Electric): Wiggles one way (e.g., Up/Down).

(Magnetic): Wiggles the other way (e.g., Left/Right).

Impedance ():

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For nonmagnetic media: →

For free space:

Intensity

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5.5 ABSORPTION AND DISPERSION

A. Absorption

To describe "loss" or "friction" in physics, we use Complex Numbers:

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Complex Susceptibility and Wavenumber

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: Propagation constant (determines phase velocity).

: Absorption coefficient (determines attenuation).

The wave amplitude decays as and intensity as

If is large, the material is opaque (like ink). If is 0, it is transparent.

Refractive Index in Absorptive Media

The real part of the wavenumber, , determines the phase velocity of the wave. We define the effective refractive index such that

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By substituting (5.5-4) back into (5.5-3), we get the fundamental relationship connecting the macroscopic optical properties () to the material properties ():

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The impedance of the medium also becomes complex:

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Weakly Absorbing Media

Most optical materials (glass, air, water) are "weakly absorbing," meaning the imaginary part of the susceptibility is very small compared to the real part ().

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Absorption is proportional to the imaginary part of the susceptibility (). Since absorbing media have , must be negative.

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Strongly Absorbing Media

In materials like metals or plasmas, the imaginary part is dominant (). The approximation becomes:

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B. Dispersion

Dispersion means that the susceptibility , and therefore the refractive index , depends on the frequency (and wavelength ).

Consequence: Different colors travel at different speeds.

Example: A prism bends blue light (higher ) more than red light (lower ).

Kramers-Kronig Relations

If you know the absorption spectrum of a material across all frequencies, you can calculate its refractive index exactly, without measuring it. This proves that dispersion is a necessary consequence of absorption.

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C. The Resonant Medium (Lorentz Oscillator Model)

The Differential Equation for the Electron Position:

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: The displacement of the electron from its equilibrium position.

: The acceleration of the electron. Multiplied by mass 0, this is the inertial force.

(Zeta): The damping coefficient. This represents friction or energy loss. As the electron moves, it loses energy (colliding with other atoms or radiating energy away). This term is proportional to velocity ().

: The resonance angular frequency. This is the natural frequency at which the electron wants to vibrate, determined by the "stiffness" of the atomic bond (the spring constant ). Specifically, .

: The driving force. In optics, this force is provided by the Electric Field of the light wave. The force on a charge is .

: The mass of the electron.

Linking Microscopic Motion to Macroscopic Polarization

If there are atoms per unit volume, the total Polarization is:

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Then rewrite (5.5-16) in terms of and

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: The low-frequency susceptibility. This constant groups together the atomic constants:

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Solving for the Frequency Response

We use phasor notation (complex amplitudes), ():

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Substituting these into equation (5.5-15):

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Now finally define as the ratio of Polarization to Electric Field ():

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: The complex susceptibility at frequency $\nu$.

: The frequency of the incoming light.

: The resonance frequency of the atom ().

: The linewidth of the resonance, related to damping by . A large damping leads to a large , meaning a "broad" resonance.

Separating Real and Imaginary Parts

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When (below resonance), is positive ().

When (at resonance), (The index passes through unity).

When (above resonance), is negative ().

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Notice the negative sign. In this convention, a negative implies energy absorption.

This function peaks sharply at . This means the material absorbs light most strongly when the light frequency matches the electron's natural frequency (resonance).

Approximations Near and Far from Resonance

1. Near Resonance () → Lorentzian function:

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This describes the shape of a spectral line (absorption line). The parameter $\Delta\nu$ is the Full Width at Half Maximum (FWHM). It describes how "fat" the absorption spike is.

2. Far From Resonance ():

If the light frequency is far away from the resonance, the damping term (imaginary part) becomes negligible. The material becomes transparent (absorption ).

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* Relationships in Dilute Media (like a gas, or sparse dopants in a solid)

Assuming the susceptibility is small (), we use the approximation :

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Absorption is directly proportional to the imaginary part of the susceptibility.

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The change in refractive index is proportional to the real part.

* The Sellmeier Equation

Real materials (like fused silica or optical glass) are not made of a single type of oscillator. They have multiple resonances—some electrons vibrate at UV frequencies, some ions vibrate at Infrared frequencies.

We use the relation (for non-magnetic media). Combining the contributions of multiple resonances (indexed by ):

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This equation mathematically describes Material Dispersion. It explains why $n$ decreases as increases (normal dispersion) in visible glass.

When : The term scales as .

When : The term becomes constant.

Singularity: If you try to calculate exactly at , the denominator becomes zero and the equation blows up. This is expected because the Sellmeier equation assumes we are far from resonance (zero absorption). It cannot be used right at the absorption peak.