Questions: Dispersion Relations for Electromagnetic Waves

Question 1

A light pulse containing a range of frequencies is launched into a long optical fiber. After traveling thousands of kilometers, the pulse has spread out in time and can no longer be distinguished from adjacent pulses. What is the fundamental cause of this broadening?

• A. The pulse loses energy to absorption, which stretches its duration
• B. Different frequency components travel at different phase velocities, arriving at the receiver at different times
• C. The group velocity increases along the fiber, causing later components to catch up and overlap with earlier ones
• D. Reflection at the fiber walls mixes frequency components together

Answer: (B)

In a dispersive medium, the dispersion relation ω(k) is nonlinear, so phase velocity v_phase = ω/k varies with frequency. A pulse is a superposition of many frequency components; in a dispersive fiber these components 'walk apart' as they travel because each propagates at a slightly different speed. What was a sharp pulse at the transmitter arrives as a smeared-out pulse at the receiver. Absorption (option A) reduces amplitude but doesn't cause spreading. This pulse broadening directly limits fiber data rates because adjacent pulses must be spaced far enough that they don't overlap after spreading.

Question 2

In a dispersive medium, which velocity determines how quickly a signal or information is transmitted?

• A. Phase velocity (ω/k), because it describes how fast the wave crests propagate
• B. Group velocity (dω/dk), because it describes how fast the envelope of a wave packet propagates
• C. The arithmetic average of phase and group velocity
• D. The speed of light in vacuum, which all signals must obey

Answer: (B)

Phase velocity is the speed of a particular crest in an infinite sinusoidal wave — a mathematical abstraction that doesn't carry information. Group velocity (dω/dk) is the speed at which a localized wave packet (the superposition of nearby frequencies) moves, and crucially, it is the speed at which energy and information propagate. In vacuum, phase and group velocity both equal c. In dispersive media they differ, and it is the group velocity that matters for signal engineering. Option D is incorrect: phase velocity can exceed c in anomalous dispersion regions without violating relativity, because it does not carry information.

Question 3

In vacuum, a light pulse containing many different frequencies will spread out over time because each frequency travels at a slightly different speed.

• T. True
• F. False

Answer: False

In vacuum, the dispersion relation is simply ω = ck — a linear relationship between angular frequency and wavenumber. This means phase velocity v_phase = ω/k = c and group velocity v_group = dω/dk = c for all frequencies. Every component of the pulse travels at the same speed c, so the pulse maintains its shape indefinitely. Dispersion and pulse spreading only occur in media where ω(k) is nonlinear — i.e., where the index of refraction varies with frequency.

Question 4

Phase velocity and group velocity can differ significantly in a dispersive medium, and the group velocity is the physically meaningful quantity for the transmission of energy and information.

• T. True
• F. False

Answer: True

This is the central distinction in dispersion theory. In a dispersive medium, the nonlinear ω(k) causes v_phase = ω/k to differ from v_group = dω/dk. Phase velocity can even exceed c in regions of anomalous dispersion without violating special relativity, because no information travels at the phase velocity. All signal transmission, energy flow, and information encoding occur at the group velocity, which is constrained to be ≤ c in physical systems.

Question 5

Explain why pulse broadening in optical fibers limits data transmission rates, and what property of the dispersion relation determines how severe the broadening is.

Model answer: A data pulse is not monochromatic — it spans a range of frequencies. In a dispersive fiber, different frequency components travel at slightly different group velocities (because v_group = dω/dk varies with frequency). Components that were synchronized at launch arrive at the receiver at different times, spreading the pulse. If adjacent pulses spread enough to overlap, the receiver cannot distinguish separate bits. The severity of broadening is determined by group velocity dispersion (GVD), proportional to d²ω/dk² — the rate at which group velocity varies with frequency. At a wavelength where GVD ≈ 0 (the zero-dispersion point), broadening is minimized, and dispersion-managed fiber design exploits this.

This is why fiber-optic systems operate near specific wavelengths and use dispersion-compensating fiber: engineering the dispersion profile is as important as minimizing loss. Higher data rates require shorter pulses, which span broader frequency ranges, which suffer more from dispersion — creating a fundamental engineering trade-off directly governed by the dispersion relation.