With the previous problem in mind, now consider things...

With the previous problem in mind, now consider things spread across space at a given moment in time. Each wave separately would result in an irradiance distribution $I_{1}$ and $I_{2}$. Plot both on the same space axis and then draw their sum $I_{1}+I_{2}$. Discuss the meaning of your results. Compare your work with Fig. $7.16 .$ What happens to the net irradiance as more waves of different frequency are added in? Explain in terms of the coherence length. Hypothetically, what would happen to the pattern as the frequency bandwidth approached infinity?

Key Concepts

Frequency Bandwidth

The frequency bandwidth of a light source indicates the range of frequencies (or wavelengths) present. As more waves with different frequencies are added, the overall coherence of the light decreases because the phase relationship between components becomes less defined. In the limit of infinite bandwidth, the coherence length approaches zero, and any interference pattern is averaged out, leaving a uniform irradiance distribution without distinct interference fringes.

Coherence Length

Coherence length is a measure of the distance over which a wave maintains a predictable phase relationship. In the context of adding waves of different frequencies, a shorter coherence length implies that the relative phases between contributions vary rapidly in space. This rapid variation diminishes the persistence of clear interference patterns when many distinct frequencies contribute, effectively washing out the fringe visibility.

Wave Interference

When two or more waves are present, their coherent addition can result in an interference pattern characterized by alternating regions of high and low irradiance due to phase differences. This pattern is sensitive to the relative phases and amplitudes of the contributing waves, so plotting individual intensities alongside their sum can reveal how the spatial arrangement of these regions arises from interference effects.

Superposition Principle

This concept underlies the idea that when multiple waves overlap in space and time, their resulting effect is the sum of their individual contributions. In optical systems, this means that the net irradiance pattern is obtained by algebraically adding the fields (or intensities, under certain conditions) from each wave, leading to phenomena such as constructive and destructive interference.

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