A filter passes light with a mean wavelength of λ̅0=500 nm. If the emerging wavetrains are rough

step 1 Hello students in this question a filter passes light with mean wavelength lambda not bar of 500

A filter passes light with a mean wavelength of λ̅0=500 nm....

A filter passes light with a mean wavelength of $\bar{\lambda}_{0}=500 \mathrm{nm}$. If the emerging wavetrains are roughly $20 \bar{\lambda}_{0}$ long, what is the frequency bandwidth of the exiting light?

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Key Concepts

Wavelength-Frequency Relationship

Light’s frequency and wavelength are inversely related by the equation c = ? · f, where c is the speed of light. This fundamental relationship allows conversion between a given wavelength (e.g., 500 nm) and its corresponding frequency, forming the basis to analyze any changes or spreads in the light’s spectral distribution.

Temporal Duration and Frequency Bandwidth

A finite pulse or wavetrain of light, such as one spanning roughly 20 wavelengths in space, corresponds to a pulse of finite duration in time. This limited temporal extent leads to an intrinsic uncertainty in the light’s frequency, often estimated by the reciprocal of the pulse duration, emphasizing that shorter pulses produce broader spreads in frequency.

Fourier Transform and Time-Frequency Uncertainty

The Fourier transform provides the mathematical framework that connects a time-localized signal to its frequency spectrum. The time-frequency uncertainty principle, derived from this transform, states that the product of the temporal duration of a signal and its frequency bandwidth is approximately constant. This principle explains why a wavetrain of finite length inherently exhibits a nonzero frequency bandwidth.

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