A pulse of light is created by the superposition of many...

A pulse of light is created by the superposition of many waves that span the frequency range $f_{0}-\frac{1}{2} \Delta f \leq f \leq f_{0}+\frac{1}{2} \Delta f,$ where $f_{0}=c / \lambda$ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of $600 \mathrm{nm}$ and lasts a mere $6.0 \mathrm{fs}\left(1 \mathrm{fs}=1 \text { femtosecond }=10^{-15} \mathrm{s}\right)$ a. What is the center frequency of this pulse of light? b. How many cycles, or oscillations, of the light wave are completed during the $6.0 \mathrm{fs}$ pulse? c. What range of frequencies must be superimposed to create this pulse? d. What is the spatial length of the laser pulse as it travels through space? e. Draw a snapshot graph of this wave packet.

A pulse of light is created by the superposition of many waves that span the frequency range $f_{0}-\frac{1}{2} \Delta f \leq f \leq f_{0}+\frac{1}{2} \Delta f,$ where $f_{0}=c / \lambda$ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of $600 \mathrm{nm}$ and lasts a mere $6.0 \mathrm{fs}\left(1 \mathrm{fs}=1 \text { femtosecond }=10^{-15} \mathrm{s}\right)$
a. What is the center frequency of this pulse of light?
b. How many cycles, or oscillations, of the light wave are completed during the $6.0 \mathrm{fs}$ pulse?
c. What range of frequencies must be superimposed to create this pulse?
d. What is the spatial length of the laser pulse as it travels through space?
e. Draw a snapshot graph of this wave packet.

Key Concepts

Superposition of Waves

The principle of superposition involves adding together multiple waves of different frequencies to form a resultant wave packet. This is fundamental to understanding short pulses since a limited time duration pulse is created by combining waves with a spread of frequencies, leading to constructive and destructive interference in time.

Center Frequency

This concept refers to the main or ‘average’ frequency about which the frequencies of a wave packet are distributed. In optics, the center frequency is often determined by the relationship f = c/?, where c is the speed of light and ? is the wavelength of the light, representing the dominant oscillation frequency of the pulse.

Pulse Duration

Pulse duration is the time span over which the pulse exists, usually measured in femtoseconds or another unit of time. It is an essential concept because it determines how many cycles of the waveform occur within the pulse and tightly relates to the time localization of the pulse, which in turn has implications for its spectral content.

Spectral Bandwidth

Spectral bandwidth is the range of frequencies that are combined to form the pulse. A broader frequency range (larger ?f) corresponds to shorter pulse durations due to the time-frequency uncertainty principle, and it defines the limits between which the individual frequency components must lie to construct the desired pulse shape.

Spatial Length of a Pulse

The spatial length of a pulse is determined by multiplying the pulse duration by the speed of light in the medium. It represents the physical distance over which the pulse is spread as it propagates through space, linking the temporal properties of the pulse with its spatial characteristics.

Wave Packet Representation

A wave packet is a visual and mathematical representation of a pulse of light showing both the fast oscillatory behavior of the electromagnetic wave (the carrier wave) and the slower varying envelope that describes the overall shape of the pulse. The snapshot graph typically shows a high-frequency oscillation modulated by an envelope that represents the pulse's amplitude and temporal profile.

Transcript

00:01 So we are given that a pulse of light is created by the superposition of a lot of waves.

00:10 That's just the usual stuff.

00:15 And then that we have a pulse that its wavelength is 600 nanometers and lasts 6 .50 seconds.

00:29 And with this, we want to know the center frequency of this pulse of light.

00:32 So that's quite simple.

00:35 So that's a.

00:41 And the center also, it's f0.

00:44 That's just light velocity over the wavelength.

00:51 So this is three times 10 to the 8 meters per second.

00:58 And the wavelength is 600 times 10 to the 9 meters.

01:04 This will give 5 times 10 to the 14.

01:12 That's the central frequency.

01:15 Okay, so then we want to know how many oscillations.

01:21 The wave completes during the pulse and the pulse is 6 femtose seconds long.

01:29 So we just need to take into account that for b, well, this is a pulse, it's a wave, so it fulfills the fact that the bandwidth times the duration is of the other of 1, okay? and therefore, knowing that the central frequency is this, we know the time period is simply one over the central frequency, which is around 2 times, exactly.

02:16 2 times 10 to the minus 15 seconds.

02:23 It's really fast.

02:26 And the number of cycles in the pulse is simply the duration over this time period.

02:34 So we split the duration in periods.

02:38 And we have delta t over t.

02:42 And this is the 6 times 10 to the minus 15 over 2 times 10 to the minus 15 seconds.

02:49 It's just three cycles.

02:51 Cycles or oscillations.

02:56 Then for c we want to know the range of frequencies...

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