Maxwell's Rainbow — Question 4

Question 4

Equal cycle counts do not imply equal pulse durations across the spectrum

Two approximately monochromatic vacuum pulses each contain N=120 carrier cycles. Pulse A has wavelength 2.5 mm; pulse B has wavelength 500 nm. Treat the envelope length as $N\lambda$ and use $c=3.00\times10^8$ m/s.

1. Find each pulse duration and spatial length.

2. Find the duration ratio and the number of B cycles in the duration of pulse A.

Question 4 - Solution

Equal cycle counts do not imply equal pulse durations across the spectrum

(a) Method and calculation

$$\tau=N/f=N\lambda/c,\quad\ell=c\tau=N\lambda.$$$$\boxed{\tau_A=1.00\,\mathrm{ns},\ \ell_A=0.300\,\mathrm m;\quad\tau_B=0.200\,\mathrm{ps},\ \ell_B=60.0\,\mu\mathrm m}.$$

(b) Interpretation and further calculation

$$\boxed{\tau_A/\tau_B=\lambda_A/\lambda_B=5000}.$$$$\boxed{N_{B\text{ in }\tau_A}=f_B\tau_A=6.00\times10^5}.$$

Physical check

A finite pulse has a bandwidth; the carrier-cycle description is an approximation suitable for a many-cycle pulse.