Maxwell's Rainbow — Question 3

Question 3

A wavelength passband reverses its endpoints when converted to frequency

An ideal filter transmits vacuum wavelengths from 500 nm through 600 nm. Use $c=3.00\times10^8$ m/s and define fractional frequency bandwidth as $\Delta f/f_c$, with arithmetic frequency center fc.

1. Find the transmitted frequency interval and its width.

2. Find fc and the fractional bandwidth; compare fc with the frequency corresponding to 550 nm.

Question 3 - Solution

A wavelength passband reverses its endpoints when converted to frequency

(a) Method and calculation

$$f_{\min}=c/(600\,\mathrm{nm})=5.0\times10^{14}\,\mathrm{Hz},\quad f_{\max}=6.0\times10^{14}\,\mathrm{Hz}.$$$$\boxed{\Delta f=1.0\times10^{14}\,\mathrm{Hz}}.$$

(b) Interpretation and further calculation

$$\boxed{f_c=5.5\times10^{14}\,\mathrm{Hz},\quad\Delta f/f_c=2/11=0.1818}.$$$$\boxed{c/(550\,\mathrm{nm})=5.45455\times10^{14}\,\mathrm{Hz}\ne f_c}.$$

Physical check

The longest transmitted wavelength corresponds to the lowest transmitted frequency.