Maxwell's Rainbow — Question 9

Question 9

Two ideal filters transmit the intersection of their spectral windows

A beam has uniform power per unit vacuum wavelength from 450 nm to 650 nm and zero outside. Filter A passes 450–650 nm; filter B passes frequencies from $4.0\times10^{14}$ to $6.0\times10^{14}$ Hz. Use $c=3.00\times10^8$ m/s and unit transmission in each passband.

1. Find the wavelength interval transmitted by both filters.

2. Find the fraction of incident power transmitted and determine whether reversing filter order changes it.

Question 9 - Solution

Two ideal filters transmit the intersection of their spectral windows

(a) Method and calculation

$$\text{Filter B: }c/f_{\max}=500\,\mathrm{nm},\quad c/f_{\min}=750\,\mathrm{nm}.$$$$\boxed{[450,650]\cap[500,750]=[500,650]\ \mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\boxed{P_{\mathrm{out}}/P_{\mathrm{in}}=(650-500)/(650-450)=3/4}.$$$$\mathcal T_A(\lambda)\mathcal T_B(\lambda)=\mathcal T_B(\lambda)\mathcal T_A(\lambda)\Rightarrow\boxed{\text{order independent}}.$$

Physical check

The width ratio gives power only because the incident spectrum is uniform per unit wavelength, as specified.