Finding Total Number of Optical Field Modes for Visible Light

[freddie123]

Homework Statement

Calculate the total number of optical field modes per unit volume for visible light (i.e. in the range 400-700nm).

Relevant Equations

ρ_kdk = k^2/π^2 dk
k=2π/λ

ρ_kdk = k^2/π^2 dk is the density of field modes (what we are trying to solve for here), and as ρ_kdk = ρ_λdλ, and k=2π/λ, we can rearrange this to get ρ_λdλ = 8π/λ^4dλ
This is where my confusion lies. I am not sure what to do next. I know this equation physically means the number of modes per volume for frequencies between λ and λ+dλ, so do we just take λ as 400nm and then dλ is 300nm? Or are we meant to integrate to get
-8π/3[(1/(700x10^-9)^3 - 1/(400x10^-9)^3)] = 1.06x10^20 number of modes??

I don't know how to use this equation to get the number of optical field modes per unit volume for the given frequency range.

Help would be much appreciated.

Thanks!

 

[DrClaude]

Yes, you need to integrate. Your result appears correct.

 

[freddie123]

Yes, you need to integrate. Your result appears correct.

Awesome. Thank you!