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- 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!

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!