Question

Prove that the coherence length is related bu bandwidths following equation:-
L=c/∆v= λ²/∆λ

Answer 1

The frequency and wavelength are related through

$$v=\frac{c}{\lambda} \longrightarrow \frac{dv}{d\lambda} = -\frac{c}{\lambda^2}.$$

Differentiating the latter we can find the frequency width $\Delta v$ from the wavelength width $\Delta \lambda$

$$\frac{\Delta v}{\Delta \lambda} \approx \left| \frac{dv}{d\lambda} \right|=\frac{c}{\lambda^2} \longrightarrow \Delta v=\frac{c \Delta \lambda}{\lambda^2}.$$

The coherence time is

$$\Delta t=\frac{1}{\Delta v}.$$

The coherence length is

$$L=c\Delta t=\frac{c}{\Delta v}=\frac{\lambda^2}{\Delta \lambda}.$$