Answer to Question #109187 in Optics for Tanya

Question

Can fermats principle be used to derive Rayleigh Criteria of Resolution

Answer 1

Fermat Principle:- A ray of light transverse from on point to another point by a rout which takes least time.

momentum of photon $(p)=\dfrac{h\nu}{c}$

Rayleigh Criteria: The Rayleigh criterion stated in the equation $θ=1.22λD$ gives the smallest possible angle θ between point sources.

The intensity of the expression for grating for peak at the phase $(\delta)=2m\pi$

The closest minimum $(\delta)=\pi/2$

Change in phase $\triangle \delta=\dfrac{2\pi}{\lambda}b\sin\theta$

Now, taking the partial differentiation,

$d\delta=\dfrac{2\pi}{\lambda}b\cos \theta d\theta$

For maximum condition,

$b\sin\theta = m\lambda$

differential equation,

$b\cos\theta d\theta=md\lambda$

now substituting the value in the above equation,

$\dfrac{2\pi}{N}=\dfrac{2\pi}{\lambda}md\lambda$

Now, by rearranging,

$R=\dfrac{\lambda}{\Delta\lambda}=mN$