Multiple beam reflection in Fabry-Perot cavity

[anthony27149]

Homework Statement

Just having some trouble with a question about Fabry-Perot intereferometers. Here's the question:

a) Consider a beam of light undergoing multiple reflections in a Fabry-Perot cavity between two surfaces, both with reflectance R, and with no absorption.

Homework Equations

The Attempt at a Solution

So my problem is incorporating N into my final answer. I've derived a general equation of the electric amplitude of the reflected waves inside the cavity (as shown in this diagram), which I found to be:

EN = A·τ·ρ'N·e^i(ωt-Nδ),

where ρ is the Fresnel reflection coefficient and τ is the transmission coefficient [Not sure if the equation is entirely correct]

And by geometric series:

Ec = A·τ·e^i·ω·t/(1-sqrt(R)·e-iδ)

Using a power density function for both inside the cavity Sc and the original beam S:

Sc = 1/2·nf·c·ε(Ec*·Ec),

where E* is a complex conjugate

S = 1/2·n·c·ε·A2

I get: Sc/S = (nf·τ2 )/(n·(1-sqrt(R))2 ·(1+C·sin2 (δ/2))),

where C is 4·sqrt(R)/(1-sqrt(R))2

So, as you can see, this ratio doesn't have the number of reflections N, and I'm not sure where to go from here.

 

[Charles Link]

Homework Statement

Just having some trouble with a question about Fabry-Perot intereferometers. Here's the question:

a) Consider a beam of light undergoing multiple reflections in a Fabry-Perot cavity between two surfaces, both with reflectance R, and with no absorption.

Homework Equations

The Attempt at a Solution

So my problem is incorporating N into my final answer. I've derived a general equation of the electric amplitude of the reflected waves inside the cavity (as shown in this diagram), which I found to be:

EN = A·τ·ρ'N·e^i(ωt-Nδ),

where ρ is the Fresnel reflection coefficient and τ is the transmission coefficient [Not sure if the equation is entirely correct]

And by geometric series:

Ec = A·τ·e^i·ω·t/(1-sqrt(R)·e-iδ)

Using a power density function for both inside the cavity Sc and the original beam S:

Sc = 1/2·nf·c·ε(Ec*·Ec),

where E* is a complex conjugate

S = 1/2·n·c·ε·A2

I get: Sc/S = (nf·τ2 )/(n·(1-sqrt(R))2 ·(1+C·sin2 (δ/2))),

where C is 4·sqrt(R)/(1-sqrt(R))2

So, as you can see, this ratio doesn't have the number of reflections N, and I'm not sure where to go from here.

The (sum of the ) geometric series that you (correctly) used assumes an infinite number of reflections, of which the E amplitude for the Nth reflection contributing to the sum gets smaller and smaller. The derivation for this can also be found in the Optics Textbook by Hecht and Zajac if you wish to check your algebra.