Interference of Two Waves (Interferometer)

[roam]

Homework Statement

The input signal to the interferometer shown in the picture below is given by:

$$E_{\text{in}}=\sqrt{P}\exp\left(j\omega_{0}t+\frac{jD(t)\pi}{2}+j\varphi_{p}(t)\right). \tag{1}$$

$P$ is the power that is received. The delay present in one arm of the interferometer is $T_b$.

I want to calculate the intensity/optical power at the output of the interferometer.

Homework Equations

For two waves $\sqrt{I_{1}}\exp(j\varphi_{1})$ and $\sqrt{I_{2}}\exp(j\varphi_{2})$, the resulting interference equation is:

$$I=I_{1}-I_{2}+2\sqrt{I_{1}I_{2}}\cos\varphi \tag{2}$$

where $d$ is the delay distance, and $\varphi = \varphi_2 - \varphi_1$.

The Attempt at a Solution

So from the interference equation:

$$P_{\text{out}}=2P\left[1+\cos\left(\frac{2\pi d}{\lambda}\right)\right]=2P\left[1+\cos\left(\frac{2\pi cT_{b}f}{c}\right)\right]$$

$$=2P\left[1+\cos\left(2\pi T_{b}f\right)\right]=2P\left[1+\cos\left(T_{b}\omega\right)\right]. \tag{3}$$

Is this correct?

The context of this problem is in telecom. $\omega_0$ represents the carrier frequency, $D(t)$ is the actual data that has to be recovered. However $D$ is absent from (3) so I am thinking that my result is probably not correct.

Any explanation would be greatly appreciated.

 

[Charles Link]

One article you might find useful is the following Insights article that I authored: https://www.physicsforums.com/insights/fabry-perot-michelson-interferometry-fundamental-approach/ It seems somewhat difficult to find a thorough treatment of interferometric principles, even in the Optics textbooks. Hopefully the Insights article is helpful.