 Problem Statement
 I am trying to understand how the three equations below were solved simultaneously to get to the equation shown below. This equation was presented in a paper and it relates the input and output fields of a FabryPerot etalon.
 Relevant Equations

This is the equation I am trying to derive:
$$\frac{E_{2}}{E_{1}}=\exp\left[i\left(\pi+\varphi\right)\right]\frac{\taur\exp\left(i\varphi\right)}{1r\tau\exp\left(i\varphi\right)}$$
Here are three equations:
$$E_{2}=rE_{1}+itE_{3} \tag{1}$$
$$E_{4}=rE_{3}+itE_{1} \tag{2}$$
$$E_{2}=\tau\exp\left(i\varphi\right)E_{4} \tag{3}$$
I started by substituting Eqn. 3 into Eqn. 2,
$$\frac{E_{2}}{\tau}\exp\left(i\varphi\right)=rE_{3}+itE_{1}, \ \therefore E_{3} = \frac{1}{r}\left[\frac{E_{2}}{\tau}\exp\left(i\varphi\right)itE_{1}\right],$$
and then I substituted this result into Eqn. 1 to get
$$E_{2}=rE_{1}+it\frac{1}{r}\left[\frac{E_{2}}{\tau}\exp\left(i\varphi\right)itE_{1}\right]$$
$$\left[1\frac{it}{\tau r}\exp\left(i\varphi\right)\right]E_{2}=\left[r+\frac{t^{2}}{r}\right]E_{1}\Rightarrow\frac{E_{2}}{E_{1}}=\frac{\left[r+\frac{t^{2}}{r}\right]}{\left[1\frac{it}{\tau r}\exp\left(i\varphi\right)\right]}.$$
After applying the complex conjugate I got
$$\frac{r+\frac{t^{2}}{r}+i\left[\frac{t^{3}}{\tau r^{2}}+\frac{t}{\tau}\right]\exp\left(i\varphi\right)}{1+\left(\frac{t}{\tau r}\right)^{2}\exp\left(i2\varphi\right)}.$$
I couldn't manipulate this further to get to the desired equation. Am I on the right track? :
Also, how did the authors introduce the ##\pi## into the argument of the exponential?
Any help is greatly appreciated.
$$E_{2}=rE_{1}+itE_{3} \tag{1}$$
$$E_{4}=rE_{3}+itE_{1} \tag{2}$$
$$E_{2}=\tau\exp\left(i\varphi\right)E_{4} \tag{3}$$
I started by substituting Eqn. 3 into Eqn. 2,
$$\frac{E_{2}}{\tau}\exp\left(i\varphi\right)=rE_{3}+itE_{1}, \ \therefore E_{3} = \frac{1}{r}\left[\frac{E_{2}}{\tau}\exp\left(i\varphi\right)itE_{1}\right],$$
and then I substituted this result into Eqn. 1 to get
$$E_{2}=rE_{1}+it\frac{1}{r}\left[\frac{E_{2}}{\tau}\exp\left(i\varphi\right)itE_{1}\right]$$
$$\left[1\frac{it}{\tau r}\exp\left(i\varphi\right)\right]E_{2}=\left[r+\frac{t^{2}}{r}\right]E_{1}\Rightarrow\frac{E_{2}}{E_{1}}=\frac{\left[r+\frac{t^{2}}{r}\right]}{\left[1\frac{it}{\tau r}\exp\left(i\varphi\right)\right]}.$$
After applying the complex conjugate I got
$$\frac{r+\frac{t^{2}}{r}+i\left[\frac{t^{3}}{\tau r^{2}}+\frac{t}{\tau}\right]\exp\left(i\varphi\right)}{1+\left(\frac{t}{\tau r}\right)^{2}\exp\left(i2\varphi\right)}.$$
I couldn't manipulate this further to get to the desired equation. Am I on the right track? :
Also, how did the authors introduce the ##\pi## into the argument of the exponential?
Any help is greatly appreciated.