Reflection coefficient in a microwave amplifier
what is it and what is it based on?

The reflection coefficient is a measure of how different an impedance is to a &quot;standard&quot; impedance. By &quot;standard&quot; impedance we usually mean 50 Ω but, we could also mean 75 Ω or any value that is used in the transmission of a signal by a cable. So, if the cable is designed to have a characteristic impedance of (say) 50 Ω we might  be interested in knowing how different a load impedance might be compared to 50 Ω.
We might be interested in this &quot;load mismatch&quot; because if the cable is of significant length, then we might get an unreasonable signal reflection level from a mismatched load. Reflection coefficient (in its simplest non-complex form) is a number between -1 and +1 that represents how different our load resistance is to the transmission line characteristic impedance (usually a resistive value).
To get to the root meaning of reflection coefficient we can set up an experiment: -

Assume a transmission line (t-line) of characteristic impedance \$Z_0\$
Assume an applied voltage (\$V_F\$) at one end of the line
From the above, the current (\$I_F\$) that (initially) flows equals \$V_F\$ divided by \$Z_0\$

When the voltage and accompanying current reach the end of the t-line and meet \$Z_L\$ there will be a violation of ohm's law if \$Z_0\$ does not equal \$Z_L\$. We have to fix this violation. The mathematical process of fixing of this violation uncovers the meaning of the reflection coefficient.
For instance, if \$Z_L\$ &gt; \$Z_0\$ we have to consider a mechanism that prevents the violation of ohm's law. As a thought experiment we can: -

Somehow make the voltage arriving at \$Z_L\$ a bit bigger and, at the same time
Somehow make the current arriving at \$Z_L\$ a bit smaller
The modified voltage and current is &quot;adjusted&quot; in such a way so as to produce a ratio suitable for \$Z_L\$

Or, algebraically we could say: -
$$\dfrac{V_F + \delta V_F}{I_F - \delta I_F} = Z_L$$
$$\therefore \dfrac{V_F}{I_F}\cdot \dfrac{1 + \delta}{1 - \delta} = Z_L\longrightarrow Z_0\cdot \dfrac{1 + \delta}{1 - \delta} = Z_L$$
$$\text{Hence,}\hspace{1cm}\delta Z_0 +\delta Z_L = Z_L - Z_0$$
$$\text{And,}\hspace{1cm}\delta = \dfrac{Z_L-Z_0}{Z_L+Z_0}$$
But, of course, we call \$\delta\$ by it's usual name (reflection coefficient) \$\Gamma\$. \$\delta\$ is just a device I invented to get through the thought experiment.
However, the important subtlety that prevents an ohm's law violation is the &quot;bit&quot; we add to voltage and the &quot;bit&quot; we subtract from current (\$\delta V_F\$ and \$\delta I_F\$). If we examined their ratio we would find it is \$Z_0\$. This means that they can naturally flow (together) back into the transmission line because they have the perfectly correct ratio to do so.
That is called a reflection and travels from load to source.
Clearly, if \$V_F\$ and \$I_F\$ were originally of a ratio that matched the load (\$Z_0\$) impedance (right from the start), we wouldn't need to set up the algebra that figured out how to deal with the unwanted signals and, there would be no thoughts of violating ohm's law nor talk of reflections.