How can I make an amplifier such that its gain increase proportional to the input difference
Think about what you are actually asking.  A normal amplifier does:

&nbsp; &nbsp; OUT = A * IN

where A is the gain.  You want the gain to be proportional to IN, so:

&nbsp; &nbsp; OUT = IN * IN = IN2

In other words, you want a quadratic response, not a linear one.

The problem with this is that multiplying analog signals is difficult to do with good accuracy.  There are such things as analog multipliers.  You could put a analog multiplier in front of a fixed-gain amplifier to give you what you want.

You also need to decide whether you want 1 quadrant or 2 quadrant operation.  It's not clear whether your input signal can go negative.  That would make things more complicated.

There is a possible simplification since you are squaring a single signal instead of multiplying two arbitrary signals, and because the result is monotonic.  Back in the Pleistocene when you were stuck with only analog processing, diode function generators were used to create various monotonic functions with single inputs and single outputs.

However, back in today's real world, you should seek to do this kind of operation digitally if possible.  Determine what speed you need.  If a few 100 kHz sample rate is sufficient, then doing it digitally is a no-brainer.  That can be done in various common microcontrollers.  If over 1 MHz, you may need something like a high speed DSP.  Faster is possible with dedicated high speed A/Ds and a FPGA.