The input voltage V1 is the output of the phototransistor front-end.
It swings between 1V and 11V, with a middle value of 6V.
This input is immediately buffered by a voltage follower op-amp that isolates
the sensing front-end circuit from the amplifying circuit.
The middle of the diagram shows a similarly buffered virtual 6 volt
reference, for comparison with the input signal.
The IC1-C op-amp amplifies the difference between V1 and V2.
If r = R4/R3, the amplified difference V3 is
V3 = V2 + (V2 - V1)*r.
For this circuit r = 10.
The relation is obtained from expressing the op-amp draws no current:
(V2 - V3)/R4 = (V1 - V3)/(R3 + R4).
(R2 has no real role since the op-amp, to first approximation,
draws no current.)
The amplified signal (difference) drives an LED.
The top of the diagram is the feedback circuit.
(The feedback does not use the amplified signal.)
The L1 coil is positioned inside the probe, between the lower magnet
stack and the magnet float.
Its role is to counteract the movement of the float to the extent that it
can. Because the float signal is amplified, the float swing must be kept
small or else it will bring the amplifier to saturation.
The main benefit of keeping the probe movement small is that this
will keep it in the light sensitive region, where slight movements either way
still affect the amount of light transmitted.
The current through the coil now becomes the best measure of the forces
trying to move the float.