Will a matching impedance network also act as a filter?
A pi network, as described, is a filter and it can sometimes be used to match impedances. However, a straight impedance matcher is normally only resistive (such as for the termination of cables to prevent reflections at RF).
I'm using a pi network and am wondering if I should also add a filter
The pi network will act like a filter and if you are not happy about its performance as a filter then add a filter but, the filter will need to match the impedances you may be seeking to maintain.
Why not use a CommonBase device, letting the reverse-biased photodiode dump its current into the emitter. Try the 2SC5646A, a 12.5GHz device.
For 2.5GHz bandwidth, you need about 80 picoseconds collector tau. With 1PF total capacitance on the collector, you can use 82 Ohms Rcollector.
Run the CB stage on 3 volt, have 82 ohms in emitter to GND, have 82 ohms in collector to +3volts. Use cermet pot, 1KOhm to base, and bypass the base to GND with 1,000pF tiny tiny SurfaceMount capacitor. Use a ground plane under the circuit.
Buffer the collector, tied to base of a second 2SC5646A, its collector to +3volts, and emitter to GND thru 150 ohm resistor. Emitter is your output; do not short it.
Adjust the pot's wiper for 820 milliVolts across the 82_ohm Rcollector, thus 10mA flows in the CommonBase first device. Voltage gain will be GM * Rc;
GM will be Ic/0.026 or Ie/0.026, thus GM is 0.4. Product of GM * RC is 0.4 * 82, or Av = 32X (about 30dB).
Use a 2-sided PCB, and under the first device's collector node, use a Dremel to remove the GND foil, giving a useful boost to the bandwidth.
Best Answer
What you are describing in your question is a loss-less, high-pass, L-pad impedance matching circuit as per this example: -
The L-pad is used to match a lower impedance on the left with a higher impedance on the right. Either end can be source of course. In other words; you can match a higher source impedance to a lower load impedance or vice versa.
The formulas for L and C are dependant on the operating frequency of interest (\$\omega\$), the input impedance (\$R_{IN}\$) and the output impedance (\$R_L\$): -
$$C = \dfrac{1}{\omega\cdot R_{IN}}\cdot\sqrt{\dfrac{1}{\frac{R_L}{R_{IN}}+1}}$$
$$L = R_{IN}\cdot R_L\cdot C$$
Here you can find a calculator that saves you crunching the numbers by hand: -
That website also provides proofs for the formulas.
Well, you can make a low-pass version like this: -
Both work the same; at the frequency of interest, you can provide loss-less impedance matching as opposed to a wideband lossy impedance matcher like this: -
It provides loss-less impedance matching i.e. it makes the input impedance looks resistive and the output impedance looks resistive at the desired operating frequency. Either side of the mid-operating frequency, it's not quote perfect, but, it's good enough for most signals just like an antenna is not quite perfect either side of it's target mid-point frequency.