My first question is what type of bias-Tee is that L and R form and what is the purpose of them?
At RF, the inductor will look like an open circuit (assuming you choose a high enough RF frequency for the inductor being used). Then looking into the RF port you just see the ~45 ohm resistor in series with the laser diode. Since the laser diode has a low differential resistance (1-5 ohms is not uncommon), you have a pretty good match for a 50 ohm source.
I would not add the 50-ohm external resistor you show in your proposed diagram for two reasons:
You now have a ~100 ohm termination, not a good match.
Your external diode will have to be connected via some wires or traces, and the inductance of these traces could be significant at RF frequencies. If you're using a K connector, presumably you're working in the 20-40 GHz range, and designing clean terminations is not trivial at these frequencies.
If I want to pulse this diode using a precision function generator (with 50 Ω output impedance) by 5 V pulse train with 1ns pulse width and 50 ns period how should I achieve that?
Attach the pulse generator to the RF port of the laser using coaxial cable.
Use an external DC block in the RF path so that the RF generator doesn't disturb the DC bias point of the laser.
But how to bias the diode in that case it seems there is no separate pin for RF and bias input.
Use the laser anode as a common RF and DC ground. Apply the DC bias to the through the inductor. This will be a negative voltage and your source will need to sink current.
Your pulsed source will also need to pulse negative to increase the laser output power, and go positive to reduce the laser output power. So you will set your pulsed source to be high for 49 ns and low for 1 ns out of each 50 ns period.
I want to pulse this diode ... by 5 V pulse train
You should work out what current you need to drive through the laser to get the optical power you want. Then work out the voltage from that.
Don't start by making an arbitrary choice of what the pulse voltage should be.
5 V into a 50 ohm termination will give you 100 mA pulse current through the laser. If that's what you want then you're fine. If you want more or less current than that, you need to adjust your pulse source voltage, not try to change the laser to work with 5 V pulses.
why second one has same node for RF and bias?
Pin 3 is for DC bias. Pin 12 is for RF. Pins 11 and 13 are for the common (ground) voltage.
I'm not sure why you're saying RF and bias are the "same node".