but you have to add a voltage source on the "+" IN of the OP-AMP
The circuit on the right is wrong for me.
I wonder, why these simple photodiode cicrcuits are often causing confusion?
You can, of course, reverse bias the photodiode. This way, you reduce the junction capacitance and slightly increase
the speed, but also get increased reverse leakage current. For a DC or low frequency application (e.g. chopper modulated
photometer), zero bias is the preferable operation mode in my opinion.
By reversing the photodiode polarity, you get an inversed output signal. As said, it only works with dual supply. There
are however possible reasons to use a dual supply with a high performance I/V converter anyway. E.g. the fact, that the
best available OPs are probably needing a positive common mode voltage. Or the requirement of an output signal scale
down to true zero.
As another point, some popular photodiodes for photometric applications have the cathode connected to the case. They
would be preferably operated in the right circuit, with dual supply and a negative signal output.