@ Cjdelphi... Good Luck in the contest.. The one thing apparent is that you have no direct knowledge of Photodiode operation.
The output current is in the order of uA.. If that part was a phototransistor a second transistor might work... Not however with a photodiode.... The capacitor typically is in the order of 10 to 100 pF.. It's used to compensate for the photodiode junction capacity.
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from both Datasheets. http://www.datasheetcatalog.com/datasheets_pdf/1/N/5/4/1N5406.shtml
and the Analog Devices web site where I captured that quote.
The photodiode is reversed and that it works at all is a wonder.. For two reasons, with the + and - inputs biased as they are that Op-Amp should be hard against one rail... At a minimum the + input should go to a 10K pot to set the comparator threshold.
However there is no data sheet for the Op-Amp.. which is all the more reason to suspect a phototransistor.. There are many IR phototransistors available. Notice I didn't say IR Detector.. The three leaded type? they are two completely different devices..
For all this is worth a "Regular" Phototransistor... would in all probability work as well as that kludge you've wired up.
The old National Semiconductor analog app notes books have a great section on the why's and wherefore's of using Photodiodes properly...
BTW for Really Fast pulses it isn't unusual to see 100 or more volts on the cathode of the photodiode.. The high voltage is used to increase the avalanche current for fast particles, which because of their speed don't cause much avalanche current to flow..