I see what you are saying. I assumed that something was wrong because of the way you responded to me saying that the ref design used a CMOS op amp. Now I see that I misinterpreted your response as a functional issue but rather you were addressing that the AD8541 family is just not well suited for use in the front end because of the inherent noise performance for CMOS based amplifiers.
I have just attempted removing C16, R8, C17, R9 replacing the resistors with jumpers and leaving he caps as opens. The signal is no longer meaningful in the slightest. I am not sure if I have caused an amp to go into oscillations or if it is some other problem but for some reason the output is swinging all around with no source present at the input.
Two of the frustrations I have had with this circuit is that the signal out of the first stage is not large enough to see by direct probing, and the circuit is so noise sensitive that I have to completely shield it between each change. So I have been dependent on the 2nd and 3rd stages working for me to see the anything.
I'm the new guy at ADI, but it looks to me like you decided to use this discrete JFET front end based on assumptions that may not be true. First off, this is an ac application since you want pulses out of the photo-diode. That means that you should not bias the photo-diode with a resistor and capacitively couple it to the amp. The best deal about using an amplifier is that you can run the photodiode across a virtual ground so that no voltage forms across it as it generates a photo-current. My first instinct would be to toss the JFET and the Linear Tech part. Unlike the 1983 bipolar or 1993 CMOS process that LT has used for decades, ADI stays way up on the process curve, so the JFETs we put right inside our amps are darn good ones and may achieve the voltage noise you want.
The noise is tricky. Yes, capacitive coupling throws out all the 1/f noise by dc, but it is also letting the photo-diode generate a voltage and the pole created by the photo-diode capacitance is killing the signal. The approach to use is to make the photo-diode amp first stage a dc-coupled wide-band front-end. Then you ac couple to your second stage where you can also filter the top end. Narrow bandwidth means low noise.
So my approach would be to find a fast JFET amp. We are the kings of that, although Burr-Brown makes one too. All FETs and JFET amps have low current noise, so look for low voltage noise, the thing that you were trying to get from that discrete JFET. The AD825 seems dandy. I like the AD8620. It has 6nV/rtHz. It's a JFET, has 25MHz bandwidth, 50V/uS slew rate. Ibias is great but not important to you-- all bias current does is make a dc offset that you are going to filter out. Same goes for offset voltage. You don't care about the dc error it will make.
Here is the tricky thing about photo-diode amps. First off, you might not want to take all the gain you need, since it will also limit your bandwidth. You might want to take less gain (smaller Rf) and lower the feedback cap so the response is faster. Then when you have a nice low-impedance signal coming out of the photo-diode amp, then take the rest of the gain, at the same time you can capacitive couple, and filter. That second op amp can be much more of a "jellybean". If you really are at the limits of physics and really want to use discrete JFETs you can also look at Bob Pease's circuit you mentioned.
Bob had the same problem as Linear Tech, National makes no fast JFET amps. The LF411 sure won't do your job. The LM6171 is a pretty decent high-voltage bipolar part and pretty cheap, but I would stick with ADI parts since they will all have better tolerances than any National part. That way you won't get the call from manufacturing in a year when the next batch gets built.
If you do use discrete JFETs, maybe parallel ones like Bob Pease did in his Figure 6, the amplifier becomes much less critical. You are welcome to use a bipolar part or anything else. I like ADI's FastFet parts, like I said, we really are ahead on process, so we can get you a fast amp that slews fast to represent the pulses. Same deal as with a single amp though, you may be better off reducing the gain to get good bandwidth, and then getting the rest of the gain in the second stage. Once you have the general architecture, then you might look at increasing the photo-diode amp feedback capacitance to lower the bandwidth, but if you can get away with that, it might mean we have selected too fast a JFET amp and we can find another one with even lower noise.
So, to sum up--
1) Most important, stop letting a voltage appear across the photo-diode. Put the cathode into the minus pin of an op amp and pull the anode down to minus whatever does not avalanche the diode. Or you can put anode to the minus pin and pull the cathode up, only the output pulses will now go negative. Run the op amp off as much voltage as you can, and please use a negative supply. Use the money you save on JFETs for a ADM660 charge pump and an ADI low-dropout regulator.
2) Try that fast ADI JFET amp, and you should be amazed at the nice pulse response. Lower the feedback resistor to get as much bandwidth as you need.
3) Capacitively couple the photo-diode amp to the next stage, yea, no 1/f noise. Filter the output to toss out as much high-frequency noise as you can. You get diminishing returns at higher filter orders, but maybe a 4-pole (2 amps, you can use any decent dual op amp). If you need gain, use multiple feedback, otherwise Sallen-Key is fine.
4) If you still have too much noise, well then look at the discrete JFETs, but bias the photo-diode like Pease did, heck just copy his circuit. You should pull down the photo-diode anodes to a negative voltage to reduce the diode capacitance. The big deal is to not let the photo-current generate any voltage across the photo-diode. And try to use an ADI amp, like I said, any decent high-speed amp will work as long as it has low voltage noise. ADI makes dozens of amps better than the LM6171.
5) Oh, for discrete JFETs don't uses separate discretes and hand-match them like Pease did. Just buy a dual from Linear Systems. http://www.linearsystems.com/SiliconixFairchildXREF.pdf
6) If one matched pair does not do it, you can use 2, like Pease did in figure 6, or even go to 5 pairs before the diminishing returns make it senseless. The deal here is that the signals from the JFETs are adding arithmetically, but that noise is adding RMS. This is because the noise is uncorrelated in the various JFETs and random noise adds RMS. You will get really good current noise in any JFET, the deal is then to lower the bias resistor until you burn so much current through them, you also lower the voltage noise. I hope you are willing to burn 5 or 10 mA per pair. Physics is physics, so low voltage noise means high-current. But that is why you should look to a JFET amp. It will buy you a lot of low noise for much less current than discrete JFETs. We have really good IC designers here at ADI.