Hi there,
A basic 10x scope probe will have a large series resistor in parallel with a small capacitor.
The idea is to build two voltage dividers into the same circuit that are in parallel, one that is made from the circuit resistances and the other made from the circuit capacitances. As you probably know, the output of a resistive voltage divider is Vo=Vin*R2/(R1+R2) and that is clearly not frequency dependent. Well guess what? The output of a capacitive voltage divider is almost the same: Vo=Vin*C1/(C1+C2), and it's not frequency dependent either even though we are using capacitors, which by themselves are inherently frequency dependent!
The idea then is to take both of these and make a circuit that attenuates by a factor of 10, where the capacitor C1 provides a voltage divider ratio same as the resistive part. The reason we want a factor of 10 is both because it's a convenient divider ratio for scope viewing and also it allows for a nice small probe capacitance that is quite a bit lower than the scope input.
The Details:
The resistive part of the scope is probably 1 Megohm, and so the resistor has to be 9 times that, which is 9 Megohms. Since capacitive reactance is an inverse, C1 must be equal to the scopes input capacitance DIVIDED by 9, so if the scope has 18pf input the probe parallel cap has to be 2pf.
Of course if we take the line capacitance into account, we will have a higher 'input' capacitance so the parallel cap will come out bigger. If the line capacitance is 18pf/foot and we used 3 feet that would add 18pf times 3 to the total input cap, which would come out to 4 times 18pf, so the parallel cap would have to be 8pf. It's usually made variable to make up for irregularities in the line or scope and connectors.
Also, if we could find better line than 18pf/foot that would reduce the parallel cap requirement, so we would end up with less probe capacitance.
Theoretically because in the end analysis all the frequency components go away with this kind of design we end up with a totally flat response:
Vout=Vin/10
and it's as simple as that
To design a nice 1:1 probe, a well chosen unity gain op amp built into the probe body and powered with separate lines through the cable would do the trick really nicely, keeping the distance from the op amp input terminal to the circuit under test under one or two inches.