the increase in forward voltage across the diode above the barrier potential is due to the voltage drop across the internal dynamic resistance.
There are 2 components to VI curve:
a) The intrinsic exponential current of If vs Vf
b) a bulk series resistance, Rs where the curve turns almost linear from ~ 1/3 to 100% rated current @ 25'C.
While the If increases exponentially with Vf, is evident on all diode VI curves in datasheets This incremental linear slope of the curve is due to the fixed interface resistance of electrodes and the crystal interface. This is inversely related to size and thus power rating of a PN junction or LED.
I have found by comparing hundreds of datasheets that diodes fall into groups of constant k= Rs*Pd(max rated at some temp) = depending on the design and chemistry.
This means for part of the rated current you can see an exponential rise to 0.7V for Silicon diodes then after this the slope follow a more linear resistor Rs=k/Pmax
For example in epoxy encapsulated LEDs k = 1 +50% /-25% ( e.g. 65mW 5mm LEDs have incremental Rs near 15 Ohms, while SMD power LEDs are closer to k=0.5 and in high power diodes k = 0.25 nom. New LED's tend to have tighter tolerances such as +25%/-13% on Vf typ. due to design/process improvements.
A 100W silicon diode will be < 10 mOhm but > 2mOhm for the Rs value in the upper current range.
So a small signal diode rated for 100mW might have an Rs of 2~10 Ohms so after 0.7V including this resistance so you can extrapolate the pulsed voltage knowing this.