Instead of using a resistor in series with the LED and simply turning the voltage on and off, you can drive the LED by switching a current source, and additionally switching the LED off by applying a reverse voltage (be cautious, most LEDs cannot withstand a reverse voltage of more than -5 V).  But this will not diminish the switching time by orders of magnitude.

For signalling purposes, you will probably get the best results by constantly biasing the LED with DC current, and applying the signal via a capacitor in such a way, that the LED is never completely switched off but only modulated. Of course, the resulting signal has no DC component.

With common low power LEDs, you might reach some 100 kHz. In contrast, with laser diodes, you can reach some GHz, but I wouldn't call the necessary circuit exactly simple.

Hope this helps.

The following application note ( https://docs.broadcom.com/docs/pub-005893 ) gives one driving circuit - including part values' calculation.

This is just ONE example. Broadcom stores more application notes of such circuits - a heritage from the times when these components were produced by Hewlett-Packard. The data sheets of the HFBR components hold such driving circuits as well.

Common to most if not all of these circuits is a switch nearly shorting the LED (some residual current is left) and a capacitor to inject resp. extract additional current during switching. A side effect is that the current consumption in the LED leg is nearly constant - irrespective whether the LED is on or off.

Hope this helps ...

I agree with  Fricke and Dreher.   

Good comments.

I only wish to add that a simple BJT transistor

can drive the LED pulsed mode

in the Audio KiloHertz range.  

In my audio project, I use an OpAmp to drive a BJT to pulse LED devices. 

So, now you have ideas for Slow LED ,

and Audio speed LED, and High speed LED.

In my own project I chose to use the Slow speed LED mode, initially.

I plan to replace this with a LM3914 logrithmic scale LED driver .

Most of my circuit stages have special requirements,

which are very specific to the goals of my project.

So, 

being aware of the variety of possible solutions is important.

HTH.

https://www.geocities.ws/glene77is

Basically I tried to use the inductor in series with resistor mentioned in this article http://pdfserv.maximintegrated.com/en/an/AN3790.pdf 

but I don't see any improvement, any suggestion?

Do you use the MAX3967?

You did not (yet) give away your speed requirements. I suppose the components in the MAXim application are dimensioned for 270 MHz. Depending on your speed requirements for this circuit some re-dimensioning of the component values is required.

I drive it with FET, working frequency is 20MHz. I don't wan to change the driver, my current fall time is 25ns. I want to reduce it based on mentioned pdf in previous comment. since I don't know the circuit model for LED, it even makes it worse.

OK - for 20 MHz the parallel circuit of MAXIM is certainly not appropriate. I'd really suggest to use one of the HP/Agilent/Broadcom circuits (shunting the current, eventually paired with some cap to speed-up switchoff/swotchon) that do not depend that heavily on the working frequency.

Do you have any link for that? then I can look at it?

One link is in my first answer (answer #2), abother circuit in the following data sheet: https://www.broadcom.com/products/fiber-optic-modules-components/industrial/eval-kits-accessories/industrial-eval-kits/hfbr-0400

More is available - as already stated - on the broadcom web sitel.

BTW: Do you have a single FET or a push-pull output stage? Regarding your current Tf of 25 ns the "20 MHz" seems a bit of wishful thinking. And what is your Tr?