Hi, I would also like to learn more about driving the backlight leds.
The "LQ035NC111" datasheet says 18.6V - 21.0V and max 20mA.
I found many solutions on the internet with pros and cons:
1) Using a boost converter aka constant voltage and adjusting the voltage to adjust the brightness.
Pros: Easy circuitry
Cons: No constant brightness, varies with led temperature. Led can be destroyed because there is no current limiting
2) Using a boost converter and a current limiter and adjusting the current to adjust the brightness.
I found an easy circuit for diy at: http://www.radio-electronics.com/info/c ... source.php
Pros: Constant brightness, constant with temperature, led wont be driven over the max allowed current at no circumstances
Cons: Complicated circutery
3) Using a ready made LED backlight driver, like: http://www.ti.com/product/tps61160
Pros: Can be controlled digitally via arduino or raspberry to set the brightness
Cons: I didn't found any fully soldered breakout boards, so we have to solder SMD and design a PCB