Bi-Directional Rheostat/Potentiometer
Dimming AC-powered bulbs isn't really the simplest task, especially when you want a custom brightness curve like that.

I would suggest you use bright white LEDs instead (and not the AC light bulb replacement types, but simple DC ones powered from about 3&ndash;5 volts), and control the dimming from an Arduino. The Arduino IDE already comes with two very relevant and quite simple examples: LED dimming, and reading the position of a potentiometer. All you would need to do on the software side is combine the examples so that the position is turned into a brightness value. On the hardware side, you would probably need to add a simple transistor ciruit (such as just a single logic-level FET and a current-limiting resistor) to drive the current to the LED, since bright ones probably need more than the Arduino can directly supply.

This would be a pretty cheap and simple solution, and there would be no chance of frying yourself, unlike when working with mains AC circuits.