1 Watt LEDs are widely used in Emergency lights and LED lights. Most of the circuits use 4 volt battery having 0.5 to 4 Ah capacity. The 1 watt LED is rated 3.6 volts so we need the correct value resistor for LED to get maximum brightness. Let us see how the LED resistor value is determined.

Generally the ordinary LEDs require only 20-30 mA current to get good brightness. But 3.6 V LED requires high current more than 100 mA. So when we use it in 4 volt battery, very low value resistor is needed. Select the resistor as per the formula.
Vs-Vf / If
Vs is the battery voltage, Vf is the Forward voltage of 1 watt LED, If is the current required to pass through LED to get moderate to maximum brightness.
Battery voltage = 4 volts
1 Watt LED forward voltage = 3.6 V max
Current fixed through LED = 40 mA.
So , Vs-Vf / If = 4 V – 3.6 V / 0.04 A = 10 Ohms
So if we use 10 Ohms resistor in series with 1 watt LED at 4 V input, it consumes 40 mA current. But the brightness will be very low since the LED needs more than 100 mA current. So select Resistor as per the following ready recknor
4 Vs – 3.6 Vf / If 4V – 3.6 V = 0.4 V
0.4 / 0.58 = 6.8 ohms – With 6.8 Ohm resistor LED passes 58 mA current – Minimum brightness
0.4 / 0.071 = 5.6 ohms – With 5.6 Ohm resistor LED passes 71 mA current – Moderate brightness
0.4 / 0.085 = 4.7 ohms – With 4.7 Ohm resistor LED passes 85 mA current – Moderate brightness
0.4 / 0.102 = 3.9 ohms – With 3.9 Ohm resistor LED passes 102 mA current – Good brightness
0.4 / 0.120 = 3.3 ohms – With 3.3 Ohm resistor LED passes 120 mA current – Good brightness
0.4 / 0.140 = 2.8 ohms – With 2.8 Ohm resistor LED passes 140 mA current – Maximum brightness
0.4 / 0.180 = 2.2 ohms – With 2.2 Ohm resistor LED passes 180 mA current – Maximum brightness
0.4 / 0.200 = 2 ohms – With 2 Ohm resistor LED passes 200 mA current – High brightness
0.4 / 0.350 = 1 ohm – With 1 Ohm resistor LED passes 350 mA current – Very high brightness
If the current through LED is more than 85 mA, it needs a good Heat sink to dissipate heat. If the current is more than 140 mA, life of LED reduces and generates lot of heat.
So it is better to fix LED current between 102 mA to 140 mA for good brightness as well as long life of LED.