The Cmos 555 has a typical output high current of only 2ma when its supply is 3.6V.
Some ICs will have less output current and the current is less when the battery voltage runs down.
Then the LED will be extremely dim.

 

You could always go with another AA, or 9V. The heat build up is a valid point.

Look for one of the current LED flashlights with the 3 AAA adapters. You need the extra volt.

The Cmos 555 has a typical output high current of only 2ma when its supply is 3.6V.
Some ICs will have less output current and the current is less when the battery voltage runs down.
Then the LED will be extremely dim.

Check the specs of this specific CMOS 555, it can go to 100ma down to 3V. I was suprised when I saw it. Here is the datasheet.

 

LEDs are non-linear current-driven devices. Trying to modulate that beast
as a sine-wave will be tricky! You might be able to approximate it by using
the square wave as input to charge a cap with the right CR time constant
so that you get something that looks like a triangle wave. Prototype it
or simulate it first?

 

Right now we need to figure out how to a> feed a LED that drops between 3.2 to 3.8 volts 20ma with a 3.6V battery or b> get a small battery that won't cause a lot of waste heat from excessive voltage. Once these are solved then we can move on.

Just from what the OP has said I think we can PWM modulate the 555 with a sine wave, but first things first. What ever scheme is used space it tight.

 

Right now we need to figure out how to a> feed a LED that drops between 3.2 to 3.8 volts 20ma with a 3.6V battery or b> get a small battery that won't cause a lot of waste heat from excessive voltage. Once these are solved then we can move on.

Just from what the OP has said I think we can PWM modulate the 555 with a sine wave, but first things first. What ever scheme is used space it tight.

I am happy with LED modulation circuit acheived thanks to you and other guys on the forum. I am going to stick with square wave, and the LED is dropping 3.15V across it, which is fine. I used 2 x 12V batteries in series to do so. I have decided to use a heat sink to solve the heat issue.

 

OK, cool. I suspect some of your distand issue could be an LED that is barely turning on though. If you decide to try the experiment you might try an extra cell and see if that extends the distance. Hope you're having fun, I am.