Why the intensity of light is more around the region of hot air around the flame of candle in the shadow cast by the light from the collimated source ?
Most of the articles mention it is due to the refraction of light through hot air around the candle's flame. But I'm unable to understand how refraction make this happen.
The column of warm air has lower density and thus lower refractive index. This makes a cylinder of low refractive index which acts as a lens wi h a negative focal length.
Light that would have passed through the column is bent outwards, where it is added to the light that was already there. So there is a dark region directly behind the cylinder, and a brighter rim right outside it.
I hope the following diagram makes it clear:
Incidentally, the refractive index of air is quite small: 1.0002772 (at 15 °C, 1 atm). You can expect this to scale roughly with density, so if you heat the air to about 300 °C, the density drops by a factor 2x and the change in $n$ is roughly 0.00014, or 0.014%. We can estimate the focal length of a column of air that is 2 mm diameter and 300 °C:
The focal length is approximately given by
$$f = \frac{R}{\Delta n}$$
which is a simplification of the lensmaker's equation
Putting in the numbers I estimated above, we find $f = \frac{1 mm}{0.00014}= 7 m$
This means that the bright fringe will be best observed if the screen is a few meters away from the candle; if you get much beyond the focal length, the effect will start to diminish again. If the column of hot air is smaller, the focal length will come closer; the same will happen if the air is hotter.
Note that if the incident source is extended (that is, light is arriving from multiple directions), then this tends to "smear out" any effect - which is why this is most visible if you use a point (or slit) source.
My gut reaction for the answer as to what causes this phenomena is that the candle's flame adjusts the index of refraction around the candle in two ways.
For one, heat is well known to cause refraction in a medium- you need only compare it to a mirage to see the similarities. As in the hazy image you'll find in a mirage caused by layers of hot air, roughly the same convective processes are going on in the heat around a burning candle- a gradient in the refractive index of the medium around the candle and the candle's surroundings. Lower index near the flame, higher surrounding the candle.
Additionally, flame needs oxygen to burn, and this intake of fuel removes some physical material- if you had pure oxygen around the flame, you could roughly create a vacuum in this way, after removing the products of combustion. I don't know exactly what effect this has, but would be willing to bet it would also lower the refractive index around the candle's flame, if only mildly.
