To really understand light and lumens, we must first realize that artificial light cannot keep its strength as it moves further away from its original source. In fact, light emitted from even the best bulbs will decrease, exponentially, as the distance increases between light source and garden.
When a bulb is turned on, the bulb will emit light with a certain amount of power. That power will stream out equally in all directions, forming an ever-increasing spherical wave. While the total power of that wave will remain the same, it will be spread over an increasing surface as the light moves away from the source. The light intensity, which is the power per unit area, will diminish as the distance from the source increases (see Figure 1).
Figure 1: The energy twice as far from the source is spread over four times the area, hence one-fourth the intensity
The Inverse-Square Law for Light Intensity dictates to us the exact proportions for which our light will diminish over space. If you have one lumen shining on one square foot at a height of one-foot from above, then at two-feet from above that lumen is now only a quarter of a lumen (1/4), because that single lumen will diminish by traveling a greater distance and is now also spread out over a greater area. Not only do we have to contend with the light losing strength, but also with the increased surface area that light must now cover.
As most of the mathematical relationships we encounter are linear, many of us would guess that doubling the distance from a light source would result in a halving of intensity. Of course, we know otherwise. The inverse square law tells us that the intensity would fall to the square of one half or one fourth. At three times the distance, it would fall to the square of one third or one ninth, and so on. This sharp rate of fall-off has many consequences.
Considering the facts stated here, a good conclusion would be that keeping your lights close to plants is your best option for utmost efficiency. But, as always, there are even more considerations. While keeping your lights lower help your plants to catch maximum lumens, they will only be catching those lumens that are emitted towards your plants. This is why a good reflector can make a big difference for your plants. Reflectors are often used in the lights to maximize light efficiency. Many gardeners also cover the walls of their grow-room with a reflective material, or alternatively, white paint to maximize efficiency.