# Unsuccessful concentration of electric light via a lens

Why can a lens not focus an electric lamp's illumination to a much smaller area than the area of the lens, as it apparently can do with sunlight?

A lens can "intensify" sunlight via focusing it to a much smaller area ( causing a spot of concentrated heat ) than the area of the lens. But this is not similarly possible when the light source is electrical, as my own experiments with either a magnifying glass or a fresnel lens have both failed to demonstrate.

Note also that the lens of a movie projector doesn't seem to focus the lamp light smaller, at any distance from the lens. However, it's able to merely inhibit the spreading of illumination; i.e. within a movie theater it limits the maximum spread of the projector's light to the perimeter of the display screen.

An incandescent bulb having a clear ( non-coated with white diffuser ) glass globe is a reasonable single point source of light for comparison to the sun. These are found in movie projectors and are also marketed for use with vanity mirrors.

I merely wish to shrink the visible illumination from a lamp to a much smaller, brighter spot than the area of that light source. I'm not trying to generate heat.

Similar questions have already been thoroughly answered, albeit unsatisfactory, often digressing to thermodynamics. Have the answerers missed the point ? Surely the correct explanation must be simpler, no?

To expand on @knzhou 's comment some. The sun's rays are very parallel, but not exactly parallel. Most of the time we consider them parallel but really the disk of the sun is about 0.5 degrees. However when you are focusing the sun's light you are not imaging that disk, instead the rays that are within the numerical aperture of the lens (essentially all that within the diameter of the lens) are collected and you are focusing them to the spot.

On the other hand you light bulb is an extended source, meaning that if it is 5 cm diameter, and you are 50 cm away, the disk spans about 5.7 degrees. So from an angular perspective it is much much larger than the sun.

For most experiments you are probably trying to either image the light from the on screen, or collimate the light (make a beam) using the light as a source.

If it is an extended source, you can see from above whatever height it has will be magnified to its new height by the lens. If you put an aperture at the bulb and make its extent smaller to have a very small angular size, you will get a smaller spot.

If instead of a light bulb you use a high powered LED and do the experiment you can usually image the LED and you will see a sharp square that is the LED chip and maybe even a shadow from the wire bond. If you have a light bulb with a filament, you can also image the filament.

This is a common problem, for example getting light into a single mode optical fiber that has a very small core is much easier with a laser since the rays can be made parallel than an LED where the source looks extended and you can't get the imaged spot to be smaller than the core of the optical fiber.

Focusing the light from the source on a screen means that you get an image of the source. For a clear incandescent bulb you get an image of the filament. This is when the image of the source is "focused" on your screen. The filament is not a point-like source at the distance you have usually between the bulb and the lens. It looks like a spiral or double spiral. The size of the image of the source depends on the distances between the lens and the source and image. For the sun, the object distance is almost infinite and the image is practically in the focal point of the lens. The magnification is almost zero so the image of the huge object which isthe sun would be almost point-like in the ray-optics approximation. For a lightbulb the distance to the lens is much less and the image is farther from the lens than the focal distance. These two distances determine the magnification of the image and it will tell you how big the image of the filament will be. But it will be the same order of magnitude as the source filament so it may be much larger than the image of the sun.(for a single lens, in the paraxial approximation, the magnification is equal to the ratio between the image and object distances). In order to build a projector with filament lamps you need a lot more components than just one lens. The fact that you can project a nice spot with a projector does not prove that you can do the same thing with a single lens and an ordinary filament bulb. The spot you get from a flashlight on a wall or screen is not actually "focused". If it were you will see the filament of the bulb or an image of the LEDs pattern in the newer flashlights or lamps. You can do this (the actual focusing) with either a lens or a concave mirror and see for yourself. For flashlights or spot lights the purpose is not to focus but to make a parallel beam or at least not very divergent beam.

There are several ways to look at this. 1) Thermodynamics: An extended diffuser has a certain area A and from each point in that area the rays are emitted in totally random directions. The light has very high disorder or ENTROPY. Transforming the light to an area <A to make it more intense is forbidden BY ANY LENS including diffractive optics, metamaterials and mirrors since that would mean the entropy is reduced without energy input or an entropy increase somewhere else. The smaller spot (if it were not impossible), would have the same randomness from each point but crammed into a smaller area than A, so its entropy would be lower. The second law of thermodynamics totally forbids this reduction of entropy, ever. It is a very solid fundamental law and there will be no exceptions to it in this universe !! It's not a digression to use this thermodynamic argument. Remarkably, Rudolf Clausius realised this is why diffuse light cannot be concentrated back in 1863 , but today many still expend energy in vain trying to squeeze diffuse light into fibres , never understanding why it's impossible. 2) Geometrical Optics: A large diffuser can be transformed down to a smaller image of that diffuser by DEMAGNIFICATION. Unfortunately this requires the lens to be further away from the diffuse surface than it is from the image plane, so it can't capture as much light from the diffuser as if it were close by. If it were close it would be MAGNIFYING and the intensity at the image plane would be lower. It's almost like you can't win. Because you can't.... It turns out the product of the collected solid angle and the energy throughput is fixed , so for a wide angle the intensity transmitted has to decrease. In fact the best you can ever do is a 1:1 imaging system (no magnification) or to not imager the diffuser at all ! This limitation is embodied in constraints like the Sine-Optic Condition, Lagrange invariant and other geometrical lens-design rules. In other words if you want to get diffuse light into a fibre optic you might as well butt the fibre directly against the diffuser and waste a load of its emitting area, as introducing ANY TYPE OF LENS will reduce the power transmitted to the fibre compared to this maximum case. 3) ECONOMICS: If you could squeeze light from a cheap large area LED into a tiny fibre optic , everyone would be doing it (it's such an obvious idea even though it's one that's impossible to realise) , and there would 10Watt point sources that only cost \$5 to make. Lasers, on the other hand, can be launched into tiny fibres with lenses since their beams are almost parallel to begin with - ie LOW ENTROPY and LOW DIVERGENCE ANGLE. So you can trade off beam diameter against intensity at the focal spot. Lasers are fancy and expensive compared to bulbs and LEDs, and their special property of highly ordered light output (with low entropy=low divergence) allows them to be manipulated by optics with much lower losses and with much greater versatility, and with very high power densities possible at the focal spot. This allows them to be used in grandiose attempts to bisect temporarily restrained British spies, or be strapped to sharks to make them even more frickking dangerous - plus many other practical uses. To make matters more awkward (since people often ask this question wondering why you can't squeeze diffuse light into an optical fibre) the fibre also has its own limited acceptance angle for rays called the NUMERICAL APERTURE. Demagnifying increases the angle of rays into the fibre , anything outside the numerical aperture can't be transmitted down the fibre. This further limits how much of that diffuse light you can transmit. Annoying, isn't it ?? Annoying, but true. As Clint Eastwood said, "a man has to know his limitations". In years gone by they used to teach this stuff at universities, and students were kindly asked to try and remember it. Those were the days....