How does the power of a lens affect its ability to focus light?

[Meron35]

So I've been doing some experiments, using bottles filled with various liquids and timing how long they take to burn paper which is placed at the focal point (where the light was focused). I've noticed that using denser liquids with high refractive indexes lower the focal length, and thus increase the lens power. This causes the bottle to be able to achieve a faster burn. Specifically, using glycerol in comparison to water in the same bottle allowed me to halve the time it took to achieve a burn.

My question is, why? I've searched frantically over the net, and have found two answers I need explanation and elaboration on.

1. The inverse sq Law - this means that light being focused over a longer distance will diminish in intensity. However, I do not believe that this applies to this context.

2. A larger focal length means a larger focal spot, thus reducing intensity of the light - I simply don't understand this. Don't all convex lenses converge light onto one small spot? Why does the size of this spot change? Isn't this related to aberrations/refractive errors instead?

Please, if possible include sources in your responses.

 

[jerromyjon]

http://en.wikipedia.org/wiki/Focal_length

The focal length of an optical system is a measure of how strongly the system converges or diverges light... A system with a shorter focal length has greater optical power than one with a long focal length; that is, it bends the rays more strongly, bringing them to a focus in a shorter distance.

You have all the physical facts you need, since it is optics it is obvious to "see" where the power lies. What do you see when you look through the lenses and focus an image? I suggest you look at text on paper and report your observations...

 

[Meron35]

Unfortunately I haven't been able to look through the lens and focus a very good image (bottles aren't very good), though I can tell it magnifies to an extent. I've been able to focus an image of the outside (window) however, and the image is inverted on both axes.

Still, I don't understand how the focal length affects it's ability to converge more or less light. How does focusing light at a closer distance focus more light than at a farther distance? Shouldn't all light that enters the lens be focused at the focal point?

 

[jerromyjon]

Since your lenses are crude you can't "clearly" see what I was trying to point out, but when you focus the lenses, for instance on a character of text, if the optical power is greater, the text "appears" larger. When the focal length is longer, the text looks smaller (farther away). What does this imply about the inverse square law?

 

[jerromyjon]

If it were a concentrated beam of light (like a laser) in a vacuum, would it matter how far away the target was?

 

[mfb]

It can matter, but for laser beams the distances where this becomes relevant are huge.

The sun is not a laser beam - it has a finite size. If the focal length is too long this becomes relevant. You might be able to focus the light emitted by "a point" on the sun to "a point" on paper - but not all points from the sun to the same point on the screen.

 

[jerromyjon]

1. The inverse sq Law - this means that light being focused over a longer distance will diminish in intensity. However, I do not believe that this applies to this context.

Fact checking #1 first...

 

[Drakkith]

So I've been doing some experiments, using bottles filled with various liquids and timing how long they take to burn paper which is placed at the focal point (where the light was focused). I've noticed that using denser liquids with high refractive indexes lower the focal length, and thus increase the lens power. This causes the bottle to be able to achieve a faster burn. Specifically, using glycerol in comparison to water in the same bottle allowed me to halve the time it took to achieve a burn.

My question is, why? I've searched frantically over the net, and have found two answers I need explanation and elaboration on.

1. The inverse sq Law - this means that light being focused over a longer distance will diminish in intensity. However, I do not believe that this applies to this context.

2. A larger focal length means a larger focal spot, thus reducing intensity of the light - I simply don't understand this. Don't all convex lenses converge light onto one small spot? Why does the size of this spot change? Isn't this related to aberrations/refractive errors instead?

Number 1 doesn't apply. The inverse square law only applies to light that is diverging in a spherical wavefront. A good example is a laser vs a light bulb. Laser light doesn't diminish in intensity according to the inverse square law because it has been focused into a beam. But light from a light bulb does obey the law because it spreads out in a spherical pattern, not a beam like a laser.

Number 2 is the correct reason. Light is an electromagnetic wave, and you cannot focus a wave down to a single point. Instead, the wave is focused down to a finite sized "spot". The underlying reasons why the spot size is different for different power lenses is actually kind of complicated and I'm not sure I can explain it very well, but it has to do with how different parts of a converging wavefront interfere with each other. With longer focal lengths, the different parts of the wavefront will constructively interfere over a large area at the focal point, which spreads out the energy and forms a larger spot. With a shorter focal length, the different parts of the wavefront constructively interfere in a smaller area, which leads to a small spot.

When talking about images, the physics lead to a smaller image at the focal plane when the focal length is shorter, which focuses all of the energy into a smaller area, quickly heating up the paper compared to a longer focal length which has a larger image at the focal plane.

See the following link for a very good (but also very in depth and complicated) source for understanding optics. It's geared for telescope optics, but the same principles apply to all optics.
http://www.telescope-optics.net/

 

[jerromyjon]

Giving answers doesn't teach how to learn...

 

[mfb]

Laser light doesn't diminish in intensity according to the inverse square law because it has been focused into a beam.

Sure it does, just not in the lab. For very long distances it follows (to a very good approximation) the inverse square law as focussing cannot be perfect.

 

[Drakkith]

Sure it does, just not in the lab. For very long distances it follows (to a very good approximation) the inverse square law as focussing cannot be perfect.

If it doesn't follow it in the lab, and only begins to approximate it at large distances, that seems to imply that it doesn't follow the inverse square law.

 

[Drakkith]

Giving answers doesn't teach how to learn...

We're not here to teach people how to learn. We're here to teach people about physics.

 

[jerromyjon]

Then point the OP to a link on QED and let him "figure it out".

 

[Drakkith]

Then point the OP to a link on QED and let him "figure it out".

Why would we do that? This is about optics, not QED.

 

[jerromyjon]

Since gravity has very little effect on optics, this should be covered far more precisely by quantum mechanics.

 

[jerromyjon]

And the OP wasn't here to "learn physics", the OP was here to understand the results of the experiment that was performed.

 

[Drakkith]

Since gravity has very little effect on optics, this should be covered far more precisely by quantum mechanics.

Sure, if we were wondering about the quantum description of light and its interaction with the matter of the lens. But we aren't. That's far too in depth.

And the OP wasn't here to "learn physics", the OP was here to understand the results of the experiment that was performed.

Nonsense. The two are one and the same.

Since your lenses are crude you can't "clearly" see what I was trying to point out, but when you focus the lenses, for instance on a character of text, if the optical power is greater, the text "appears" larger. When the focal length is longer, the text looks smaller (farther away). What does this imply about the inverse square law?

It says nothing about the inverse square law, since it doesn't apply here. Also, remember that looking through a lens with your eye is not the same as taking that same lens and focusing light to a focal point with it. Looking through a lens at an object involves a complex optical system that uses multiple optical elements to form an image on your retina.

Still, I don't understand how the focal length affects it's ability to converge more or less light. How does focusing light at a closer distance focus more light than at a farther distance? Shouldn't all light that enters the lens be focused at the focal point?

Think of a "focal plane" instead of a "focal point". The image of whatever you are looking at is focused along a 2d plane known as the focal plane. The image formed at the focal plane of a lens with a short focal length is smaller than the image formed by a lens with a longer focal length. See the following images:

Note how the light from each point on the arrow is focused to a different point at the focal plane. While this picture only shows 3 points, there are actually an infinite number of them in every image. The light a the focal plane is the combined light from every point. In the next picture, the light rays are emanating from P1 in the above image. The light rays from P2 and P3 are not shown.

See how the shorter focal length lens at the top forms a smaller image than the longer focal length lens at the bottom? If we imagine the light from P2 and P3 being placed in this image, you can see that the points are spread further out in the bottom image. Since every image is composed of an infinite number of points, spreading them out means that we are also spreading the light out and the image formed is dimmer. The total amount of light is gathered by the lens and deposited at the focal plane for both cases. The only difference is that the amount of light deposited per unit of surface area is different, being higher for the top case and lower for the bottom case.

 

[Dadface]

Since gravity has very little effect on optics, this should be covered far more precisely by quantum mechanics.

This problem is better dealt with using geometrical optics not quantum mechanics. Indeed how would you use quantum mechanics?
The system being investigated is not a spherical lens, it's a cylindrical lens and as such a good starting point would be to research the theory behind such lenses.

 

[mfb]

If it doesn't follow it in the lab, and only begins to approximate it at large distances, that seems to imply that it doesn't follow the inverse square law.

The same is true for the light bulb as long as your distance to it is not significantly larger than the light bulb. It is just a quantitative difference, not a qualitative one.

 

[Drakkith]

The same is true for the light bulb as long as your distance to it is not significantly larger than the light bulb.

How so?

 

[mfb]

A light bulb is not a point source.

Assuming the light bulb is a sphere* with a uniform surface brightness and uniform emission in all directions, the intensity you receive is proportional to the apparent solid angle of the light bulb. For very small distances, this is nearly constant (2 pi), then gradually falls off and approaches an inverse square law for distances much larger than the size of the bulb.

*good approximation for a non-transparent light bulb, otherwise consider the hot wire itself.

 

[Drakkith]

Okay, I see what you're saying. I was under the assumption that you take a point on the surface of an emitter (the frosted glass of the light bulb or the aperture of the laser) and see if the light emitted from that point follows the inverse square law.

 

[Meron35]

Sorry for the late reply.

So, to re-iterate and summarize (correct me if I'm wrong)

When focusing the light onto the paper, an image of the sun is formed. This image changes in size as the focal length does, with a longer focal length producing a larger image, in accordance with the laws of refraction for convex lens.

However, no matter how big or small the image is, the amount of light energy the image has is the same. This means that a smaller image is better, because it has the same amount of energy, but over a smaller area, leading to an easier scorch.

 

[Drakkith]

Exactly.

 

[Khashishi]

The sun is so far away that the focal length of the lens doesn't really matter. The demagnification is essentially infinite and the size of the focused image has more to do with imperfections in the optics (especially for a crude lens such as a jar) than the focal length. My guess is that a longer focal length will give a better focused spot even though the demagnification is less pronounced (closer to 1), because I think the imperfections will be less important. I wouldn't be surprised either way.

 

[Drakkith]

The sun is so far away that the focal length of the lens doesn't really matter. The demagnification is essentially infinite and the size of the focused image has more to do with imperfections in the optics (especially for a crude lens such as a jar) than the focal length. My guess is that a longer focal length will give a better focused spot even though the demagnification is less pronounced (closer to 1), because I think the imperfections will be less important. I wouldn't be surprised either way.

No, that's not correct. The size of the image is determined almost entirely by the focal length of the lens/optical system. I deal with this regularly in astrophotography. A long focal length telescope is great for small objects, as it magnifies them on the image plane, but when you want to get a wide-field image you need to use a short focal length scope.

 

[Khashishi]

The poster is using a bottle as a lens, not a telescope. For an ideal lens, you are right, and the size of the image is easily calculated using geometric optics. For a bottle, the "spot" will be a smear which has very little to do with the image of the sun.

 

[Drakkith]

The poster is using a bottle as a lens, not a telescope. For an ideal lens, you are right, and the size of the image is easily calculated using geometric optics. For a bottle, the "spot" will be a smear which has very little to do with the image of the sun.

Re-reading your other post, it seems like you're saying that the size of the Sun's image depends heavily on the aberrations introduced by the bottle. Is that correct?

 

[Khashishi]

Yeah, although I probably should have not used the term image. The spot size depends on aberrations. It can hardly be called an image at all.

 

[Drakkith]

Well, if the aberrations are more important than the focal length, why is the OP burning through paper faster with a shorter focal length lens when the aberrations are even more pronounced at short focal lengths?

 

[Khashishi]

Well then my guess was wrong, but I'm not surprised.

 

[Drakkith]

Well, you're correct in that you need to take the aberrations into account to determine how well the image is focused at the focal plane, which will affect how fast the OP burns through paper.

 

[NascentOxygen]

So I've been doing some experiments, using bottles filled with various liquids and timing how long they take to burn paper which is placed at the focal point (where the light was focused).

This is an interesting investigation---is it something you thought up, or is it in a science program you are studying at school?