[syhprum (OP)]

When I focus the light from the Sun onto my kindling with a lens to start a fire I find that the smaller the F number of the lens I use the better.
I take it that the temperature of the focused spot cannot exceed the 5820°K surface temperature of the Sun but assuming I am focusing onto a black body how does the temperature vary with the F number and if I was making a thermal Solar power plant what F number optics should I use. 

[JP]

This is a rough guess, so someone who knows about solar concentrators might do better.

The power per unit area incident on your lens is going to be roughly equivalent to the power per unit area of the central peak of the diffraction spot (an Airy disk) that the lens produces.  The area of your lens is:

A_lens=π(d/2)²,

where d is the lens diameter.  The area of the focused spot is roughly

A_spot=1.22λf/d,

where λ is the wavelength of the light.  Now if you know the incident power per unit area on the lens, call it p_lens, and you want to calculate the power per unit area, p_spot at the focal spot, you get

1.22λf/d p_spot=π(d/2)² p_lens.

Therefore, roughly

p_spot=(π/4.88λ) (d³/f) p_lens.

Now, if you know the power incident per unit area of a black body, you can use the Stefan-Boltzmann law to figure out its equilibrium temperature, since the power radiated per unit area of a black body is:

p_bb=σT⁴,

where σ is Stefan's constant. Equating the power emitted by a black body and the power incident, you can solve for the equilibrium temperature:

T=[π/(4.88 σλ) (d³/f) p_lens]^1/4.

So there's two factors at work: (1) the size of the focal spot, which is governed by the f-number and (2) the light-gathering power of the lens, which is governed by its diameter.  You're best off increasing the diameter and decreasing f, so that you get the smallest-size focal spot coupled with the largest diameter lens possible. 

[LeeE]

Hmm... the aperture of a lens shouldn't affect the focus of a lens (although in practice, lenses aren't perfect and using a smaller proportion of the lens can reduce the degree of error due to imperfections: most camera lenses give their best quality images when stopped down two or three stops, although stopping down the lens then means that an increased exposure time must be used).

A smaller f-stop number equates to a larger aperture, and a greater light collecting area, so I would expect smaller f-numbers to produce better results here.

[syhprum (OP)]

Can the temperature of the focused light spot ever exceed that of the Suns surface ?, can we re-focus the light emitted from the the light spot and obtain a still higher temperature ?, I feel that there must be a barrier somewhere

[JP]

The restriction is that you can't capture more energy than the sun is emitting into the direction of your lens.  But if you put all that energy onto a tiny point, you should be able to heat it up higher than the surface of the sun.  I don't know if this is possible on the earth's surface, however.  You're better off using a giant mirror or a bunch of mirrors rather than a lens, however, since lenses are thick objects and making large ones that focus well is tough.  Mirrors are much simpler to make.

[lightarrow]

I thought the effect noticed by the OP was caused by cromatic aberration...

[LeeE]

I don't think that chromatic aberration is likely to be an issue with the lens being used by the OP.  As it has an adjustable aperture it's likely to be a fairly high quality compound camera lens which minimises optical aberrations.

[syhprum (OP)]

Considering the temperature problem, as the Sun is a source of radiation 0.5° wide there is a limit to how small the focused spot can be, When I spoke of a lens I included reflective optics where F numbers of 0.25 can easily be achieved but I still do not think temperatures exceeding that of the Suns surface can be achieved.
With a laser light source of course the situation is different, the effective diameter of the source is near zero and the focused spot can have a very high power density.
The only solar thermal power station that I know of uses reflective optics of about F 0.5 and an operating temperatures of about 700°K

[LeeE]

Considering the temperature problem, as the Sun is a source of radiation 0.5° wide there is a limit to how small the focused spot can be

The size of the sun's 'spot' in the refracted image is just dependent upon the focal length of the lens.  As there is, in theory, no minimum limit to the focal length then the size of the 'spot' is equally unlimited.

When I spoke of a lens I included reflective optics where F numbers of 0.25 can easily be achieved...

This is difficult to accept, at least for a single element lens or reflector.  For a solid lens it would mean that the focal point would be inside the lens and for a reflector it would require a very deep parabolic reflector.  I'm just not aware of any such devices ever being made.  In solar heating systems it may be possible to incorporate many separate elements of focal length l such that the diameter of the effective total area exceeds l but when you look at how they're arranged you end up back at a parabola.  There's also the issue that the standard f-number assumes a circular aperture, being based upon the diameter of the aperture, but if you use a non-circular aperture then you can end up with any f-number you like depending upon which measurement you use.

...but I still do not think temperatures exceeding that of the Suns surface can be achieved.
With a laser light source of course the situation is different, the effective diameter of the source is near zero and the focused spot can have a very high power density.
The only solar thermal power station that I know of uses reflective optics of about F 0.5 and an operating temperatures of about 700°K

Lasers are not point sources.  Rather, they act as an area light-source, where the light emitted from every point in the area is coherent with the light emitted from every other point in the area.  The area has a finite size and can be focused down into a smaller area, increasing the intensity at that point.

But are we talking about heat or temperature here?  Heat is the quantity of temperature, so for example, at high altitudes the temperature of the air molecules you'll find will be higher than those you'll find at sea-level, but because there will be a lot less of them it will feel colder.

[JP]

Considering the temperature problem, as the Sun is a source of radiation 0.5° wide there is a limit to how small the focused spot can be

The size of the sun's 'spot' in the refracted image is just dependent upon the focal length of the lens.  As there is, in theory, no minimum limit to the focal length then the size of the 'spot' is equally unlimited.

Exactly.  Also, the f-number is a ratio of the focal length to lens diameter.  You can have differently sized lenses or mirrors with the same f-number if they are different sizes.  The f-number is what tells you the size of your focal spot.  To figure out how much light power you're putting on that spot, you also need to specify the size of the lens, which tells you how much total light you can put onto that focal spot.  Clearly if you have both a big lens and small lens with the same f-number, their focal spots will be the same size, but the amount of light being put on the spot will be different.

[syhprum (OP)]

I have found an account by a university group of a project to pump a laser via a 40cm f2 mirror and an additional acrylic lens.

http://www.nap.edu/openbook.php?record_id=1838&page=100

[LeeE]

The f-number is what tells you the size of your focal spot.

I can't quite see this, although I might be misinterpreting what you mean by 'focal spot'.  Are we talking about the size of the focussed image of the sun here?

For example, if I use an 800mm f8 lens to capture an image of the sun and compare the size of the sun's image captured with a 6mm lens at f8, the image of the sun will be much larger with the 800mm lens than with the 6mm lens.  The intensity of light within the captured image will be the same though i.e. with the same exposure time, both will be correctly exposed.

I'd just like to add a correction re sub f1 lenses:  Lenses with apertures < f1 are perfectly feasible, although I wouldn't say that they're quite 'easy' to achieve[make].  No idea what I was thinking there; apologies.

[lightarrow]

I don't think that chromatic aberration is likely to be an issue with the lens being used by the OP.  As it has an adjustable aperture it's likely to be a fairly high quality compound camera lens which minimises optical aberrations.

I don't understand how the size of the focal spot, given the wavelenght, can depend on focal lenght, in absence of lens aberrations...

[JP]

The f-number is what tells you the size of your focal spot.

I can't quite see this, although I might be misinterpreting what you mean by 'focal spot'.  Are we talking about the size of the focussed image of the sun here?

I mean the diffraction limit on the size of the focused blob of light, here.  I'm not worrying about trying to image anything.  In other words, the central peak of the airy disk made by a circular lens: http://en.wikipedia.org/wiki/Airy_disc

[syhprum (OP)]

LeeE

Although precision lenses of less than F1 are difficult to make if you are only interested in heating Fresnel lenses do fine, with reflectors F .25 is no problem and I think some car headlamps go down a low as F .1

The link I posted at 01/11/2009 21:40:44 answered my question, it is possible to obtain very large intensities with a two stage system.   

[LeeE]

The f-number is what tells you the size of your focal spot.

I can't quite see this, although I might be misinterpreting what you mean by 'focal spot'.  Are we talking about the size of the focussed image of the sun here?

I mean the diffraction limit on the size of the focused blob of light, here.  I'm not worrying about trying to image anything.  In other words, the central peak of the airy disk made by a circular lens: http://en.wikipedia.org/wiki/Airy_disc

Uup! - sorry, yes.  Got it now []

[LeeE]

LeeE

Although precision lenses of less than F1 are difficult to make if you are only interested in heating Fresnel lenses do fine, with reflectors F .25 is no problem and I think some car headlamps go down a low as F .1

The link I posted at 01/11/2009 21:40:44 answered my question, it is possible to obtain very large intensities with a two stage system.   

Good point re lenses aimed solely at heating/illumination; I've been only been thinking about imaging lenses - doh!