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Imagine the sliding part of the mirror is controlled by computer and opens on intervals.

Is it possible to increase the power of the beam by making it bounce between the mirrors thus going through the lens and then releasing it resulting in beam with more power ?

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    $\begingroup$ Isn't that what a telescope does? Don't point one to the sun though, it will end badly. $\endgroup$ – John Alexiou Dec 25 '13 at 22:11
  • $\begingroup$ People has been using this technology to build telescope to collect the light from the stars. $\endgroup$ – Everett You Dec 25 '13 at 22:13
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    $\begingroup$ You mean a catadioptric telescope with a lens in the middle? I don't see the direct benefit of placing a lens in the middle. $\endgroup$ – jinawee Dec 25 '13 at 23:16
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    $\begingroup$ That's a pretty cool gif. $\endgroup$ – Kyle Kanos Dec 26 '13 at 2:48
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    $\begingroup$ Lenses do not increase power. Also, if that first mirror is particularly reflective, it will reflect the same from both sides. You cannot create a mirror that's only reflective from one direction. $\endgroup$ – Colin K Dec 26 '13 at 3:36
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Is it possible to focus the sun in such way?

Yes, as others have pointed out, all of the ideas in your sketch are already used in existing designs - perhaps excepting the shutter (which actually performs no useful purpose so far as I can see).

As Chris White commented - "this exact design (with the shutter permanently open) is a Schmidt-Cassegrain telescope, probably the most popular high-end consumer scope these days."

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Is it possible to increase the power of the beam by making it bounce between the mirrors

No, focusing a beam of light, or reflecting it, does not increase the power. The amount of power is the amount of light energy entering the system per second. That is limited by the diameter of the entry pupil. Energy is conserved.

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    $\begingroup$ Perhaps worth editing into this answer to make it 100% complete - this exact design (with the shutter permanently open) is a Schmidt-Cassegrain telescope, probably the most popular high-end consumer scope these days. $\endgroup$ – user10851 Dec 26 '13 at 8:59
  • $\begingroup$ @Chris: Thanks, I've edited your comment in with attribution, I hope that's OK. If not, please make any edits/tweaks (or suggestions) you see fit. $\endgroup$ – RedGrittyBrick Dec 26 '13 at 11:50
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Fun fact: You cannot focus the beams from the sun create a hotter area than the surface of the sun itself. This would break the second law of thermodynamics. 6000 °C is hot enough for many spectacular destructive applications, though, a youtube search on "melting steel with sun light" offers hours of entertainment.

If you want to make something that potentially focuses more energy than the surface of the sun, head for a solar pumped laser.

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