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Suppose a parabolic mirror of a telescope is large enough (or equivalently that the camera exposure time is short enough) that you need to take into account the transit time of the light signal from hitting the mirror to reaching the focal point.

What shape should the mirror be to have the same effect as a parabolic mirror that doesn't take into account light transit time ?

Added after comments:

Suppose you want to view surface details on an exoplanet with an array of space mirrors spanning millions or even billions of km, also redirecting photons towards a detector at a focal point. The property to preserve is photons from an event on the exoplanet arriving at the focal point at the same time.

If the transit time from hitting the mirror to reaching the focus is long enough then given that objects aren't ideal or at infinity, then any discrepancy at all in arrival times is magnified by the huge transit-time so if the object being viewed changes fast it can change significantly during the discrepancy.

In talking about shutter speed I was thinking about viewing an object whose properties change very quickly (say clouds moving on a giant exoplanet). If there is a discrepancy in the time it takes for photons leaving the object to reach the focal point and the object changes during that time then the image would be blurred (temporally aberrated).

As Ross Millikan points out distances to the focus point should be the same, so I was misunderstanding things, however looking at Wikipedia exact focus only exists for point sources otherwise there is coma aberration. The article also says there is something called bestform or aplantic lenses to minimise this, but you can't apply a lense to an array of mirrors spanning billions of km, so the original question stands but with a changed understanding.

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If I could clarify, you are seeking a shape such that if two photons strike the mirror at different points but at the same time, then they will arrive at the focus at the same time. Is that correct? –  cobaltduck Dec 1 '10 at 12:55
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two questions. (a) Physically, why would you want that? A property of the parabolic mirrors is that photons coming from an ideal source at infinity will arrive at the focal point at the same time; that is, the the different transit times from the mirror to the focal point is exactly balanced out by the different transit times from the plane-wave source to the mirror (b) Mathematically, can you be more precise about what you mean by "the same effect"? What is the precise property of the parabolic mirror you want to preserve? –  Willie Wong Dec 1 '10 at 13:05
    
@Willie Wong: I've expanded the question to address these points. –  Msw Dec 1 '10 at 14:53
    
@Wade: That's what I originally thought, but actually it's about photons that leave the object at the same time, arrive at the focus at the same time. –  Msw Dec 1 '10 at 14:59
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1 Answer

As Willie Wong points out, this is not an issue for a parabolic mirror. In fact, another way to think about optical focus is that it makes constructive interference at the image point. This means that all light rays coming from the subject take the same travel time to reach the focus or else the interference wouldn't be constructive.

Note that shutter speed doesn't matter unless you are thinking about exposing different parts of the sensor at different times (like the old 35mm curtain cameras) and the subject is moving. Focus is focus for any shutter speed.

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@Ross Milikan: In talking about shutter speed I was thinking about viewing an object whose properties change very quickly. If there is a discrepancy in the time it takes for photons leaving the object to reach the focal point and the object changes during that time then the image would be blurred (temporally aberrated). –  Msw Dec 1 '10 at 14:48
    
@Msw: you are missing Willie Wong's point. Long travel time does not affect focus. It is just additive, like the fact that 1001-1000 is the same as 11-10. One definition of a parabola is that the length of rays emanating perpendicular from a far plane to the parabola and then to the focus is the same. –  Ross Millikan Dec 1 '10 at 15:06
    
@Ross Millikan: Suppose you have a 1m diameter mirror and there is a 1 microsecond discrepancy. The discrepancy caused by nonparallel rays travelling slightly different distances (?). If the mirror is enlarged then the slightly different distance is also enlarged hence the time discrepancy is enlarged. –  Msw Dec 1 '10 at 15:26
    
@Msw: Focus has the property that distances are the same. The non-parallel rays still travel the same distance. That is how they "decide" where to focus-to make the distance the same. It is also why interference patterns work with only one photon at a time. –  Ross Millikan Dec 1 '10 at 15:30
    
Ok, but looking at Wikipedia exact focus only exists for point sources otherwise there is coma aberration. –  Msw Dec 1 '10 at 15:42
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