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I should start out by saying... I'm really wondering why you want an 800mm focal length. Not many photographers would use this and more focal length is not necessarily better. It comes with a lot of limitations. The most popular focal length range for photographers who want "long" lenses tends to be the various 150-600mm zoom lenses.
You'll find that what you are looking for at the $250 budget doesn't exist... but there's a good reason for it.
Lenses have to maintain something called a "focal ratio". The ratio is simply the focal length of the lens (in your case 800mm) vs. the diameter of clear aperture (the area where light can pass through and into the camera.)
For example, if we had a lens with a 100mm focal length, but the largest possible clear aperture in that lens was 25mm across, then that lens has a ratio of 1:4 but it's usually written as f/4,
They can't effectively just increase the focal length to 800 and do nothing with the aperture size... because a longer lens at 800mm without changing the aperture size (leaving it at 25mm) would result in a best case focal ratio of f/16... which doesn't allow much light to pass into the camera (it could only be used in maybe sunny daytime conditions).
About this time, you might be thinking... just make the lens have a bigger diameter. It turns out that's basically what the camera company has to do... so to maintain (say that f/4) focal ratio, they'll scale up the diameter. But now each piece of otptical glass is 4x wider... but it's ALSO a lot thicker.
This thickness creates a new problem... light "bends" when it passes through the glass (thats how the lens focuses the image). It turns out different "wavelengths" of light (colors of light) bend by a different amount. "blue" light bends more than "green" light which bends more than "red" light. This creates a new problem where not all the colors of light come into focus at the same distance and you end up with an poor quality image.
This problem of the lens acting like a prism and splitting the light up into different wavelengths is called "dispersion" and it results in an image in which objects have color fringing ... they call that "chromatic aberration" (often abbreviated as "CA").
So to fix this, the camera maker has to do a few things.
#1 they try to find types of "glass" that still have excellent refractive properties... but minimize the "dispersion" problem.
These low-dispersion types of glass are sometimes made out of exoctic materials (fluorite crystal is popular) which drive up the cost. But what really drives up the cost is that you can't find fluorite crystal in a pure enough form in nature to produce a lens. But you can "grow" crystals in a kiln. But it takes MONTHS to grow a crytal large enough to produce a lens. If you try to speed up the process, you end up with optical imperfects imbedded within the crystal structure which makes it worthless for use as an optical lens. So this has to be done slowly.
#2 they add in even more glass elements. As the initial lens element starts to focus the light (but causes optical problems), additional glass elements in the lens can try to re-merge the wavelengths back together again. But this means you have a lens with many more elements than the smaller size lens.
What you're left with is a combination of an extra big lens... made with exoctic glasses which take a very long time to produce AND you need many more of these elements. The lenses become very expensive and that creates an economic problem becuase all the equipment/machinery, factor floor space, people, etc. are producing the more common lenses for millions of customers... these massive lenses have even more expensive machinery but are only producing the lenses for a handful of customers. That means a much larger chunk of the cost of overhead has to go into the price of each lens... making them even MORE expensive.
Canon's 800mm f/5.6 lens is.... $12,999.00!! (no kidding)
So that's obviously not in the budget.
Sigma makes a 300-800mm f/5.6 zoom lens which has the bargain price of merely $7,999.00. That's a $5000 savings off the price of the Canon lens... but still obviously out of budget.
You're not necessarily stuck without any options. There are some alternatives.
One option is to change your requirements.... the most common "long" focal length lenses tend to be 150-600mm zoom lenses. Both Sigma and Tamron make these lenses... the "cheap" ones tend to be around $800... the "expensive" ones tend to be around $2000. That's over your budget (but a LOT less expensive than the 800mm lenses.)
There is a category of lens commonly called a "mirror lens" because of the optical design. The lens primarily focuses the light using a spherical mirror at the back of the lens (with a hole in the middle) and this focuses light inward (and back to the front), where it hits yet another mirror... that further focuses the light and sends it back through that hole and into the camera. There is a single glass element called a "corrector plate". Since the lens has 1 "glass" lens element and 2 "mirror" elements, it's called a "compound" lens (you'll find the word "catadioptric" which just means it uses both lenses and mirrors to focus the light).
These things are shockingly cheap ... but they come with limits.
1. They must be MANUALLY focused (there is no auto-focus).
2. You cannot adjust the aperture... it's a fixed aperture lens.
I am not specifically aware of any of these with a focal length above 500mm... not in a camera lens design anyway.
But the advatnage is these things tend to be in the $100-250 price range... so NOW we're in a category that fits in your budget (but it's not 800mm and it's manual and you can't control the aperture.)
You can buy a catadiatropic telescope spotting scope and attach a camera to it. But these will likely be even more than 800mm (e.g. a Celestron C90 is in your price range but it's 1250mm)
BTW, at 800mm you absolutely must use a tripod. You'd never be able to hold the beast steady enough to get a clean shot that didn't have blur caused by camera movement during the shot.
I would not buy that lens. It is a cat lens. It is manual everything. Unless you know what you are doing this lens will be difficult to use.
However it can be done. Lots of folks do use them. IQ will be poor but that is almost a given with its price point.
Save your money for one of the Sigma or Tamron 150-600mm super zooms. Even look used if money is too tight.
If you just want a prime look at the Canon EF 400mm f/5.6L USM Lens. It is light-years better than the one you are considering. Fantastic IQ and auto everything. It is a super popular lens so the used market on it is great. Way better idea.
That's absolutely true... but they are cheap. So if you wanted the cheapest way to get to 500mm to image distant objects... it would be an inexpensive option.
My other concern is that at high focal lengths, if an object in flight (such as a bird or plane) gets out of the frame... it can be very difficult to get it back in the frame fast enough (before it's gone). Sometimes a long focal length lens is not the best option.
Considering these mirror lenses are completely manual focus... that would be especially challenging. It would work for the moon because the moon isn't racing across the sky and you can take all the time you need to focus. For an airplane... you'd be trying to track the plane and keep in the frame of view WHILE trying to focus at the same time and THEN try to grab a photo before it's gone. That's quite a challenge and likely to result in mostly a lot of out of focus or missed shots where the plane wasn't in the frame.
"My other concern is that..."
The extremely ugly bokeh. Its great if you love donuts!
IMHO, absolutely a poor option, except for the price it has nothing going for it. The Bower 500mm f/8 Manual Focus Telephoto T-Mount Lens is, again, light-years better choice. Best choice is to find a used, even a very used, ugly rating, Canon EF 400mm f/5.6L USM Lens. You will be well better off.
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