At 300mm focal length at a distance of 200 yards away what F stop will give me edge to edge sharpness in my photos?
The edge-to-edge sharpness of your lens is a function of the optical design much more than f-stop. What I would be more concerned with is depth of field. The important consideration is what the closest and farthest point you want to be in focus. Here are some figures that might help. For a 300mm lens focused at 200 yards (600 feet), mathematical calculations yield the following (0.03 mm circle of confusion in all cases):
f/22
Near limit = 256' 3", far limit = infinity, total depth = infinity, hyperfocal distance = 447' 5"
F/16
Near limit = 303' 9", far limit = 24,350' 8", total depth = 24,046' 11", hyperfocal distance = 615' 2"
f/11
Near limit = 359' 2", far limit = 1,821' 3", total depth = 1,462' 1", hyperfocal distance = 894' 9"
f/8
Near limit = 403' 4", far limit = 1,172' 2", total depth = 767' 10", hyperfocal distance = 1,230' 4"
f/5.6
Near limit = 447' 4", far limit = 910' 12", total depth = 463' 8", hyperfocal distance = 1,757' 7"
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