How far would reflected light penetrate into the interior of a McKendree cylinder from windows at each end

A McKendree cylinder is a rotating space station that is 920 km (580 miles) in diameter, and 4600 km (2900 miles) in length.

This version would orbit the sun at 1 AU (at the same distance as earth). It would be oriented transversely ("sideways") relative to the sun, as seen in the diagram, with large mirrors used to reflect light through transparent windows at each end. (Assume windows are approximately 50% of the diameter of the cylinder itself.)

Is it possible to calculate how far light in the visible portion of the spectrum would penetrate into the cylinder? Would it be dim in the center? Or is it too difficult to tell due to the large number of variables? I'm trying to understand if a setup such as this would run into any issues with lighting the interior.

I should maybe point out that the atmosphere inside the cylinder hugs the walls, so that the axial center is nearly a vacuum. That should help light penetration because there is no air attenuation there.

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How far would reflected light penetrate into the interior of a McKendree cylinder from windows at each end

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