How do submarine periscopes work without using mirrors?

For many decades, Submarine optical periscopes have used prisms instead of mirrors, for both reflection and magnification. The simplest way to describe it is 2 telescopes pointed at each other. Prisms are used instead of mirrors because they are more rugged, and unlike mirrors, do not require special coatings to keep them from being impaired by environmental factors.

Traditional periscope masts are very long, over 30' in most cases. They are raised and lowered using the ship's main hydraulic systems. All modern submarines have at least 2 periscopes - one for attack (low profile) and the main periscope, which typically has many other functions built into it. Something that's not well known is that the Navigator can hook a Sextant to a Periscope and get a sunline or star fix if the need were to arise.

More recent newer classes of Submarines (e.g., U.S. Virginia and British Astute classes) use a device called a Photonics Mast which replaces the old optical type periscope system. Since it doesn't have to be contained within the pressure hull or raised/lowered from the Control Room like a traditional scope, it means the Control Room can be located in an area not directly under the sail. Photonics masts are essentially digital camera arrays, which have increased functionality over their optical counterparts, and of course since they don't penetrate the hull, if damaged the chance of a flooding casualty is much less.