Which structure controls how much light passes through the specimen

The diaphragm.

Explanation:

A diaphragm is a thin non transparent structure with an aperture at its center. Aperture is the opening in a lens through which light passes to enter the camera. Diaphragm controls the passage of light through specimen. It stops the passage of light except for the light passing through aperture. It also limits the brightness of light reaching the focal plane.

The diaphragm is placed close to the lens, where objects are defocused to the maximum in order to pass every ray from the object through the lens. Diaphragm discards some of those rays but allows multiple rays to move through to produce an image. This means that the size of the aperture controls the amount of light that passes through the lens. The center of the aperture coincides with optical axis of the lens.  Iris diaphragm is an example. It is used in modern cameras.

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