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I noticed that human body conducts red visible light quite well. A hand placed over a powerful red lamp seems to be semi-transparent while over any other color it does not.

Possibly the light conductivity in near infra-red range is even better.

So my question is why (infra-)red light is not used for diagnostics instead of harmful X-rays?

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Although IR light penetrates tissue much more easily than visible (much lower absorption), it is still scattered quite quickly (within a few hundred microns from the surface). This makes it impossible to form an image with it in the same fashion as with x-rays. However, it does find use in medicine with techniques such as optical coherence tomography and two-photon microscopy.

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Also in pulse-oxygenation monitors. –  Colin K Jan 16 '12 at 3:46

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