The given question asks about the Mughal Emperor known for popularizing the pietra dura (stone inlay technique) and the double dome in Indian architecture.

Let's analyze the options one by one:

1. Babur: The founder of the Mughal Empire in India, Babur was known more for his military conquests and establishing the empire rather than his contributions to architecture.
2. Akbar: Akbar, one of the most famous Mughal emperors, did innovate in architecture by blending Persian, Indian, and Central Asian styles, but he wasn't specifically associated with the pietra dura technique or the double dome.
3. Shah Jahan: Shah Jahan is the most renowned Mughal Emperor when it comes to architecture. He built many famous structures including the Taj Mahal, where pietra dura (stone inlay) was extensively used. The technique involves creating artwork by inlaying semi-precious stones into marble, which was prominently used in the Taj Mahal. He is also known for popularizing the use of the double dome, as seen in his constructions. Thus, Shah Jahan is the correct answer.
4. Aurangzeb: Aurangzeb's reign was marked by military expansion and orthodox policies. His contributions to architecture were not as significant as Shah Jahan's.

Based on this analysis, the correct answer is Shah Jahan.

Conclusion: Shah Jahan is the Mughal Emperor known for popularizing the pietra dura, or stone inlay technique, and the double dome in Indian architecture, with the most famous example being the Taj Mahal.