This is a great question. By the process of elimination, we can say that the best two answers are 18th and 19th centuries. The Renaissance did not produce too much technology, which brought immigrants to Europe. Moreover, transportation was still rather primitive. The Reformation was more a religious movement. To be sure, there were some very important innovations such as the movable type face, but it did again not bring too much immigration.
The 18th century brought great growth to Europe, particularly England, through the beginning of the industrial revolution. The population of England and Europe began to grow. This would continue and accelerate in the 19th century. There were great advancements in the steam engine, internal combustion engine, and the electric engine. These made productivity great and these in turn created large factories where people were needed.
Also in the 19th century, Europeans would come in droves to America searching for new opportunities.