The foreign traveller who extensively discussed diamonds and diamond mines of India was Jean Baptiste Tavernier. He was a French explorer and trader who traveled to India during the 17th century and spent a considerable amount of time studying and trading diamonds.

Tavernier`s observations and descriptions of the diamond mines in India are detailed in his book "Travels in India" which he published in the mid-17th century. In this book, he gave an account of the diamond mines in Golconda, which he visited several times during his travels. He also described the methods of mining diamonds, the process of sorting and grading them, and the diamond trade in India.

Tavernier`s knowledge of diamonds and diamond mines in India was highly regarded by the rulers of his time. In fact, he was invited to meet Emperor Shah Jahan, who commissioned the famous Taj Mahal. Tavernier sold many of the diamonds he acquired in India to the Mughal emperors and other wealthy individuals in Europe.

In contrast, Francois Bernier was a French physician and traveler who visited India during the 17th century, but his focus was mainly on the culture, geography, and politics of the region. Jean de Thevenot was also a French traveler who visited India during the 17th century, but his interests were more diverse and included natural history, languages, and customs. Abbe Barthel Carre was a French missionary who traveled to India during the 18th century, but there is no evidence to suggest that he discussed diamonds or diamond mines in his writings.

In conclusion, the correct answer to the question is Option 2: Jean Baptiste Tavernier, who provided extensive details on diamonds and diamond mines in his book "Travels in India".