The second caliph is Abu Jafar :)
The Muslim Empire established by second Rashidoon Caliph Hazrat Umar RAU, the Ummayad Caliphate, The Abbasid Caliphate in Baghdad, and the Ummayad Caliphate in Spain
The Muslim Empire in Baghdad reached its climax during the reign of Caliph Haroon ur Rashid. The Muslim Empire in Spain reached its climax during the reign of Abdul Rahman III. It took about less than 150 years .
It was during the reign of second Rightful Caliph Hazrat Umar r.u.a.
When the Abbaside were in power the greatest Arab empire was the Abbasid Caliphate (750-1258). It was the second largest Muslim (not just Arab) empire in history, second only to the Umayyad Caliphate (661--750) which preceded it.
The Muslim rulers after prophet Muhammad death in year 632 AD were called Caliphs. The first Caliph was Abou Bakr. The Caliph after death of the Caliph Abou Bakr was Omar Ibn Al-Khattab. That is why he was the second Caliph.
The first Muslim Empire was located in Saudi Arabia Under the second Rightful Caliph Hazrat Umar (May Allah be pleased with him). Its capital was Madina Munawara. Later Muslim Empires had their capitals at Kufa, Baghdad, Cordova, Istambol.
The Muslims, the Companions of Prophet Hazrat Muhammad SAW , elected Hazrat Abu Bakr RAU their Ruler, the First Rightful Caliph. He was succeeded by Hazrat Omar RAU who founded the Muslim Empire.
Answer: The second Caliph Omar ibn Al-Khattab.
It may be debatable, however, considering the spiritual and worldly achievements together, we can safely say that the Muslim Empire under the rule of second righteous Caliph Hazrat Umar Farooq (RAU) made the greatest achievements.
The Muslims had to face the enmity of the neighboring countries. They conquered them one by one. In this way, the Muslim State became Muslim Empire during the reign of the Second Rashidoon Caliph Hazrat Omar (RA).
Caliph Omar/Omer/Umar (RA), the second Rashidun Khalifa or Rightly-Guided Caliph, reigned from August 23, 634 to November 7, 644, when he was killed by a Persian prisoner.
There is no Islamic empire. Islam is a religion not a physical entity. The first Islamic Empire was established in the reign of second Rightful Caliph Hazrat Umar (May Allah be pleased with him) with its capital at Madina Munawara-Saudi Arabia. Later Muslim Empires were located in Iraq, Spain, and Turkey.