This part of the Ottoman Empire refers to how they affected culture of other societies, rather than their own culture.
The Ottoman Empire was culturally and religiously tolerant. After conquering an area, they took in other cultures and allowed the people in the millet system to maintain there traditions. By taking in these cultures, the Ottoman expanded their own culture, and further expanded other cultures, and so on.
One of the biggest effects on other cultures in the world came from The Ottoman Empire's siege on Constantinople. When the city was captured, it ended the Byzantine's 1100 year reign. The Ottoman's siege on Constantinople caused many scholars to flee to Italy. Because of this, the Renaissance era was brought along with ease. Additionally, the scholar's flee increased trade relations in the east.
The capture of Constantinople nearly started an additional crusade, aiming to regain control of the city. The Pope at the time had tried to get an army together, but was unsuccessful.