given this post has gotten some interest without a response, lets see if i can summon potential flaired users like /u/DelayedTrain or /u//u/PaxOttomanica
I tried not to make a top comment since I believe my knowledge on the issue is inadequate, but since the question didn't get any responses at all, maybe my poor comment could be tolerable.
Although the Ottoman Empire's response to Napoleon 'after' the French revolution is asked in the question, here's a couple of fun facts about Napoleon and the Turks:
According to his own memoirs, as a young artillery officer, Napoleon was about to be employed by the Ottoman Empire as an army reformer. Napoleon managed to prove himself 'worthy of keeping' to his higher-ups 'right before' he set sail for Istanbul.
As an officer, Napoleon invaded Egypt which was the Ottoman Empire's territory at the time. After the invasion, he ruled Egypt as the absolute ruler of a newly founded state, and due to his policies towards popular support, he was widely revered by the Egyptian masses. Among his policies, probably the most prominent one was radicalizing the Mamluks (which were ruling the land in the Ottoman Empire's name. This Turkish speaking elite has a very interesting history and we can discuss them more under this thread if interested) and promoting the French as staunch and historical ally of the Caliph in Istanbul, of the Ottoman Empire (with historical references to Francis I and Suleyman the Magnificent) and Islam. As a matter of fact, he frequently promoted himself as a muslim and the French as the defenders of Islam. According to his companions, this was nothing but a strategy to deceive the masses, though. Napoleon himself didn't really believe in any religion, but he strongly believed in usefulness of religions for properly ruling over masses.
When he went back to France after his defeat before the Turks and the British, he brought Mamluk cavalries with himself and formed a light cavalry military class based on their fighting styles and tactics. They would be another Turkish influenced part of European armies, which were a lot especially in the French army.
For the French Revolution, obviously enough, the Ottoman Empire read the incident as an "anti-monarchic movement" (they initially could not realize the French Revolution would fuel nationalism and eventually collapse their multi-ethnic multi-religious empire) and did not recognize the government in France. Same goes for Napoleon, despite repeated demands from Napoleon himself (one of his letters read as "Most high...and invincible prince, the great Emperor of the Muslims, Sultan Selim, in whom all honor and virtue abound... You, descendant of great Ottomans, emperor of one of the greatest empires in the world, do you cease to reign? Why do you suffer the Russians to dictate to you? If Russia has 15,000 men at Corfu, do you think that they are there against me? Armed vessels have the habit of hastening to Constantinople. Your dynasty is about to descend into oblivion... Trust only your true friend, France"), Selim III refused to recognize the 'usurper' as the legitimate French ruler, until his tremendous victory in Ulm and Austerlitz (though some claim Selim III wanted to recognize Napoleon earlier, but was afraid of a confrontation against the British in the sea and the Russians on the land. Though there is truth in such claims as the Ottoman Empire at that time was hardly a match against a united front of Russian and British empires, there's no reason for Selim III to favor the French as it was Napoleon himself who invaded Egypt just a couple of years ago, during Selim III's reign). After that, Selim III did not ratify the alliance treaties with the British and the Russians, and recognized Napoleon as the French emperor which caused the British and the Russians to wage war against the Empire.
After the Crimean War, the Ottoman Empire also joined the Concert of Europe which was formed after the Napoleonic Wars.
Just like almost every other country in Europe, Napoleon's successes created a trend of imitating his reforms and strategies in the Ottoman Empire too. Mahmud II finally managed to get rid of troublesome Janissaries which were the most important obstacle before enacting effective reforms, and introduced conscription along with a great deal of other "Western" or "Napoleonic" reforms.
In the Ottoman Empire (and in most Turkic/Islamic states in general, actually), anyone had the chance to climb up the ladders of social strata, and a slave had the chance to become a high ranking general one day. However, Napoleon's rise to throne and his successes as a "one man army" inspired young Turkish officers greatly after Napoleon became a legend (which, I believe, happened somewhere after the Crimean War/establishment of the Second French Empire). The Young Turks organisation, and then Committee of Union and Progress were all founded according to the principles of the French Revolution, and were influenced by the ways of Napoleon (and also Robespierre, as the later CUP coup shows). Even their motto "Hurriyet, Musavat, Uhuvvet, Adalet!" was a translation of "Liberte, Egalite, Fraternite!" with the addition of "Justice!".
One can even evaluate that Mustafa Kemal's rise against invading powers (which three of four were the Great Powers the British Empire, France and Italy) and the Ottoman Sultan (though this is rather controversial. There is convincing information that Mustafa Kemal was initially sent by the Sultan himself in order to organise an armed resistance, as he was already experienced in Libya fighting in same fashion against the Italian army); and his victory over these much powerful nations armies are inspired by Napoleon's rise to throne and victories prior to Leipzig.