Where did the police badge originate? How did it evolve?
I haven’t tried to research this very much, but I don’t want the question to be lost below the fold (nothing apepared in the archives). Hopefully, others will opo in with more detailed info, or correct me if I lead you astray 
The first modern police force, as we know it today, originated in London in the 1820’s. The model was adopted in big cities in the United States shortly thereafter. New York established a police force in 1844, with Boston and Philadelphia soon afterwards. Police forces were locally organized, so there were many variations in uniforms. Different emblems or insignia were used to identify the various ranks - buttons, shields, stars, hats, et al. The United Stats’ oldest badge manufacturing company - Blackinton - began operations in 1852 in Massachusetts, http://www.blackinton.com Unfortunately, their web page doesn’t provide a history of their company. I was able to corroborate that they were the first, though.
So to answer your question, badges probably originated in some form between the mid 1840’s and 1850 in the Northeastern US, likely New York. (I couldn’t find any info that indicated the use of badges in England - it seems they used buttons?) In checking, it seems that Chicago did not use badges until the 1880’s (again, could be wrong, though). I know it’s not much, but it’s the best I could dig up at this hour
As to evolution … there are dozens of books on the subject. Also, there is quite a community of police memorabilia collectors. Run a google search and start visiting some sites.
FWIW the English coppers don’t have badges, they have laminated plastic ID’s called Warrant Cards.
I always assumed they simply evolved from the tradition of soldiers carrying coats of arms. The badges are just little coats of arms. That’s why the slang for badge is “shield,” after all.