How Far Does The First Amendment Extend?

For the lawyers and potential lawyers among us, I present the following:

Heated words fly in N.M., and legal trouble ensues

New Mexico man is facing the possibility of a year in jail for carrying a picket sign that called a police officer a "liar" and a "dirty cop."

...The case is the most recent in a small but steady stream of prosecutions under state laws, some dating to the 19th century, that criminalize false, damaging statements about officials or private citizens.

Note that the argument here is not about criticizing the government, per se, or an elected official, but a government employee, in this case a police officer.

Now, I know I may be missing something here. I'm pretty sure e..gage knows more about First Amendment law than I do.

Does the state have the authority to limit free speech in this manner? By the same token, how much abuse to our public officials have to accept? I'd like to hear your thoughts.