Chutkan issued the gag order on Monday after a two-hour hearing, but filed the written order today.
She cited court precedents in countering Trumps argument that a gag order is in violation of the First Amendment.
Judge Tanya Chutkan said that her order also will apply to court personnel.

Donald TrumpScott Olson/Getty Images
No other criminal defendant would be allowed to do so, she said.
She said that her order was not about whether or not I like the language Mr. Trump uses.
Instead, she said, the order was about language that presents a danger to the administration of justice.

Violators of the order will be subject to sanctions, she said.
Trump reacted to the gag order by vowing to appeal and writing on Truth Social, WITCH HUNT!
He also wrote, A TERRIBLE THING HAPPENED TO DEMOCRACY TODAY GAG ORDER!
Lauro said, I will advise my client not to make a statement like that.
But he said that he does not think such a statement would be a threat.
Chutkan, though, appeared to be not buying it.
To write in all caps DEATH about a potential witness in this case?
Doesnt that go too far?
Such behavior puts court staff at tremendous risk of harassment, she said.
Lauro tried to argue that the restrictions that Chutkan has already put in place is working.
I am going to have to take issue with that.
At one point she commented to Lauro, You obviously have an audience other than me in mind.
Trump also has repeatedly called Smith deranged, and his office cited a report calling a prosecutor really corrupt!
Prosecutors want an order that pose a substantial likelihood of material prejudice to this case.
Chutkan has scheduled the trial to start on March 4, one day before Super Tuesday.