The phrase "alarmed by how" is correct and usable in written English.
It can be used when expressing concern or surprise about a particular situation or behavior.
Example: "I was alarmed by how quickly the situation escalated."
Alternatives: "worried about how" or "concerned by how".
Many fathers have signed the petition, alarmed by how disturbing this is for their daughters.
The K.D.H.E. was alarmed by how many people said they planned to stay.
Already alarmed by how well Facebook's adverts seem to know you?
I don't have a clear answer, but I'm alarmed by how much the capping-deductions option is catching on.
"They're hurting you, aren't they," she said quickly, alarmed by how ill-looking and thin I'd become.
These inventors are alarmed by how easy it is to wind up with mismatched pairs of socks.
Probably because: –You're alarmed by how much you have to spend on gas.
To see how that this is so, just listen to then-Senator Joe Biden in 2006 attack the NSA for collecting phone records: he does criticize the program for lacking FISA court supervision (which wasn't actually true), but also claims to be alarmed by just how invasive and privacy-destroying that sort of bulk record collection is.
Instructors were especially alarmed by his request to learn how to fly large jet aircraft but not how to take off or land.
It is thought that one of the reasons Greece's lenders are downplaying the possibility of Athens producing a sizable surplus is that they are alarmed by the debate in Greece about how this amount will be allocated and whether social spending could be increased.
Just think of how many parents, alarmed by the bogus link between vaccines and autism, have left their children unprotected against serious disease.
