no need for photo documentation. call the Sheriff's office and tell them you need to make a report for property damage to livestock. Be sure to have them include in the report that it has happened before. They will talk to the owner of the dog and inform them of YOUR rights and also give them a bill for damages. Be sure not only to include the birds killed, but any expenses for replacing fencing or other damages from the dog.
Here, there is no leash law so it is legal for dogs to roam where ever. However, it is also legal for me to shoot them if I have good reason to believe that they are a danger to livestock. That's why I always file a report and include a description of the dog. That way, I don't have to wait until they start killing birds to shoot them. Of course, my neighbor down the street has killed about 20 dogs in the time I've lived here, so most of them don't show up more than once at my place. Now if we could just get the idiots to stop replacing them with more dogs then we might make some progress.
Be careful though. What you did (following the dog and then continuing to beat it) could get you at least a warning for animal cruelty. Next time, just shoot the dog and be done with it.