From what I've read, currently, there is no way to determine programmatically that a particular calculation has completed.
The backfill information is internal to the analysis engine and PSE can access that data, however, that data has not been exposed for general consumption.
Can you provide insight as to if, and if so, when this functionality will be available?
You can use windows performance counter to check runtime status but will not provide any other details. You need to check analysis logs recalculation logs for pending ones.
Performance counters :
PI Analysis Service_Recalculation Requests Completed
PI Analysis Service_Recalculation Requests Queued
Next version : PI AF 2018 SP2 will be released soon and will have analysis runtime information methods.
As Lal suggested you can look at the Performance Counters or you could even read from this file: %PROGRAMDATA%\OSIsoft\PIAnalysisNotifications\RecalculationProcessor.RequestedRecalculations.xml
Don't write to the file as this would prevent backfills/recalculations to work.
I'm not sure if the the backfill/recalc status are exposed in the AF SDK in 2018 SP2.