Flight Crew Air Transport Pilot (ATP)
Confusion; Fatigue
Deviation / Discrepancy - Procedural Clearance; Ground Incursion Taxiway
Air Traffic Control Issued New Clearance; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Became Reoriented
Airport; Human Factors
We we cleared to Line up and wait on Runway 1 and the taxi line and lights made our turn onto J1 very obvious and the only choice.  I looked right into what seemed like a black hole and it appeared to me that this taxiway was the end of the runway; however as soon as we took the runway it became clear that there was still another taxi way at the very end of Runway 1 which I did not see!  We both had our Commercial airport diagram out at the time.  I quickly contacted Tower and they stated 'no problem' and we cleared the runway at taxiway J so as not to be rushed.  The Tower had no issue as they were very busy with arrivals.  We did a 180 turn on Taxiway J and waited for takeoff clearance without further issue.  The third leg of a very long day and the visibility and lack of lighting to taxiway J were poor to nonexistent.  The airport lighting in this area was very poor and even though I was looking towards Taxiway J; it looked like a dead end.  The obvious choice was Taxiway J1 which seemed like the only path to the runway end.  Tower did not specify which taxiway to take and had they stated J specifically; we would have stopped and not entered at J1.  I was unfamiliar with the airport and can see how easily this mistake was made.
B737 First Officer reports confusing Taxiway J1 and J at the approach end of Runway 1 at DCA.  Reporter cited fatigue and poor lighting as factors and resulted in entering the runway at J1 instead of J.