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Let μ_M,D be a self-affine measure generated by an expanding real matrix M=(arraycca&e f&barray ) and the digit set D= {(0,0)^t, (1,0)^t, (0, 1)^t, (1, 1)^t}. In this paper, we consider that when does L^2(μ_M,D ) admit an infinite orthogonal set of exponential functions? Moreover, we obtain that if e=f=0 and a, b{pq,p,q2Z+1}, then there exist at most 4 mutually orthogonal exponential functions in L^2(μ_M,D ), and the number 4 is the best possible. , number=3, journal=Journal of Mathematical Study, year=2022, month=Sept., pages=327–336