  Phase estimation is a major mission in quantum metrology. In the
finite-dimensional Fock space the NOON state ceases to be optimal when the
particle number is fixed yet not equal to the space dimension minus one, and
what is the true optimal state in this case is still undiscovered. Hereby we
present three theorems to answer this question and provide a complete optimal
scheme to realize the ultimate precision limit in practice. These optimal
states reveal an important fact that the space dimension could be treated as a
metrological resource, and the given scheme is particularly useful in scenarios
where weak light or limited particle number is demanded.
