We introduce the multi-dimensional permutation-modulation format in coherent optical communication systems and analyze its performance, focusing on the power efficiency and the spectral efficiency. In the case of four-dimensional (4D) modulation, the polarization-switched quadrature phase-shift keying (PS-QPSK) modulation format and the polarization quadrature-amplitude modulation (POL-QAM) format can be classified into the permutation modulation format. Other than these well-known modulation formats, we find novel modulation formats trading-off between the power efficiency and the spectral efficiency. With the increase in the dimension, the spectral efficiency can more closely approach the channel capacity predicted from the Shannon's theory. We verify these theoretical characteristics through computer simulations of the symbol-error rate (SER) and bit-error rate (BER) performances. For example, the newly-found eight-dimensional (8D) permutation-modulation format can improve the spectral efficiency up to 2.75 bit/s/Hz/pol/channel, while the power penalty against QPSK is about 1 dB at BER=10(-3).
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