Mode Division Multiplexed Coherent Optical Transmission in Time Domain by Using Higher-order Hermite-Gaussian Pulses

We propose a new mode-division-multiplexing (MDM) technique in time-domain using higher-order Hermite-Gaussian pulses. 32-QAM, 450-km MDM transmission was successfully demonstrated with HG<inf>0</inf>, HG<inf>1</inf>, HG<inf>2</inf>, and HG<inf>3</inf> pulses, where the time-domain orthogonality was used for demultiplexing.


Introduction
Higher-order transverse modes in a few-mode fiber, for example LP11, LP21 and LP02 modes, have been successfully transmitted with the aim of increasing the transmission capacity in the spatial domain as a space division multiplexing (SDM) technique [1], [2].However, although the space (wavenumber and position) and the time (frequency and time) are interchangeable, there has yet to be a report on a mode-division-multiplexing (MDM) technique in the time domain where there are many different pulses (modes) in the same time slot.In the present paper, we propose and demonstrate an MDM coherent transmission in the time domain with the use of the Hermite-Gaussian (HG) eigenmodes.
So far, it has been difficult to generate higher-order HG pulses directly from a laser, but we recently demonstrated that such pulses can be generated from an FM mode-locked laser by incorporating an FM modulation function into the master equation [3], [4].When performing the experiment on the HG pulse generation, we considered that since these HG pulses are also orthogonal to each other in the time domain, we may be able to use this HG pulse orthogonality for demultiplexing many different HG modes in the same time slot.It should be noted that the present method is different from conventional optical time division multiplexing (OTDM), which adopts the interleaving of a pulse where only one pulse exists in one time slot.
In this paper, we demonstrate a new MDM technique in the time domain by carrying out 32 QAM multi-mode Hermite-Gaussian coherent pulse transmission over 450 km, where the time-domain orthogonality of HG pulses was used for demultiplexing the transmitted HG pulses.

Fundamental Property of Higher-order Hermite-Gaussian Pulses
The mth-order HG pulses, aHGm(t), are well-known as eigen solutions of the following Schrödinger equation, Here, m is the mth eigenvalue, which can be replaced with m = 2m+1 due to the energy quantization of the harmonic oscillator.The solution is given by 2 As easily confirmed, we have the following important orthogonality relation between aHGm(t) and aHGq(t).When m ≠ q, ( ) ( ) 0 HGm HGq a t a t dt This orthogonality plays a very important role in demultiplexing a single eigenmode from the multiplexed (superimposed) HG pulses in the time domain.
When we use the definition Cm = 2 ( )   m i , a general HG spectrum can be written as First, we numerically describe the propagation of a single HGm pulse.Figure 1 shows the waveforms (a-1)~(a-4), corresponding spectra (b-1)~(b-4), and constellations (c-1)~(c-4) of 16 QAM signals after a 450 km transmission when HG0, HG1, HG2, and HG3 pulses are transmitted individually in a single channel.Here, the simulation was performed using the same parameters as those in the experiment presented below.The red and black curves are analogue HG waveforms after transmission and the theoretical input HG waveforms, respectively.We can see clean time-domain HG pulses, the corresponding spectrum, and QAM constellations.Here, the average input power was optimized for each HG pulse to minimize the error vector magnitude (EVM).

Coherent QAM Transmission Experiment and Result on Mode-division Multiplexed Hermite-Gaussian Pulses in the Time Domain
The experimental setup for a 4-mode (HG0~HG3) MDM transmission over 450 km is shown in Fig. 2. In the present experiment, we used a combination consisting of a comb generator and an LCoS filter to generate various HG pulses.An external-cavity laser-diode (ECLD) with an 8 kHz linewidth [5] and a 10 GHz optical comb generator [6] were used as a pulse source, and the output signal was split into 5 arms.The wavelength of the ECLD was 1550 nm.Four LCoS filters and IQ modulators were used simultaneously to generate QAMmodulated HG0~HG3 pulses.The IQ modulators were driven by arbitrary waveform generators 1 and 2 (AWG1 and 2).These signals were combined after matching all the pulse timing with appropriate optical delay lines, and then pol.mux.operation was achieved by using a polarization division multiplexer.The 9th harmonic of the optical comb signal, which was 90 GHz down-shifted from the center frequency of the signal, was simultaneously extracted with a narrow optical filter and used as a pilot tone for the optical phase-locking of the LO HG pulse.
The transmission line we used consisted of six 75 km spans with a 50 km super large area (SLA) fiber (dispersion: 20 ps/nm/km, dispersion slope: 0.07 ps/nm 2 /km) and a 25 km inverse dispersion fiber (IDF, dispersion: -40 ps/nm/km, dispersion slope: -0.14 ps/nm 2 /km) so that the second-and third-order dispersions were compensated  for simultaneously.The average loss was 17 dB/span including a splicing loss of 2 dB, which was compensated for with an EDFA.At the receiver, the transmitted signal was split into 3 arms, where the MDM, pilot tone, and 10 GHz clock signals, respectively, were extracted.The extracted MDM signal was input into a coherent receiver consisting of a 90-degree hybrid and a balanced photo detector (B-PD) after polarization demultiplexing.We used the injectionlocked LD and a comb generator to generate phase-locked HGm pulses.The time-domain MDM signal was homodyne-detected with the LO HG pulse.Then, the detected signal was A/D-converted by a digital oscilloscope and demodulated by using a digital signal processor (DSP).In the DSP, the MDM signal was demultiplexed by calculating the overlap integral given by eq.(4-1), and the demultiplexed QAM signal was demodulated with a vector signal analyzer (VSA) software [7].
The intensity waveforms of HG0 ~ HG3 pulses generated by the comb generators are shown in Fig. 3. (a-1) ~ (d-1), respectively, where an optical sampling oscilloscope (OSO) with an 800-fs resolution was used for the measurement.It should be noted that the phase of the HG pulse is inverted each time when the waveform crosses zero, but the OSO has a positive value since it detects the signal intensity.The blue dashed lines correspond to the theoretical curves, which agree well with the experimental curves shown by the black lines.It can be seen that highquality HG pulses were successfully prepared.
Figures 4 and 5 show the demodulation performance of polarization-multiplexed 10 Gbaud, 32 QAM, 4-mode MDM (400 Gbit/s) transmission under a back-to-back condition and after a 450 km transmission, respectively.It is clearly seen that we could achieve error-free MDM transmission in the time domain by employing a 7 % overhead FEC threshold (2 x 10 -3 ) for all HG pulses even after a 450 km transmission.There is no difference between the BER degradation with different HG mode.