Eigen-function division multiplexed coherent optical transmission in time domain by using higher-order Hermite-Gaussian pulses

Masataka Nakazawa; Masato Yoshida; Toshihiko Hirooka

doi:10.1364/OE.510189

1. Introduction

The total information capacity worldwide has been rapidly increasing at a rate of 40% per year, which indicates that in 20 years we will need a new communication infrastructure that can handle an information capacity 1000 times larger than today [1]. To realize such an ultrahigh capacity network, 3M technology consisting of multi-level modulation, multi-core fiber, and multi-mode control has been intensively studied [2,3,4]. Recently, as a result of combining all these 3M techniques, peta bit/s transmission experiments [5,6] have been reported although the transmission distances were less than 50 km.

Higher-order transverse modes in 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 [7,8]. Fan-in and fan-out devices [9,10] and a multi-input and multi-output (MIMO) technique [11] play important roles in SDM.

It is well-known that the transverse higher-order modes in a gas laser cavity are given by the HG shapes in the spatial domain [12], where the HG modes are special eigenmodes that satisfy the paraxial wave equation or a slowly varying envelope approximation of Maxwell’s wave equation in a cartesian coordinate system [13]. It is also independently well-known that a harmonic oscillator or particle trapping in a parabolic potential well can be described by the Schrödinger equation in the spatial domain, where the solutions (eigen-functions) are given by the HG distribution in a spatial plane [14]. Although the physics behind them are different, it is interesting to note that they have the same HG waveforms with the same orthogonality between eigenmodes in space.

On the other hand, in the time domain approach, when the AM or FM mode-locking equation of the laser that was developed by Haus [15,16] for the generation of an optical pulse train was parabolically approximated in the time and frequency regions, the equation also became a Schrödinger equation. This means that there are HG eigen-solutions in the time domain. Although it was difficult to generate higher-order HG pulses directly from a laser, we recently demonstrated that such pulses can be generated from an FM mode-locked laser [17,18]. When performing the experiment, 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 eigen-functions in the same time slot. This is the key issue in the present paper, where we call this technique the eigen-function division multiplexing (EDM) of higher-order HG pulses and demultiplexing in the time domain. It is also important to note that orthogonality integration can realize both demultiplexing and demodulation simultaneously.

Code division multiplexing (CDM) and the present method are similar in that multiple signals at the same wavelength can be shared in the same time slot, but a fundamental difference from the proposed method is that CDM realized by assigning an individual optical code to each channel uses the code orthogonality rather than the eigen-function orthogonality used in the present case, and therefore demultiplexing is performed simply with code correlation. Although our method can realize complete demultiplexing by using waveform orthogonality, CDM is generally susceptible to interference from other channels, making it difficult to demodulate a signal with a high signal-to-noise ratio.

In addition, in previous studies, the application of HG functions to OFDM subcarriers in a wireless transmission has been proposed and numerically studied [19,20,21], where subcarriers whose shape is given by the HG function are digitally generated and multiplexed. Recently, the use of an HG function in a nonlinear eigenvalue transmission has also been proposed [22,23], where a time-domain waveform is converted to an HG function as a function of the real part of the eigenvalue through an inverse scattering transformation. They are different from the present method where information is directly encoded on HG pulses that are generated, multiplexed and demultiplexed in the optical domain.

It should be noted that the present method is also different from conventional optical time division multiplexing (OTDM), which adopts the interleaving of a pulse where only one pulse exists in one time slot. Even in a Nyquist pulse transmission, although there is a large overlap between adjacent pulses, we use only one pulse in one time slot [24,25,26]. There is Nyquist pulse orthogonality given by the fact that there is output between the Nyquist pulses in the same time slot but there is no output in different time slots. This makes it possible to demultiplex the Nyquist pulse in the different time slot, which we call coherent OTDM Nyquist demultiplexing [27]. However, there are no higher-order Nyquist pulses that have orthogonality between them. In the present paper, it is important to note that there are higher-order HG pulses, which can be superimposed and demultiplexed in the same time slot. This makes EDM possible in the time domain, resulting in a further increase in the maximum transmission capacity even after minimizing the time slot with OTDM. Although HG-EDM in the time domain can thus be combined with an OTDM technique, it will constitute future work.

This paper is composed as follows. In section 2, we describe the fundamentals of the HG pulses and their superimposition in the time domain. In section 3, we propose an EDM technique in the time domain and present numerical analyses of coherent QAM transmission over 450 km, where the time-domain orthogonality is used for demultiplexing the transmitted HG pulses. In section 4, we prove experimentally that the present EDM in the time domain is possible by undertaking 16∼64 QAM coherent EDM transmission experiments over 450 km. Finally, we summarize the work in section 5.

2. Fundamental property of higher-order Hermite-Gaussian pulses

The mth-order HG pulses, a_HGm(t), are well-known as eigen solutions of the following Schrödinger equation,

(1)$$\frac{{{d^2}}}{{d{t^2}}}{a_{HGm}}(t) + ({{\lambda_m} - {t^2}} ){a_{HGm}}(t) = 0.$$

Here, λ_m is the mth eigenvalue, which can be replaced with λ_m = 2m + 1 due to the energy quantization of the harmonic oscillator. The solution is given by

(2)$${a_{HG\textrm{m}}}(t) = {H_m}(t){e^{ - {t^2}/2}},$$

where H_m(t) is called the Hermite polynomial function. It is known that H_m(t) can be generally expressed in the following polynomial form by using the Maclaurin expansion of the generating function [28]. When m is an even number, H_m(t) is expressed in the form

(3)$$ \begin{aligned} H_m(t) & =m ! \sum_{n=0}^{m / 2} \frac{(-1)^n}{n !(m-2 n) !}(2 t)^{m-2 n} \\ & =2^m t^m-\frac{2^{m-2} m !}{(m-2) !} t^{m-2}+\frac{2^{m-4} m !}{2 !(m-4) !} t^{m-4}-\frac{2^{m-6} m !}{3 !(m-6) !} t^{m-6}+\ldots+(-1)^{m / 2} \frac{m !}{\left(\frac{m}{2}\right) !}. \end{aligned} $$

When m is an odd number, H_m(t) is expressed in the form

(4)$$ \begin{aligned} H_m(t) & =m ! \sum_{n=0}^{(m-1) / 2} \frac{(-1)^n}{n !(m-2 n) !}(2 t)^{m-2 n} \\ & =2^m t^m-\frac{2^{m-2} m !}{(m-2) !} t^{m-2}+\frac{2^{m-4} m !}{2 !(m-4) !} t^{m-4}+\frac{2^{m-6} m !}{3 !(m-6) !} t^{m-6}+\ldots+(-1)^{\frac{m-1}{2}} \frac{2 \cdot m !}{\left(\frac{m-1}{2}\right) !} t.\end{aligned} $$

Thus, we obtain

(5)\begin{align}{H_0}(t) &= 1,\end{align}

(6)\begin{align}{H_1}(t) &= 2t,\end{align}

(7)\begin{align}{H_2}(t) &= 4{t^2} - 2,\end{align}

(8)\begin{align}{H_3}(t) &= 8{t^3} - 12t,\end{align}

(9)\begin{align}{H_4}(t) &= 16{t^4} - 48{t^2} + 12,\end{align}

(10)\begin{align}{H_5}(t) &= 32{t^5} - 160{t^3} + 120t,\end{align}

(11)\begin{align}{H_6}(t) &= 64{t^6} - 480{t^4} + 720{t^2} - 120,\end{align}

…

As easily confirmed, we have the following important orthogonality relation between a_HGm(t) and a_HGq(t). When m ≠ q,

(12)$$\int\nolimits_{-\infty }^\infty a_{HGm}(t)a_{HGq}(t)dt = 0$$

or

(13)$$\int\nolimits_{-\infty }^\infty H_m(t)H_q(t)e^{-t^2}dt = 0.$$

This orthogonality plays an important role in demultiplexing a single eigen-solution from the multiplexed (superimposed) HG pulses in the time domain as we will see in sections 3 and 4.

Since the Fourier transformation of the Schrödinger equation of Eq. (1) is also the Schrödinger equation, the Fourier transformation of an HG pulse is also the HG function with a certain coefficient change. That is, the Fourier transformation of ${H_m}(t){e^{ - {t^2}/2}}$ is given by ${H_m}(\omega ){e^{ - {\omega ^2}/2}}$ multiplied by $\sqrt {2\pi } {( - i)^m}$. A factor of (−i)^m means that the spectrum of an HG pulse changes with a coefficient of 1, −i, −1, and i periodically. That is, the spectrum alternates between real and imaginary values. Therefore, when we use the definition ${C_m} = \sqrt {2\pi } {( - i)^m}$, a general HG spectrum can be rewritten as

(14)$${A_{HGm}}(\omega ) = {C_m}{H_m}(\omega ){e^{ - {\omega ^2}/2}}.$$

Figure 1(a)∼(c) show a_HGm(t) and the corresponding A_HGm(ω) for m = 0, 1, and 2, respectively, where the waveforms in the time domain given on the left hand side have the same shape as the corresponding spectral profile given on the right hand side with changes in the ± sign and the real and imaginary values. Similarly, a_HGm(t) and the corresponding A_HGm(ω) for m = 3, 4, and 5 are given in Fig. 2(a)∼(c), respectively. From these figures, we see that the waveform in the time domain alternates between even and odd functions, where the corresponding spectrum of the even function in the time domain is given by a real even function, and that of the odd function by a pure imaginary odd function.

Fig. 1. Hermite-Gaussian waveforms a_HGm(t) in the time domain (left) and the corresponding spectra A_HGm(ω) (right) for m = 0, 1, and 2. (a)∼(c) correspond to m = 0, 1, and 2, respectively.

Abstract

1. Introduction

2. Fundamental property of higher-order Hermite-Gaussian pulses

3. Proposal of eigen-function division multiplexing and demultiplexing in the time domain by using HG pulses

3.1 Numerical model

3.2 Transmission performance of a single higher-order HG pulse

3.3 Nonlinear effects in the time-domain EDM transmission

3.4 Comparison of transmission performance of time-domain EDM with different number of eigen-function and QAM multiplicity

4. Coherent transmission experiments on eigen-function division multiplexed Hermite-Gaussian pulses in the time domain

4.1 Experimental setup for time-domain EDM transmission

4.2 Generation of higher-order HG pulses and their IQ modulation

4.3 Polarization-multiplexed 10 Gbaud, 16 QAM, 4 eigen-function EDM transmission

4.4 Relationship between receiver bandwidth and sampling rate to obtain improved orthogonality integration

4.5 High-performance 16∼64 QAM time-domain EDM transmission by incorporating 256 GSa/s digitization with a 40 GHz bandwidth

5. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (25)

Equations (15)

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