As data-traffic demands grow in areas such as the Internet of Things and artificial intelligence, it is important to increase the spectral efficiencies (SEs) of optical-fiber transmission systems [1, 2]. With the advent of coherent optical (CO) communication systems [3, 4], the SEs of optical communication systems have grown remarkably, greater than 10 bit s⁻¹ Hz⁻¹. The CO communication systems can discriminate the differences in polarizations, amplitudes, frequencies, and phases of received optical signals.

The SE can be enhanced when CO communication systems use the orthogonal frequency-division multiplexing (OFDM) technique [5]; these are called CO-OFDM systems. OFDM uses many closely spaced subcarriers that are modulated independently. CO-OFDM systems have produced high SE records [6, 7], because the CO-OFDM subcarriers are orthogonal to each other and form a complete set. Usually the modulation speed for subcarriers is low, which helps to overcome the optical fiber’s dispersion. However, CO-OFDM suffers from a high peak-to-average power ratio (PAPR) and is sensitive to the phase noises of laser diodes. Most of all, it requires heavy use of digital signal processing (DSP) circuits, which makes its real-time operation difficult [6, 7].

Recently it has been suggested to use a linear sum of optical receiver modes (ORMs) as an optical signal, called an ORM signal [8, 9]. We refer to this kind of multiplexing as ORM division multiplexing (ORMDM). Let’s assume that ORMDM is used in CO-ORMDM systems. Then, the ORMs are inherent modes of the coherent optical receiver (COR) in the CO-ORMDM system [10–13]. They are orthogonal to each other and form a complete set as well. Thus the CO-ORMDM can yield high SE values.

In a CO-ORMDM system the optical channel is a linear sum of ORM subchannels, as will be explained in Section II. The ORM subchannels have wider spectra and higher baud rates than those of CO-OFDM systems. Thus the foregoing PAPR and phase noise problems can be mitigated, as in wavelet OFDM [14, 15]. In this case, how is one to detect each ORM subchannel separately, in real time? To this end, we propose a new COR that uses an ORM subcarrier as its local oscillator (LO).

Let us consider the i^th optical channel in a CO-ORMDM system. The electric field of this optical channel can be written as e(t) = Re{E(t)exp( jω_it)}, where ωi is the center angular frequency of the i^th optical channel, and E(t) is the complex electric field amplitude (CEFA). For a single ORM signal localized at t = 0, E(t) is a linear sum of ORM mode functions ψn(t){n = 0, 1, 2, …} [8]

(1)$$\begin{array}{l}E(t)={\displaystyle \sum _{n=0}^{M-1}{a}_n{\psi }_n(t)}\\ \end{array}$$

where M is the number of ORMs used for the ORMDM. The complex mode coefficient a_n includes the data to be transmitted. We call the t = 0 time the origin of this ORM signal [8, 9]. The mode functions are real and complete, satisfying the orthogonality relation [8]

(2)$${\displaystyle {\int }_{-\infty }^{\infty }d{t}_{}}{\psi }_m(t){\psi }_n(t)={\delta }_{mn}$$

where δ_mn is the Kronecker delta function.

Taking the Fourier transform of both sides of Eq. (1), we have an alternative form of the ORM signal that is more convenient than Eq. (1) in many cases: $\epsilon (\omega )={\displaystyle {\sum }_{n=0}^{M-1}{a}_n{\varphi }_n(\omega )}$, where ε(ω) and ϕ_n(ω) are the CEFA and the n^th ORM mode function in the optical frequency domain respectively [8]. The mode functions ϕ_n(ω){n = 0, 1, 2, …} are also complete and satisfy the orthogonality relation ${\displaystyle {\int }_{-\infty }^{\infty }d{\omega }_{}}{\varphi }_m^{*}(\omega ){\varphi }_n(\omega )=2\pi {\delta }_{mn}$ ∫^∞−∞ dωϕ^*m(ω)ϕn(ω) = 2πδmn.

With all ORM signals present, the full expression for E(t) can be written as

(3)$$E(t)={\displaystyle \sum _{n=0}^{M-1}{\displaystyle \sum _{l=-\infty }^{\infty }{a}_{n,l}{\psi }_n(t-lT)}}$$

where T is the period of the ORM signal, and we use the simplification a_n,0 = a_n. The optical channel is a linear sum of ORM subchannels. The CEFA of the m^th ORM subchannel is ${\displaystyle {\sum }_{l=-\infty }^{\infty }{a}_{m,l}{\psi }_m(t-lT)}$. The m^th ORM subchannel before its modulation is the m^th ORM subcarrier, the CEFA of which is given by ${\displaystyle {\sum }_{l=-\infty }^{\infty }{\psi }_m(t-lT)}$. The CEFA of the ORM subcarriers and the origin times are shown in Fig. 1, for the three lowest-order ORMs.

Figure 1. Complex electric field amplitudes of the three lowest-order optical receiver mode subcarriers.

In Fig. 2, we illustrate the proposed COR that detects the m^th ORM subchannel of the i^th optical channel. This COR will be denoted as COR_i,m. Let us first discuss the characteristics of the direct-detection unit (DDU) within the dotted box in Fig. 2. The DDU defines the ORM set, and is the key element of the COR. The optical filter (OF) demultiplexes the received optical wavelength division multiplexing (WDM) channels. The photodetector (PD) is an ideal one, and its frequency response is absorbed by the electrical filter (EF). The EF filters out beat noises generated by the amplified spontaneous emission (ASE) optical noise. The ORMs and related quantities are dependent on the optical channel index i, but we will not use the optical channel index here explicitly, except in COR_i,m.

Figure 2. The COR_i,m that detects the m^th ORM subchannel within the i^th optical channel. COR, coherent optical receiver; ORM, optical receiver mode; WDM, wavelength division multiplexing; LO, local oscillator; LD, laser diode; MOD, optical modulator; DDU, direct-detection unit; OF, optical filter; PD, photodetector; EF, electrical filter; DSP, digital signal processing.

If the CEFA at the DDU input is εDD(ω), then the output voltage of the DDU can be evaluated as [10, 13]

(4)$$y_{DD}(t)=\frac{k}{8{\pi }^2}{\displaystyle {\int }_{-\infty }^{\infty }d\omega {\displaystyle {\int }_{-\infty }^{\infty }d\omega }{\text{'}}_{}}\exp \{j(\omega \text{'}-\omega )t\}\cdot {\epsilon }_{DD}^{*}(\omega )K(\omega ,\omega \text{'}){\epsilon }_{DD}^{}(\omega \text{'})$$

The kernel is given by K(ω, ω') = H^*_o(ω)H_e(ω' − ω)H_o(ω'); it has the Hermitian property K(ω, ω') = K^*(ω', ω). H_o(ω) is the transfer function of the OF between the CEFA input to the OF and the CEFA output from the OF, in the optical frequency domain. H_e(ω) is the transfer function of the EF between the current input to the EF and the voltage output from the EF, in the optical frequency domain. k is a proportionality constant. ϕ_n(ω) satisfies the following homogeneous Fredholm integral equation of the 2^nd kind with real eigenvalue λ_n:

(5)$${\varphi }_n(\omega )={\lambda }_n{\displaystyle {\int }_{-\infty }^{\infty }d\omega \text{'}}K(\omega ,\omega \text{'}){\varphi }_n(\omega \text{'})$$

Exact solutions of Eq. (5) are often not available, in which case ϕ_n(ω) and λ_n are found numerically [10]. The eigenvalues are positive in general, and increase indefinitely as n increases: λ0 ≤ λ1 ≤ λ2 ≤ ….

As for the COR_i,m in Fig. 2, the LO’s output light is proportional to the m^th ORM subcarrier having the same center wavelength as the i^th optical channel. Only the in-phase detection part is shown, for brevity. The quadrature-phase detection part is hidden, and has the same structure except for the 90° phase shift for the LO [3]. For ease of explanation, we also assume that the received optical WDM channels and the LO have the same polarization.

The transmitted optical WDM channels are applied to the 3-dB coupler, along with the LO light. The two outputs of the 3-dB coupler are directed to the two DDU inputs. The two DDUs use the same kinds of devices and have the same ORM set. Let us assume that the CEFAs at the upper and the lower DDU inputs in the optical frequency domain are ε₁(ω) + ε₁(ω) and ε₁(ω) − ε₂(ω) respectively [3]. ε₁(ω) is from the optical WDM channels and ε₂(ω) is from the LO. If we denote the voltages at the upper and the lower DDU outputs as y₊(t) and y₋(t) respectively, then using Eq. (4) we obtain

(6)$$y_{\pm }(t)=\frac{k}{8{\pi }^2}{\displaystyle {\int }_{-\infty }^{\infty }d\omega {\displaystyle {\int }_{-\infty }^{\infty }d\omega }{\text{'}}_{}}\exp \{j(\omega \text{'}-\omega )t\}\cdot \{{\epsilon }_1^{*}(\omega )\pm {\epsilon }_2^{*}(\omega )\}K(\omega ,\omega \text{'})\{{\epsilon }_1^{}(\omega \text{'})\pm {\epsilon }_2^{}(\omega \text{'})\}$$

Then the differential output voltage y(t) = y₊(t) − y₋(t) is

(7)$$y(t)=\frac{k}{4{\pi }^2}{\displaystyle {\int }_{-\infty }^{\infty }d\omega {\displaystyle {\int }_{-\infty }^{\infty }d\omega }\text{'}\cdot \left[{\epsilon }_1^{*}(\omega )K(\omega ,\omega \text{'}){\epsilon }_2^{}(\omega \text{'})\exp \{j(\omega \text{'}-\omega )t\}\right.\left.+{\epsilon }_2^{*}(\omega )K(\omega ,\omega \text{'}){\epsilon }_1^{}(\omega \text{'})\exp \{j(\omega \text{'}-\omega )t\}\right]}$$

We exchange the integration variables ω and ω' within the first term of Eq. (7) to find

(8)$$y(t)=\frac{k}{2{\pi }^2}{\displaystyle {\int }_{-\infty }^{\infty }d\omega {\displaystyle {\int }_{-\infty }^{\infty }d\omega }\text{'}\cdot \mathrm{Re}\left[{\epsilon }_2^{*}(\omega )K(\omega ,\omega \text{'}){\epsilon }_1^{}(\omega \text{'})\exp \{j(\omega \text{'}-\omega )t\}\right]}$$

where we have used the Hermitian property of K(ω', ω). Equation (8) is the result for the in-phase detection. Changing ε₂(ω) to jε₂(ω) in Eq. (8), we can also find the result for the quadrature-phase detection. With these two detection results, we can build a complex received signal for the COR_i,m as follows:

(9)$$Y_m(t)=\frac{k}{2{\pi }^2}{\displaystyle {\int }_{-\infty }^{\infty }d\omega {\displaystyle {\int }_{-\infty }^{\infty }d\omega }\text{'}}\cdot {\epsilon }_2^{*}(\omega )K(\omega ,\omega \text{'}){\epsilon }_1^{}(\omega \text{'})\exp \{j(\omega \text{'}-\omega )t\}$$

Now, we assume that only a single ORM signal of the i^th optical channel is received, and exclude any crosstalk for the reception of this ORM signal. We use ${\displaystyle {\sum }_{n=0}^{M-1}{a}_n{\varphi }_n(\omega )}$ as the signal part of ε₁(ω). The ASE noise part of ε₁(ω) can be expressed as ${\displaystyle {\sum }_{n=0}^{\infty }{\chi }_n{\varphi }_n(\omega )}$. Also, we use a single ORM waveform for the LO: ε₂(ω) = c_mϕ_m(ω), where c_m is a complex constant. Then we obtain

(10)$$Y_m(t)=4c_m^{*}\left\{{\displaystyle \sum _{n=0}^{M-1}(a_n}+{\chi }_n)y_{mn}(t)+{\displaystyle \sum _{n=M}^{\infty }{\chi }_n{y}_{mn}(t)}\right\}$$

where ymn(t) is defined as [8, 9]

(12)$$Y_m(0)=k{c}_m^{*}(a_m+{\chi }_m)/\pi {\lambda }_m$$

Since ymn(0) = δmnk / 4πλm, we have

(12)$$Y_m(0)=k{c}_m^{*}(a_m+{\chi }_m)/\pi {\lambda }_m$$

which is our main result. It tells us that at t = 0 we can find the transmitted data of the m^th ORM subchannel directly, without the heavy use of DSP circuits. Note that this property is unique to the CO-ORMDM, because of the kernel K(ω, ω') in Eq. (11). The t = 0 time point is equal to the origin of the ORM signal Eq. (1). If we regard the maximum point of the zeroth ORM waveform as the center of the ORM signal, each origin time in Fig. 1 has an offset from the center of the corresponding ORM signal, reflecting the delay of the DDU filters.

Let us decompose the mode coefficient a_m as a_m = γζ_mz_m, where z_m is the transmitted data. γ is a proportionality constant, and ζ_m is a dimensionless real constant related to the optical power of the ORM subchannel. The mode coefficient χ_m is decomposed as χ_m = γn_m, where n_m is a dimensionless zero-mean complex Gaussian random variable. All of the real and imaginary parts of {n₀, n₁, n₂, ...} are mutually independent of each other, having identical variance [10]. With these decompositions we obtain

(13)$${\overline{Y}}_m(0)=z_m+n_m/{\zeta }_m$$

where Y¯_m(t) = Y_m(t)πλ_m / c^*_mγζ_mk. Only the m^th ORM contributions appear in Eq. (13), for both the signal and the ASE noise. In addition, we need to include the effects from other ORM signals in the same and adjacent optical channels, and we need to use the full subcarrier expression for the LO. These modifications will add crosstalk terms to the right-hand side of Eq. (13). The phase and the polarization diversities can be included as per [3].

In this section we present the waveforms of Y_m(t) for the foregoing case of single-ORM-signal input. We assume that H_o(ω) and H_e(ω) are Gaussian [13]. The eigenvalue of the m^th ORM is λm = λ₀ / q^m , where q is a positive quantity smaller than 1. The expressions for λ0 and q are in [13]. From Eq. (12), Y_m(0) is proportional to q^m. To use higher-order ORMs, we need to use high q values, which can be done by decreasing the EF bandwidth compared to the OF bandwidth [13]. Thus we choose 3-dB bandwidths |H_o (ω)|² and |H_e(ω)|² as 100 GHz and 10 GHz respectively, which gives q = 0.754. For simplicity, we set ζm = 1 for all m. Neglecting the ASE, we have from Eq. (10)

(14)$${\overline{Y}}_m(t)={\displaystyle \sum _{n=0}^{M-1}{z}_n}{y}_{mn}(t)/q^m{y}_c$$

where y_c = k / 4πλ₀ [8]. We show the real and imaginary parts of Y¯_m(t) in Fig. 3, denoted as R_m(t) and I_m(t) respectively. We set M = 3 with z₀ = 1 + j, z₁ = −1 + 3j, and z₂ = 3 + j. Note that Y¯_m(0) = z_m for m = 0, 1, 2, 3, where z₃ = 0 naturally.

Figure 3. Real and imaginary parts of the waveforms of Y¯_m(t), denoted as R_m(t) and I_m(t) respectively. The three lowest-order optical receiver mode subchannels are transmitted (M = 3) with z₀ = 1 + j, z₁ = −1 + 3j, and z₂ = 3 + j. (a) m = 0, (b) m = 1, (c) m = 2, (d) m = 3.

The proposed COR can be applied to conventional optical communication systems, to upgrade their SEs. From the ORMDM point of view, the optical signal of a conventional optical channel, excluding the OFDM, can be regarded as an ORM signal with a_n {n = 0, 1, 2, …} dependent on each other. If one of the ORMs (say the m^th ORM) has an independent am, we have more freedom and can increase the SE. In this case we need two CORs. One uses the m^th ORM subcarrier as its LO, while the other uses the sum of the rest of the ORM subcarriers as its LO. This procedure can be extended to other ORMs, until the ORMDM is fully utilized.

The COR can be used in multiple-access networks [16]. If only one ORM subchannel is to be dropped at an optical node, for example, we use one COR at that node.

If we detect multiple optical channels and ORM subchannels, we use a WDM demultiplexer (DMUX) and place optical splitters after the WDM DMUX, as shown in Fig. 4. We place one COR at the end of each optical-splitter arm. The optical splitters can be integrated [17], and their losses can be compensated by the optical amplifier (OA) before the WDM DMUX. We can also use gain materials for the optical splitter [18]. If we use the WDM DMUX as the DDU’s OF in Fig. 2, we may remove all OFs from within the CORs. Then the modulation signal to obtain the LO in Fig. 2 is modified to make the LO light seem as if it has passed the OF already.

Figure 4. Receiver side of a CO-ORMDM system to detect multiple optical channels and ORM subchannels. CO, coherent optical; ORM, optical receiver mode; ORMDM, ORM division multiplexing; OA, optical amplifier; WDM, wavelength division multiplexing; DMUX, demultiplexer; COR, coherent optical receiver.

We perform numerical simulations for the foregoing CO-ORMDM system. The 256-quadrature amplitude modulation (QAM) code is used for the modulation of the ORM subcarriers. The optical channel power is fixed here by allocating smaller optical power evenly to the ORM subchannels as M is increased from 1 to 12. Thus we set ${\zeta }_m=1/\sqrt{M}$ in Eq. (13). For M = 1, we assume that the ASE noise term in Eq. (13) is below 5% of the minimum constellation distance (MCD). If we want to get error-free results in this case, the crosstalk fluctuations should be kept below 45% of the MCD. Under this condition, we have a SE of 6.0 bit s⁻¹ Hz⁻¹. For M = 6, the ASE noise term in Eq. (13) is below 8.7% of the MCD, and we have a SE of 9.8 bit s⁻¹ Hz⁻¹. Similarly, for M = 12 SE is 11.0 bit s⁻¹ Hz⁻¹ and exhibits saturation behavior. If the ORM subchannels have unequal optical powers and unequal QAM codes, we can further increase the SE. Also, with forward error-correction methods the SE can be increased even further [4, 19].

In contrast to CO-OFDM, real-time operation is possible in our CO-ORMDM system. The speed of the CORs in our CO-ORMDM system is not limited by the DSP circuits, as Eq. (12) shows. As for the transmitter of the CO-ORMDM system, we may use a single in-phase/quadrature optical modulator driven by two digital-to-analog converters (DACs) to produce one or a few subchannels. Then the DSP circuit limits can be avoided, and real-time operation can be attained. The digital inputs to the DACs can be modified to pre-compensate for the optical-fiber dispersion, etc. As a reference, a 100 GS/s DAC can generate electrical signals that are about 25 GHz in bandwidth [20].

To increase the channel bandwidth beyond the limit of the DACs, we can use optical arbitrary-waveform generators (OAWGs) to obtain ORM subcarriers from mode-locked laser diodes [21^–23]. Modulating the ORM subcarriers, we get ORM subchannels. Also, there is no limit from the DSP circuits in this case.

To make the proposed CO-ORMDM system more practical, we could integrate photonic devices. Similar works have been done for conventional COR [24, 25].

Building each optical channel as a linear sum of multiple ORM subchannels, CO-ORMDM systems can attain high SEs. To detect the ORM subchannels selectively, we have introduced a new COR that is fast and does not require heavy use of DSP circuits. Thus, in contrast to CO-OFDM, real-time operation is possible in our CO-ORMDM system. In addition, we can use OAWGs to increase the channel bandwidth beyond the limit of the DACs. With photonic integration, CO-ORMDM systems using the proposed CORs can be made simple and practical.

The author declares no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2019.

The author received no financial support for the research, authorship, or publication of this article.