Optical-fiber communication is one of the key technologies that enable heavily data-oriented world of the current Internet [1, 2]. As the data quality obtainable from the Internet becomes more and more personal and intelligent, the amount of data to be transmitted over optical fibers grows exponentially. It is more cost-effective to increase the spectral efficiencies (SEs) of optical-fiber transmission systems [3–5] than to install new fibers.

Recently, a new kind of optical multiplexing called optical-receiver-mode-division multiplexing (ORMDM) has been proposed [6–8]. Using ORMDM, an optical channel is produced as a linear sum of optical-receiver-mode (ORM) subchannels modulated independently [8]. The ORMs are optical modes of a direct-detection (DD) optical receiver determined by the optical-receiver filters [9–12]. The ORM mode functions form a complete set; Thus ORMDM can give high SE values, similar to orthogonal frequency-division multiplexing (OFDM) [13–15].

After the optical transmission, we use a coherent-optical receiver (COR) whose local oscillator (LO) is modulated to produce an ORM subcarrier. This kind of optical communication system has been proposed in [8] and is called the coherent-optical (CO)-ORMDM system. In CO-ORMDM systems, the data can be read instantly, and real-time operation is possible without heavy digital-signal-processing (DSP) circuits. In OFDM systems, however, heavy DSP circuits are needed at the optical receivers to find the transmitted data.

In this paper, we introduce a new method of crosstalk analysis to estimate the spectral efficiencies (SEs) of CO-ORMDM communication systems. Using this method of crosstalk analysis, we can maximize the SE by allocating quadrature-amplitude-modulation (QAM) codes and QAM step sizes unevenly over ORM subchannels.

The following sections 2 and 3 summarize the ORMDM optical channel and the COR for the ORMDM optical channel given in [8].

For a given optical channel in an ORMDM system, the complex electric field amplitude (CEFA) of its n^th ORM subchannel can be expressed as ${\displaystyle {\sum }_{k=-\infty }^{\infty }{a}_{nk}{\psi }_n(t-kT)}$ [8]. The complex mode coefficient a_nk includes the data loaded into the n^th ORM subchannel. T is the ORM signal period; the ORM signal will be explained below. ψ_n(t) is the n^th ORM mode function having λ_n as its eigenvalue, which is real. Its Fourier transform is φ_n(ω), the n^th order ORM mode function in the optical frequency domain. The mode functions ψ_n(t) {n = 0, 1, 2, …} are real and complete [6]. They satisfy the orthogonality relation, ∫^∞_−∞ dtψ_m(t)ψ_n(t) = δ_mn, where δ_mn is the Kronecker delta function.

The CEFA of the optical channel is obtained by aggregating all of the ORM subchannels as [8]

where M is the number of ORMs used for the optical channel. For fixed k, Eq. (1) represents an ORM signal. For example, the CEFA of the ORM signal for k = 0 is given by $E(t)={\displaystyle {\sum }_{n=0}^{M-1}{a}_n{\psi }_n(t)}$, where we abbreviate a_n0 = a_n. The CEFA for the n^th ORM subcarrier is defined as ${\displaystyle {\sum }_{k=-\infty }^{\infty }{\psi }_n(t-kT)}$.

We show in Fig. 1 the COR in the CO-ORMDM system [8]. It incorporates balanced detection [2, 16] with two identical direct-detection unit (DDU) arms. The DDUs determine the ORMs, where a single DDU is composed of an optical filter for the selection of the optical channel, a photodetector, and an electrical low-pass filter [8].

Figure 1. The COR for reception of the m^th ORM subchannel within the i^th optical channel. LD, laser diode; MOD, modulator; LO, local oscillator; WDM, wavelength-division multiplexing; DDU, direct-detection unit. The LD has the same center wavelength as the i^th optical channel.

For brevity, only the in-phase part of the COR is shown, which is very similar to the quadrature part. We denote the CEFAs in the optical frequency domain at the upper and lower DDU inputs as ε₁(ω) + ε₂(ω) and ε₁(ω) − ε₂(ω) respectively [2], where ε₁(ω) is from the received optical channels and ε₂(ω) is from the local oscillator (LO). Let us assume, for example, that the COR is for reception of the m^th ORM subchannel within the i^th optical channel. Then the center wavelength of the LO is the same as that of the i^th optical channel and ε₂(ω) is proportional to the the m^th ORM subcarrier. Using the outputs from the in-phase and quadrature parts, we build the complex received signal [8] as follows:

Here, we present the method for crosstalk analysis of the CO-ORMDM system. The crosstalk included in this analysis comes from adjacent ORM signals within the same or adjacent optical channels. The effects of the amplified-spontaneous-emission (ASE) noise are also included, but optical reflections are not included [17].

In Fig. 2 we show the nine ORM signals used in our analysis. Every single rectangle represents an ORM signal that uses M QAM-modulated ORMs. To distinguish these ORM signals, we use an index l whose values are written inside the rectangles. The l = 0 ORM signal is to be detected currently. The other eight ORM signals, l = 1–8, generate crosstalk for this detection. Note that the horizontal axis is the optical-channel number. To suppress the crosstalk, the ORM signals in the (i ± 1)^th optical channels, l = 3–6, are delayed by T/2 with respect to the ORM signals in the i^th optical channel. The l = 7 and 8 ORM signals have no time delay. Their contributions become important when the channel spacing becomes so small that the l = 7 and 8 ORM signals penetrate into the l = 0 ORM signal.

Figure 2. Optical-receiver-mode (ORM) signals used for the crosstalk analysis. Every rectangle is an ORM signal that carries M quadrature-amplitude-modulation (QAM)-modulated ORMs. The l = 0 ORM signal is the one curently under detection.

ε₁(ω) in (2) is composed of signal and ASE-noise terms, ε₁(ω) = ε_signal(ω) + ε_ASE(ω). Including all of the l = 0–8 ORM signals of Fig. 2, the signal part can be written as

When l = 5, for example, we have ω₅ = 2πf_s and τ₅ = T/2. f_s is the optical-channel spacing in the frequency domain. The ASE noise ε_ASE(ω) is generated by the optical amplifiers during transmission. It can be expanded using the ORM mode functions as ${\epsilon }_{ASE}(\omega )={\displaystyle \sum _{n=0}^{\propto }{\chi }_n{\varphi }_n(\omega )}$, where χ_n is a complex constant.

We decompose the mode coefficient a_nl as a_nl = γζ_nz_nl, where z_nl is the transmitted QAM data. Note that the k index for the mode coefficient in Eq. (1) is no longer used, being changed to l. z_nl is normalized to have odd integers as its real and imaginary parts. For example, if z_nl belongs to the 16-QAM code, its real and imaginary parts have values among {±3, ±1}. γ is a proportionality constant. ζ_n is a dimensionless real constant that determine the optical power of the n^th ORM subchannel. When l = 0, we abbreviate z_n0 = z_n. The mode coefficient χ_n is decomposed as χ_n = γn_n, where n_n is a dimensionless, zero-mean complex Gaussian random variable. All the real and the imaginary parts of {n₀, n₁, n₂, …} are mutually independent Gaussian random variables having an identical standard deviation σ [9].

As for the LO, we use three successive m^th ORM mode functions ${\epsilon }_2(\omega )=c_m{\varphi }_m(\omega )\left\{1+\exp (-j\omega T)+\exp (j\omega T)\right\}$, where c_m is a complex constant. The first term is for the detection of the l = 0 ORM signal, while the second and third terms generate crosstalk for this detection.

To evaluate Y_im(t), we introduce a new function y_mnl(t, τ_LO) that describes the crosstalk from the n^th ORM in the l^th ORM signal assuming ${\epsilon }_2(\omega )=c_m{\varphi }_m(\omega )\exp (-j\omega {\tau }_{LO})$,

y_mnl(t, τ_LO) is a generalization of y_mn(t) defined in [6, 7]

τ_LO has three values: 0 and ±T. Aggregating these three contributions, we define x_mnl(t) as x_mnl(t) = y_mnl(t, T) + y_mnl(t, 0) + y_mnl(t, −T). Then, we may write

Grouping all the crosstalk terms as X_m(t), we rearrange (6) as

where y_mn0(t, 0) = y_mn(t) and

At t = 0 we obtain

where we have used y_mn(0) = δ_mnk_c / 4πλ_m and Y_cm = c^*_m γζ_mk_c / πλ_m. Thus we can find z_m instantly at t = 0 without the heavy use of DSP circuits. As mentioned in [8], the moment t = 0 is called the origin time for the l = 0 ORM signal. ñ_m is given as ${\tilde{n}}_m=(4\pi {\lambda }_m/k_c){\displaystyle {\sum }_{n=0}^{\infty }{n}_n{x}_{mn}(0)}$, which is not Gaussian, except when T is infinite. When T is infinite, we have x_mn(0) = y_mn(0, 0). Then ñ_m becomes proportional to n_m, which is Gaussian. The constants c_m, γ, and k_c cancel out in the last term of (9).

Denoting W = X_m(0) // Y_cm, we introduce d_m, defined as $d_m=Max\left(\left|W_r\right|,\left|W_i\right|\right)$, where W_r and W_i are the real and imaginary parts of W respectively. Thus d_m is the maximum deviation of the measured z_m in magnitude, owing to the crosstalk along the real or the imaginary directions in the QAM constellation map. For proper operation of the transmission system, we impose an upper limit on d_m, denoted as D_m. If we want to obtain error-free transmission, the necessary condition is D_m ≤ 1, since the distance between the nearest QAM constellations is 2. If this condition is not met, the sampled data will be closer to other QAM constellations, resulting in decision errors. If ASE is present, the difference 1 − D_m needs to be large enough compared to σ/ζ_m, where we have assumed that the standard deviations of the real and imaginary parts of ñ_m are still σ approximately. In our calculations, we find the maximum SE by adjusting T and f_s under the condition of d_m ≤ D_m, for all m.

From now on we assume that the CORs are Gaussian [12], such that $H_o(\omega )=H_o(0)\exp (-j\omega t_o-{\omega }^2/2{\alpha }^2)$ and $H_e(\omega )=H_e(0)\exp (-j\omega t_e-{\omega }^2/2{\beta }^2)$, where H_o(ω) and H_e(ω) are the transfer functions of the optical and the electrical filters within the DDU respectively [8]. t_o and t_e are the time delays.

The mode function of the m^th ORM is given by the m^th Hermite function [6]

where H_m(ω/a) is the m^th Hermite polynomial and t_d = t_o + t_e. The parameters a, α, and β are related as α² = a²(1 + q) / (1 − q) and β² = a²(1 − q²) / 2q. q is a positive quantity smaller than 1 given by $q=1+4/r^2-2\sqrt{4/r^4+2/r^2}$, where r = 2α/β. The eigenvalue of the m^th ORM is λ_m = λ₀ / q^m with ${\lambda }_0{ }^{-1}=|H_0(0){|}^2{H}_e(0)a{\pi }^{1/2}\sqrt{1-q^2}$. Since λ_m ∝ q^−m, it is preferable to choose high r values, to increase M. This trend is true even if we use non-Gaussian filters [9].

In our Gaussian COR, the y_mnl(t, τ_LO) function of (4) can be expressed as

For simplicity, we choose the time delay such that exp(−jω_lt_d) = 1 for all calculations. The integrals of (11) can be evaluated to give

where p = 1 − q, Z₁ = pat − aτ_l + qaτ_LO + jω_s / a, Z₂ = pat + qaτ_l − aτ_LO + jqω_s / a, and

The details of this evaluation are given in the Appendix.

We show the SE values obtained from the crosstalk analysis for CO-ORMRM systems having Gaussian ORMs. Polarization-division multiplexing is also assumed [2].

At first, we use the 256-QAM code uniformly for M ORM subchannels with ζ_m = 1. D_m is the same for these ORM subchannels and denoted as D. The 3-dB bandwidths of |H_o(ω)|² and |H_e(ω)|² are chosen as 100 GHz and 10 GHz respectively, which means r = 10 and q = 0.754. Later, we will show how to upgrade the system by making the QAM code and ζ_m uneven over the ORM subchannels.

Figure 3 shows our estimated SE values versus D. The procedure was explained at the end of section IV. The SE increases as D increases from zero, since we are allowing more crosstalk. Note that the allowable ASE noise decreases in this direction; Thus the transmission distance also decreases in this direction. Owing to the square QAM modulation, the SE traces are angled. As M increases the SE also increases, although with saturation.

Figure 3. Spectral efficiencie (SE) versus D for M = 1, 3, 6, 9, and 12. The ORM subchannels are modulated using the 256-QAM code.

In Fig. 3, the optical-channel power also increases in proportion to M, since the ORM mode functions are normalized to have the same energy, ∫^∞_−∞ dtψ_n(t)2 = 1 {n = 0, 1, 2, …}. Note that the time-domain mode functions are real [6]. We can make the optical-channel power independent of M by choosing ${\zeta }_m=1/\sqrt{M}$ [8]. As was stated in section IV, ζ_n determines the optical power of the n^th ORM subchannel. Then the ASE noise term in (9) becomes proportional to $\sqrt{M}$, while the third crosstalk term remains unchanged. Thus when the ASE noise is present, we need to reduce D as M increases. To find D, we note that 1 − D is proportional to σ/ζ_m, as explained at the end of section IV. Thus we choose D using the relation 1 − D = $\sqrt{M}$ (1 − D1), where D = D₁ when M = 1. With all other conditions the same, our results are shown in Fig. 4. We have chosen D1 to be 0.1 times an integer from 1 to 10.

Figure 4. Spectral efficiencie (SE) versus D with the optical-channel power fixed, for M = 1, 3, 6, 9, and 12. The initial value of D for M = 1 is chosen to be 0.1 times an integer from 1 to 10.

We can increase the SE further by allocating the QAM code and ζ_m unevenly over ORM subchannels. The lower the mode index, the smaller the crosstalk that the ORM subchannel experiences [12]. This is because ORM mode functions having lower mode indices are more localized in the time domain. Therefore, as the mode index decreases we can use higher-order QAM codes or smaller QAM step sizes, or both, for the modulation of the ORM subchannels.

In this way, we upgrade one of the cases in Fig. 3 with D = 0.5 and M = 9. The total optical-channel power is unchanged, so that ñ_m in (9) has the same distribution after the upgrade. We use prime notation for the upgraded system. As m decreases from M − 1 to zero, we decrease ζ′_m monotonically from 1 and increase the order of the QAM code. For m = M − 1, we use 64 QAM. Assuming that 1 − D′_m is proportional to σ/ζ′_m, we find D′_m using the relation (1 − D) / ζ′_m= (1 − D′_m) . Table 1 lists our choices. This upgrade gives SE up to 12.33 bit s⁻¹ Hz⁻¹, which is much higher than the original SE of 10.37 bit s⁻¹ Hz⁻¹. The optical-channel power is unchanged within 0.4%.

TABLE 1. Upgrade example for the system in Fig. 3 with D = 0.5 and M = 9.

m	Original ORM Signal Parameters			Upgraded ORM Signal Parameters
	QAM	D	dm/D	QAM	D′m	d′m/D′m
0	256	0.5	1.3 × 10−4	2,048	0.001	0.14
1	256	0.5	8.0 × 10−4	2,048	0.001	0.83
2	256	0.5	3.2 × 10−3	1,024	0.004	0.88
3	256	0.5	1.2 × 10−2	1,024	0.012	0.96
4	256	0.5	3.6 × 10−2	1,024	0.034	0.99
5	256	0.5	8.8 × 10−2	512	0.085	1.0
6	256	0.5	0.22	256	0.18	0.97
7	256	0.5	0.5	256	0.33	0.97
8	256	0.5	1.0	64	0.5	0.99

We show in Fig. 5 the SE values versus M for r = 10, 50, and 100 with D = 0.5. We use the 256-QAM code with ζ_m = 1 for all ORMs used. The 3-dB bandwidth of |H_o (ω)|² is 100 GHz. Beyond M = 30, the SE values are almost the same. At M = 36 the SE is 12.5 bit s⁻¹ Hz⁻¹, where q^M−1 = 0.37 for r = 100. If M = 50 and r = 100, we have an SE of 12.86 bit s⁻¹ Hz⁻¹ (13.07 bit s⁻¹ Hz⁻¹ for D = 1). Upgrading this system in a similar manner as in Table 1, we obtain SE up to 15.29 bit s⁻¹ Hz⁻¹, where T = 0.82 ns and f_s = 76 GHz.

Figure 5. Spectral efficiencie (SE) versus M for r = 10, 50, and 100. We use the 256-quadrature-amplitude-modulation (QAM) code with D = 0.5 and ξ_m = 1 for all optical-receiver-modes (ORMs) used.

For comparison, SE values of around 12 bit s⁻¹ Hz⁻¹ were reported [18, 19] with conventional CO communication systems using Nyquist pulses [20]. The optical channels were modulated using the 256-QAM code. When the OFDM was used with the 256-QAM code, an SE of 14 bit s⁻¹ Hz⁻¹ was reported [15]. All of these previous results used forward-error-correction (FEC) techniques [17]. With FEC, the SE of the CO-ORMDM systems presented here would be improved further [21].

To estimate the SEs of CO-ORMDM communication systems, we have introduced a new method of crosstalk analysis. Using this method we have found the SEs of CO-ORMDM systems and their upgrades, assuming Gaussian CORs. At first, all of the ORM subchannels were modulated using the same QAM code. Then, we allocated QAM codes and QAM step sizes unevenly over ORM subchannels and obtained much higher SE values. As an example, with 50 ORMs we have obtained an SE of 12.86 bit s⁻¹ Hz⁻¹, when all the ORM subcarriers had the same optical power and were modulated using the 256-QAM code. This SE value increased to 15.29 bit s⁻¹ Hz⁻¹ when the QAM code and QAM step size were made uneven over ORM subchannels, with the total optical power fixed. If more optimized ORMs having lower crosstalk properties were used, or if FEC were used, we would obtain higher SE values, which could be verified experimentally.

To evaluate y_00l(t, τ_LO) as in (13), we apply a Gaussian integral formula to (11) with m = n = 0, given as

for Re{μ} > 0 [22]. This yields y_00l(t, τ_LO) in (13).

To evaluate y_00l(t, τ_LO) as in (12), we use the generating function for Hermite polynomials [23],

We replace H_m(ω/a) in (11) with exp(−h²₁ + 2h₁ω / a). Similarly, we replace H_n{(ω′ − ω_l) / a} in (11) with exp(−h²₂ + 2h₂(ω′ − ω_l) / a). The resultant equation can be integrated also using (A1), and the result is

From the Taylor series for (A3), the coefficient of the h^m₁ hⁿ₂/ m!n! term gives y_mnl(t, τ_LO) as in (12).

This work was supported by the Research Grant of Kwangwoon University in 2023.

Research Grant of Kwangwoon University in 2023.

The authors declare no conflict 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.