Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2024; 8(2): 156-161
Published online April 25, 2024 https://doi.org/10.3807/COPP.2024.8.2.156
Copyright © Optical Society of Korea.
Kyung Hee Seo^{1}, Jae Seung Lee^{2}
Corresponding author: ^{*}jslee@kw.ac.kr, ORCID 0000-0002-3927-9200
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Recently, a new kind of optical multiplexing called optical-receiver-mode (ORM)-division multiplexing (ORMDM) has been proposed, in which an optical channel is a linear sum of ORM subchannels modulated independently. Using coherent-optical (CO) techniques, it has been reported that CO-ORMDM communication systems can have very high spectral efficiencies (SEs). To estimate the SEs of CO-ORMDM communication systems, we introduce a new method of crosstalk analysis. Using this method, we can allocate quadrature-amplitude-modulation (QAM) codes and QAM step sizes unevenly over ORM subchannels to obtain higher SEs. With 50 Gaussian ORMs, we obtain a SE of up to 15.29 bit s^{−1} Hz^{−1}.
Keywords: Coherent optical communication, Optical fiber communication, Optical receivers, Wavelength division multiplexing (WDM)
OCIS codes: (060.0060) Fiber optics and optical communications; (060.2330) Fiber optics communications; (060.2360) Fiber optics links and subsystems; (060.4510) Optical communications
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
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
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].
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 ε_{1}(ω) + ε_{2}(ω) and ε_{1}(ω) − ε_{2}(ω) respectively [2], where ε_{1}(ω) is from the received optical channels and ε_{2}(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 ε_{2}(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.
ε_{1}(ω) in (2) is composed of signal and ASE-noise terms, ε_{1}(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 ω_{5} = 2πf_{s} and τ_{5} = 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
We decompose the mode coefficient a_{nl} as a_{nl} = γζ_{n}z_{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 {n_{0}, n_{1}, n_{2}, …} 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
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
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_{mn}k_{c} / 4πλ_{m} and Y_{cm} = c^{*}_{m} γζ_{m}k_{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
Denoting W = X_{m}(0) // Y_{cm}, we introduce d_{m}, defined as
From now on we assume that the CORs are Gaussian [12], such that
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 α^{2} = a^{2}(1 + q) / (1 − q) and β^{2} = a^{2}(1 − q^{2}) / 2q. q is a positive quantity smaller than 1 given by
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_{l}t_{d}) = 1 for all calculations. The integrals of (11) can be evaluated to give
where p = 1 − q, Z_{1} = pat − aτ_{l} + qaτ_{LO} + jω_{s} / a, Z_{2} = 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.
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
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 | d_{m}/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.
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(−h^{2}_{1} + 2h_{1}ω / a). Similarly, we replace H_{n}{(ω′ − ω_{l}) / a} in (11) with exp(−h^{2}_{2} + 2h_{2}(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^{m}_{1} h^{n}_{2}/ 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.
Curr. Opt. Photon. 2024; 8(2): 156-161
Published online April 25, 2024 https://doi.org/10.3807/COPP.2024.8.2.156
Copyright © Optical Society of Korea.
Kyung Hee Seo^{1}, Jae Seung Lee^{2}
^{1}Sogang Institute for Convergence Education, Sogang University, Seoul 04107, Korea
^{2}Department of Electronic Engineering, Kwangwoon University, Seoul 01897, Korea
Correspondence to:^{*}jslee@kw.ac.kr, ORCID 0000-0002-3927-9200
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Recently, a new kind of optical multiplexing called optical-receiver-mode (ORM)-division multiplexing (ORMDM) has been proposed, in which an optical channel is a linear sum of ORM subchannels modulated independently. Using coherent-optical (CO) techniques, it has been reported that CO-ORMDM communication systems can have very high spectral efficiencies (SEs). To estimate the SEs of CO-ORMDM communication systems, we introduce a new method of crosstalk analysis. Using this method, we can allocate quadrature-amplitude-modulation (QAM) codes and QAM step sizes unevenly over ORM subchannels to obtain higher SEs. With 50 Gaussian ORMs, we obtain a SE of up to 15.29 bit s^{−1} Hz^{−1}.
Keywords: Coherent optical communication, Optical fiber communication, Optical receivers, Wavelength division multiplexing (WDM)
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
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
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].
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 ε_{1}(ω) + ε_{2}(ω) and ε_{1}(ω) − ε_{2}(ω) respectively [2], where ε_{1}(ω) is from the received optical channels and ε_{2}(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 ε_{2}(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.
ε_{1}(ω) in (2) is composed of signal and ASE-noise terms, ε_{1}(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 ω_{5} = 2πf_{s} and τ_{5} = 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
We decompose the mode coefficient a_{nl} as a_{nl} = γζ_{n}z_{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 {n_{0}, n_{1}, n_{2}, …} 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
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
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_{mn}k_{c} / 4πλ_{m} and Y_{cm} = c^{*}_{m} γζ_{m}k_{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
Denoting W = X_{m}(0) // Y_{cm}, we introduce d_{m}, defined as
From now on we assume that the CORs are Gaussian [12], such that
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 α^{2} = a^{2}(1 + q) / (1 − q) and β^{2} = a^{2}(1 − q^{2}) / 2q. q is a positive quantity smaller than 1 given by
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_{l}t_{d}) = 1 for all calculations. The integrals of (11) can be evaluated to give
where p = 1 − q, Z_{1} = pat − aτ_{l} + qaτ_{LO} + jω_{s} / a, Z_{2} = 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.
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
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 | d_{m}/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.
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(−h^{2}_{1} + 2h_{1}ω / a). Similarly, we replace H_{n}{(ω′ − ω_{l}) / a} in (11) with exp(−h^{2}_{2} + 2h_{2}(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^{m}_{1} h^{n}_{2}/ 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.
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 | d_{m}/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 |