Complex field reconstruction of optical OFDM signals based on temporal transport-of-intensity equation

Masayuki Matsumoto

doi:10.1364/OE.438149

1. Introduction

A number of studies have been recently devoted to field reconstruction in direct-detection (DD) optical receivers. Such receivers, which allow modulation of both amplitude and phase of optical signals and enable electrical dispersion compensation, realize cost-effective and yet high-speed medium- and short-distance data transmission. Widely studied field reconstruction DD receivers include Kramers-Kronig receivers [1], self-coherent receivers using signals having spectral gap between the carrier and the signal [2], iterative signal-signal beat interference cancellation receivers [3], and self-coherent Stokes-vector receivers [4]. Recently, carrier-assisted differential detection (CADD) receivers that are compatible with double-sideband (DSB) optical signals have been proposed [5]. More general computational phase retrieval approaches have also been applied to field reconstruction of carrier-less optical signals [6,7].

In [8], field reconstruction by solving temporal transport-of-intensity equation (TIE) has been studied. This approach allows non-iterative computation of the signal phase and can be used for reconstruction of DSB signals. It has been pointed out, on the other hand, that the approach requires relatively high carrier powers for accurate phase reconstruction and is sensitive to electrical noise at detection. In [8,9], field reconstruction for single carrier (SC) Nyquist 16QAM signals was studied. Most recently, the TIE-based reconstruction for optical orthogonal frequency division multiplexing (OFDM) signals was studied, and it was shown that subcarriers closer to the carrier have worse performance [10]. In [10], however, the important issue of the influence of electrical noise introduced at detection was not explicitly considered.

The temporal TIE is a second-order differential equation with respect to time. Its solution involves integration with time, which naturally enhances low-frequency error and noise. This leads to the dependency of the reconstruction error of OFDM signals on the subcarrier frequency relative to the carrier. The issue of low-frequency noise amplification in phase reconstruction based on TIE has been known in its application to two-dimensional phase imaging [11,12]. In this paper, field reconstruction of optical OFDM signals using temporal TIE is studied under the influence of electrical noise at detection with an attention being paid to how the reconstruction error is generated in the solution of TIE.

2. TIE-based phase reconstruction

In dispersive media, power and phase of optical signals obey

(1)$$\frac{\partial }{{\partial t}}\left( {P\frac{{\partial \phi }}{{\partial t}}} \right) = \frac{1}{{{\beta _2}}}\frac{{\partial P}}{{\partial z}},$$

which is called temporal TIE [13], where β₂ is the group-velocity dispersion coefficient of the medium. By integrating the TIE, phase of the signal ϕ(t) is obtained in terms of P(t,z) and its derivative with respect to distance ∂P(t,z)/∂z. The derivative is approximately evaluated by a finite difference formula. When forward or central difference approximation is used, the derivative is, respectively, given by [P(t,z + d)-P(t,z)]/d or [P(t,z + d)-P(t,z-d)]/(2d), where d is a small distance increment. This indicates that when P(t,z) and either or both of P(t,z + d) and P(t,z-d) are available, the phase ϕ(t) is obtained by solving the TIE. When electrical noise including thermal, shot, and quantization noise at detection is considered, the finite difference is expressed as

(2a)$$\frac{{P({t,z + d} )- P({t,z} )}}{d} = \frac{{\partial P}}{{\partial z}} + \frac{{{n_1}(t )- {n_0}(t )}}{d} + O(d ),$$

or

(2b)$$\frac{{P({t,z + d} )- P({t,z - d} )}}{{2d}} = \frac{{\partial P}}{{\partial z}} + \frac{{{n_1}(t )- {n_{ - 1}}(t )}}{{2d}} + O({{d^2}} ),$$

where n_±1 and n₀ are noise added to P(t,z ± d) and P(t,z), respectively. When the noises, n₁, n_-1, and n₀ are uncorrelated with each other, contributions of the noise in the right-hand sides (RHSs) of (2a) and (2b) are enhanced as smaller distance increment d is chosen for better finite difference approximation. Because the finite difference approximation error is in second order of d in the central difference (2b), larger d can be used with consequently reduced contribution from noise when the central difference is used. The central difference is assumed in the analysis in this paper. The receiver structure using the central difference approximation is shown in Fig. 1. It is noted that the received signal is detected after dispersions -β₂d, 0, and β₂d are applied to the signal in Fig. 1. Phase reconstruction can be performed using other combinations of dispersions 0, β₂d, and 2β₂d, for example.

Fig. 1. A direct-detection receiver using TIE-based field reconstruction.

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The TIE (1) can be solved by different methods. Originally it was solved after being discretized to a set of linear equations having a tridiagonal symmetric coefficient matrix [8]. In [9], (1) was solved by using discrete Fourier transform (DFT), which is computationally efficient when fast Fourier transform (FFT) algorithm is used. In this paper, the latter approach is used. With a procedure described in Appendix and in [9], the solution of (1) is given by

(3)$$\boldsymbol{\mathrm{\phi }}\, = \,{{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\bf FP}^{\prime}{{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\mathbf F{\textbf g}} - ({d^{\prime}/d^{\prime\prime}} ){{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\bf FP}^{\prime}{\bf 1}\, + \,{c_2}{\bf 1}$$

in a vectorial form, where g = [g(0), g(Δt),…,g((M-1)Δt)]^t is a vector whose components are the RHS of (1), g(t)=[P₁(t)-P_-1(t)]/(2β₂d), sampled at t = mΔt (m=0, 1,…., M-1), and [•]^t indicates transpose. The total time width over which the functions are defined is T = MΔt. P is a diagonal matrix whose diagonal component P_j is given by the inverse of the power P(t), P_j=1/P(jΔt) (j=0, 1, …, M-1), D is also a diagonal matrix corresponding to integration with respect to time in the frequency domain,

(4)$$D{^{\prime}_j} = \left\{ {\begin{array}{cc} 0&{({j = 0} )}\\ {\dfrac{i}{{j{\omega_0}}}}&{\left( {1 \le j \le \dfrac{M}{2} - 1} \right)}\\ { - \dfrac{i}{{({M - j} ){\omega_0}}}}&{\left( {\dfrac{M}{2} \le j \le M - 1} \right)} \end{array}} \right.,$$

where ω₀=2π/T and i is the imaginary unit. 1 is the all-ones vector. F and F ^-1 are coefficient matrices appearing in the DFT and inverse DFT expressions, respectively. d and d are direct current (DC) components of P F ^-1D F g and P 1, respectively. c₂ is an integration constant. Derivation of (3) is detailed in Appendix.

3. Analysis of complex filed reconstruction of optical OFDM signals

3.1 Optical OFDM signals

In this numerical study, the OFDM symbol duration T and number of subcarriers M are fixed at 9.1429 ns and 256, respectively, with which subcarrier frequency spacing and total bandwidth are Δf=1/T=109.38 MHz and M/T=28 GHz, respectively. Data are encoded in 16QAM format. No cyclic prefix is used for the purpose of studying fundamental performance of OFDM signal reconstruction in this paper. Transmission fiber dispersion is compensated for by frequency-domain equalization after signal reconstruction.

For successful field reconstruction by solving the TIE, signal power P(t) at the receiver needs to stay well away from zero. To fulfill this condition, carrier is added to the signal at the transmitter. The carrier is located at the center of the signal spectrum, which forms a DSB signal as shown in Fig. 2. Compared with transmission schemes using single-sideband (SSB) signals, systems using DSB signals have an advantage that radio-frequency (RF) oscillators and/or an optical frequency shifter are not needed in the transmitter [14] or it avoids wasting the resolution of transmitter digital-to-analogue converter (DAC) caused by adding virtual carriers [15]. Use of tight optical filters to make the signal SSB at the receiver is also not needed. In the field reconstruction by solving TIE, however, the DSB spectrum of the signal does not reduce the bandwidth of the receiver. This is because the signal-signal beat should be fully captured and relatively wide electrical signal bandwidth must be preserved in solving the TIE. In this study, a sampling rate of 112 GHz (equivalent to 4 samples per symbol as used in the SC transmission [9] and also in the OFDM system [10].) is employed. This sampling rate is similar to that used in typical Kramers-Kronig receivers [1].

Fig. 2. Spectrum of a DSB OFDM signal.

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3.2 Comparison between single-carrier and OFDM signal reconstruction

Firstly 28 Gbaud Nyquist 16QAM SC and OFDM signal reconstruction performances are compared. The Nyquist SC signal has the roll-off factor of zero so that its spectral width is the same as that of the OFDM signal shown in Fig. 2. Different transmission distances are considered with a group velocity dispersion of D=17ps/nm/km. Transfer function of the fiber is given by H(ω) =exp(iβ₂ω²z/2) where β₂= -λ²D/(2πc), λ, c, and z are the wavelength, the light velocity in vacuum, and transmission distance. The transmission fiber nonlinearity is not considered. All the other optical and electrical components are linear and their frequency responses are flat and ideal for both SC and OFDM signals. Amplified spontaneous emission (ASE) noise is added to the signal before detection, which defines optical signal to noise ratio (OSNR). The reconstruction performance is evaluated by the bit error rate (BER) after the field reconstruction followed by dispersion compensation. The bit error rates are numerically evaluated under the use of Gray code mapping. Theoretical BER determined by the additive Gaussian noise for the Gray-coded 16QAM signal is given by

(5)$$BER = \frac{3}{8}\textrm{erfc}\left( {\sqrt {\frac{{OSN{R_e}\cdot{B_{\textrm{ref}}}}}{{10B}}} } \right).$$

In (5), erfc (•) is the complementally error function, OSNR_e is the effective OSNR defined as P_s/(NB_ref) where P_s, N, and B_ref are the signal power, noise power density, and reference noise bandwidth (12.5 GHz), respectively. B is equal to the bandwidth of the ASE added to the signal B_ASE in the case of SC transmission or to the product of subcarrier number M and the subcarrier frequency spacing Δf in the case of OFDM transmission. In the numerical study in this paper, both of B_ASE and MΔf are chosen to be 28 GHz, so that the theoretical BERs for SC and OFDM signals are equal.

Figure 3 shows BERs as a function of the carrier to signal power ratio (CSPR) for (a) SC and (b) OFDM transmissions for different transmission distances when OSNR_e is equal to 21 dB. The OSNR where the carrier power is included in the signal power is given by OSNR = (1+CSPR) OSNR_e. Solid curves in Fig. 3(c) show the dependency of the required CSPR for the BER equal to 3 × 10⁻³, a target for systems using hard-decision forward error correction (FEC) codes. (The CSPR requirements are relaxed by about 0.8-1.1 dB when the target BER is 2 x 10⁻², a threshold for 20% soft-decision FEC.) In Fig. 3(c), peak-to-average power ratios (PAPRs) excluding the carrier for SC and OFDM signals are also shown in dotted curves. The SC transmission requires larger CSPR as the transmission distance is increased because PAPR grows as the distance [16]. For the OFDM signal, PAPR, so the required CSPR, are almost constant in the distance range considered in this calculation. This is consistent with the result reported in [17], where the impact of the Kerr nonlinearity in transmission fibers on the PAPR performance of optical OFDM signals was studied.

Fig. 3. BER versus carrier to signal power ratio (CSPR) for (a) SC and (b) OFDM transmission of different distances. Horizontal dash-dotted lines indicate theoretical BER (5) in the absence of reconstruction error. (c) Solid curves: CSPR at which BER is 3 × 10⁻³ versus transmission distance for SC and OFDM signals, dotted curves: PAPR of the signals without carrier. OSNR_e=21 dB.

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Then, the finite difference approximation error in the RHS of TIE and the electrical noise added to the detected signals are considered. Real uncorrelated Gaussian noises n₀, n₁, and n_-1 are added to the detected powers P₀, P₁, and P_-1, respectively, before phase reconstruction calculations. Assuming AC (alternating current)-coupled photodetection [18], where averaged value is subtracted from the detected electrical signal, the SNR is defined as

(6)$$ S N R=<\left[P_{i}(t)-< P_{i}(t)>\right]^{2}>/<n_{i}^{2}(t)>(i=0,1, \text { or }-1). $$

where <•> represents an average. It is noted that the AC-coupled detection reduces the required resolution of the following analog-to-digital converters (ADCs) while separate estimation of the DC component of the electrical signal (P₀(t)) is needed in the electrical signal processing as discussed in [18,19]. (For P₁(t) and P_-1(t), estimation of their DC components is not needed because they are cancelled with each other in calculating their difference in evaluating ∂P/∂z.) Fig. 4 shows BER as a function of the dispersion Dd = -2πcβ₂d/λ² (c is the light velocity in vacuum and λ is the wavelength) given to the signal in the receiver for (a) SC and (b) OFDM transmission. Transmission distance, CSPR, and OSNR_e are 100 km, 11 dB, and 21 dB (OSNR = 33.4 dB), respectively. Finite difference approximation errors in the evaluation of ∂P/∂z in the TIE, degrade the performance as large dispersion Dd is used as shown in the black solid curves in Fig. 4. When noise is considered, the TIE-based phase reconstruction fails at smaller dispersion approaching to zero because difference in powers P₁-P_-1 becomes concealed by the noise. In Fig. 4, it is seen that OFDM signal transmission gives somewhat better performance than SC transmission. This can be attributed to the frequency selective nature of the phase reconstruction error, which will be discussed in the next section. The SC signals suffer from the reconstruction error as a whole, while the errors are concentrated on low-frequency subcarriers in the case of OFDM signals. The symbol error averaged over all the subcarriers is thus expected to be smaller in the OFDM transmission.

Fig. 4. Bit error rates versus dispersion Dd given to the signal in the receiver for the evaluation of intensity derivative with respect to distance. (a) SC and (b) OFDM signal transmission. Horizontal dash-dotted lines indicate theoretical BER (5) in the absence of reconstruction error under no electrical noise addition.

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3.3 Subcarrier dependence of OFDM signal reconstruction error

The BERs of OFDM signals shown in Fig. 4(b) are those averaged over all the subcarriers. Dependence of the BER on subcarrier number is shown in Fig. 5 for three values of dispersion at the electrical SNR of 70 dB. Transmission distance, CSPR, and OSNR_e are again 100 km, 11 dB, and 21 dB, respectively. The subcarrier number n_sc indicates the subcarrier location with respect to the carrier at the center of the spectrum. The subcarrier frequency is given by n_scΔf. These curves show that the reconstruction is severely degraded as the subcarrier frequency approaches to zero. For the dispersion Dd as large as 200ps/nm, BER becomes large also at the spectral edges with n_sc = ±128. Constellation diagrams for n_sc = 10, 60, and 126 at Dd=200ps/nm are shown in Fig. 6. The BER dependency on the subcarrier number has already been discussed in [10].

Fig. 5. Dependency of BER on subcarrier number of OFDM signal. Dispersion Dd given to the signal in the receiver is either 60, 130, or 200ps/nm. Electrical SNR of the detected signal is 70 dB.

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Fig. 6. Constellation diagrams of subcarriers n_sc=10, 60, and 126 at Dd=200ps/nm.

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3.4 Sources of signal reconstruction error

To identify the source of the reconstruction error, we analyze the phase and field reconstruction errors in the solution process of the TIE. When ∂P/∂z in the RHS of TIE is approximated by the central finite difference, the TIE can be written as

(7)$$\frac{\partial }{{\partial t}}\left( {P\frac{{\partial \phi }}{{\partial t}}} \right) = \frac{1}{{{\beta _2}}}\frac{{\partial P}}{{\partial z}} + \left( {\frac{{{P_1}(t )- {P_{ - 1}}(t )}}{{2{\beta_2}d}} - \frac{1}{{{\beta_2}}}\frac{{\partial P}}{{\partial z}}} \right) + \frac{{{n_1} - {n_{ - 1}}}}{{2{\beta _2}d}}.$$

The second term in the RHS shown in parentheses represents the finite difference approximation error and the third term is the electrical noise. Figure 7(a) and (b) show the spectra of P₀(t) and ∂P/∂z(t) in the absence of electrical noise. ∂P/∂z is calculated also by the central finite difference but with small distance increment d (Dd=0.1ps/nm is used in the numerical evaluation of the true intensity derivative in this section. No appreciable change in the intensity derivative is observed when smaller Dd is used.). In Fig. 7(a) frequency components appearing in the range outside of the signal spectrum correspond to the signal-signal beat. Direct current component is zero in the spectrum of ∂P/∂z in Fig. 7(b). This is explained by that the signal energy, or the time average, of the signal power is conserved with distance in dispersive media. The spectrum of ∂P/∂z has peaks at the edges of the signal spectrum. This is because the oscillation in distance of the beat between the carrier and subcarrier is faster for larger subcarrier frequency.

Fig. 7. Spectra of (a) signal power P₀(t), (b) ∂P/∂z, and (c) inverse of signal power 1/P₀(t).

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Figure 8(a) shows the spectrum of the finite difference approximation error

(8)$$\mathrm{\Delta }{\mathbf g} = \frac{{{P_1} - {P_{ - 1}}}}{{2{\beta _2}d}} - \frac{1}{{{\beta _2}}}\frac{{\partial P}}{{\partial z}}$$

given by the second term in the RHS of the TIE. Three dispersion values Dd=100, 150, and 200ps/nm are assumed. The error grows as the dispersion Dd is increased and has sharp peaks at the edges of the signal spectrum as the spectrum of ∂P/∂z does. Figure 8(b) shows the spectra of integration of the finite difference approximation error with respect to time

(9)$${{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\mathbf F}\mathrm{\Delta }{\mathbf g} = \smallint \left( {\frac{{{P_1} - {P_{ - 1}}}}{{2{\beta_2}d}} - \frac{1}{{{\beta_2}}}\frac{{\partial P}}{{\partial z}}} \right)dt.$$

It can be seen that the part of the spectra close to zero frequency is enhanced as compared with Fig. 8(a). This is because the time integration enhances the low-frequency components of the integrand. Then, the spectra of (9) divided by P(t)

(10)$${\mathbf P}^{\prime}{{\mathbf F}^{ - 1}}{\mathbf D}{^{\prime}}{\mathbf F}\mathrm{\Delta }{\mathbf g} = \frac{1}{{{P_0}}}\smallint \left( {\frac{{{P_1} - {P_{ - 1}}}}{{2{\beta_2}d}} - \frac{1}{{{\beta_2}}}\frac{{\partial P}}{{\partial z}}} \right)dt$$

are shown in Fig. 8(c). The spectra, Fig. 8(c), are convolution of the spectra shown in Fig. 8(b) and that of 1/P₀, which is shown in Fig. 7(c), along the frequency axis. Finally, the spectra of phase error obtained by integration of (10) with respect to time again

(11)$$\Delta \boldsymbol{\mathrm{\phi }} = {{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\mathbf {FP}}^{\prime}{{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\mathbf F}\Delta {\bf g}$$

are shown in Fig. 8(d).

Fig. 8. Spectra of phase errors generated by the finite difference approximation error in the RHS of TIE. Spectra of (a) finite difference approximation error, (b) its integration with respect to time, (c) integration of the finite difference approximation error divided by P₀(t), and (d) generated phase error.

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The corresponding spectra in the phase error evolution originated from the electrical noise appearing in the RHS of the TIE, that are (n₁-n_-1)/(2β₂d), $\smallint ({{n_1} - {n_{ - 1}}} )/({2{\beta_2}d} )dt$, $P_0^{ - 1}\smallint ({{n_1} - {n_{ - 1}}} )/({2{\beta_2}d} )dt$, and the noise-induce phase error, are shown in Fig. 9(a), (b), (c), and (d), respectively, where the SNR of the electrical noise is 70dB and the dispersion values are Dd=100, 150, and 200ps/nm. Larger dispersion reduces the influence of the electrical noise. The noise source which is flat in spectra gives rise to phase errors having sharp peak at zero frequency, which is caused by the integration process in solving the TIE. Vertical axes in corresponding panels in Figs. 8 and 9 are common, which enables comparison of contributions to the total phase error from the two error sources. These features in phase reconstruction error result in the dependence of OFDM performance on the subcarrier frequency as shown in Fig. 5.

Fig. 9. Spectra of phase errors generated by the electrical noise in the RHS of TIE. Spectra of (a) electrical noise, (b) its integration with respect to time, (c) integration of the electrical noise divided by P₀(t), and (d) generated phase error.

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3.5 Modified performance of OFDM signal reconstruction

Because the most serious reconstruction errors appear in the low-frequency region in the signal spectrum, avoiding using the low-frequency subcarriers close to the carrier increases the averaged performance of OFDM signal reconstruction, which has been practiced in [10]. Dashed, solid, and dotted curves in Fig. 10 represent averaged BERs when all the 256 subcarriers are used, 10, and 20 most low-frequency subcarriers are not used, respectively, as a function of dispersion Dd. Transmission distance, OSNR_e, and CSPR are 100 km, 21 dB, and 11 dB, respectively. By not using 10 subcarriers (3.9 percent of 256 subcarriers), optimum averaged BER is reduced from 7.9 × 10⁻³ to 1.6 × 10⁻³ when electrical SNR is 70 dB.

Fig. 10. BER performance of OFDM signals for different amounts of electrical noise. BERs are averaged over all subcarriers. Solid and dotted curves are obtained when 10 and 20 low-frequency subcarriers immediately close to the carrier are not used, respectively. Dashed curves are BERs when all the subcarriers are used. Transmission distance is 100 km. CSPR and OSNR_e are 11 dB and 21 dB, respectively.

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4. Approach to reduce the noise sensitivity of the signal reconstruction

As discussed in the previous sections, the signal reconstruction based on solving temporal TIE is sensitive to the electrical noise introduced at detection. In the last numerical example not using 10 low-frequency subcarriers, the required electrical SNR for the BER to be smaller than 2 × 10⁻² is 58.0 dB at Dd∼220ps/nm. The required electrical SNR of 58 dB is considerably large. When it is assumed that the electrical noise is generated by quantization at the analog-to-digital converter (ADC), although other noises such as thermal and shot noise may not be neglected in practice, the required effective number of bits (ENOB) of the ADC is estimated to be (SNR [dB]-1.76)/6.02 > 9.4, which is demanding.

When the received optical signal power is sufficiently large, the quantization noise will dominate the electrical noise. Under this condition, one approach to remedy the sensitivity to the quantization noise in measuring P₁(t)-P_-1(t) in the finite difference approximation of the RHS of TIE is to use analog subtraction before ADC using the receiver structure shown in Fig. 11. For measuring the small difference between the power waveforms P₁(t) and P_-1(t), group delay difference between the delays in the upper and lower dispersive elements in Fig. 11 needs to be compensated either in the optical or electrical domain before subtraction. Use of optical delay is assumed in Fig. 11. Δτ shown in Fig. 11 should be chosen as Δτ = τ_g,1 - τ_g,-1, where τ_g,1 and τ_g,-1 are the group delays in the upper and lower dispersive elements, respectively. When τ_g,-1 > τ_g,1 is satisfied, the delay element should be placed in the upper, instead of the lower, dispersion branch. After subtraction in the electrical domain and low-noise amplification, ADC is performed to measure the power difference. It is expected that this scheme can avoid the difference signal from being masked by the quantization noise in the ADC. The SNR in the detection of P₁(t)-P_-1(t) is now defined as SNR=<[P₁(t)-P_-1(t)-<P₁(t)- P-₁(t)>]²>/<n²(t) > . Figure 12 shows BER versus dispersion for various SNRs. Numerical parameters are the same as those used in Fig. 10 with 10 out of 256 subcarriers not used. Because the SNR is defined as the ratio of the power difference to the noise, deterioration of BER as the dispersion is decreased for fixed SNR as shown in Fig. 10 is not observed. Noise tolerance is somewhat improved by this detection method: required SNR for the BER to be smaller than 2 × 10⁻² becomes 52.8 dB (corresponding ENOB of 8.5).

Fig. 11. A receiver structure where the subtraction between P₁(t) and P_-1(t) is performed in the analog electrical domain before ADC.

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Fig. 12. BER versus dispersion Dd given to the signal in the receiver shown in Fig. 11. 10 out of 256 subcarriers are not used. P₁(t) and P_-1(t) are subtracted in the analog electrical domain before ADC.

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5. Conclusion

Field reconstruction of optical OFDM signals in a direct detection receiver using temporal transport of intensity equation is presented. It is shown that the phase reconstruction based on TIE generates large errors in the low frequency spectral region close to the carrier. This leads to better reconstruction performance for OFDM signals than for SC signals especially when the low-frequency subcarriers are not used. Sensitivity to electrical noise of the method based on solving the TIE is still large. Considering its advantageous features that the solution process is non-iterative and it is compatible to DSB signals, further studies to reduce the noise sensitivity and required carrier power are highly expected.

Appendix

In this study, the differential equation of the form

(12)$$\frac{\partial }{{\partial t}}\left( {P\frac{{\partial \phi }}{{\partial t}}} \right) = g,$$

where P(t) and g(t) are known functions, while ϕ(t) is unknown function to be obtained, is solved using DFT. g(t) in (12) represents (∂P/∂z) /β₂ in (1).

The first step in solving (12) is to integrate (12) with respect to time t. When g(t) is expressed in the form

(13)$$g(t )= \mathop \sum \limits_n {h_n}{e^{ - in{\omega _0}t}} = {h_0} + \mathop \sum \limits_{n \ne 0} {h_n}{e^{ - in{\omega _0}t}},$$

where ω₀=2π/T, T being the time duration over which g(t) is defined, its integration is given by

(14)$$\mathop \smallint \nolimits_{}^t g({t^{\prime}} )dt^{\prime} = \mathop \sum \limits_{n \ne 0} \frac{i}{{n{\omega _0}}}{h_n}{e^{ - in{\omega _0}t}} + {h_0}t + {c_1},$$

where c₁ is a constant. Discretizing t into t_m=mΔt, with m=0, 1, 2, …, M-1 and Δt = T/M, g(t) at t_m is expressed as

(15)$${g_m} = g({t = m\mathrm{\Delta }t} )= \mathop \sum \limits_n {h_n}{e^{ - in{\omega _0}m\mathrm{\Delta }t}} = \mathop \sum \limits_n {h_n}\textrm{exp}\left( { - \frac{{i2\pi mn}}{M}} \right),$$

which is a form of the inverse discrete Fourier transform (IDFT). (15) can be written in a matrix form as g = F ^-1h, where the coefficient matrix F ^-1 is composed of elements exp (-i2π m n /M). The corresponding transform in the forward direction from the time to frequency domains is h = Fg, which is the DFT. Using matrices F ^-1 and F, the discretized representation of (14) is expressed as

(16)$${{\mathbf F}^{ - 1}}{\bf D}^{\prime}{\bf Fg}\, + \,{h_0}{\bf t}\, + \,{c_1}{\mathbf 1},$$

where matrix D and vectors t and 1 are defined in the main text. h₀ is the DC component or the time average of g(t)= (∂P/∂z)/β₂. When the medium is a lossless dispersive medium as assumed in this study, the signal energy or the time average of the signal power is invariant with distance so that h₀ is equal to zero.

Then, the integration of g(t), (14), is divided by the power P(t). The resulting expression in the discretized vectorial form is

(17)$${\bf P}^{\prime}{{\mathbf F}^{ - 1}}{\bf D}^{\prime}{\bf Fg}\, + \,{c_1}{\bf P}^{\prime}{\bf 1},$$

where P is a diagonal matrix whose diagonal component is given by P _m=1/P(t = mΔt).

Now the solution of (12), the phase ϕ, is obtained by the integration of (17) with respect to time again. The same matrix operations as used in deriving (16) results in the expression of the phase ϕ in a vectorial form

(18)$$\boldsymbol{\mathrm{\phi }} = {{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\mathbf F}({\bf P}^{\prime}{{\mathbf F}^{ - 1}}{\mathbf D}^{\prime}{\bf Fg} + {c_1}{\bf P}^{\prime}{\bf 1}) + {d_0}{\mathbf t} + {c_2}{\bf 1},$$

where d₀ is the time average of P F ^-1D F g + c₁P 1 and c₂ is an integration constant. In this analysis, it is assumed that the complex field $\sqrt P \textrm{exp}({i\phi } )$ has the periodicity in time with a period of T. In the numerical solution of (12), it is further assumed that ϕ is also periodic in time with the period T. This assumption imposes that d₀ is equal to zero and leads to $d^{\prime}+c_{1} d^{\prime \prime}=0$, or $c_{1}=-d^{\prime} / d^{\prime \prime}$, where $d^{\prime}$ and $d^{\prime\prime}$ are DC components of P F ^-1D F g and P 1, respectively. This gives the expression of the phase solution of (3).

Funding

Japan Society for the Promotion of Science (20K04464); Support Center for Advanced Telecommunications Technology Research Foundation.

Disclosures

The author declares no conflicts of interest.

Data availability

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

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Complex field reconstruction of optical OFDM signals based on temporal transport-of-intensity equation

Abstract

1. Introduction

2. TIE-based phase reconstruction

3. Analysis of complex filed reconstruction of optical OFDM signals

3.1 Optical OFDM signals

3.2 Comparison between single-carrier and OFDM signal reconstruction

3.3 Subcarrier dependence of OFDM signal reconstruction error

3.4 Sources of signal reconstruction error

3.5 Modified performance of OFDM signal reconstruction

4. Approach to reduce the noise sensitivity of the signal reconstruction

5. Conclusion

Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Equations (19)

Optics Express