Influence of the device parameters in ICRBD on SSB-OOFDM signal with reduced guard band

Jianxin Ma

doi:10.1364/OE.22.029636

1. Introduction

Orthogonal frequency division multiplexing (OFDM) has been extensively used in both wired and wireless broadband communication systems due to its resistance to the inter symbol interference (ISI) in the multipath channel [1–7]. Now, it is increasingly being considered as an advanced modulation format for optical communication due to its high dispersion tolerance and enhanced spectral efficiency (SE). Coherent optical OFDM (CO-OFDM) with higher- level QAM has achieved an SE of 14bit/s/Hz [8]. However, the lasers with very narrow linewidth (<100kHz) are needed at both transmitter and receiver along with some additional processing to solve the frequency offset and phase noise. Direct detection optical OFDM (DD-OOFDM) requires a simple receiver structure because only a square-law photodiode (PD) can recover the transmitted OFDM signal [9, 10] without the requirement of compensation for the frequency and phase offset. This is because the OOFDM signal and the optical carrier come from the same laser with the synchronous frequency and phase offset, and thus the frequency and phase offset are cancelled out as they are heterodyne beating in the PD. Single-sideband optical OFDM (SSB-OOFDM) is promising in that it cannot only overcome the inherent chromatic dispersion induced fading effect associated with double-sideband OOFDM, but also improve the SE. However, signal-signal beat interference (SSBI) is generated with the desired OFDM signal during the DD by PD, and degrades the received OFDM signal. Several methods have been proposed to minimize the penalty due to SSBI [9–23]. In [10], a sufficient guard band (GB) is used to avoid the spectrum overlap of the SSBI and the desired RF-OFDM signal. Since the minimum GB should be no less than the bandwidth of OOFDM signal, the SE is half comparing with the CO-OFDM system. Another interleaved SSB-OOFDM scheme without GB can eliminate the impact of SSBI by bearing data only on the even subcarriers, but its SE is also reduced [11]. Although the SE can be further improved by directly reducing the GB of the SSB-OOFDM signal, the system performance of the DD is degraded due to the spectral overlap of the SSBI and the desired OFDM signal. Some research works have been done to address this issue [12–26]. Cao et al. have proposed to use the turbo coding to compensate for the SSBI [13]. An iterative detection is also proposed to reduce the SSBI [14], which requires large carrier-to-signal power ratio (CSPR) and at least 4 iterations are needed at the optimal CSPR of 4dB. Yang et al. have proposed a digital signal processing algorithm based on the nonlinear system of equations to solve both CD and SSBI [15]. Recently, blockwise signal-phase-switching (SPS) is proposed to cancel SSBI of the specially designed OOFDM signal with two iteration schemes [16]. In [17, 18], based on the symmetry of the interleaved OFDM symbol in the time domain, the half-cycled OFDM signal with the reduced GB is used to overcome the SSBI without the SE reduction, but two electrical OFDM transceivers are required in the real time system. Moreover, in [19], the optical carrier is separated from the OOFDM signal and amplified by an EDFA to suppress the SSBI. The EDFA as well as the optical hybrid used to couple the two optical tones makes the scheme very complex. The receiver in [20] using an optical filter to abstract the optical carrier from the SSB-OOFDM signal is simple, but the two optical paths between the two OCs have different lengths, so the coherency between the optical carrier and the OOFDM is reduced. In [21, 22], a beat interference cancellation receiver (BICR) is proposed and investigated in detail to mitigate the SSBI in the reduced GB SSB-OOFDM systems via one optical filter and one balanced receiver with improved tolerance to both phase noise and PMD. However, half optical power is used to generate the SSBI without contribution to the received OFDM signal, so the receiver sensitivity is reduced. Moreover, the receiver configuration is not fully symmetric although a balanced photodiode (BPD) is used. In [24], a simple SSBI cancellation receiver (ICRBD), consisting only of an optical interleaver (IL), a 2 × 2 3dB optical coupler (OC) and a BPD, has been proposed to eliminate the SSBI, but the parameters of the devices in the ICRBD have ideal values and the analysis of the receiver parameters has not been performed in detail. Moreover, the noise in optical domain is not considered. In the real system, the device parameters in the ICRBD usually deviate away from the ideal values due to their tolerable error. These deviations destroy the coherent addition for both the signal and the noise, and so lead to not only the reduced enhancement of the received OFDM signal, but also the imperfect cancellation of the beating interferences and noises. The nonideal values of the parameters have great influence on the received signal, so it is meanful to explore how the deviations of these parameters influence the received signal performance, and how much the errors can be tolerated in the ICRBD.

In this paper, the influence of the device parameters in the ICRBD on the received signal converted from the noised SSB-OOFDM signal with the reduced GB is investigated in detail theoretically and by simulations. The conditions to achieve the better reception of the SSB-OOFDM signal with the reduced GB are given. This paper is organized as follows. In Section 2, the principle of ICRBD for receiving the noised SSB-OOFDM signal with the reduced GB is described mathematically. In Section 3, the effect of the receiver performance is analyzed for the main parameters of the devices in detail. In Section 4, the simulations are conducted to verify our theoretical results, and the simulation results are discussed in detail. At last, conclusions are drawn in Section 5.

2. Principles

Figure 1 shows the schematic of the proposed SSBI cancellation receiver based on balanced detection (ICRBD) for receiving the noised SSB-OOFDM signal with the reduced GB. The noised SSB-OOFDM signal and each device in the ICRBD are described mathematically.

Fig. 1 The principle diagram of SSBI cancellation receiver with the balanced detection (ICRBD) for the noised SSB-OOFDM signal. IL: interleaver; OC: optical coupler; PD: photodiode.

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2.1 The noised SSB-OOFDM signal

The SSB-OOFDM signal may be generated by optical double sideband modulation of RF-OFDM signal with optical SSB filtering [25], as shown in the inset of Fig. 1, and can be expressed with the optical noise as

\begin{array}{l} E (t) = \hat{θ} (t) {E_{C} e^{j ω_{o} t} + E_{S} \sum_{i = - \infty}^{\infty} [\sum_{n = - N / 2}^{N / 2 - 1} c_{n i} Π (t - i T_{s}) e^{j (ω_{o} + ω_{R F} + ω_{n}) t}] + w (t)} \\ \equiv E_{C} (t) + E_{S} s (t) + w (t) \end{array}

Here E_C and ω_o = 2πf_o are the field amplitude and angular frequency of the optical carrier, respectively;

\hat{θ} (t)

denotes the polarization unit vector of the generated SSB-OOFDM signal. E_S is the field amplitude of the optical OFDM signal; c_ni is the complex signal carried by nth subcarrier in the ith OFDM symbol with N subcarriers; П(t) is pulse shaping function, which is 1 in [0,T] and 0 otherwise; T is the OFDM symbol duration;

s (t) = \sum_{i = - \infty}^{\infty} [\sum_{n = - N / 2}^{N / 2 - 1} c_{n i} Π (t - i T_{s}) e^{j ω_{n} t}]

is the baseband OFDM signal with normalized amplitude, and n is the integer; f_RF = ω_RF/2π is the frequency of the RF-OFDM signal; f_n = ω_n/2π = n/T is the frequency of the nth subcarrier of the baseband OFDM signal. Since −N/2≤n<N/2, the bandwidth of the OFDM signal is W_S = N/T and the GB between the OOFDM signal and the optical carrier is W_G = f_RF−W_S/2. w(t) is the optical noise in time domain which has flat broadband spectrum on both polarizations. Since the OOFDM signal and its optical carrier come from the same laser with linear polarization and are transmitted over the same optical path, they are always rotated synchronously and keep parallel polarizations even if the rotation is random.

2.2 Mechanism of the ICRBD on the noised SSB-OFDM signal

As shown in Fig. 1, the ICRBD consists of an IL, a 2 × 2 OC and a BPD. The three devices are bridged in turn by the parallel optical paths. After passing though the IL, OC and BPD along with the connecting optical paths, the SSB-OOFDM signal is converted to the electrical RF-OFDM signal with the suppressed SSBI and noise. Here the mechanism of the ICRBD is analyzed in detail with the influence of the device parameters and optical path characteristics on the received OFDM signal.

In the ICRBD, the noised SSB-OOFDM signal E(t), as given by Eq. (1), is firstly demultiplexed as the OOFDM signal and the optical carrier by the IL along with the sliced noise in frequency domain. If the IL has flat passband and stopband with sharp edges for its two output ports, the transmission functions in frequency domain can be expressed as

H_{1} (ω) = {\begin{matrix} H_{1} (ω_{0}) \approx 1 & in passband \\ H_{1} (ω_{o} - ω_{R F}) \approx ρ_{1} ~ 0 & in stopband \end{matrix},

H_{2} (ω) = {\begin{matrix} H_{2} (ω_{0}) \approx ρ_{2} ~ 0 & in stopband \\ H_{2} (ω_{o} - ω_{R F}) \approx 1 & in passband \end{matrix} .

Here the insert loss of IL is neglected, ρ_{1, 2} is the suppression depth of the residual component in stopbands, and correspondingly, I_{1, 2} = −20log(ρ_{1, 2}) is defined as the isolation of IL. Usually, ρ_{1, 2} has a smaller and almost equal value. Typically, ρ is 0.1 for the isolation of 20dB.

After the IL, the OOFDM signal and the optical carrier can be expressed in time domain as

\begin{array}{l} {E^{'}}_{S} (t) = F^{- 1} {H_{2} (ω) F {E (t)}} \\ \approx H_{2} (ω_{o} + ω_{R F}) {E_{S} \sum_{i = - \infty}^{\infty} [\sum_{n = - N / 2}^{N / 2 - 1} c_{n i} Π (t - i T_{s}) e^{j ω_{n} t}] \hat{θ} (t) + w_{S} (t)} e^{j (ω_{o} + ω_{R F}) t} \\ + H_{2} (ω_{o}) [E_{C} \hat{θ} (t) + w_{C} (t)] e^{j ω_{o} t} \\ = [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j (ω_{o} + ω_{R F}) t} + ρ_{2} [E_{C} \hat{θ} (t) + w_{C} (t)] e^{j ω_{o} t} \\ \approx [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j (ω_{o} + ω_{R F}) t} \end{array}

\begin{array}{l} {E^{'}}_{C} (t) = F^{- 1} {H_{1} (ω) F {E (t)}} \\ \approx H_{1} (ω_{o} + ω_{R F}) {E_{S} \sum_{i = - \infty}^{\infty} [\sum_{n = - N / 2}^{N / 2 - 1} c_{n i} Π (t - i T_{s}) e^{j ω_{n} t}] \hat{θ} (t) + w_{S} (t)} e^{j (ω_{o} + ω_{R F}) t} \\ + H_{1} (ω_{o}) [E_{C} \hat{θ} (t) + w_{C} (t)] e^{j ω_{o} t} \\ = ρ_{1} [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j (ω_{o} + ω_{R F}) t} + [E_{C} \hat{θ} (t) + w_{C} (t)] e^{j ω_{o} t} \\ \approx [E_{C} \hat{θ} (t) + w_{C} (t)] e^{j ω_{o} t} \end{array}

Here F{·} and F⁻¹{·} are the forward and inverse Fourier transform, respectively. The first approximation is decent for the IL with the filter flat pass and stop bands while the second is right as the isolation of the IL is higher enough for both equations. It can be seen that (1) there is residual optical carrier along with the OOFDM signal due to the finite isolation of the IL, so does the optical carrier, while the residuals reduce with the increase of the IL isolation; (2) the noise in optical domain is sliced by the IL on both polarizations. Although the IL slices the noise power spectrum into two periodically interleaved tributaries due to its periodical transmission function, for simplification, the noise within one period is considered here; (3) when the transmission function of the IL has a nonflat passband, the IL distorts both the OOFDM signal and the optical carrier, but more harm will be done to the former because of its broader bandwidth. The mathematical analysis of the influence of the transmission function on the transmitted OOFDM signal is very complicated. For simplifying the mathematical deduction, the rectangle transmission function with constant suppression depth of ρ_{1, 2} is considered here, and the influence of the titled edges of the IL will be discussed based on the simulations below. The lightwave fields after the IL are expressed in matrix vector as

\begin{array}{l} E^{'} (t) = (\begin{matrix} {E^{'}}_{S} (t) \\ {E^{'}}_{C} (t) \end{matrix}) = (\begin{matrix} [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j (ω_{o} + ω_{R F}) t} + ρ_{2} [E_{C} \hat{θ} (t) + w_{C} (t)] e^{j ω_{o} t} \\ ρ_{1} [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j (ω_{o} + ω_{R F}) t} + [E_{C} \hat{θ} (t) + w_{C} (t)] e^{j ω_{o} t} \end{matrix}) \\ = (\begin{matrix} 1 & ρ_{2} \\ ρ_{1} & 1 \end{matrix}) (\begin{matrix} [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j ω_{R F} t} \\ [E_{C} \hat{θ} (t) + w_{C} (t)] \end{matrix}) e^{j ω_{o} t} \end{array}

After the IL, two parallel optical paths guide the OOFDM signal and the optical carrier separately to a 2 × 2 OC. The optical paths may introduce phase shift between the two tones and rotate their polarization directions. The length difference of the two optical paths between IL and OC, referred as ∆l₀, will introduce a relative time delay of ∆τ₀ = n_o∆l₀/c between the OOFDM signal and the optical carrier. Since the optical carrier is not modulated by the signal and has much narrow linewidth, this time delay can be represented by a phase difference, φ = 2πn_o∆l₀/λ_c. Here n_o is the efficient refractive index, and λ_c is the central wavelength of the optical carrier. If the length difference is much smaller than coherent length of the optical carrier, the coherency degradation of the SSB-OOFDM signal is minor, and so the intensity noise converted from the beating phase noise is small enough to be neglected. Meanwhile, the polarization states of the OOFDM signal and the optical carrier, which are parallel originally, may be changed by the separate optical paths even if they have identical length. Since the optical paths in the ICRBD are fixed in stable condition, here we assume the time-invariant polarization rotations ${\hat{θ}}_{2}$ and ${\hat{θ}}_{1}$ are introduced on the OOFDM signal and the optical carrier, respectively. If both the time delay and the polarization rotation are considered simultaneously, the transmission matrix of the parallel optical paths connecting IL and OC can be expressed as

T_{L} = (\begin{matrix} {\hat{θ}}_{2} & 0 \\ 0 & {\hat{θ}}_{1} e^{j φ} \end{matrix})

For the 2 × 2 OC with the additional loss of α_x and coupling coefficient (CC) of c, its transmission matrix is

T_{O C} = α_{x} (\begin{matrix} \sqrt{1 - c} & j p \sqrt{c} \\ j p \sqrt{c} & \sqrt{1 - c} \end{matrix})

where p is the sign, which is 1 or −1. After the OC, the OOFDM signal and the optical carrier are combined together again, and two recombined SSB-OOFDM signals are obtained,

\begin{array}{l} E^{″} (t) = (\begin{matrix} E_{2} (t) \\ E_{1} (t) \end{matrix}) = T_{O C} T_{L} E^{'} (t) \\ \approx α_{x} (\begin{matrix} {\hat{θ}}_{2} \sqrt{1 - c} [E_{S} s (t) + w_{S} (t)] e^{j ω_{R F} t} + {\hat{θ}}_{1} j p e^{j φ} \sqrt{c} [E_{C} \hat{θ} (t) + w_{C} (t)] \\ {\hat{θ}}_{2} j p \sqrt{c} [E_{S} s (t) + w_{S} (t)] e^{j ω_{R F} t} + {\hat{θ}}_{1} e^{j φ} \sqrt{1 - c} [E_{C} \hat{θ} (t) + w_{C} (t)] \end{matrix}) e^{j ω_{o} t} \end{array}

Here the approximation is decent as the isolation (I = −20logρ) of IL is higher since ρ<0.1 (I>20dB). From Eq. (9), it can be seen that ± 90° additional phase shifts are introduced between the OOFDM signal and the optical carrier for the two recombined SSB-OOFDM signals from the two output ports of OC, respectively. Moreover, there is another 90° phase shift between the two SSB-OOFDM signals from the two tributaries. If an ideal 3dB optical coupler is used, viz. c = 0.5, the OOFDM signals output from the two ports have equal amplitude, so do the optical carriers.

Similar to the above, the parallel optical paths connecting the OC with the BPD can also cause the time delay difference and asynchronous polarization rotation for the two recombined SSB-OOFDM signals. If we assume that the time delay difference caused by the two optical paths from the OC to the BPD is ∆τ₁ = n_o∆l₁/c, and the polarization rotations of the two recombined SSB-OOFDM signals are ${\hat{ϑ}}_{1}$ and ${\hat{ϑ}}_{2}$ , respectively, the lightwave fields injected in the BPD become

(\begin{matrix} E_{2 i n} (t) \\ E_{1 i n} (t) \end{matrix}) = (\begin{matrix} E_{2} (t) {\hat{ϑ}}_{1} \\ E_{1} (t - Δ τ_{1}) {\hat{ϑ}}_{2} \end{matrix})

Since the two PDs in BPD usually have much symmetrical configuration and similar parameters, we assume they have identical sensitivity μ, and their dark current noise and the thermal noise generated by the two PDs are denoted by w_D₁(t) and w_D₂(t). The photocurrents generated by the PDs in the BPD can be expressed as

(\begin{matrix} I_{2} (t) \\ I_{1} (t) \end{matrix}) = μ (\begin{matrix} {| E_{2 i n} (t) |}^{2} \\ {| E_{1 i n} (t) |}^{2} \end{matrix}) + (\begin{matrix} w_{D 2} (t) \\ w_{D 1} (t) \end{matrix}) = μ (\begin{matrix} {| E_{2} (t) |}^{2} \\ {| E_{1} (t - Δ τ_{1}) |}^{2} \end{matrix}) + (\begin{matrix} w_{D 2} (t) \\ w_{D 1} (t) \end{matrix})

Although the PDs are insensitive to the polarization of the injected optical signals, the beating photocurrent between the OOFDM signal and the optical carrier depends on their polarization angle. Since each optical path between the OC and BPD rotates the OOFDM signal along with the optical carrier synchronously, the angle between them keeps unchanged, and so the generated photocurrents are insensitive to the polarization rotation of the recombined SSB-OOFDM signals by the optical paths between the OC and BPD.

The length difference of the RF waveguides between the PDs and the subtractor in the BPD also has influence on the coherent addition of the two photocurrents. If we assume that the time delay difference caused by the two RF waveguides in electrical domain is ∆τ₂, the output photocurrent from the BPD can be expressed as

\begin{array}{l} I (t) = I_{2} (t) - I_{1} (t - Δ τ_{2}) = μ {| E_{2} (t) |}^{2} + w_{D 2} (t) - μ {| E_{1} (t - Δ τ_{1} - Δ τ_{2}) |}^{2} - w_{D 1} (t - Δ τ_{2}) \\ = μ α_{x}^{2} [1 - c + ρ_{1}^{2} c - 2 p {\hat{θ}}_{1} \cdot {\hat{θ}}_{2} ρ_{1} \sqrt{(1 - c) c} \sin φ] {| E_{S} s (t) \hat{θ} (t) + w_{S} (t) |}^{2} \\ ​ - μ α_{x}^{2} [c + ρ_{1}^{2} - ρ_{1}^{2} c + 2 p {\hat{θ}}_{1} \cdot {\hat{θ}}_{2} ρ_{1} \sqrt{(1 - c) c} \sin φ] \cdot {| E_{S} s (t - Δ τ_{1} - Δ τ_{2}) \hat{θ} (t) + w_{S} (t - Δ τ_{1} - Δ τ_{2}) |}^{2} \\ + μ α_{x}^{2} [ρ_{2}^{2} - ρ_{2}^{2} c + c - 2 p {\hat{θ}}_{1} \cdot {\hat{θ}}_{2} ρ_{2} \sqrt{(1 - c) c} \sin φ] {| E_{C} \hat{θ} (t) + w_{C} (t) |}^{2} \\ - μ α_{x}^{2} [1 - c + ρ_{2}^{2} c + 2 p {\hat{θ}}_{1} \cdot {\hat{θ}}_{2} ρ_{2} \sqrt{(1 - c) c} \sin φ] \cdot {| E_{C} \hat{θ} (t - Δ τ_{1} - Δ τ_{2}) + w_{C} (t - Δ τ_{1} - Δ τ_{2}) |}^{2} \\ + 2 μ α_{x}^{2} Re {[ρ_{2} (1 - c) + ρ_{1} c + j p {\hat{θ}}_{1} \cdot {\hat{θ}}_{2} \sqrt{(1 - c) c} (ρ_{1} ρ_{2} e^{j φ} - e^{- j φ})] [E_{C}^{*} \hat{θ} (t) + w_{C}^{*} (t)] \\ \cdot [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j ω_{R F} t}} \\ - 2 μ α_{x}^{2} Re {[ρ_{2} c + ρ_{1} (1 - c) + j p {\hat{θ}}_{1} \cdot {\hat{θ}}_{2} \sqrt{(1 - c) c} (e^{- j φ} - ρ_{1} ρ_{2} e^{j φ})] \\ \cdot [E_{C}^{*} \hat{θ} (t - Δ τ_{1} - Δ τ_{2}) + w_{C}^{*} (t - Δ τ_{1} - Δ τ_{2})] \\ \cdot [E_{S} s (t - Δ τ_{1} - Δ τ_{2}) \hat{θ} (t - Δ τ_{1} - Δ τ_{2}) + w_{S} (t - Δ τ_{1} - Δ τ_{2})] e^{j ω_{R F} (t - Δ τ_{1} - Δ τ_{2})}} \\ + w_{D 2} (t) - w_{D 1} (t - Δ τ_{2}) \end{array}

Equation (12) gives the photocurrent detected by the ICRBD without electrical amplification. It can be seen that the received OFDM signal, the beating interference and other noises are all related with the IL, the OC, and the optical and electrical paths. While the polarization rotation of the recombined SSB-OOFDM signals caused by the optical paths between the OC and the BPD has no influence on the received OFDM signal.

In [24], the ideal case is considered where ρ₁ = ρ₂ = 0, ∆τ₀ = ∆τ₁ = ∆τ₂ = 0, ${\hat{θ}}_{1}$ = ${\hat{θ}}_{2}$ = 0, c = 0.5, ${\hat{ϑ}}_{1}$ = ${\hat{ϑ}}_{2}$ = 0 without the optical noise, and the IL has the ideal rectangle filters, so the signal photocurrent is simplified as $I (t) = - 2 p μ α_{x}^{2} Im {E_{C}^{*} E_{S} s (t) e^{j ω_{R F} t}}$ . While, for the noised SSB-OOFDM signal detected by the ICRBD with ideal parameters, Eq. (12) becomes

\begin{array}{l} I (t) = - 2 p μ α_{x}^{2} Im {[E_{C}^{*} \hat{θ} (t) + w_{C}^{*} (t)] \cdot [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j ω_{R F} t}} \\ = - 2 p μ α_{x}^{2} Im {[E_{C}^{*} E_{S} s (t) + E_{C}^{*} \hat{θ} (t) \cdot w_{S} (t) + E_{S} s (t) \hat{θ} (t) \cdot w_{C}^{*} (t) + w_{C}^{*} (t) \cdot w_{S} (t)] e^{j ω_{R F} t}} \end{array}

Figure 1 demonstrates the principle of constructive or destructive additions of both the signal and noise by the ICRBD with the ideal parameters for the noised SSB-OOFDM signal. It can be seen that although the self beating interferences and noises |E_C(t)|², |E_S(t)|², |w_S(t)|², |w_C(t)|², and some cross beatings E_C(t) × w_C(t), E_S(t) × w_S(t) can be cancelled out completely by the ICRBD, the other cross beating noises E_C(t) × w_S(t), E_S(t) × w_C(t), w_S(t) × w_C(t) are constructively added and coexist with the OFDM signal in frequency domain. These beating noises limit the improvement of the received OFDM signal. Fortunately, these beating noises are usually smaller for the SSB-OOFDM signal with higher optical signal to noise ratio (OSNR).

Based on the theoretical deduction, here we will extensively investigate the influence of the parameters on the received OFDM signal to obtain the determined factors on the performance of the ICRBD and the conditions to achieve the better reception of the SSB-OOFDM signal with the reduced GB.

3. Influence of the device parameters in ICRBD on the received OFDM signal

According to Eq. (12), the time delays ∆τ₁ and ∆τ₂ make the analysis of the received OFDM signal more complex to study, so we divide the issue into two cases: (1) the case with ∆τ₁ + ∆τ₂ = 0, and (2) the case with ∆τ₁ + ∆τ₂≠0. If we assume ρ₁ = ρ₂ = ρ and the angle between ${\hat{θ}}_{1}$ and ${\hat{θ}}_{2}$ has a constant value of θ in the real system, the received OFDM signal photocurrent for the case (1) with ∆τ₁ + ∆τ₂ = 0 can be simplified as

\begin{array}{l} I (t) = μ α_{x}^{2} [(1 - 2 c) (1 - ρ^{2}) - 4 p ρ \sqrt{(1 - c) c} \cos θ \sin φ] {| E_{S} s (t) \hat{θ} (t) + w_{S} (t) |}^{2} \\ ​ ​ ​ ​ + μ α_{x}^{2} [(1 - 2 c) (ρ^{2} - 1) - 4 p ρ \sqrt{(1 - c) c} \cos θ \sin φ] {| E_{C} \hat{θ} (t) + w_{C} (t) |}^{2} \\ + 4 p μ α_{x}^{2} \sqrt{(1 - c) c} \cos θ Re {j (ρ^{2} e^{j φ} - e^{- j φ}) [E_{C}^{*} \hat{θ} (t) + w_{C}^{*} (t)] [E_{S} s (t) \hat{θ} (t) + w_{S} (t)] e^{j ω_{R F} t}} \\ + w_{D 2} (t) - w_{D 1} (t - Δ τ_{2}) \\ = 4 p μ α_{x}^{2} \sqrt{(1 - c) c} \sqrt{1 + ρ^{4} - 2 ρ^{2} \cos 2 φ} \cos θ Re {E_{C}^{*} E_{S} s (t) e^{j [ω_{R F} t - \tan^{- 1} (\frac{1 + ρ^{2}}{1 - ρ^{2}} \tan φ) + \frac{π}{2}]}} - - S_{R} (t) \\ \begin{array}{l} + μ α_{x}^{2} [(1 - 2 c) (1 - ρ^{2}) - 4 p ρ \sqrt{(1 - c) c} \cos θ \sin φ] {| E_{S} s (t) \hat{θ} (t) + w_{S} (t) |}^{2} \\ + μ α_{x}^{2} [(1 - 2 c) (ρ^{2} - 1) - 4 p ρ \sqrt{(1 - c) c} \cos θ \sin φ] {| E_{C} \hat{θ} (t) + w_{C} (t) |}^{2} \\ + 4 p μ α_{x}^{2} \sqrt{(1 - c) c} \sqrt{1 + ρ^{4} - 2 ρ^{2} \cos 2 φ} \cos θ \\ \cdot Re {[E_{C}^{*} \hat{θ} (t) \cdot w_{S} (t) + E_{S} \hat{θ} (t) \cdot w_{C}^{*} (t) s (t) + w_{S} (t) \cdot w_{C}^{*} (t)] e^{j [ω_{R F} t - \tan^{- 1} (\frac{1 + ρ^{2}}{1 - ρ^{2}} \tan φ) + \frac{π}{2}]}} \\ + w_{D 2} (t) - w_{D 1} (t - Δ τ_{2}) \end{array}} - - N_{R} (t) \\ = S_{R} (t) + N_{R} (t) \end{array}

Here the signal and noise photocurrents are

S_{R} (t) = 4 p μ α_{x}^{2} \sqrt{(1 - c) c} \sqrt{1 + ρ^{4} - 2 ρ^{2} \cos 2 φ} \cos θ Re {E_{C}^{*} E_{S} s (t) e^{j [ω_{R F} t - \tan^{- 1} (\frac{1 + ρ^{2}}{1 - ρ^{2}} \tan φ) + \frac{π}{2}]}},

\begin{array}{l} N_{R} (t) \approx μ α_{x}^{2} [(1 - 2 c) (1 - ρ^{2}) - 4 p ρ \sqrt{(1 - c) c} \sin φ \cos θ] \cdot {| E_{S} s (t) |}^{2} \\ ​ ​ ​ ​ + μ α_{x}^{2} [(1 - 2 c) (ρ^{2} - 1) - 4 p ρ \sqrt{(1 - c) c} \sin φ \cos θ] \cdot {| E_{C} |}^{2} \end{array}

According to Eq. (15) and (16), the magnitudes of the received OFDM signal and the noise depend on the isolation of IL, the CC of OC, and the relative time delay and asynchronous polarization rotation between the OOFDM signal and the optical carrier caused by the two optical paths from the IL to the OC.

The transmission spectrum outline of IL shears the spectra of both the OOFDM signal and the noise, so it has great influence on the received OFDM signal. Usually the IL with proper bandwidth and sharper edge cannot only do little harm on the OOFDM signal, but also suppress more optical noise, while the IL with too narrow bandwidth or titled edge often shrinks the outside subcarriers of the OOFDM signal. Here we check the influence of the IL on the received OFDM signal only with a rectangle transmission function analytically. According to Eq. (14), the isolation of IL, I = −20logρ, and the phase shift between the OOFDM signal and the optical carrier, φ, vary the magnitude of the OFDM signal in a combined pattern as $K = \sqrt{1 + ρ^{4} - 2 ρ^{2} \cos 2 φ}$ , as shown in Fig. 2. The received OFDM signal varies periodically with the phase shift φ, and gets the maximum at φ = (q + 0.5)π and the minimum at φ = qπ, here q is the integer. Although the magnitude of the received OFDM varies periodically with the phase shift φ, its influence on the signal magnitude becomes minor as the isolation of IL increases. For the IL with I>20dB, the magnitude deviation is less than 1%.

Fig. 2 The influence of the isolation of IL, I = −20logρ, and the phase difference, φ, caused by the optical paths between the IL and OC on the magnitude of the received OFDM signal.

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Equation (15) also shows that the asynchronous polarization rotation of the OOFDM signal and the optical carrier reduces the signal magnitude by cosθ. The signal gets the maximum magnitude as the optical carrier and the OOFDM signal have parallel polarization states (θ = 0) and disappears as they have vertical polarization states. Since the slope of cosine varies slowly as θ is close to zero, smaller polarization angle between the OOFDM signal and the optical carrier does not cause larger reduction of the received OFDM signal.

According to Eq. (15), the CC of the OC shrinks the received OFDM signal magnitude by the factor $χ = \sqrt{(1 - c) c}$ , as shown in Fig. 3, and the signal gets maximum magnitude as c = 0.5, so an ideal 3dB OC is preferred in the real system. The reduction of the signal magnitude is slow as the CC is deviated away from the ideal value of 0.5, and the signal is reduced by only 1% and 8.5% when the CC is deviated from 0.5 by 0.1 and 0.2, respectively. So the received OFDM signal is insensitive to the small error of the power splitting ratio of the 3dB OC.

Fig. 3 The influence of the coupling coefficient of the OC, c, on the magnitude of the received OFDM signal.

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From Eq. (15), it can be seen that the phase of the received OFDM signal is also shifted by $Φ = \tan^{- 1} (\frac{1 + ρ^{2}}{1 - ρ^{2}} \tan φ)$ . Fig. 4 shows the phase shift versus φ at different isolations of the IL. If I>20dB, namely, ρ<0.1, the phase caused by the difference of the optical paths can be approximated as Φ ≈φ. It increases almost linearly with the relative phase shift between the OOFDM signal and the optical carrier, φ, caused by the length difference of the optical paths between the IL and the OC, as shown in Fig. 4. The phase of the received signal can keep stable since the isolation of the IL and the delay caused by the optical paths are time invariant.

Fig. 4 The influence of the phase difference φ caused by the length difference of the optical paths between the IL and OC on the phase of the OFDM signal at different isolations of IL, I = −20logρ.

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It can be seen from Eq. (16) that the noise comes from three aspects: the self beating of the noised optical carrier, |E_C(t) + w_C(t)|² and that of the noised OOFDM signal, |E_S(t) + w_S(t)|²; the cross beating noise E_C(t) × w_S(t), E_S(t) × w_C(t) and w_S(t) × w_C(t); and the noise from the BPD, w_D₁(t), w_D₂(t). The self beating |E_C(t) + w_C(t)|² consists of |E_C(t)|², |w_C(t)|² and E_C(t) × w_C(t), and has much small power spectrum density within the frequency band of the OFDM signal since |E_C(t)|² is the dc component although it has consider magnitude; |w_C(t)|² and E_C(t) × w_C(t) are small. Moreover, they are further suppressed by the factor $[(1 - 2 c) (ρ^{2} - 1) - 4 p ρ \sqrt{(1 - c) c} \sin ϕ \cos θ]$ according to Eq. (16). Among the three self beating noise |E_S(t) × w_S(t)|² of |E_S(t)|², |w_S(t)|² and E_S(t) × w_S(t), the SSBI, viz., |E_S(t)|², has considerable magnitude ranging from the dc to W_S in frequency domain and overlaps with the received OFDM signal for the SSB-OOFDM signal with the reduced GB. In the ideal case with c = 0.5, φ = 0, the SSBI is cancelled out completely by the factor $[(1 - 2 c) (ρ^{2} - 1) - 4 p ρ \sqrt{(1 - c) c} \sin φ \cos θ]$ . If the condition is not met, the SSBI appears with considerable magnitude and contaminates the received OFDM signal. The beating noises, E_C(t) × w_S(t), E_S(t) × w_C(t), w_S(t) × w_C(t), have the spectra overlapped completely with the received OFDM signal and varies in the same way as the received OFDM signal by the factor $\sqrt{(1 - c) c} \sqrt{1 + ρ^{4} - 2 ρ^{2} \cos 2 φ} \cos θ$ . Moreover, the beating noises w_S(t) × w_C(t) include two beating components from the two polarizations. These noises cannot be suppressed by the ICRBD, but we can reduce them by improving the OSNR of the SSB-OOFDM signal. So the proper filtering in optical domain to enhance the OSNR of the SSB-OOFDM signal is crucial to suppress the beating noises w_S(t) × w_C(t). The noises from the BPD usually come from the dark current noise and thermal noise, and can be cancelled out by the subtractor. The cancellation degree depends on their correlativity. Extremely, the noise power spectrum density will be doubled if they are irrelative completely. Usually its density is also smaller to be neglected if the signal power is considerable.

Generally, the SSBI has dominant impact on the degradation of the received OFDM signal, and other noises become considerable only as the SSBI is almost cancelled out completely. In the real system, what we desired is that the signal is as large as possible, while the noise is as small as possible since the electrical signal to noise ratio (SNR) of the received OFDM signal is related with its EVM by SNR~EVM⁻² [26]. So, the influence of the SSBI as well as the noises on the signal can be described by SNR or EVM. For the 16-QAM signal with EVM of 12.5%, the equivalent BER is about 1.3 × 10⁻⁴, which is below the FEC threshold of 2 × 10⁻³.

For the case with ∆τ₁ + ∆τ₂≠0, namely, the relative time delay caused by the optical paths between the OC and BPD is not balanced by the relative time delay of the electrical paths between the photodiode pair and subtractor. According to Eq. (12), the combined time delay (∆τ₁ + ∆τ₂) has great influence on the output signals. When the time delay is much smaller than the sample period, it can be equalized as a phase shift, which may change or even reverses the coherent addition of both the signal and the noise. It means that the received OFDM signal, which should have maximum magnitude by constructive addition, is reduced, while the noise, which should be cancelled out by destructive addition, may appear with a considerable magnitude. If the time delay is comparable to or bigger than the sample period, it not only changes the coherent additions of both the signal and the noises, but also causes the ISI and makes the error symbol rate increase rapidly, which degrades the system performance seriously. So the ICRBD is very sensitive to this combined time delay due to the fast variation of the waveform of the OFDM signal in time domain.

4. Simulation setup and results

To verify our theoretical analysis above, a simulated optical back-to-back SSB-OOFDM link with the ICRBD is built, as shown in Fig. 5. The procedure is similar to that in [24]. In the transmitter, the baseband OFDM signal is generated by IFFT module. The pseudo random binary sequence (PRBS) with the word length of 2¹²−1 is mapped to 16-QAM data with I- and Q-branches, and then they are input into the IFFT module for serial-parallel conversion, inverse fast Fourier transform (IFFT) and parallel-serial conversion. The IFFT size is 256. Among the 256 subcarriers, 128 subcarriers are allocated for bearing the signals while the others are zero-padded at the edges for oversampling. No cyclic prefix and pilot subcarrier are added because no fiber transmission is conducted. After digital-to-analog conversion, the baseband OFDM signal is upconverted by the RF with the frequency of 10GHz. The 10GHz bandwidth RF-OFDM signal with the 10GHz RF is shown by the spectrum in Fig. 5(a) with the resolution of 100MHz. Then, the RF-OFDM signal is modulated on the lightwave from a CW laser diode with the central frequency of 193.1THz and linewidth of 100MHz, as shown in Fig. 5(b). The RF-OFDM signal drives an optical Mach-Zehnder modulator (MZM) in pull-push pattern with a peak-to-peak voltage swing of 0.5V_π to reduce the nonlinear distortion. The carrier-to-signal power ratio (CSPR) can be adjusted flexibly via the dc bias voltage of MZM. The generated double sideband optical OFDM signal is filtered by a tunable optical filter for suppressing the lower sideband to produce the SSB-OOFDM signal. By properly adjusting the dc bias voltage, the bandwidth of the optical filter and the pump optical power, noise source power, the SSB-OOFDM signals with the CSPR of 1.2dB and the optical power of 4.4dBm are generated with the GB of 5GHz. Figure 5(c) is the optical spectrum of SSB-OOFDM signal with the bandwidth of 10GHz and the noise floor of −43dBm at the resolution of 100MHz. After detected by the ICRBD, the typical spectrum of the received OFDM signal is given as Fig. 5(d). Based on the simulation platform, we will check the influence of the IL, the OC and the optical and electrical paths connecting them in the ICRBD on the received OFDM signal in turn as follows.

Fig. 5 The simulation link of the SSB-OOFDM signal with reduced GB and detected by the ICRBD. The insets are the spectra of (a) the transmitted RF-OFDM signal and (d) the received RF-OFDM by ICRBD, (b) the optical spectrum of CW laser diode and (c) the transmitted SSB-OOFDM signal at the resolution of 100MHz. CW LD: continuous wave laser diode; LO: local oscillator; MZM: Mach-Zehnder modulator; TOF: tunable optical filter; EDFA: Erbium doped fiber amplifier; IL: interleaver; OC: optical coupler; PD: photodiode.

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4.1 Interleaver (IL)

Here we firstly investigate the influence of the IL in the ICRBD on the received OFDM signal. In the ICRBD, the IL with the rectangle and Gaussian filter at different bandwidths and orders is used while the other devices keep the ideal parameters. The passband profiles of different IL with the bandwidth of 15GHz are shown in Fig. 6(a). One of the edges of the IL at the isolation of 20dB is fixed at the middle of the GB, viz., at 193.1025THz, which corresponds to zero frequency in Fig. 6(a). The other edges are adjusted by the bandwidth of the filter and the tilt of the edge is tuned by varying the order of the Gaussian optical filter. The influences of the bandwidth and outline of the filter in the IL on the received OFDM signal are shown in Fig. 6 (b) by the curves of the EVM versus the bandwidth of the rectangle filter and Gaussian filters with different orders. It can be seen that, for the rectangle and higher order Gaussian filters, the EVM reduces rapidly to 12.5% and reaches a floor as the IL bandwidth increases from 12.5GHz to 13GHz since the damage of the upper edge of the filter on the highest frequency subcarriers is avoided. For the rectangle and 16-order Gaussian filters, the EVM floors keep flat even as the IL bandwidth is increased to 100GHz, and the IL with the bandwidth from 13 to 100GHz does no harm to the received OFDM signal since it can always keep a flat transmission spectrum within the OFDM signal spectrum. While, for the IL with 10 and 6-order Gaussian filters, the EVMs maintain flat floors before its bandwidth is increased to 70 and 40GHz, respectively. This is because the lower edge of the lower-order Gaussian filters is expanded and overlapped with the OOFDM signal with the increase of the IL bandwidth. Consequently, the titled edge of the IL shrinks the lower frequency subcarriers. Extremely, the two titled edges of the filter damage the OOFDM signal on both sides, so the penalty is minimized only when the OFDM signal is located at the middle of the passband symmetrically. As can be seen from Fig. 6(b), the EVM of the IL with 2-order Gaussian filter gets minimum of 13.1% at the bandwidth of 15GHz.

Fig. 6 (a) the passband profiles of the IL with rectangle and different-order Gaussian filters at 15GHz bandwidth, and (b) the EVM versus the bandwidth of the IL with the rectangle and Gaussian optical filters at different orders for the SSB-OOFDM signal with the GB of 5GHz, CSPR of 1.2dB, optical power of 4.4dBm and noise floor at −43dBm.

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To check the influence of the isolation of IL on the performance of ICRBD, a typical 25/50GHz IL with 10-order Gaussian filter is used, and the isolation of the IL, I, varies from 0.5 to 40dB. The EVM versus the isolation of IL along with the received signal and noise powers is shown in Fig. 7. It can also be seen that, with the decrease of the IL isolation, the received signal power reduces, which agrees well with the theoretical prediction in Eq. (15). This attributes to the fact that the leaped optical carrier and OOFDM signal from the two ports increase with the decrease of I and generate another beating OFDM signal, which adds destructively to the main signal. However, the EVM maintains nearly 12% as I>20dB and increases slowly to 12.25% at I = 3dB, almost keeps constant. This is because the SSBI is cancelled out completely, and the beating noises E_C(t) × w_S(t), E_S(t) × w_C(t) and w_S(t) × w_C(t) experiment the same variation as the received OFDM signal E_C(t) × E_S(t) according to Eq. (16). The noises, w_D₁(t) and w_D₂(t) from the two PDs, have much small power spectrum density. So the SNR keeps almost constant as I>3dB, as verified by simulated results in Fig. 7. With the further decrease of I below 2dB, the signal power reduces rapidly, and the noises, w_D₁(t) and w_D₂(t), become dominant due to their incomplete cancellation by the BPD, consequently, the SNR begins to reduce, which increases the EVM of the received OFDM signal.

Fig. 7 The EVM, received signal and noise powers versus the isolation of IL (I) for the SSB-OOFDM signal with the GB of 5GHz, CSPR of 1.2dB, optical power of 4.4dBm, and the noise floor at −43dBm detected by the ICRBD with the 25/50GHz IL.

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4.2 Parallel optical paths between IL and OC

The parallel optical paths between the IL and OC have influence on the received OFDM in two aspects: introduction of the phase shift between the OOFDM signal and the optical carrier because of their optical length difference, and the asynchronous polarization rotation of the OOFDM signal and the optical carrier, as described by the matrix in Eq. (7). To check the influence of the phase shift between the OOFDM signal and the optical carrier on the received OFDM signal, the simulation platform above is used with an additional phase shifter in one of the two parallel optical paths to emulate the phase shift, φ, caused by their length difference. Based on the results in the subsection above, a 25/50GHz IL with 10-order Gaussian filter at the isolation of 10, 15, 20, 25 and 30dB is used here, and other devices have ideal parameters. Figure 8 shows the EVM of the received OFDM signal versus the phase shift φ at different isolations of IL along with the signal and noise power. It can be seen that the EVM as well as the signal and noise powers varies periodically with φ, and they get maximum and minimum at φ = (q + 0.5) × 180° and q × 180°, respectively, here q is the integer. Their swing ranges reduce with the increase of I since the leaped optical carrier and OOFDM signal from the two ports decrease. When φ = q × 180°, the EVM gets minimal and becomes insensitive to the isolation of IL simultaneously. While, as the phase difference deviates away from q × 180° to (q + 0.5) × 180°, both the signal and noise powers increase, and their increments also increase as I reduces. However, since the SSBI increases more rapidly than the signal, the SNR of the received OFDM signal decreases, and the signal performance, represented by the EVM in Fig. 8, does not improve but degrades as the signal power increases. The phenomenon reaches extreme as φ = (q + 0.5) × 180°, which is verified by the RF spectra and the constellation in Fig. 8. It can be seen from Fig. 8 that the SSBI in the GB reduces according to the RF spectra and the constellation diagrams become clear as the isolation of the IL increases from 10dB to 30dB. This is in accord with the prediction by Eq. (15) and (16). Of course, if the isolation of IL is 20dB or larger, the EVM and magnitude of the received OFDM signal suffer little from this relative phase shift between the OOFDM signal and the optical carrier.

Fig. 8 The EVM, received signal and noise power versus the phase shift φ between the OOFDM signal and the optical carrier caused by the optical paths between the IL and OC at different isolations of IL. Inset: the RF spectra and constellations of the received OFDM signal with φ = 90 ° and different isolation of the IL.

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Figure 9 demonstrates the influence of the asynchronous polarization rotation of the OOFDM signal and the optical carrier. In the simulation platform, a polarization controller (PC) is introduced on the optical carrier to tune the polarization angle θ between the OOFDM signal and the optical carrier. The isolation of IL is set to 20dB and the phase difference φ is set to 0° and 90° at the two extremes. It can be seen from Fig. 9 that the EVM curves keep flat within 2% variation over the range of θ from −50° to 60° while the signal power along with the noise reduces rapidly as the polarization angle |θ|. increases. This is because the main noise from the SSBI changes in the same way as the OFDM signal according to Eq. (16), which can also be observed from the RF spectra and the constellation diagrams in Fig. 9. Since the noise from the BPD keeps a small value, it causes only a small reduction of the SNR of the received signal, and thus the EVM increases slowly as the polarization angle |θ| increases. But, as |θ| goes beyond 50°, the noise from the BPD is comparable to the reduced SSBI, so the SNR of the received signal begins to reduce rapidly. Generally, the EVM of the received OFDM signal is insensitive to the asynchronous polarization rotation of the OOFDM signal and the optical carrier over a wide range, and the EVM can keep flat at the polarization angle θ from −50° to 60° although the received signal magnitude varies greatly.

Fig. 9 The EVM, received signal and noise powers versus the polarization angle θ between the OOFDM signal and the optical carrier caused by the optical paths between the IL and OC with the phase shift φ of 0° and 90°. Inset: the RF spectra and constellations of the received OFDM signal with different phase shift φ and polarization angle θ.

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According to the EVM curves as well as that of the signal and noise powers for the cases with the phase shifts φ of 0° and 90° in Fig. 9, the asynchronous polarization rotation of the OOFDM signal and the optical carrier has almost the same influence on the received OFDM signal. This is associated with the simulated results in Fig. 8 at the IL isolation of 20dB.

4.3 Coupling coefficient of optical coupler (OC)

To check the influence of the OC on the received OFDM signal, the simulation platform above is used based on the SSB-OOFDM signal with the phase shifts φ of 0° and 90° and the polarization angles θ of 0°, 30° and 60° between the OOFDM signal and the optical carrier. The CC of the OC is varied from 0.1 to 0.9 and the isolation of the 25/50GHz IL is fixed at 20dB. Figure 10 shows the EVM curves as well as the signal and noise power curves. For the case with θ = 0° and φ = 0°, the EVM gets the minimum of 12.1% at the CC of 0.5, where the signal gets the maximum, and the noise, coming mainly from the SSBI, gets the minimum. As the CC deviates away from the optimal value, the received signal power reduces while the noise increases, which results in the increase of the EVM. But the EVM can be kept below 14% (BER = 5.3 × 10⁻⁴ for 16QAM) even for 1dB deviation of the CC. On the other hand, the polarization angle θ has great influence on the received OFDM signal. Since the signal power reduces a little more rapidly than the noise power with the increase of θ as the CC is close to 0.5, the minimal EVMs increase slowly from 12.1% to 12.4% and to 14.4% when the polarization angle is increased from 0 to 30° and to 60° as φ = 0°, respectively, as given in Fig. 10(a). With the deviation of the CC away from 0.5, the signal power reduces more slowly than that of the noise, and so the EVM increases rapidly. This trend increases as θ increases. The noise power curves in Fig. 10(a) also show that, at the three polarization angles, the noise gets equal power at the CC of about 0.15 and 0.85 symmetrically.

Fig. 10 The EVM, the received signal and noise powers versus the coupling coefficient (c) of the OC at the phase difference of (a) φ = 0° and (b) φ = 90° with the polarization angle of θ = 0°, 30°, 60° between the OOFDM signal and the optical carrier, respectively.

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Figure 10(b) shows the case at the other extreme of φ = 90°, the curves of the signal and noise power versus the CC have similar but asymmetrical outline compared to the case of φ = 0° in Fig. 10(a). The signal gets a little larger maximum at c = 0.5 than that in Fig. 10(a), and the noise gets the minimum at c = 0.4 where the SSBI can be cancelled out completely. Correspondingly, the minimal EVMs of 12.1% and 12.6% are shifted to the CC of 0.4 for the cases of φ = 0° and 30°, respectively. The minimal EVM keeps at c = 0.5 but increases to 15.2% as φ = 60°. Compared to the case of φ = 0° in Fig. 10(a), the CC, at which the noise has the same power for the three polarization angles, is shift to 0.2. The outline variation of the EVM curves results mainly from the noise, which is described well by Eq. (16).

4.4 Optical/electrical paths between OC and subtractor

After the OC, the recombined SSB-OOFDM signals with different phase difference between the OOFDM signal and the optical carrier are injected into the two PDs of the BPD via two parallel optical paths, and then the converted electrical signals are transmitted to the subtractor over the parallel electrical paths. The asynchronous polarization rotation and the time delay difference of the two recombined SSB-OOFDM signals are emulated with a second PC and a second tunable optical time delay, respectively, on one optical path. The time delay difference caused by electrical paths between the BPD and electrical subtractor is emulated with a tunable electrical time delay. The curves of the EVM, the signal and noise power of the received OFDM signal versus the polarization angle ϑ are shown in Fig. 11. The simulation platform with the same parameters as above is used except that a PC is introduced into one of the two optical paths to adjust the polarization angle ϑ between the two recombined SSB-OOFDM signals. The simulation results in Fig. 11 show that the EVM keeps constant at 12.1% as the polarization angle ϑ varies over 180° for both the received signal and the noise keep constant powers. It means that the polarization rotation of the recombined SSB-OOFDM signals caused by the optical paths between the OC and BPD has no influence on the received OFDM signal as the theoretical prediction above.

Fig. 11 The EVM, the received signal and noise powers versus the polarization angle ϑ between the two recombined SSB-OOFDM signals caused by the optical paths between the OC and BPD.

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Figure 12 shows the influence of the length difference of the optical paths between the OC and the BPD, as well as the length difference of the electrical paths between the BPD and electrical subtractor, on the received OFDM signal. Here the length differences of the optical paths between the OC and the BPD, Δl₁ and the electrical paths between the BPD and electrical subtractor, Δl₂, are represented equivalently by the relative time delays of Δτ₁ and Δτ₂, respectively, in the simulation platform. It can be observed from Fig. 12 that the ICRBD is very sensitive to the length differences of the optical path pairs. For the case with the additional relative phase shift φ of 90° caused by the optical paths between the IL and OC, the EVM gets minimum of 13.1% at Δl₁ = 0 and then increases symmetrically with this length difference of the optical paths, \Δl₁\. This attributes to the fact that the noise power gets the minimum and the signal gets maximal power as the two optical paths have the equal length, viz., Δl₁ = 0. With the increase of the relative time delay caused by the optical path difference, the received OFDM signal power reduces due to the decline of the constructive addition of the signals from the two tributaries, while the noise power increases because of the decline of its destructive addition. From the EVM curves in Fig. 12, it can be seen that the two optical paths between the OC and the BPD can tolerate the relative time delay of ± 9ps with the EVM below 16.3% (BER = 2.3 × 10⁻³ for 16QAM), which corresponds to the fiber length difference of 2mm. According to the transmission matrix of the OC in Eq. (8), the OC introduces a 90° phase shift between the two recombined SSB-OOFDM signals by the imagine unit (j). This phase shift is compensated for by the 90° relative phase shift between IL and OC in the case of φ = 90° above, so the EVM, signal and noise power curves have symmetrical outlines, and the minimal noise power and maximal signal power occur as the two optical paths between the OC and BPD have equal length, viz., Δl₁ = 0. However, if there is no such a relative phase shift between the IL and OC, viz., φ = 0°, the 90° phase shift caused by the OC passes over the consequent optical paths, and shifts the minimal EVM to the case with Δτ₁ = −6ps. Only when the length difference Δl₁ is −1.2mm (Δτ₁ = −6ps), this phase shift is compensated for and the signals from the two branches are constructively added perfectly, and so the ICRBD outputs the maximal signal power. Although the noises from the two branches are destructively added at this point, in fact, the output noise is minimized as the relative time delay of Δτ₁ = −1ps.

Fig. 12 The EVM, the received signal and noise power versus the time delay caused by the optical paths between the OC and BPD (Δτ₁), and the electrical paths between the BPD and the subtractor (Δτ₂) for the cases without and with 90° phase shift between the OOFDM signal and the optical carrier caused by the optical paths between the IL and OC, namely, φ = 0° and 90°, respectively.

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The influence of the length difference of the electrical paths between the BPD and the subtractor is also simulated with identical outline of the curves as that of the optical paths between the IL and OC, as shown in Fig. 12. The curves of the EVM, the signal and the noise power versus Δτ₁ and Δτ₂ are overlapped completely. This means that the relative time delay, Δτ₂, caused by the length difference of the electrical paths between the BPD and the subtractor is equivalent to that of the optical paths between the OC and BPD, Δτ₁, and their impact on the received OFDM signal is completely identical. Since the recombined SSB-OOFDM signals and its down-converted RF-OFDM signals are transmitted over two individual optical/ electrical paths from the OC to the subtractor separately, and each includes the optical path, the PD and the electrical path, the total time delay (Δτ₁ + Δτ₂) of the two down-converted RF-OFDM signals in each path works according to Eq. (12). So the relative time delay Δτ₁ caused by the optical paths can be compensated for by the time delay Δτ₂ in electrical domain, vice versa. The simulation is conducted as the time delay caused by the optical paths between the OC and PD is varied with both optical and electrical paths matched (Δτ₁ + Δτ₂ = 0). The simulated EVM, signal and noise powers keep constant values over the range over 90ns as we maintain both optical and electrical paths matched by an optical or electrical tunable time delay line. This means that the length difference of optical path pair between the OC and BPD can be compensated for neatly by that of the waveguide pair; and there is no penalty if the optical and electrical paths are well matched.

5. Conclusion

In the paper, we have investigated the influence of the device parameters in the ICRBD on the output electrical OFDM signal. The theoretical analysis and simulations results demonstrate that the received OFDM signal and noise power are associated with the parameters of the devices in the ICRBD, and their deviation away from the ideal values degrades the signal performance. The IL with the sharp edges and proper bandwidth does little harm to the subcarriers and maintains the received OFDM signal with low EVM. The degradation caused by the residual OOFDM signal and the optical carrier due to the finite isolation is minor for the conventional IL with the isolation of 20dB. For the 3dB OC, since the degradation caused by the deviation of the CC away from the ideal value is limited, the ICRBD can tolerate 1dB deviation of the CC with the EVM below 14% (BER = 5.3 × 10⁻⁴ for 16QAM). The length difference of the parallel optical paths between the IL and OC makes little degradation of the received signal especially when the isolation of IL is larger than 20dB. Moreover, since the polarization angle between the OOFDM signal and the optical carrier can reduce the OFDM signal and the SSBI noise synchronously, the EVM is insensitive to polarization deviation within the polarization angle of ± 50° though the signal power reduces obviously. The received OFDM signal is sensitive to the relative total time delay between the OC and the subtractor in the BPD although the relative time delay less than 9ps can keep the EVM below the FEC threshold of 16.3%. Fortunately, the time delay in optical domain can be compensated for by the electrical delay neatly, or vice versa, and the EVM can keep constant as we maintain both optical and electrical paths matched. In addition, the received signal is immune to the asynchronous polarization rotation of the two recombined SSB-OOFDM signals in the optical paths between the OC and BPD since the photodiodes are polarization insensitive.

Acknowledgments

This work is supported part by Program for New Century Excellent Talents in University under Grant NECT-11-0595, and the Fundamental Research Funds for the Central University of China under Grant 2013RC0209.

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Influence of the device parameters in ICRBD on SSB-OOFDM signal with reduced guard band

Abstract

1. Introduction

2. Principles

2.1 The noised SSB-OOFDM signal

2.2 Mechanism of the ICRBD on the noised SSB-OFDM signal

3. Influence of the device parameters in ICRBD on the received OFDM signal

4. Simulation setup and results

4.1 Interleaver (IL)

4.2 Parallel optical paths between IL and OC

4.3 Coupling coefficient of optical coupler (OC)

4.4 Optical/electrical paths between OC and subtractor

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (12)

Equations (16)

Optics Express