Self-phase modulation (SPM) effect is analyzed in a dispersion-compensated transmission using optical BPSK single sideband (SSB) modulation. The effect was evaluated numerically using both waveform degradation and spectral degradation, clarifying that waveform degradation is induced dominantly by peak power of the quadrature component of a Hilbert-transformed signal. Eye-opening degradation of BPSK-SSB is induced by lower fiber input power than the conventional double sideband (DSB) case because the SSB-homodyne system is sensitive to phase error resulting from SPM. Spectral degradation from SPM has two phases with increasing fiber input power. In the first phase, the sideband in the suppressive frequency region expands with increasing optical power. In the second phase, the spectral envelope in the non-suppressive frequency region becomes broad, and its shape is somewhat varied.
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In the optical fiber communications field, channel bandwidth efficiency becomes quite an important factor because fiber transmission bandwidth is currently limited by the gain bandwidth of optical amplifiers. Various modulation formats and their transmission systems have been studied recently for high spectral efficiency fiber transmission . Carrier-suppressed return-to-zero (CS-RZ) formats  and optical duo-binary signal  can conserve the modulation bandwidth of each signal pulse. Recently, polarization-multiplexed multi-level signaling in the optical domain has been the subject of numerous investigations. Orthogonal frequency division multiplexing (OFDM) techniques [4,5] have received much attention because of their tailored spectrum using many frequency channels with narrow bandwidth. These modulation techniques are categorizable into double sideband (DSB) modulations. Use of a single sideband (SSB) modulation signal engenders high bandwidth efficiency because the SSB signal occupies half of the channel bandwidth of intensity modulation (IM) . The narrow channel bandwidth of optical SSB modulation is promising for dense wavelength division multiplexing.
Yonenaga et al. demonstrated optical SSB modulation using a ring resonator for sideband suppression . Optical SSB modulation of this type is called the filtering method, which has evolved into vestigial sideband modulation [8,9]. For this method, the main issue is sideband rejection filter performance, such as the frequency response of transmittance and the center frequency of filters, taking account of the signal spectrum. It might be difficult to design optimum filters. Moreover, a filter should be exchanged if the signal format is changed.
Phase-shift method  for SSB requires no sideband rejection filter. The SSB signal in this method is created by addition of in-phase components modulated by data signal and quadrature components modulated by a Hilbert-transformed data signal. Hilbert transformation is defined as a one-half π phase shift for overall frequency. Sieben and Davis et al. [10,11] demonstrated a phase-shift method for optical SSB modulation. Those studies used an optical intensity modulator based on an LN Mach–Zehnder (MZ) interferometer, an LN phase modulator and tapped delay filters as Hilbert transformers. Their SSB signal has an optical carrier in the spectrum (SSB-Emitted Carrier, SSB-EC). Consequently, conventional direct detection (DD) receivers can detect it. Shimotsu et al. fabricated SSB modulators by integrating LN modulators . The SSB modulator comprises two parallel MZ interferometers presented in Fig. 1 . Using the modulator, Higuma et al. demonstrated optical SSB modulation for a base-band binary data signal with microwave hybrid couplers as Hilbert transformers . Their SSB modulator is for a carrier-suppressed optical SSB (SSB-Suppressed Carrier, SSB-SC). It is also called binary phase shift-keying SSB (BPSK-SSB) when the base-band signal is binary code. The optical power density in a fiber is mitigated because no carrier exists. The details of how optical SSB signals propagate through optical fibers must be clarified for long haul fiber transmission.
Our previous paper reported fiber transmission characteristics such as waveform degradation from group velocity dispersion of transmission fibers, comparing the optical BPSK-SSB with the SSB-EC, which means On–Off-keying SSB (OOK-SSB) . The BPSK-SSB modulation was found to have less residual sideband in the SSB suppressive frequency region than the OOK-SSB under the assumption of the same modulation depth. It was also revealed that the BPSK-SSB is more tolerant to fiber dispersion than the OOK-SSB in electrically dispersion-compensated transmission such as the use of microstrip line in receivers. To produce long-fiber transmission, one method is a boost in transmitter output or in repeater output. The larger fiber input power tends to induce the fiber nonlinear effect, for example, the stimulated Brillouin scattering and self-phase modulation (SPM). We reported waveform degradation from SPM in BPSK-SSB fiber transmission quantitatively [15,16], but spectral degradation from SPM has not yet clarified. This paper describes the waveform degradation and spectral degradation from SPM. Results clarified that both the mechanism of waveform degradation and spectral broadening attributable to SPM in BPSK-SSB transmission differ from that of the intensity modulation (IM) case.
Section 2 presents a description of in detail the optical BPSK-SSB modulation and detection system. The assumptions for fiber transmission are also described. In Section 3, the simulation results are shown for the influence of SPM effect both in the time domain  and in the frequency domain with emphasis on the case of NRZ coded optical BPSK as a base-band signal. Waveform degradation from SPM can be mitigated by using other codes, such as RZ code and Manchester code. The results are shown in the first half of Section 4 with a short review of our previous paper . The influence of the SPM effect on the optical spectrum of the both codes is described in the second half of Section 4. Section 5 presents concluding remarks.
2. Optical BPSK-SSB modulation and simulated system
Optical sideband suppression can be made by addition of the quadrature component to the base-band signal . It is achieved by the two parallel LN-MZIs : the upper MZI in Fig. 1 modulates light intensity by base-band signal d(t); the lower MZI modulates light by quadrature signal dH(t), which is created by Hilbert transformation defined as a π/2 phase shift for all frequency components of the base-band signal . When each MZI is biased at the transmission null point, a carrier-suppressed optical SSB (SSB-Suppressed Carrier, SSB-SC) can be created. If the base-band signal is a binary code, we can call it optical BPSK-SSB.
To evaluate SPM effects from fiber transmission, a dispersion-compensated single-channel system with homodyne detection was employed, as presented in Fig. 1. The wavelength of the signal light source (LD) was 1550 nm. Both the wavelength and the phase of a local oscillator (LO) were identical to those of the LD. The power of the LO was a thousand times larger than the received averaged optical power in suppressing spurious components resulting from transmission nonlinearity of MZIs . The modulation depth was 5%. A standard single-mode optical fiber (SMF) and a dispersion compensated optical fiber (DCF) were used and connected in this order. Table 1 presents each fiber parameter. The SMF length LSMF was 100 km in this simulation. The DCF length LDCF, 21.3 km, was determined to compensate the SMF dispersion completely at the LD wavelength. Fiber transmission was calculated using the nonlinear Schrödinger equation with the split-step Fourier method .
Recent coherent detection techniques or digital coherent receivers do not need optical dispersion compensation devices such as DCF. However, the digital coherent receiver equalizes dispersion after O/E conversion with digital signal processing including many system parameters. In our evaluation system, we employed DCF for dispersion compensation because its simplicity and technical maturity. We have already checked that fiber nonlinear effect at the DCF was too small to result in signal degradation because optical power at DCF was quite small through the SMF fiber loss.
3. SPM effect in optical BPSK-SSB fiber transmission with NRZ code
3.1 Waveform degradation
Figure 2 presents detected eye diagrams by the homodyne receiver with changing averaged fiber input power of Pin,ave when the base-band signal was used with non-return-to-zero (NRZ) code with 10 Gbit/s pseudo-random sequence (PN: 7-stage). Relative large fiber input power induces asymmetrical eye and degrades the eye-opening. In Fig. 2(d), we can find major eye opening degradations at the four pattern lines. Figure 3 presents the detected signal waveform (a), optical quadrature component, which is emitted from the lower MZI (b), and the base-band signal waveform (c) in the case of the averaged fiber input power of + 8 dBm. The four arrows in Fig. 3(a) indicate the timing at which eye opening is degraded, as presented in Fig. 2(d). The quadrature component power is remarkably high at those timings. Thereby, the waveform degradation from SPM can be dominated by the peak power in the quadrature component. This waveform degradation attributable to SPM is a unique phenomenon of optical SSB modulation. This phase shift is, in other words, the cross-phase modulation between the in-phase component and the quadrature component through fiber nonlinearity. The quadrature component is inferred to have higher power at the transition timing between the long mark-bit (1) and the long space-bit (0) in the NRZ base-band signal. This results from the edge detection feature of Hilbert transformation .
The waveform degradation from SPM was evaluated as the eye opening penalty (EP), which was calculated using the following equation:
The box plot portrayed in Fig. 4 shows EP when non-return-to-zero (NRZ) code was used with 10 Gbit/s pseudo-random sequence; EP increased concomitantly with increasing averaged fiber input power because of waveform degradation from SPM. The SPM threshold Pin,1 dB is defined as the fiber input power at which EP is equal to 1 dB. The threshold was 4.4 dBm, which is 6.8 dB lower than that of IM-DD (dashed line of Fig. 4).
3.2 Spectral broadening
Figure 5 shows optical power spectra of BPSK-SSB signal at the receiver input with changing averaged fiber input power of Pin,ave. For comparison, gray lines represent the transmitter output spectra. Simulation conditions were identical to descriptions in the previous section. For Pin,ave = 2 dBm, optical power was suppressed in the SSB suppressive frequency region, which is a higher frequency region than that of the optical carrier in this case. The sideband suppression was, however, slightly degraded compared with the transmitter output spectra. When the fiber input power is increasing, the sideband suppression tends to be degraded. Degradation was observed when the averaged fiber input power was greater than −6 dBm, which is an almost equal value to that at which waveform degradation is induced, as portrayed in Fig. 4.
In the non-suppressive frequency region, the spectral envelope was broadened. The envelope shape could not be maintained when input power was increased from 10 dBm. For comparison, the receiver input power spectra of the conventional intensity modulated signal (IM) are depicted in Fig. 6 . For the IM signal, the spectrum is broadened by increased fiber input power, but the envelope shape is similar to that of the base-band signal.
By analogy with the IM-DD case, it is apparently strange that SPM threshold of SSB-homodyne system, described in Section 3.1, is much different from optical power inducing spectral broadening. To clarify the low SPM threshold, we checked the theorem of SSB homodyne detection. In the homodyne systems, the detected photo current i(t) can be written as the following equation :Eq. (2). For this reason, the SSB homodyne system is more sensitive to phase shift from SPM than the IM-DD system is.
From Section 3.2, we can find spectral degradation of two kinds from the SPM effect in optical SSB fiber transmission: (1) Destructive sideband rejection, Pin,ave. > −6 dBm; (2) Sideband envelope degradation, Pin,ave. > 10 dBm. The first phenomenon results from the temporal phase shift from SPM because the peak fiber input power is 5.8 dB larger than the averaged fiber input power for the NRZ case. For example, Pin,ave. = −6 dBm means the optical peak power is 0 dBm. Figure 3(b) shows that the peak does not appear frequently. The temporal phase change at the peak power does not strongly influence the spectrum, but results in the degradation of sideband suppression. However, when Pin,ave. > 10 dBm, the SPM effect induces the phase shift, pulse broadening and interaction with fiber dispersion at every signal timing. Then the spectrum envelope varies greatly.
4. SPM effect in optical BPSK-SSB fiber transmission with RZ code and Manchester code
4.1 Waveform degradation of RZ/Manchester coded SSB signal
Previous Section 3 shows that waveform degradation from SPM tends to occur when symbol changes after signal retain the same level because Hilbert transformation, which generates the quadrature component of SSB signal, is a kind of edge detection filter . The SSB modulation, then, prefers the code without keeping level. In this section, we employ two signal codes, such as RZ code and Manchester code.
RZ code has almost half time duration of mark bit, comparing with NRZ code. Time duration of the same level of Manchester code is only one time slot. The optical Manchester code  is obtainable by direct modulo-2 addition of the NRZ signal and a clock component. In this code, a transition exists at the center of each bit interval. A negative-going transition indicates a mark-bit; a positive-going transition means that a space-bit was sent. Figure 7 shows that longest duration keeping the same level in optical Manchester code is limited a time slot.
Figure 8 shows the eye opening degradation of Manchester coded SSB (Man-SSB) and RZ coded SSB (RZ-SSB). Pin,1dB is 11.5 dBm and 8.2 dBm for Man-SSB and RZ-SSB, respectively. The Manchester code is tolerant 7.1 dB more than NRZ-SSB format from the eye penalty viewpoint. Optical peak power of Manchester and RZ coded SSB signal was 11.7 mW and 19.2 mW, respectively. These are less than that of NRZ coded SSB of 24.2 mW, as shown in Fig. 3(b). The less peak power was induced by the less output power from the Hilbert transformer. Generally, Hilbert transformers output high power when the input waveform has level transition after long identical level. Manchester code, then, has more tolerant to SPM because Manchester code has relative short time to keep the same level.
4.2 Spectral broadening of RZ/Manchester coded SSB signal
Figure 9 and Fig. 10 respectively show optical power spectra of RZ-coded and Manchester coded BPSK-SSB signal at receiver input with changing averaged fiber input power. For comparison, gray lines represent the transmitter output spectra. Simulation conditions were identical to the descriptions in the previous section. We also find the destructive sideband rejection and sideband envelope degradation in non-suppressive frequency region, as well as NRZ code. However, sideband envelope degradation for RZ and Manchester code is not remarkable compared with the NRZ code because the both code have equivalent optical peak power in respective time slots.
The SPM effect from standard single-mode fiber was analyzed in a single-channel BPSK-SSB transmission with dispersion compensation. Waveform degradation is induced dominantly by peak power of the quadrature component of the Hilbert-transformed signal. The SPM threshold of optical BPSK-SSB is low because the SSB-homodyne system is sensitive to the phase error between the received light and the local oscillator. Manchester or RZ coded SSB is more tolerant to SPM than NRZ code because the peak output power of Hilbert transformers depends on the input signal code.
Spectral degradation from SPM in optical BPSK-SSB fiber transmission with NRZ code has two phases with increasing fiber input power. The first phase is destructive sideband suppression. In this phase, the sideband in the suppressive frequency region expands with increasing optical power. This phenomenon occurs at the averaged fiber input power Pin,ave. > −6 dBm. The second phase is spectral envelope change in the non-suppressive frequency region. The change contains spectral broadening and spectral shape variation. It results from averaged fiber input power greater than 10 dBm. We have analyzed spectral degradation for other codes of RZ code and Manchester code to mitigate waveform degradation from SPM. Both codes have similar spectral change to that of the NRZ code.
This work was supported in part by SCOPE (No. 092102033) and MEXT (KAKENHI No. 21360177).
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