Abstract

We demonstrate an extended-cavity (1-km round trip) transmitter employing a reflective-semiconductor optical amplifier (RSOA) self-seeded by spectrally-sliced passive modulation-averaging reflector. We show that using modulation averaging reflectors in self-seeded transmitters improves link margin, allows a wider range of bias conditions for the RSOA by removing the modulation in the seeding light and consequently allows operation with higher extinction ratios. We furthermore demonstrate 47 km transmission at 1.25 Gbps with a 16-channel fully passive remote node. This type of transmitter is suitable for application in colorless WDM-PON systems.

© 2012 OSA

1. Introduction

Wavelength division multiplexing in passive optical networks (WDM-PON) is actively investigated as a next-generation optical network architecture meeting the future cost and performance needs [14]. Wavelength-agnostic transceivers at the user end in colorless WDM-PON systems offer significant cost reduction as each of the optical network units (ONU) is identical. The cost and complexity of the system is further reduced by using self-seeded WDM-PON architectures in which a reflector at a remote node allows the amplified-spontaneous-emission (ASE) signal emitted from the reflective semiconductor optical amplifier (RSOA) to seed itself [5]. In this paper we experimentally demonstrate that the link margin in such self-seeded systems can be further improved using a modulation-averaging reflector located in the remote node as was theoretically predicted [6]. The modulation-averaging reflector changes the amplitude probability density distribution of the modulated seeding light from a sharply bound distribution into a near-normal distribution which then alters the error statistics and results in improved performance. Most importantly, the modulation-averaging reflector is a passive optical device and therefore the remote nodes in our proposed WDM-PON system remain passive, with no power needed to accomplish this improvement.

2. Modulation averaging reflector

Figure 1 shows a proposed self-seeded spectrally-sliced optical transmitter which includes a modulation-averaging reflector (AR). The time delay between the mirrors is τ/2 so that the round-trip between reflections at two adjacent mirrors is τ, which defines the AR design line rate Bτ = 1/τ. All the mirrors distributed along the fiber are semi-transparent with power reflection coefficient r except the last one which has reflectivity close to unity.

 

Fig. 1 Schematic view of the demonstrated WDM transmitter employing a modulation-averaging reflector with n mirrors. The last layer is terminated with unit reflectivity (rHR = 1), while all other mirrors are semi-transparent (r < 1).

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3. System design considerations

We constructed AR prototypes with commercially available Fiber-Bragg-Gratings (FBG) integrated into a single fiber segment. The FBG center wavelength was 1550 nm and the separation between partially reflective mirrors was designed for Bτ = 125 Mbps, which makes the FBG separations on the fiber about 80 cm. The test symbol rate of 1250 Mbps is an integer multiple of the design rate which does not fundamentally alter the expected AR performance. The measured FBG reflectivity and loss per pass at the center wavelength were r ≈6% and a ≈1.5%, respectively. Three segments, one with 10 and two with 20 FBGs, were produced. Using these segments, we were able to assemble 10, 20, 30, 40, and 50-mirror reflectors, but we performed all the experiments reported here using a 20 mirror AR as it offered best performance. Longer designs include additional connectors which add more loss. The end-termination hard reflector (HR) was constructed from a ceramic ferrule coated with high-reflectivity interference coating (rHR ≈97%). Previously demonstrated self-seeding architectures [5] had no means for removing the modulation in the seeding light and are equivalent to a link with just the HR (HR case).

The active optical element was a CIP Technologies SOA-R-C-7S-FCA RSOA. According to the specification, the polarization dependent gain (PDG) ratio of this RSOA is around 20 dB. The device is modulated and biased through a 50-Ω input impedance SMA port and has a direct modulation bandwidth of 1.2 GHz. We used seven-bit pseudorandom binary sequence (PRBS7) for all experiments. In PRBS7, the bit sequence repeats after 127 bits and the maximum run length (MRL) is 7 bits. This short test sequence was used to mimic 8b10b encoding used in Gigabit Ethernet (MRL = 5 bits). The averaging is expected to improve with the increasing ratio of the expected delay in the reflector ED (expressed in bits) to the MRL of the bit stream since in most AR designs, the increase in the number of mirrors in the AR proportionally increases the ED [6]. In the opposite limit, when MRL >> ED, the AR converges to a HR. The expected delay of the AR used in these experiments is ~20 bits, hence MRL < ED. The AWG is a 16 channel flat-top filter with Δλ = 1.2 nm (σλ = 0.37 nm) and channel separation of 200 GHz. The distance from AWG to ONU (RSOA) is L1 = 500 m. In our experiment with smaller optical bandwidth, we used a DWDM filter with Δλ = 0.6 nm (σλ = 0.17 nm; 100 GHz spacing). Based on these optical bandwidths, the coherence time τcoh of the incoming light beam is significantly shorter than the round trip time τ: τcoh << τ as is required by the design [6]. The cavity length (L1) of this light-source is significantly larger than both the coherence length and the bit length, conditions satisfied in a laser cavity. Consequently, this is an incoherent (thermal) light source. All measurements were performed at room temperature.

4. Transmitter performance

We assembled the system shown in Fig. 1 and performed experiments with the intent to confirm the theoretically-expected link-margin improvement and understand the performance limitations. We investigated the effects of (a) the output coupling ratio C, and (b) the optical bandwidth on the extended-cavity source performance.

Output-coupling ratio C is varied to optimize between reflected power used for seeding and output optical power. Smaller output coupling (smaller C) increases the amount of self-seeding power and thereby decreases the current needed for the RSOA to reach self-seeding threshold. This is visible in Fig. 2(a) where we show measured output power from the extended-cavity source versus RSOA drive current for three different output couplers: C = 1%, 10%, and 50%. Clearly, larger output coupling C also provides higher output power, which is preferred, as long as the reflected power is sufficient to enable the link to operate. The change of optical bandwidth (Δλ) is expected to produce a dramatic effect on the link performance because spectrally-sliced thermal light (spontaneous emission) exhibits amplitude variance (σSP2) that is proportional to the square of the optical power (P¯) rather than being proportional to the optical power as it is for coherent light (laser output) [7]. This proportionality places a fundamental limit on the optical signal to noise ratio (Q0) for polarized thermal light to

Q0P¯/σSP=M,
where M is the number of degrees of freedom and is equal to M = T/τc when T >> τc. Here τc is the coherence time, and T is the integration time equal to bit duration [7]. To quantify this effect, we tested back-to-back performance of the extended cavity light source with two different optical bandwidths. We define back-to-back as L1 = 500 m and L2 = 0 in Fig. 1. From Eq. (1) the expected Q0 values for the two cases are 7.7 and 10.9. The bit-error rate measurement results are shown in Fig. 2(b). We fitted both curves to
Q=P¯/Pth2+(P¯/Q0)2,
where P¯is the average optical power, Pth is the fitted thermal noise and Q0 is the fitted optical signal to noise ratio. The fitted values were Q0 ≈5.8 for Δλ = 0.6 nm and Q0 ≈8.1 for Δλ = 1.2 nm configuration (we estimated this from the slight change in curvature at the bottom of the curve). The value of the receiver thermal noise Pth = 2.1 μW remained unchanged for both curves. We neglect the fact that spontaneous-emission exhibits amplitude fluctuations that are slightly broader than a normal distribution [7]. The measured Q0 values are smaller than the theoretically predicted values because in this simple estimate we did not account for the finite extinction ratio (as in [8]) and increased noise power arising from the modulation converted into noise on the AR [6]. However, most importantly we observe that larger optical bandwidth does increase the Q0, i.e., reduces the error floor, a trend predicted by Eq. (1). We measure Q1.2/Q0.6 ~1.4, which is in excellent agreement with 2 improvement predicted by Eq. (1).

 

Fig. 2 (a) Output powers from extended cavity for splitter ratios C = 50%, 10%, and 1%. (b) The influence of optical bandwidth on the performance of extended cavity optical source at 1250 Mbps (Ibias = 38 mA, VRF = 2 Vpp).

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It has been previously observed that insufficient seeding power in HR architecture leads to an error floor or an inoperative link [5]. Our measurements on the HR architecture confirm this, but also find that the averaging provided by the AR architecture removes this limitation. The back-to-back test conditions were as follows: lower line-rate (155 Mbps) in order to avoid influence of RSOA electrical bandwidth (specified at 1.2 GHz) and constant average receive power of Pavg = −15.6 dBm set via variable attenuator. The line-rate of 155 Mbps is not an integer multiple of design line-rate and it was chosen because it was the lowest line-rate supported by our clock-and-data recovery (CDR) unit (Maxim MAX3872). This discrepancy should make the performance of the AR worse, but should have no effect on the HR case.

In both cases (AR and HR) a range of bias conditions were tested (from 15 to 35 mA with 5 mA step). For HR only 25, 30 and 35 mA are shown as the link was not functional with lower bias currents (we could not lock the CDR). For AR case, only currents of 15 and 20 mA are shown as with higher bias current the AR provided no errors for 3 minutes of transmission (equivalent to BER < 10−10 with confidence level = 94%). We do not show those points in Fig. 3 . For each bias current the modulation voltage (ΔV) was varied between 0.5 and 2 Vpp.

 

Fig. 3 BER vs modulation voltage (ΔV) for HR and AR seeding.

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Figure 3 illustrates the difference in seeding performance of the HR and AR systems. For the HR case, the performance improves when the RSOA drive in the logical zero is increased: either by increasing DC bias current or decreasing the modulation depth (ΔV). This is because there always has to be sufficient self-seeding power returning to the RSOA to maintain a working link; hence providing sufficient power in the logical zero at all times becomes imperative. The system with the AR behaves differently: The logical zero can be very low (even below pn-junction threshold), and only the average power must be above some limit for the system to be steadily seeded. The increase in performance due to increase in ΔV at same DC bias can be explained by highly nonlinear relation between optical power and applied current around self-seeding threshold (see Fig. 2(a)). In this way, as the modulation voltage (ΔV) is increased, the average optical power is also increased. Therefore, using AR indeed allows for a wider range of bias conditions for the RSOA relative to the HR case and provides for a potential increase in the optical modulation amplitude of the transmitter.

Back-to-back performance at 1250 Mbps line-rate for HR and AR architectures was measured and the results are shown in Fig. 4 . The modulation voltage was kept at 2 Vpp and the bias current was varied which has a direct impact on the extinction ratio. For both the HR and the AR cases, the receiver sensitivity for BER of 10−3, 10−6 and 10−9 was measured as a function of bias current. Measurements were carried out for two splitter configurations: C = 50% and C = 10%. What is not visible in Fig. 4, is that the output power is ~6 dB higher with the C = 50% configuration (see Fig. 2(a)). These measurements show that, relative to the HR case, the transmission with the AR offers a larger link margin at given RSOA operating conditions. Additionally, even though several dB of improvement in the link margin can be obtained for given RSOA drive current, the most significant fact, visible in Fig. 4, is that AR system produces stable operation at a range of RSOA drive currents for which the HR case cannot close a link and consequently allows operation with higher extinction ratios.

 

Fig. 4 Back-to-back receiver sensitivity for BER = 10−3, 10−6 and 10−9 vs Ibias for HR and AR cases. The VRF = 2 Vpp in all cases.

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The output beam from the optical source in all of these experiments was linearly polarized due to high polarization dependent gain of the RSOA. We observed neither RSOA seeding instability [9] nor a need to control the polarization [5]. Experiments with a depolarizer placed in front of the array reflector in location 1 in Fig. 1 provided similar results as when no depolarizer was present. For these reasons we omitted the depolarizer in the experiments reported. A method to provide unpolarized light at the output of the optical source as well as undepolarized seeding light would be to insert a depolarizer inside the extended cavity at location 2 in Fig. 1.

5. Transmission experiments

We performed transmission measurements over a range of standard G.652 fiber lengths of up to 47 km (L1 + L2) for two optical bandwidths (Δλ = 0.6 nm and Δλ = 1.2 nm). The measurement system is shown in Fig. 1 and the measurement results in Fig. 5 . A fully passive remote node was used with C = 50%. The data shown in Fig. 5 exhibit dispersion penalty approximately consistent with the optical bandwidth and fiber length. Read from the far left limit of the data, dispersion penalties were 0.3 dB and 1 dB for Δλ = 0.6 nm and Δλ = 1.2 nm, respectively. In the high power limit, we observe an error floor that depends on both the optical bandwidth and the fiber length. In spite of the error floor, even after 47 km of transmission it is still possible to obtain BER < 10−4 in both configurations, which is sufficient for data extraction using RS(255,239) FEC code [10].

 

Fig. 5 Transmission performance over various fiber lengths of standard G.652 fiber with a fully passive remote node for two optical bandwidths (Ibias = 35 mA, VRF = 2 Vpp).

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The measured curves from Fig. 5 are fitted to Eq. (2) and Q0 value was extracted for each fiber length and optical bandwidth tested. The extracted values were plotted versus the link length L1 + L2 in Fig. 6 . This peculiar behavior was found to reasonably fit to the following function:

QLXQ0X(1+bX2L2)1/2,
where X refers to system with Δλ = 0.6 nm or Δλ = 1.2 nm, and Q0X to the limiting Q values for back-to-back performance. The coefficient bX increases with the optical bandwidth. The graph showing this fit is given in Fig. 6. There are two dispersion-related phenomena responsible for the reduction of signal-to-noise ratio with the fiber length: The optical modulation amplitude reduces, while the noise increases due to chromatic dispersion. Self-seeded architectures inherently include gain saturation in the active element (RSOA) which compresses intensity fluctuations present in the input light [11]. The propagation though the fiber adds excess noise arising from the beating between spontaneous-emission spectral components that were delayed by different times due to dispersion [8].

 

Fig. 6 Measured Q0 (markers) fitted by Eq. (3) (full lines) for two optical bandwidths.

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Figure 6 demonstrates that although larger optical bandwidth improves the back-to-back error floor of a self-seeded link, using a smaller optical bandwidth actually may make the link less sensitive to fiber length and provide more robust performance of the link. We showed that closing a link over 40 km [4] is possible with either one of the optical filters used.

6. Conclusion

We address the problems observed with colorless self-seeded architectures in WDM-PON. Such systems suffer from reduced signal-to-noise ratio originating from modulation present in the seeding light. By introducing a new component, a modulation-averaging reflector, we show that the link margin is improved and wider range of bias conditions for the active element can be used. The modulation-averaging reflector effectively converts amplitude penalty (originating from modulation present in the seeding light) into noise penalty. Furthermore, the reflector, placed in the remote node, is a passive device; a key requirement for use in PON. The suitability of proposed architecture for next generation of optical access networks is demonstrated by a 47 km transmission over standard G.652 fiber at 1.25 Gbps.

Acknowledgments

The authors gratefully acknowledge Pavle Sedić, Marcus Nebeling, and Bernd Hesse for their technical advice and help with the equipment. This work would not have been possible without the generous support from Business Innovation Center of Croatia (BICRO) and Fiber Network Engineering, Inc. of Foster City, CA, USA.

References and links

1. C.-H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the home using a PON infrastructure,” J. Lightwave Technol. 24(12), 4568–4583 (2006). [CrossRef]  

2. C. F. Lam, Passive Optical Networks: Principles and Practice (Academic Press, 2007).

3. B. Kim and B.-W. Kim, “WDM-PON development and deployment as a present optical access solution,” Conf. on Opt. Fiber Comm., San Diego (2009).

4. P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012). [CrossRef]  

5. E. Wong, K. L. Lee, and T. B. Anderson, “Directly modulated self-seeding reflective semiconductor optical amplifiers as colorless transmitters in wavelength division multiplexed passive optical networks,” J. Lightwave Technol. 25(1), 67–74 (2007). [CrossRef]  

6. T. Komljenovic, D. Babić, and Z. Sipus, “Modulation-Averaging Reflectors for Extended-Cavity Optical Sources,” J. Lightwave Technol 29(15), 2249–2258 (2011). [CrossRef]  

7. J. W. Goodman, Statistical Optics (Wiley Classics Library, 2000).

8. A. J. Keating and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. 15(1), 53–61 (1997). [CrossRef]  

9. M. Presi and E. Ciaramella, “Stable self-seeding of R-SOAs for WDM-PONs,” Conf. on Opt. Fiber Comm. and Expo., Pisa, (2011).

10. ITU-T Recommendation G.975: Forward Error Correction for Submarine Systems (10/2000).

11. A. D. McCoy, P. Horak, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Noise suppression of incoherent light using a gain-saturated SOA: implications for spectrum-sliced WDM systems,” J. Lightwave Technol. 23(8), 2399–2409 (2005). [CrossRef]  

References

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  1. C.-H. Lee, W. V. Sorin, and B. Y. Kim, “Fiber to the home using a PON infrastructure,” J. Lightwave Technol. 24(12), 4568–4583 (2006).
    [CrossRef]
  2. C. F. Lam, Passive Optical Networks: Principles and Practice (Academic Press, 2007).
  3. B. Kim and B.-W. Kim, “WDM-PON development and deployment as a present optical access solution,” Conf. on Opt. Fiber Comm., San Diego (2009).
  4. P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
    [CrossRef]
  5. E. Wong, K. L. Lee, and T. B. Anderson, “Directly modulated self-seeding reflective semiconductor optical amplifiers as colorless transmitters in wavelength division multiplexed passive optical networks,” J. Lightwave Technol. 25(1), 67–74 (2007).
    [CrossRef]
  6. T. Komljenovic, D. Babi?, and Z. Sipus, “Modulation-Averaging Reflectors for Extended-Cavity Optical Sources,” J. Lightwave Technol 29(15), 2249–2258 (2011).
    [CrossRef]
  7. J. W. Goodman, Statistical Optics (Wiley Classics Library, 2000).
  8. A. J. Keating and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. 15(1), 53–61 (1997).
    [CrossRef]
  9. M. Presi and E. Ciaramella, “Stable self-seeding of R-SOAs for WDM-PONs,” Conf. on Opt. Fiber Comm. and Expo., Pisa, (2011).
  10. ITU-T Recommendation G.975: Forward Error Correction for Submarine Systems (10/2000).
  11. A. D. McCoy, P. Horak, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Noise suppression of incoherent light using a gain-saturated SOA: implications for spectrum-sliced WDM systems,” J. Lightwave Technol. 23(8), 2399–2409 (2005).
    [CrossRef]

2012 (1)

P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[CrossRef]

2011 (1)

T. Komljenovic, D. Babi?, and Z. Sipus, “Modulation-Averaging Reflectors for Extended-Cavity Optical Sources,” J. Lightwave Technol 29(15), 2249–2258 (2011).
[CrossRef]

2007 (1)

2006 (1)

2005 (1)

1997 (1)

A. J. Keating and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. 15(1), 53–61 (1997).
[CrossRef]

Anderson, T. B.

Babic, D.

T. Komljenovic, D. Babi?, and Z. Sipus, “Modulation-Averaging Reflectors for Extended-Cavity Optical Sources,” J. Lightwave Technol 29(15), 2249–2258 (2011).
[CrossRef]

Chanclou, P.

P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[CrossRef]

Cui, A.

P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[CrossRef]

Geilhardt, F.

P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[CrossRef]

Horak, P.

Ibsen, M.

Keating, A. J.

A. J. Keating and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. 15(1), 53–61 (1997).
[CrossRef]

Kim, B. Y.

Komljenovic, T.

T. Komljenovic, D. Babi?, and Z. Sipus, “Modulation-Averaging Reflectors for Extended-Cavity Optical Sources,” J. Lightwave Technol 29(15), 2249–2258 (2011).
[CrossRef]

Lee, C.-H.

Lee, K. L.

McCoy, A. D.

Nakamura, H.

P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[CrossRef]

Nesset, D.

P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[CrossRef]

Richardson, D. J.

Sampson, D. D.

A. J. Keating and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. 15(1), 53–61 (1997).
[CrossRef]

Sipus, Z.

T. Komljenovic, D. Babi?, and Z. Sipus, “Modulation-Averaging Reflectors for Extended-Cavity Optical Sources,” J. Lightwave Technol 29(15), 2249–2258 (2011).
[CrossRef]

Sorin, W. V.

Thomsen, B. C.

Wong, E.

IEEE Netw. (1)

P. Chanclou, A. Cui, F. Geilhardt, H. Nakamura, and D. Nesset, “Network operator requirements for the next generation of optical access networks,” IEEE Netw. 26(2), 8–14 (2012).
[CrossRef]

J. Lightwave Technol (1)

T. Komljenovic, D. Babi?, and Z. Sipus, “Modulation-Averaging Reflectors for Extended-Cavity Optical Sources,” J. Lightwave Technol 29(15), 2249–2258 (2011).
[CrossRef]

J. Lightwave Technol. (4)

Other (5)

C. F. Lam, Passive Optical Networks: Principles and Practice (Academic Press, 2007).

B. Kim and B.-W. Kim, “WDM-PON development and deployment as a present optical access solution,” Conf. on Opt. Fiber Comm., San Diego (2009).

M. Presi and E. Ciaramella, “Stable self-seeding of R-SOAs for WDM-PONs,” Conf. on Opt. Fiber Comm. and Expo., Pisa, (2011).

ITU-T Recommendation G.975: Forward Error Correction for Submarine Systems (10/2000).

J. W. Goodman, Statistical Optics (Wiley Classics Library, 2000).

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Figures (6)

Fig. 1
Fig. 1

Schematic view of the demonstrated WDM transmitter employing a modulation-averaging reflector with n mirrors. The last layer is terminated with unit reflectivity (rHR = 1), while all other mirrors are semi-transparent (r < 1).

Fig. 2
Fig. 2

(a) Output powers from extended cavity for splitter ratios C = 50%, 10%, and 1%. (b) The influence of optical bandwidth on the performance of extended cavity optical source at 1250 Mbps (Ibias = 38 mA, VRF = 2 Vpp).

Fig. 3
Fig. 3

BER vs modulation voltage (ΔV) for HR and AR seeding.

Fig. 4
Fig. 4

Back-to-back receiver sensitivity for BER = 10−3, 10−6 and 10−9 vs Ibias for HR and AR cases. The VRF = 2 Vpp in all cases.

Fig. 5
Fig. 5

Transmission performance over various fiber lengths of standard G.652 fiber with a fully passive remote node for two optical bandwidths (Ibias = 35 mA, VRF = 2 Vpp).

Fig. 6
Fig. 6

Measured Q0 (markers) fitted by Eq. (3) (full lines) for two optical bandwidths.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

Q 0 P ¯ / σ SP = M ,
Q= P ¯ / P th 2 + ( P ¯ / Q 0 ) 2 ,
Q LX Q 0 X (1+ b X 2 L 2 ) 1 /2 ,

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