# Cost-effective 33-Gbps intensity modulation direct detection multi-band OFDM LR-PON system employing a 10-GHz-based transceiver

**Abstract**

We develop a dynamic multi-band OFDM subcarrier allocation scheme to fully utilize the available bandwidth under the restriction of dispersion- and chirp-related power fading. The experimental results successfully demonstrate an intensity-modulation-direct-detection 34.78-Gbps OFDM signal transmissions over 100-km long-reach (LR) passive-optical networks (PONs) based on a cost-effective 10-GHz EAM and a 10-GHz PIN. Considering 0-100-km transmission bandwidth of a 10-GHz EAM, the narrowest bandwidth is theoretically evaluated to occur at ~40 km, instead of 100 km. Consequently, the performances of 20-100-km PONs are experimentally investigated, and at least 33-Gbps capacity is achieved to support LR-PONs of all possible 20-100-km radii.

Cost-effective 33-Gbps intensity modulation

direct detection multi-band OFDM LR-PON

system employing a 10-GHz-based transceiver

Dar-Zu Hsu,

1,2,*

Chia-Chien Wei,

3

Hsing-Yu Chen,

2

Wei-Yuan Li,

1

and Jyehong Chen

1

1

Department of Photonics, National Chiao-Tung University, Hsinchu, Taiwan, 300

2

Information and Communications Research Labs, Industrial Technology Research Institute, Hsinchu, Taiwan, 300

3

Department of Applied Materials and Optoelectronic Engineering, National Chi Nan University, Nantou, Taiwan,

545

*sparkle@itri.org.tw

Abstract: We develop a dynamic multi-band OFDM subcarrier allocation

scheme to fully utilize the available bandwidth under the restriction of

dispersion- and chirp-related power fading. The experimental results

successfully demonstrate an intensity-modulation-direct-detection 34.78-

Gbps OFDM signal transmissions over 100-km long-reach (LR) passive-

optical networks (PONs) based on a cost-effective 10-GHz EAM and a 10-

GHz PIN. Considering 0–100-km transmission bandwidth of a 10-GHz

EAM, the narrowest bandwidth is theoretically evaluated to occur at ~40

km, instead of 100 km. Consequently, the performances of 20–100-km

PONs are experimentally investigated, and at least 33-Gbps capacity is

achieved to support LR-PONs of all possible 20–100-km radii.

©2010 Optical Society of America

OCIS codes:

(060.2330) Fiber optics communications; (060.0060) Fiber optics and optical

communications.

References and links

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Received 5 Jul 2011; revised 6 Aug 2011; accepted 6 Aug 2011; published 22 Aug 2011

(C) 2011 OSA

29 August 2011 / Vol. 19, No. 18 / OPTICS EXPRESS 17546

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1. Introduction

With the exponentially increasing of customer needs for broadband services, passive optical

network (PON) is considered to be the most promising candidate that can economically

provide high bandwidth to end-users [1]. Recently, there have been growing interests in the

new type of optically amplified large-split long-reach PON (LR-PON) [2–6]. Integrated the

access network with the metro network within the 100-km target range, LR-PON claims to

considerably reduce the capital and operational expenditures by increasing the coverage of the

central office and consolidating the O/E/O conversion interfaces inside the existing networks.

Besides, LR-PONs can simplify the network hierarchies and thus reduce the network latency,

which is very important for the real-time broadband services.

To realize next generation LR-PON systems, several multiplexing schemes are viable

candidates to support more ONUs with higher data rate, such as time-division multiplexing

(TDM), orthogonal frequency-division multiplexing (OFDM), and wavelength-division

multiplexing (WDM). In particular, OFDM LR-PON, which boasts to offer high spectral

efficiency and flexible bandwidth allocation, has attracted a lot of attention recently [7]. Both

TDM and OFDM LR-PONs can easily support numerous ONUs by using an optical splitter

with large splitting ratio and can broadcast the aggregated data to all ONUs on a single

wavelength, thus have the benefit of simple wavelength management [6]. However, the

aggregated data rate on a single wavelength will be very high (40 Gbps or higher), and each

ONU has to process very high-speed aggregated data in order to receive and transmit a small

portion of data (e.g. <1/128) [6]. In [7], a 108-Gbps OFDM PON over single wavelength is

demonstrated employing two Mach-Zehnder modulators, polarization multiplexing and two

receivers. For cost-sensitive ONUs, the proposed scheme will be too complicated and

expensive to be a practical solution in the near future. An alternative solution is WDM LR-

PON, where each ONU can be assigned a wavelength, thus ONUs doesn’t have to process the

aggregated data and the speed requirement of each transceiver can be considerably lower.

Besides, instead of using a large-splitting-ratio optical splitter, WDM LR-PON can use a

wavelength multiplexer/demultiplexer (waveguide grating router, WGR), and the loss of

WGR is much lower and insensitive to wavelength number. Unfortunately, even though the

speed requirement of each transceiver is lower in a WDM system, numerous color

transmitters are costly and undesired. To preserve the colorless upstream architecture, an

RSOA (reflective semiconductor optical amplifier) is employed for upstream service [8,9].

Currently, employing RSOAs are still facing some technical challenges such as limited

bandwidth, ASE noise, Rayleigh backscattering needed to be overcome. Besides, the systems

fail to allocate bandwidth flexibly among ONUs of different wavelengths [4]. In particular,

since a LR-PON is needed to be coexisted with the current PONs, such as 10-Gbps PON (XG-

PON) and Gigabit PON (G-PON). As shown in Fig. 1 [10], excluding guard bands, the

a

vailable enhancement bands provide very limited bandwidth at C-band for future LR-PONs.

For the reasons stated above, a more practical solution will be a hybrid WDM/OFDM LR-

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(C) 2011 OSA

29 August 2011 / Vol. 19, No. 18 / OPTICS EXPRESS 17547

PON, which optimizes the trade-off between the number of wavelength channels and the cost

of transceivers. For instance, if a cost-effective transceiver can provide 32-Gbps capacity,

only four wavelength channels are required for each direction (downstream and upstream) to

support a LR-PON with 128 ONUs and 1-Gbps data rate per ONU.

Fig. 1. Wavelength allocations of current G-PON/XG-PON and proposed LR-PON. (DS:

downstream; US: upstream)

To reach the target transmission distance of up to 100 km, nonetheless, will require better

performance and more expensive components. To further reduce costs, intensity-modulation-

direct-detection (IMDD) is expected. Additionally, it is desirable to generate high-speed

signals by low-bandwidth cost-effective transceivers, such as commercially matured 10-GHz

directly modulated DFB lasers (DMLs) or electro-absorption modulated lasers (EMLs) and

10-GHz PIN detectors, assisted by spectrally efficient modulation format, such as quadrature

amplitude modulation (QAM). Nonetheless, one critical drawback of DMLs or EMLs is the

generated optical signal will be double-side band (DSB) and frequency-chirped, which will

result in detrimental dispersion- and chirp-related power fading. As a result, after 100-km

transmission over single-mode fiber (SMF), the bandwidth of the transmission system is

limited to a few GHz [11,12]. Thanks to the advance in digital-signal-processing (DSP)

technology, both power levels and modulation levels of OFDM subcarriers can be adaptively

allocated to efficiently utilize very limited transmission bandwidth at baseband. Accordingly,

our previous work has demonstrated that using 128-QAM and 3-GHz bandwidth to achieve a

21-Gbps LR-PON over 100-km SMF [11]. Moreover, to increase available transmission

bandwidth, it has been proposed that modifying the chirp parameter of DMLs or EMLs to be

negative [13] or replacing the deployed fibers by the negative-dispersion fibers [14]. These

schemes, however, are too complicated and expensive to be feasible. Coded OFDM, which is

a visible scheme to overcome fading in wireless systems [15], has been used to coped with

power fading in SMF transmission [16], but at the cost of decreasing bandwidth efficiency

and increasing complexity. Nonetheless, unlike wireless system, the fiber channel response is

relatively stable, and power fading is periodic and predictable. Consequently, secondary low-

fading bands have been proposed to carry supplementary OFDM subcarriers to achieve full

utilization of transceiver bandwidth [12]. Nevertheless, within the range of 0–100 km, the

lowest supporting data rate of 7 Gbps in [12] implies that the number of 1-Gbps ONU is

restricted to only 7. Hence, to support 128 ONUs, 19 wavelength channels are required and

this will reduce the channel spacing and increase difficulties in wavelength management.

In this work, utilizing a 10-GHz IMDD transmission system, we propose a dynamic multi-

band OFDM subcarrier allocation scheme to avoid power fading effects in LR-PONs of up to

100 km. According to the different frequency responses of different SMF lengths, the power

level and modulation level of each OFDM subcarrier are dynamically modified to maximize

the transmission capacity. As such, this work successfully achieves a superior performance of

34.78 Gbps over 100-km SMF by economically using a 10-GHz EAM and a 10-GHz PIN.

Compared with the earlier work in [11], the experimental results confirmed at least 57%

increased of data rate after 100-km SMF transmission. Moreover, considering the 10-GHz

bandwidth of the EAM, the narrowest available bandwidth within 100-km SMF transmission

system does not occur at 100 km, but at ~40 km. As a result, this work also investigates the

transmission capacities over all possible cover radii of (LR-) PONs, i.e. 20–100 km, to

guarantee at least 33-Gbps obtainable capacity. Besides, subcarrier-to-subcarrier intermixing

interference (SSII), which has been theoretically revealed to be harmful after transmission in

[17], is also experimentally measured and discussed in this work. Finally, this experiment

demonstrates a cost-effective LR-PON of up to 100 km, which can provide 1-Gbps data rate

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29 August 2011 / Vol. 19, No. 18 / OPTICS EXPRESS 17548

to each of 32 ONUs. Moreover, the system could simply be upgraded to support 128 ONUs

by using only four wavelengths for each direction (downstream and upstream). Thus,

compared with the previous proposals of LR-PONs, our proposed 33-Gbps hybrid

WDM/OFDM LR-PON optimizes the trade-off between the number of wavelength channels

[2–6,8,9,12] and the cost of transceivers [7], based on cost-effective 10-GHz EML and PIN

based IMDD scheme.

2. Criteria and transmission bandwidth of feasible IMDD LR-PONs

Section

1

λ

λλ

λ

W+1

λ

λλ

λ

1,

Section

2

Section

W

Metro Edge Node

WGR

λ

λλ

λ

k

Legend:

WGR: Waveguide Grating Router

Section

k

ONU

N-1

ONU

1

WGR

λ

λλ

λ

W+k

λ

λ

λ

λ

W+2

λ

λλ

λ

2,

λ

λ

λ

λ

2W

λ

λλ

λ

W,

λ

λλ

λ

W+k

λ

λλ

λ

k,

ONU

2

10-GHz IMDD OLT

EML/

DML

PIN

128 Gbps

32 Gbps

ONU

N

10-GHz IMDD ONU

EML/

DML

PIN

λ

λλ

λ

k

λ

λλ

λ

W+k

N=32

W=4

1 Gbps

1 Gbps

Business

Mobile/

Wireless

1 Gbps

Residential

20~100 km

Optical

Splitter

Fig. 2. Proposed feasible architecture for a cost-effective LR-PON.

Based on the trade-off between wavelength channel number and transceiver cost and the

requirements of next-generation PON (NG-PON) discussed in [18], some criteria have been

contemplated for a feasible LR-PON. Though some of the proposed numbers are open to

discussion and may change with the development of technologies, these are the important

issues that need to be addressed prudently: (a) Sustainable symmetric data rate of each ONU

is 1 Gbps for supporting future broadband multimedia services [18]; (b) Power splitting ratio

(N) is 32 for a reasonable optical loss budget and the aggregated data rate on a single

wavelength is

32

≥

Gbps; (c) Supporting ONU number is

128

≥

for sharing the capital and

operational expenditures; (d) Using WDM technology to stack four (W = 4) 32-Gbps PON,

thus 128 (

128

N W

× =

) ONUs can be achieved by four wavelengths for each direction, and

(e) Cost effective transceiver design is achieved by a 10-GHz IMDD scheme. Notably, four

wavelengths for each direction (downstream and upstream) are recommended due to very

limited wavelength resources, as shown in Fig. 1. Thus, the proposed architecture of a LR-

PON and the corresponding parameters of the above criteria are illustrated in Fig. 2, and the

proposed LR-PON can meet the requirements of NG-PON [18] by using a cost-effect and

feasible architecture. Wavelength management of four channels is not technique challenging,

but the most crucial part turns out to be achieving the expected data rate of >32 Gbps via a

cost-effective transceiver. In particular, while the IMDD scheme is preferred, an optical

OFDM signal is generated in the form of DSB. Namely, an electrical OFDM subcarrier will

be translated into two conjugated optical subcarriers on both sides of an optical carrier. After

transmission in fiber link, however, the residual dispersion destroys their conjugated property

and results in a well-known power fading. Moreover, since an optical signal modulated by a

DML or an EML is chirped, the additional phase modulation also weakens the conjugated

property between subcarriers. Accordingly, both frequency chirp and fiber dispersion will

affect the received power. Considering small-signal approximation, the received power, P

r

, of

a

n OFDM subcarrier [17,19,20]

(

)

(

)

2 2 2 2 1

r 2 s

1 cos 2 tan ,

P Lf P

α π β α

−

= + − ×

(1)

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29 August 2011 / Vol. 19, No. 18 / OPTICS EXPRESS 17549

where

α

is the chirp parameter,

2

β

is the group velocity dispersion (GVD) parameter

(

2

21.66

β

= − ps

2

/km for SMF), L is the fiber length, f is the frequency of the electrical

OFDM subcarrier, and P

s

represents the received power without fading. Therefore, as the

r

esidual dispersion increases such that

2 2 1

2

| 2 tan |

Lf

π β α

−

− approaches

/ 2

π

, the received

electrical subcarrier will suffer serious fading to fail signal detection. Equation (1) also

indicates that positive chirp will make fading happen with lower positive residual dispersion

(

2

0

β

<

). In any case, frequency components that suffer from high power fading are not

suitable to carry subcarriers.

0 1 2 3 4 5 6 7 8 9 10

-30

-25

-20

-15

-10

-5

0

5

Frequency (GHz)

S

21

of the transmission system (dB)

20 km

40 km

60 km

80 km

100 km

Fig. 3. Frequency responses of 20–100-km SMF transmissions.

Via a network analyzer, Fig. 3 displays the measured frequency responses of an EAM-

based transmission system over 20–100-km SMF. Notably, as the SMF transmission distance

is increased from 20 km to 100 km, the first null point of the frequency response, f

null

, shifts

left from 10.18 GHz to 4.8 GHz and the available bandwidth of the 1st frequency band (i.e.

baseband) is decreased from 5.56 GHz to 3.5 GHz. Using traditional baseband modulation

schemes, the available bandwidth will be seriously limited to the 1st f

null

. However, allocating

multi-band OFDM subcarriers to the first and secondary low-fading bands, the total available

bandwidth is increased dramatically. While 10-GHz transceivers are technique-matured and

cost-effective, the total available bandwidth of the multiple frequency bands within 10 GHz

has not been thoroughly investigated. If 3-dB power fading is used as the criteria to determine

which frequency bands could carry subcarriers, the frequency bands which suffer from < 3-dB

power fading could be defined as passbands and the others are forbidden bands. From Eq. (1),

for a system with positive residual dispersion, the m-th 3-dB power fading will occur at

frequencies f

m

:

( )

1 2 1

2

2

2

1 tan 1 2 tan

,

2

m

m

m

f

L

π α α

π β

− −

− − + −

= (2)

where m is a positive integer and

x

denotes the largest integer less than or equal to x.

Therefore, the passbands are defined at frequencies of < f

1

, f

2

~ f

3

, f

4

~ f

5

, and so on, and the

o

ther bands are the forbidden bands. While the chirp parameter of the EAM in our experiment

measured by the method in [20] is about 0.53 at the bias voltage of –1 V, the corresponding

passbands and forbidden bands after SMF transmission are theoretically shown in Fig. 4(a).

Although the bandwidth of the 1st passband (i.e. f

1

) is reduced after longer SMF transmission,

the frequency of the 2nd passband also decreases, and it can be used to carry subcarriers

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without the requirement of high-speed (>10 GHz) electronics. Considering 10-GHz

transceivers, the total bandwidth of passbands within 10 GHz is exhibited in Fig. 4(b). The

total bandwidth increases after the distance of ~40 km due to the joining of the 2nd passband,

but it decreases after ~80-km transmission owing to the appearance of the 2nd forbidden band.

Thus, fully utilizing all the available passbands and properly allocating OFDM subcarriers can

overcome the difficulty of very limited bandwidth after transmission. The following section

will present the multi-band EAM-based transmission by allocating OFDM subcarriers in

different passbands based on the analysis results shown in Fig. 4. Since the total bandwidth is

not monotonously decreasing as transmission distance is increasing, in additional to 100-km

transmission, the capacities of 20-km, 40-km, 60-km, and 80-km transmissions are also

investigated to verify the feasibility of the proposed scheme over PONs of various sizes.

0 20 40 60 80 100

0

2

4

6

8

10

12

SMF distance (km)

Frequency (GHz)

0 20 40 60 80 100

0

2

4

6

8

10

12

SMF distance (km)

Bandwidth (GHz)

1st-passband

All passbands < 10 GHz

Fig. 4. Simulation results of (a) the passbands and forbidden bands and (b) the 1st-passband

bandwidth and the total bandwidth of passbands <10 GHz, with α of 0.53.

3. Experimental setup and results

Figure 5 illustrates the experimental setup with the insets displaying the corresponding

electrical spectra for an optical OFDM transmission system over up to 100-km SMF

transmission. The experimental setups of 20-km, 40-km, 60-km, and 80-km SMF are similar,

except that the transmission distances are different. A point-to-point transmission is adopted

to emulate a long-reach OFDM-PON, since the loss of a remote node in PONs can be

compensated by an optical amplifier. The baseband electrical OFDM signal is generated by an

arbitrary waveform generator (AWG, Tektronix

®

AWG7122) using Matlab

®

programs. The

s

ignal processing of the OFDM transmitter consists of serial-to-parallel conversion, QAM

symbol encoding, inverse fast Fourier transform (IFFT), cyclic prefix (CP) insertion, and

digital-to-analog conversion (DAC). The sampling rate and DAC resolution of the AWG are

12 GS/s and 8 bits, respectively. In order to utilize the bandwidth of both passbands as shown

in Figs. 3 and 4, the two-band electrical OFDM signal is composed of two streams of OFDM

signals with the FFT size of 512 generated owing to the limited AWG bandwidth of only 7.5

GHz. Nevertheless, since the channel number of the AWG is only 2, the signal of the 2nd

band is emulated by directly up-converting the signal from the channel 2 of the AWG, instead

of independent I- and Q-channels.

With the detailed parameters of the multi-band OFDM at different transmission distances

given at Table 1 at the end of this section, and to avoid similar scenario being repeated, the

following paragraphs only describe the parameters of the 40-km and 100-km transmissions,

because the transmission conditions of these two cases are severer than the others. That is, as

shown in Fig. 4 (b), the 40-km SMF transmission has only one passband and the available

bandwidth is the minimum, while the 100-km SMF transmission has the longest distance in

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spite of two available passbands. Thus, these two cases could be regarded as the performance

baselines of one-band and two-band OFDM signals over 20–100-km SMF transmissions.

For the case of 100-km SMF, as shown in inset (a) of Fig. 5, the channel-1 OFDM signal

is located in the 1st passband, which consists of 23.44-MSym/s 64-QAM and QPSK symbols

encoded at the 2nd–144th and 145th–180th subcarriers, respectively. Hence, the whole 179

subcarriers of the channel-1 signal occupy 4.195-GHz bandwidth (46.8 MHz – 4.22 GHz),

yielding a total data rate of 21.78 Gbps. The channel-2 OFDM signal has the same symbol

rate, but the modulation formats are 32-QAM and QPSK encoded at the 2nd–44th and 45th–

76th subcarriers, respectively, shown in inset (b) of Fig. 5. After being up-converted to 7.992

GHz (corresponding to the frequency of the 341st subcarrier), the up-converted OFDM signal

occupies the 265th–339th and 343rd–417th subcarriers (6.21 GHz – 9.77 GHz) located in the

2nd passband. Therefore, the up-converted signal with 3.52-GHz bandwidth is used to

emulate the signal of the 2nd band with the total data rate of 13 Gbps, shown in inset (c) of

Fig. 5. As shown in inset (d) of Fig. 5, through a power coupler, both two bands are then

combined to achieve a total data rate of 34.78 Gbps.

Fig. 5. Experimental setup with spectrum illustration. (a) channel 1, (b) channel 2, (c) channel 2

after up-conversion, and (d) combination of channels 1 and 2.

For the case of 40-km SMF, both the channel-1 and channel-2 OFDM signals can be only

located in the 1st passband. The channle-1 OFDM signal is consisted of 23.44-MSym/s 128-

QAM and 32-QAM symbols encoded at the 2nd–94th and 95th–135th subcarriers,

respectively. Namely, these 134 subcarriers occupy 3.14-GHz bandwidth (46.8 MHz – 3.16

GHz) to yield a total data rate of 20.06 Gbps. With the same symbol rate, 64-QAM and 32-

QAM formats are encoded at the 2nd–28th and 29th–51st subcarriers of the channel-2 signal,

respectively, and then the signal is up-converted to 4.382 GHz, i.e. the frequency of the 187

st

s

ubcarrier. Thus, by up-converting the channel-2 signal, the 136th–185th and 189th–238th

subcarriers are generated at 3.19 GHz –5.58 GHz to emulate the signal of the 2nd band with a

total data rate of 12.98 Gbps. After combining the signals of the both bands via a power

coupler, the 33.04-Gbps OFDM signal is obtained.

Furthermore, the combined OFDM signals are then sent to the EAM to generate the

optical DSB OFDM signals. After 40 km or 100 km of SMF transmission and direct-

detection, the received electrical signals are both captured by a digital oscilloscope

(Tektronix

®

DPO 71254) with a 50-GS/s sampling rate and a 3-dB bandwidth of 12.5 GHz.

An off-line Matlab

®

DSP program is used to demodulate the OFDM signals and the

d

emodulation process includes synchronization, FFT, one-tap equalization, and QAM symbol

decoding. Lastly, from the constellation, the signal-to-noise ratio (SNR) is measured and used

to calculate the bit error rate (BER).

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(C) 2011 OSA

29 August 2011 / Vol. 19, No. 18 / OPTICS EXPRESS 17552

0 1 2 3 4 5 6

-8

-6

-4

-2

0

2

4

Frequency (GHz)

Received power and SNR penalty (dB)

Relative received power

Theoretical power fading

Relative SNR penalty

0 2 4 6 8 10

-10

-8

-6

-4

-2

0

2

4

Frequency (GHz)

Received power and SNR penalty (dB)

Relative received power

Theoretical power fading

Relative SNR penalty

Fig. 6. Relative received powers and SNR penalties of each subcarrier after (a) 40 km and (b)

100 km.

Fig. 7. SNR of each subcarrier and constellations of (a) 40-km and (b) 100-km SMF

transmissions.

Figures 6(a) and (b) show the relative received powers of each subcarrier after 40-km and

1

00-km SMF, respectively. The received powers are calculated by the relative power gain of

0 1 2 3 4 5 6

14

16

18

20

22

24

26

28

Frequency (GHz)

SNR (dB)

B-to-B

40 km w/o pre-emphasis

40 km w/ pre-emphasis

B-to-B 40 km

128QAM

32QAM

32QAM

64QAM

0 2 4 6 8 10

5

10

15

20

25

30

Frequency (GHz)

SNR (dB)

B-to-B

100 km w/o pre-emphasis

100 km w/ pre-emphasis

64QAM

QPSK

32

QAM

QPSK

QPSK

B-to-B 100 km

(a)

(b)

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(C) 2011 OSA

29 August 2011 / Vol. 19, No. 18 / OPTICS EXPRESS 17553

Furthermore, comparing the SNRs before and after transmission without pre-emphasis in Fig.

7, the relative SNR penalties are also plotted by green dots in Fig. 6. The relative SNR penalty

after 40-km transmission is consistent to the relative received power shown in Fig. 6(a), and

this implies the penalty is mainly contributed by power fading. In Fig. 6(b), however, the SNR

penalties of the subcarriers at 6.2–8.2 GHz are much severer than the degradation caused by

power fading, compared with the other subcarriers. Actually, these additional penalties are

caused by the dispersion- and chirp-relevant SSII [17]. In order to clarify the influence of

SSII, the electrical spectra with and without 100-km transmission are shown in Figs. 8(a) and

(b), respectively, and only the 1st passband is used to carry subcarriers to simplify the

investigation of SSII. The blue curves are the measured electrical spectra, and the simulation

results of green curves are also drawn for comparison. Obviously, there is unexpected power

increase outside the signal band. According to the SSII theory and the small-signal

approximation given in [17], a chirped OFDM signal contains N

s

subcarriers with the same

p

ower of P

s

and the subcarrier spacing of

f

∆

at baseband, the power of the second-order SSII

at the frequency of

n f

∆

can be approximated as

( )

( )

( )

( )

( )

( )

( )

( )

2

2

2

s

SSII s

c

2 2 2

s s

2

2 2

1

8

,

1+2 cos sinc 4 4 cos sinc 2

D D D D

n

n n

P

P N

P

n n N n n N

α

θ θ θ θ

≅ + −

× + − − −

(3)

where n is a positive integer less than 2N,

D

θ

equals to

2 2

2

2

L f

π β

∆

and P

c

denotes the power

of the optical carrier. Following Eq. (3), the theoretical SSII curve is plotted by the red curve

in Fig. 8(b) to prove those additional frequency components are indeed caused b