improved expression for intensity noise in subcarrier

IJMOT-2006-10-223 © 2007 ISRAMT

Improved Expression for Intensity Noise in Subcarrier Multiplexed Fiber Networks

Xavier Fernando* and Hatice Kosek

ISRAMT, Department of Electrical and Computer Engineering

Ryerson University, Toronto, Ontario, Canada Tel: 1-416-979-5000 ext.6077; Fax: 1-416-979-5280; E-mail: [email protected]

Abstract-The relative intensity noise (RIN) plays an important role in subcarrier multiplexed multimedia over fiber (MOF) networks. The RIN is conventionally considered to be proportional to the square of the mean optical power. This is true under small signal, single channel conditions. Nevertheless, experiments by us and many others have shown that the RIN also increases with the modulation index m that reflects the power of the stochastic modulating signal s(t). Accurate characterization of the RIN is important especially in MOF systems supporting sub carrier multiplexed radio signals in addition to digital data. Modern MOF links tend to have large carrier to sideband ratio that enhances RIN. In this paper, a mathematical expression for the RIN is derived from fundamental principles that shows the dependency of RIN on modulation index m and modulating multimedia signal power E[s2(t)]. The new expression better explains the excess increment of noise power in MOF systems. The signal to noise ratio is analyzed using the new expression and numerical evaluations are done considering DOCSIS specifications. Index Terms- Cable TV networks, DOCSIS, multimedia over fiber, optical modulation, relative intensity noise (RIN), subcarrier multiplexing.

I. INTRODUCTION An optical fiber network supporting more than one type of service can be referred as multimedia-over-fiber (MOF) system. The MOF systems have become the backbone of optical access networks. For example, fiber-to-the-home (FTTH) and, less expensive fiber-to-the-curb (FTTC) systems have been rapidly deployed. 30 million people will have FTTH by 2010 in Japan only [1]. The trend is similar worldwide.

MOF will play important role in these emerging FTTH and FTTC scenarios in addition to existing hybrid fiber coaxial (HFC) networks and fiber-twisted pair networks (with DSL technology) for multimedia delivery. In these scenarios, multimedia signals such as (high definition or conventional) video, digital audio and high-speed (Internet) data signals are simultaneously transmitted over optical fiber. The video signal could be either radio frequency (RF) analog video or IP (Internet Protocol) digital video. The IP video is similar to Internet traffic with additional streaming requirement. However, the RF video signal is analog and subcarrier multiplexing is done to transmit multiple television channels. Furthermore, the RF video is a proven technology and expected to continue to play key role in the fast emerging FTTH networks as well [1]. Moreover, in addition to video, modern HFC networks promises high-speed (Internet) data transmission using cable modems. Data-over-Cable Interface Specifications (DOCSIS), for example, is the primary protocol adopted in this scenario. DOCSIS operates between 54-860 MHz range for downstream transmission of IP packets. Accurate analysis of the signal to noise ratio (SNR) of MOF systems is very important in all these cases. A directly modulated ROF link is mainly subjected to shot, RIN and thermal noise. The shot noise linearly increases with the mean optical power and is insensitive to the modulation index m provided the modulating signal s(t) has zero mean. The thermal noise is signal independent. However, the RIN is typically assumed to increase with the square of the mean

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optical power. This is true under small signal conditions. The RIN deserves special attention under large modulation conditions because it is observed to increase with the modulating RF signal as well. A widely used expression [2] gives the mean square value for the noise current IRIN due to RIN as (1) BPPI oRINRIN

222 ℜ=⟩⟨

where, PRIN is the relative intensity noise parameter typically given in dB per unit bandwidth1, B is the bandwidth of interest, ℜ is the detector responsivity and Po is the mean optical power. The parameter PRIN is generally assumed constant for a given laser diode. Note that, according to (1), the intensity noise power does not change unless PRIN, Po, B or ℜ changes. However, in our observations and many other occasions ([3], [4], [5], [6] and [7]) the RIN is found to be changing with the optical modulation index m that reflects the modulating signal power. Winston Way, investigated this behavior of laser intrinsic RIN and named a new dynamic RIN [4] under direct modulation conditions for both single and multi mode lasers. He also observed a threshold value for m above which, a sudden jump of the RIN level occurs. This is not a major concern traditionally because m used to be typically small. However, with the advent of modern optical signal processing techniques, unmodulated optical carrier can be suppressed resulting very high m [8]. Therefore in this scenario, an accurate analysis is important. In addition to laser intrinsic RIN, there is additional intensity noise in the optical link due to multiple optical reflections (Interferometric noise) and Brillouin scattering. This reflected RIN also increases with the instantaneous optical power. Shibutani et. al [5] investigated this 1 The PRIN is also widely referred as RIN and given in dB/Hz [2]. However, we use the notation PRIN to avoid confusion.

phenomenon and showed that the reflection induced RIN nonlinearly increases with m, the number of connectors and the fiber length. They also showed that the total intensity noise is the summation of reflected and intrinsic intensity noises. In this paper, we mathematically derive an improved expression for the variance of RIN using the energy conservation theory incorporating the influence of m in a directly modulated MOF link. Then we analyze the SNR performance. The derived expression is general in the sense that it does not assume device dependent parameters and is independent of modulating signal format and frequency. It better explains the typical behavior of RIN under large signal modulation conditions, typically happen with modern multimedia networks. It reduces to the conventional expression (1) for small m. II. THE FUNDAMENTAL NOISE PROCESSES

IN MULTIMEDIA OVER FIBER LINKS In this section, basic noise that exist in MOF system is analyzed. We assume that all connectors and the fiber are ideal so that no distortion occurs between the laser and the detector. We do not consider fiber chromatic dispersion. We also assume that the receiver has an electrical amplifier followed by a band-pass filter of bandwidth B centered at fc as shown in Fig.1. The transfer function of this unity gain band pass filter H(f) is shown in Fig. 2 (middle). Considering direct intensity modulation on a linear laser diode, the instantaneous optical power output P(t) from the laser in response to input electrical signal s(t) is, )]()][(1[)( tPPtmstP o ∆++= (2)

Here m is the optical modulation index, Po is the mean optical power, s(t) is the normalized modulating RF signal |s(t)| ≤ 1 ) and, is the instantaneous fluctuation term due to laser

)(tP∆

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154 INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY VOL. 2, NO. 2, APRIL 2007

Fig. 1. Block diagram of the optical receiver front end with an ideal bandpass filter. relative intensity noise. Neglecting fiber attenuation2 and dispersion3, the received optical signal at the receiver is same as P(t), which produces a detected current ID(t). ID(t) is proportional to the instantaneous optical power P(t) so that ID(t) = P(t). The well known definition for responsivity for a PIN diode is

ℜℜ

)()(

tPtI

PI D

o

D =⟩⟨

=ℜ (3)

where is the mean value of the detector current. Note that the time invariant

⟩⟨ DIℜ is the

ratio between both the instantaneous and mean values of ID(t) and P(t). Let the detected current ID(t) be filtered by a filter with a transfer function H(f) to obtain an output current I2(t) (Fig.1). Then, I2(t) is the convolution of ID(t) with the impulse response of the receiver filter h(t).

∫∫∞

∞−−

∞

∞−ℜ=−= ττττττ dthPdthItI D )()()()()(2

(4) The process I2(t) is a doubly stochastic Poisson process in the presence of the relative intensity

2 Fiber attenuation will only scale down P(t) and this will have no effect on the final expression 3 This analysis including fiber dispersion is left for future work. Furthermore, typically MOF links used in the last mile, which are short and dispersion may be ignored.

Fig. 2. Spectrum of the detector current ID(f) (up) and the first and second order frequency responses of the receiver bandpass filter, S(f) is the Fourier transform of s(t). noise )(tP∆ . The mean and variance of this doubly stochastic output process are determined by the generalized Campbell theorem as shown in [9]. I2(t) can be re-written as,

ττττ dthPPmstI o )()]()][(1[)(2 −∆+∞

∞−+ℜ= ∫ (5)

In the more convenient frequency domain, this can be written as, )()()()()(2 fHfPfHfIfI D ℜ== (6) Fig. 2 shows the spectrum of incoming signal ID(f) (up) and the transfer function of the BPF H(f) (middle). Typically, the bandwidth B of the receiver filter is greater than the bandwidth of s(t). Therefore, the filter admits the signal, however, blocks the direct current term ℜ Po. Hence, the filter output current, I2(t), depends only on the sidebands. Note that is a zero mean process. Therefore,

)(tP∆

)()()()(2 tItItsmPtI RINsho ++ℜ= (7)

)()()( tItItmsI RINshD ++⟩⟨=

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where Ish(t) is the shot noise current and IRIN(t) is the intensity noise current. The variance of I2(t) gives the shot noise when = 0 and, the variance of I2(t) gives the relative intensity noise when ≠ 0.

)(tP∆

)(tP∆ A. The Shot Noise Let us assume = 0. Under this assumption, the variance of I2(t) after the filter gives the shot noise power. The expression for this variance is given by (8). The derivation of (8) is quite involved and given in appendix C of [9] using a moment generating function of the photodiode. Note the appearance of q in (8)

)(tP∆

(8) ∫∞

−∞−ℜ= τττσ dthPq )()( 2

122

If we examine (8), the variance of I2(t) depends on the signal P(t) filtered through a hypothetical filter with an impulse response h2(t). The frequency responses of both H(f) (middle) and H2(f) = H(f) * H(f) (bottom) are shown in Fig 2. From the transfer function H2(f), it is obvious that only the DC term Po passes through this filter and both side bands are attenuated. Therefore, the shot noise variance after the filter is given by,

ℜ

(9) BIqBPqI Dshot ⟩⟨=ℜ=⟩⟨= 22 0

212

2σ

This expression for the variance of the shot noise is widely used in the literature. Note that shot or quantum noise depends on the received mean optical signal Po, therefore does not depend on the modulation conditions provided = 0. We can physically explain this phenomenon as follows: if the instantaneous optical power P(t) is below the mean level Po, then there will be less quantum noise, similarly if P(t) is above Po, then there will be more quantum noise. Statistically, if areas below and above the mean level are equal that means = 0. In this case, the average quantum noise will not change due to modulation.

⟩⟨ )(ts

⟩⟨ )(ts

B. The Relative Intensity Noise The relative intensity noise after the bandpass filter is given by the variance of I2(t) for nonzero

)(tP∆ . The variance of I2(t) is given by the general expression,

22

2212

2 )]([)]([ tIEtIE −=σ (10)

2

2

)()]()][(1[

)()]()][(1[

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∞

−∞−∆++ℜ−

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∞

−∞−∆++ℜ=

∫

∫

ττττ

ττττ

dthPPms

dthPPms

o

o

Note that )(tP∆ and s(t) are not correlated and

)(tP∆ is a zero mean process. Therefore, the expectation of the cross product term of the first term of (10) is zero. That is,

0)()](2[)](1[ 222 =−∆+ℜ ∫∞

∞−

ττττ dthPPms o

Hence,

2

122 )()](1[ ⎟⎟

⎠

⎞⎜⎜⎝

⎛−+ℜ= ∫

∞

∞−

τττσ dthPms o (11)

2

2

)()()][(1[

)()()](1[

⎟⎟⎠

⎞⎜⎜⎝

⎛−∆++ℜ−

⎟⎟⎠

⎞⎜⎜⎝

⎛−∆+ℜ+

∫

∫∞

∞−

∞

∞−

ττττ

ττττ

dthPPms

dthPms

o

However, again because ∆P(t) = 0 and ∆P(t) and s(t) are not correlated, the last term of (11) becomes,

τττ dthPsm o )(])(1[ 2222 −+ℜ ∫∞

∞−

(12)

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This will be the same as the first term of (11) after simplification. Therefore, the first and last terms of (11) are the same. Hence,

2

122 )()()](1[ ⎟⎟

⎠

⎞⎜⎜⎝

⎛−∆+ℜ= ∫

∞

∞−

ττττσ dthPms (13)

Equation (13) needs careful observation. It is interesting to note that the variance of I2(t) does not depend on Po when the input consists of the stochastic noise process . However, it does depend on m and s(t).

)(tP∆

Now, let us define an imaginary filter,

)()](1[)( τττ htmshRIN −+= ∞≤≤∞− τ (14)

2

122 )()(

⎭⎬⎫

⎩⎨⎧

−∆ℜ= ∫∞

∞−

τττσ dthP RIN (15)

In the frequency domain,

dffHfN RINRIN22

122 )()(∫

∞

∞−

ℜ=σ (16)

Here, NRIN(f) is the double sided power spectral density of the relative intensity noise. At frequencies of interest for analog optical transmission this has a constant spectrum [2]. From (14), hRIN(t) has two components, h(t) + mh(t)s(t-τ)h(t) is a bandpass filter with unity gain that encloses s(t) and centered at fc. The second term depends on s(t). Therefore, from the conservation of power, the power confined in the spectrum of |HRIN(f)|2 is same as the average square value of the term [1+ms(t)] for E[s(t)]=0. Therefore, the noise power due to RIN is given as, (17) ])(1[)(2 222

122 ⟩⟨+ℜ= tsmBfN RINσ

The double-sided power spectral density NRIN is related to the RIN parameter PRIN by [2] (18) 2/)(2)/( oRINRIN PfNHzdBP ⟩⟨=

Solving the above two equations with a knowledge that σ2

12 = in the presence of cumulative intensity noise ,

⟩⟨ RINI 2

)(tP∆ (19) ])(1[ 22222 ⟩⟨+ℜ=⟩⟨ tsmBPPI oRINRIN

When s(t) consists of n number of frequencies such as in a subcarrier multiplexed system, (19) can be rewritten as,

])(1[ 2

1

2222 ⟩⟨+ℜ=⟩⟨ ∑=

tsmBPPI i

n

iioRINRIN (20)

where mi is the modulation depth of the signal si(t). The total optical modulation index m in a MOF system is related to the per channel modulation index mi by [10]

∑=

=n

iimm

1

2 (21)

If each channel has identical modulation index, then (21) can be simplified as m = n mi. The expression we obtained in (19) more accurately explains the dependency of RIN on m and modulating signal power. When m = 0, the expression in (19) reduces to the widely used expression for the (static) RIN given in (1). Typically m ≤ 40 % and |s(t)| < 1. However, when these quantities are high, this term is not negligible.

III. THE SIGNAL TO NOISE RATIO

Considering the expression derived in (19), the signal to noise ratio also will be slightly different from conventional assumptions. From (3)

222 )](1[)(

)](1[)](1[)(

tmsItI

tmsItmsPtI

DD

DoD

+=

+=+ℜ=

The signal power is . ⟩⟨=⟩⟨ )()( 222 tsImti Dp

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The signal to noise ratio of multimedia fiber optic link considering major noise processes is given below.

noise

n

iiiD

P

tsmISNR

∑=

⟩⟨= 1

22 )( (22)

LoBtDnoise RBTKFqBIP /42 +=

])(1[ 2

1

22 ⟩⟨++ ∑=

tsmBIP i

n

iiDRIN

Shot, RIN and thermal noise terms involved in the expression are given. Thermal noise has a constant variance and depends on the receiver resistance only. The variance of the shot noise is linearly proportional to mean optical power in the fiber. Although the instantaneous optical power in the fiber fluctuates due to RF intensity modulation, the mean optical power does not change unless the DC bias current is changed. Therefore, the shot noise does not change with modulating signal power and constant for a given modulation depth m. However, the RIN changes with RF signal level. This is seen from the expression in (17). This is also logical because, the RIN is proportional to the square of the optical power. Since, the instantaneous optical power in the fiber fluctuates at radio frequency, the square of it increases with RF signal level depending on m. The following additional points are observed from the expression for signal to noise ratio:

1) The higher the bandwidth B of s(t), the lower the SNR because, the wider noise bandwidth in the optical link collects more noise.

2) The higher modulation index m yields

better SNR. This is because more power is contained in the side bands compared to the unmodulated carrier

If the thermal noise at the receiver amplifier is made small enough due to an improved design, then (22) becomes,

⎥⎦

⎤⎢⎣

⎡⟩⟨++

⟩⟨=

∑

∑

=

=n

iiiDRIN

n

iiiD

tsmBIPqB

tsmISNR

1

22

1

22

)(12

)( (23)

From (23) we further deduce that,

1) In the shot noise limited case,

qB

tsmISNR

n

iiiD

2

)(1

22∑=

⟩⟨= (24)

That is, SNR increases with the modulating RF signal power which is in sidebands.

2) In the RIN limited case,

∑

∑

=

=

⟩⟨+

⟩⟨= n

iiiRIN

n

iii

BtsmP

tsmSNR

1

22

1

22

])(1[

)( (25)

Under this condition, the SNR increases with m (RF power) first. However, when the RF power is large enough, the SNR saturates. IV. NUMERICAL EVALUATION AND

DISCUSSIONS

We assume that for the single mode laser the specified PRIN is -140 dB/Hz. Zero fiber attenuation is assumed. E[si2(t)] is taken as unity, assuming normalized base band square pulses. We consider that the receiver is a PIN diode with responsivity ℜ of 0.9 A/W. The load resistance is 50 Ω the receiver amplifier has a noise figure Ft of 3 dB. B is 6 MHz, the bandwidth allocated by each data-carrying channel in a DOCSIS system.

In Fig.3, the RIN power obtained by the conventional (1) and the improved (20) expressions are plotted. The modulation index m is 0.8 and n is 1. This figure demonstrates the

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6 7 8 9 10 11 12 13 14-92

-90

-88

-86

-84

-82

-80

-78

-76

-74

-72

Rel

ativ

e In

tens

ity N

oise

(RIN

) Pow

er [d

Bm

]

Mean Optical Power(Po) [dBm]

Conventional ExpressionImproved Expression

Fig. 3. Relative intensity noise power versus mean optical power, n=1, B=6 MHz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-103.5

-103

-102.5

-102

-101.5

-101

-100.5

-100

Modulation Index m

Rel

ativ

e In

tens

ity N

oise

Pow

er [d

Bm

]

Conventional ExpressionImproved Expression

Fig. 4. The relative intensity noise power using the conventional expression and the improved expression, n=1, B= 6 MHz additional noise power yielded by the new expression. As expected, in both cases, noise power is directly proportional to the square of Po. Fig. 4 shows the simulated RIN power using the conventional expression (1) and, the derived expression (20). The mean optical power Po is assumed to be 1 mW in the fiber and only a single channel is transmitted in the entire DOCSIS operation band. As seen in the figure, the RIN is independent of m according to (1) whereas with (20), noise power increases with m. At m=1, noise power increases by 3 dB. Fig. 5 illustrates the signal to noise ratio (SNR) at the receiver in the RIN limited case (25). In this figure, similar trend as in Fig. 4 is observed. At

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 150

55

60

65

70

75

Modulation Index, m

SN

R [d

B]

conventionalimproved

Fig. 5 SNR curves obtained by the conventional and the improved expressions in the RIN limited case, n=1, B= 6 MHz low m values, the behavior of conventional and the improved SNR plots are indistinguishable. However, with the increase in m, there is considerable difference. The SNR is reduced by 3 dB at m=1. Finally, let us look at the overall SNR expression given by (22), which considers all three shot, RIN and thermal noise. Fig. 6 shows the SNR of a single channel in the presence of varying number of adjacent channels. Note that in the simulation, each channel has identical modulation index mi and the total m is 0.8. The noise within the bandwidth (6 MHz) of the single channel increases as more number of channels is loaded into the fiber optic link. Therefore, the SNR performance of one channel drops with number of adjacent channel. This is an important phenomenon that may explain the performance degradation even before nonlinearity start to appear.

V.EXPERIMENTAL VERIFICATION We also setup an experiment shown in Fig. 7 to observe the impact of other channels on the desired channel in a subcarrier multiplexed MOF system. This is done by recording error vector magnitude (EVM). BPSK modulated RF signal (850 MHz) carrying data at 5 Mb/s is added with a second RF signal at 750 MHz at an RF power

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IJMOT-2006-10-223 © 2007 ISRAMT


0 10 20 30 40 50 6050

52

54

56

58

60

62

64

66

68

Number of channels, n

Tota

l SN

R in

[dB

] of o

ne c

hann

el

Fig. 6. SNR of a single channel (6 MHz) when varying number of adjacent channels co-exist in the DOCSIS system

Fig. 7. The block diagram of an experimental setup used in measuring the EVM of a single channel in a multichannel over fiber system combiner. The combined microwave signal is directly modulated using a distributed-feedback (DFB) low noise laser (Fiber-Span AC231T-2.5-1.3). The optical signal transmitted through a very short single mode fiber is detected by a fiber optic receiver (Fiber-Span AC231R-2.5-1.3). The receiver is a high speed, low distortion InGaAs PIN diode photo detector. Following the detection, the RF signal at 850 MHz is fed into a Sony Wireless Communications Analyzer (WCA380). Then, the EVM that reflects the noise in the link is measured. The RF power of the signal at 750 MHz is increased to emulate multiple RF channels in the system. At each power level 20 readings were taken and an average EVM was obtained. Fig. 8 shows the

0 5 10 15 20 25 301.58

1.6

1.62

1.64

1.66

1.68

1.7

1.72

1.74EVM vs. number of channels

number of channels, n

EV

M [%

rms]

Fig. 8. Average error vector magnitude (EVM) versus number of channels behavior of average EVM (in % rms) of the 850 MHz channel with the total number of channels.The total RF power was kept within the linear region of the laser diode to avoid nonlinear distortion and cross talk. The EVM of the desired channel can be seen to increase as more channels are added to the system validating the theory.

VI. CONCLUSIONS

In this paper, we derived an improved expression for the RIN that reflects the increment of the RIN with modulation index m (17) and analyzed the signal to noise ratio performance of a multimedia over fiber system. Note that the new expression is mathematically derived from the fundamental principles and it shows good agreement with experimental measurements done by us and many others ([4], [5], [6] and [7]). Simulations complying with the DOCSIS standard confirm that the dependence of RIN power on the modulation index cannot be ignored in strong modulation cases. Furthermore, in a MOF system, the noise contribution of co-existing channels has significant affect on the SNR and EVM of a channel of interest.

REFERENCES

[1] Hiromichi Shinohara, “Broadband access in Japan:

Rapidly growing FTTH market,” IEEE Communication Magazine, vol. 43, no. 9, pp.72–78, 2005.

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[2] John M. Senior, Optical Fiber Communications: Principles and Practice, Number TK5103.59 .S46. Prentice Hall, 2 edition, 1992. G. L.

[3] Xavier Fernando and Abu Sesay, “Characteristics of directly modulated ROF links for wireless access,” in Proceedings of the CanadianConference on Electrical and Computer Engineering. May 2004, vol. 4, vol. 7, no. 11, pp. 1806–1818, November 1989

[4] W. I. Way, “Subcarrier multiplexed lightwave system design considerations for subscriber loop applications,” Journal of Lightwave Technology, vol. 7, no. 11, pp. 1806–1818, November 1989.

[5] Makoto Shibutani, Wataru Domon, and Katsumi Emura, “Reflection induced degradations in optical fiber feeder for micro cellular mobile radio systems,” IEICE transactions on electronics, vol. E76-C, no. 2, pp. 287–291, February 1993.

[6] X. Lu, C. B. Su, R. B. Lauer, G. J. Meslener, and L. W. Ulbricht, “Analysis of relative intensity noise in semiconductor lasers and its effect on subcarrier multiplexed systems,” Journal of Lightwave Technology, vol. 12, no. 7, pp. 1159–1166, July 1992.

[7] K. Y. Lau and H. Blouvelt, “Effect of low frequency intensity noise on high-frequency direct modulation of semiconductor injection lasers,” Applied Physics Letter, vol. 52, pp. 694, 1988.

[8] Xijia Gu, Yifeng He, Hatice Kosek, and Xavier Fernando, “Transmission efficiency improvement in microwave fiber-optic link using subpicometer optic bandpass filter,” in Proceedings of SPIE, Photonic North, September 2005.

[9] Goran Einarsson, Principles of Lightwave Communications, Number TK 5103.59 E.36. John Wiley and Sons, West Sussex, England, 1995.

[10] Gerd Keiser, Optical Fiber Communications, Number TK5103.59.K44 in II. McGraw-Hill, 3 edition, 2000.

won both the first and second prize at Opto Canada - the SPIE regional conference in Ottawa in 2002.


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improved expression for intensity noise in subcarrier

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