Noise analysis in photonic true time delay systemsbased on broadband optical source and

dispersion components

Xiaoxiao Xue,* He Wen, Xiaoping Zheng, Hanyi Zhang, Yili Guo, and Bingkun ZhouDepartment of Electrical Engineering, National Laboratory for Information Science and Technology,State Key Laboratory on Integrated Optoelectronics, Tsinghua University, Beijing 100084, China

*Corresponding author: xxx03@mails.tsinghua.edu.cn

Received 25 September 2008; revised 7 December 2008; accepted 11 December 2008;posted 15 December 2008 (Doc. ID 102033); published 21 January 2009

The noise in photonic true time delay systems based on broadband optical source and dispersion com-ponents is investigated. It is found that the beat noise induced by the optical source begins to dominateand grows far larger than other noise terms quickly, as long as the detected optical power is above somecertain value Pthr. When the system dispersion is nonzero, the output carrier-to-noise ratio (CNR) willchange periodically with the optical bandwidth due to the noise power increment and the dispersioninduced radio frequency signal power degradation. The maximum CNR is the peak value of the firstperiod. For a set of specified system conditions, the Pthr is calculated to be −21dBm, and the optimaloptical bandwidth is 0:8nm, at which the maximum CNR is 93:3dB by considering the noise in a1Hz bandwidth. The results are verified experimentally. © 2009 Optical Society of America

OCIS codes: 270.2500, 060.5625, 280.5110.

1. Introduction

Photonic true time delay (TTD) has been underintense investigation and great development for itsapplication in optical beam-forming networks(OBFN), which is one of the most potential technol-ogies for airborne satellite communication [1], broad-band wireless access, and millimeter-wave radiolocal area networks [2,3]. OBFNs have many advan-tages compared with traditional electrical steeringsystems, such as small size, low loss, immunity toelectromagnetic interference, wide instantaneousbandwidth, and squint-free array steering.Various configurations have been proposed for

photonic TTD, of which systems employing broad-band optical source (BS) and dispersion componentshave been reported recently for their advantages ofsimplicity, high tunability, and expandability [4,5].The diagram of such a BS-based TTD network isshown in Fig. 1. The light from the broadband optical

source is modulated by a radio frequency (RF) signaland then transmits through an optical dispersioncomponent, such as a chirped fiber grating (CFG),photonic crystal fiber (PCF), and dispersion compen-sating fiber (DCF). The light after dispersion is thensplit into N channels feeding N antenna elements(see Fig. 1). The time delay of each channel can bechanged by tuning the corresponding optical tunablefilter (OTF). The relative time delay between twochannels is given by

Δτ ¼ DΔλ; ð1Þwhere D is the dispersion coefficient of the opticaldispersion component (ps=nm), and Δλ is the dis-tance of the relative central wavelength betweenthe two OTFs.

The main advantages of BS-based photonic TTDsystems over laser-based ones are their simplicityand low cost, and the feature of wide optical line-width for dispersion and detection can even mitigatethe group delay ripple effects of CFGs [4]. However,the wide optical linewidth can also lead to large

0003-6935/09/040658-06$15.00/0© 2009 Optical Society of America

658 APPLIED OPTICS / Vol. 48, No. 4 / 1 February 2009

beating noise in photodetection. In experiments theoutput carrier-to-noise ratio (CNR) of BS-based TTDsystems is at least 30dB lower than those employinga light source of lasers [6], which could greatly reducethe system performance in some applications.We examine the noise in BS- and dispersion-based

TTD systems both theoretically and experimentally.The sources of noise are investigated, and by consid-ering only the dominant noise term resulting fromthe beatings of optical source, the system noise figureand output CNR can be calculated. The relationshipbetween the CNR and the bandwidth of the OTF isalso discussed. The analysis reveals that the opticalbandwidth of the OTF can be properly designed tooptimize the system CNR.

2. Theoretical Description

The experimental setup of a BS- and dispersion-based TTD system is shown in Fig. 2. The ASE froma flat-gain erbium-doped fiber amplifier (EDFA),which is a broadband optical source, is polarizedand then modulated in a Mach–Zehnder modulator(MZM) by a RF signal from an Agilent 8257C. Themodulated light then goes through an optical circu-lator after being reflected by a CFG connected to asecond port. Then it is filtered by an optical filterand detected by a p-i-n photodiode (PD). The gener-ated signal is amplified by a low-noise microwaveamplifier and then analyzed by an electrical spec-trum analyzer (Agilent 4446A).The main noise sources in such a link may include

thermal noise of the RF signal source, thermal noiseof the modulator terminal resistance and its match-ing circuit, photocurrent shot noise, dark currentshot noise, beat noise of the BS, thermal noise ofthe photodetector resistance, and noise from theelectrical amplifier. The MZM we used has a travel-ing wave structure, so the thermal noise from the ter-minating resistor travels in the opposite direction tothe light and does not modulate the light [7]. Thep-i-n PD is connected to a 50Ω resistor inside its

package. Assuming that every electrical connectionis perfectly impedance matched, the equivalent cir-cuit of the link including the noise sources can thenbe drawn as in Fig. 3; see Table 1 for definitions of thesymbols. The noise sources include thermal noise ofthe RF signal source, photocurrent shot noise, darkcurrent shot noise, beat noise of the BS, thermalnoise of the photodetector resistance, and noise fromthe electrical amplifier. To simply the representation,we treat the thermal noise of the photodetector andthe noise from the electrical amplifier as an equiva-lent noise source denoted by i2the at the input of theamplifier. All the noise sources are statistically inde-pendent, so the equivalent total noise at the input ofthe amplifier is given by

i2n ¼ Gv2ts4R2

mþ i2the þ i2dn þ i2sn þ i2bn

¼ kTRm

BGþ 4kTRL

BFA þ 2qidBþ 2qipBþ i2bn: ð2Þ

To evaluate the noise terms, we need to derive theexpressions of link gainG and beat noise i2bn. Uniformbroadband light can be divided into numerous fre-quency elements along the frequency axis. Each isan independent optical source with power Soδν andrandom phase ϕν, where So is the power spectral den-sity (PSD). The light at frequency νi will generatethree terms after double sideband modulation by asingle-frequencymicrowave signal as shown in Fig. 4:the upper bandUi, the central bandCi, and the lowerband Li. The amplitude A and the phase ϕ of eachband after an optical filter with transfer functionHðνÞ are shown in Fig. 4 written as Aejϕ. Similarlythe light at frequency νj ¼ νi þ f e will also generatethree terms: Lj, Cj, and Uj, where f e is the modula-tion frequency.

When the modulated light passes through the sys-tem and is detected, a microwave signal of f e will begenerated through Ci −Ui beating and Lj − Cj beat-ing at the PD. For convenience, we assume here that

Fig. 1. Schematic diagram of OBFNs based on a BS and disper-sion component.

Fig. 2. Experimental configuration for testing the output CNR ofthe TTD system based on BS and CFG. PC, polarization controller;PD, photodiode; LNA, low-noise amplifier.

Fig. 3. Equivalent circuit of a BS-based TTD link including thenoise sources.

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the optical transmission loss is zero. When the opti-cal transmission loss is nonzero, So can be simply re-placed by twice the actual optical PSD at the input ofthe PD. By taking account of the system dispersionand the optical filter shape, the generated photocur-rent can be written as

δis ¼�R

�12ðm2SoHðνjÞδνÞ1=2ejðϕiþϕeÞ

�

× ½ðSoHðνiÞδνÞ1=2ejϕi �� þ R½ðSoHðνjÞδνÞ1=2ejϕj �

×�12ðm2SoHðνiÞδνÞ1=2ejðϕj−ϕeÞ

���

× e−j2πf e½Dνðνi−ν0Þþτ0�

¼ e−jf½2πf eDνðν−ν0Þþτ0�−ϕeg

× RmSo½HðνÞHðνþ f eÞ�1=2δν∣ν¼νi ; ð3Þ

where Dν is the system dispersion in s=Hz and R isthe responsivity of the PD. m is the amplitude mod-ulation index defined as m ¼ Vsπ=ð2VπÞ, where Vπ is

the half-wave voltage of the MZM and Vs is the vol-tage amplitude applied to the modulator electrodes.ϕe is the phase of the microwave source, ν0 is the cen-tral frequency of the dispersion component, and τ0 isthe group delay at ν0.

The total current is the sum of δis and is given by

is ¼ Σδis

¼ RmSo

Z þ∞

0fe−jf½2πf eDνðν−ν0Þþτ0�−ϕeg

× ½HðνÞHðνþ f eÞ�1=2dνg: ð4Þ

The beat noise can be calculated in a similar way.Since the spontaneous–spontaneous beat noise hasbeen investigated thoroughly before, we will use theresult directly from [8,9]. Since the power of the side-bands is much lower than the central bands, we willconsider only the noise resulting from the beatings ofcentral bands. The PSD of noise is given by

Pbnðf Þ ¼R2S2

o

2

Z þ∞

0HðνÞHðνþ f Þdν; ð5Þ

where f is the noise frequency. For a gate-shape filterwith

HðνÞ ¼�1 jν − νcj ≤ Bo=20 otherwise ; ð6Þ

where νc is the central frequency of the filter andBo is the optical bandwidth. The signal power isgiven by

Ps ¼is · i�s2

¼ R2m2S2oðBo − f eÞ22

sinc2�ϕ2

�; ð7Þ

where

ϕ ¼ 2πf eDνðBo − f eÞ: ð8Þ

The transfer gain can be written as

G ¼ Ps=

�V2

s

2R2m

�¼ ½πRRmSoðBo − f eÞ�2

4V2π

sinc2�ϕ2

�:

ð9Þ

The PSD of the beat noise is given by

Pbnðf Þ ¼R2S2

oðBo − f Þ2

: ð10Þ

It is worth noting that the beat noise is not white, itsPSD decreases linearly with frequency increasingfrom zero to Bo. So i2bn should be the integration ofEq. (10) within the receiver bandwidth. We let B ¼1Hz for convenience, the beat noise can then be writ-ten as

Table 1. Definitions of Mathematical Symbols

vs RF signalv2ts thermal noise of the signal sourceRs resistance of the signal source, 50ΩRm terminating resistance of the MZM, 50Ωip photocurrent

i2sn photocurrent shot noise

id dark currenti2dn dark current shot noise

i2bnbeat noise

i2theequivalent thermal noise of the photodetector,including noise from the amplifier

RL load resistor of the photodetector, 50Ωq charge of an electronG transfer gain of the link excluding the amplifierB receiver bandwidthT temperature in kelvinFA noise factor of the amplifierBo optical bandwidth in HzW optical bandwidth in nm

Fig. 4. Representation of BS and modulation.

660 APPLIED OPTICS / Vol. 48, No. 4 / 1 February 2009

i2bn ¼ Pbnðf eÞ ¼2R2P2

dðBo − f eÞB2

o; ð11Þ

where Pd is the optical power into the PD and is

Pd ¼ SoBo

2: ð12Þ

Note that we have assumed the optical transmissionloss is zero above. When the optical transmission isnot lossless and has an efficiency to, the expression ofPd should be

Pd ¼ toSoBo

2: ð13Þ

Furthermore, we let ϕ ¼ 0 in Eq. (9) and get Gm ¼G∣ϕ¼0 for noise evaluation. Actual link transfer gainwill be no larger than this value. The first threeterms of Eq. (2) account for noises from the electricalcircuit, so we treat them as one equivalent noisesource which is expressed by

i2en ¼ kTRL

��πRRmPdðBo − f eÞVπBo

�2þ 4FA

�þ 2qid: ð14Þ

The shot noise is given by

i2sn ¼ 2qRPd: ð15Þ

In the derivations above we used the ideal impe-dance match condition, which is Rs ¼ Rm ¼ RL ¼50Ω. Generally, when the optical power level is rela-tively low, electrical noise is the main noise source.As soon as the optical power is higher than some cer-tain value, the noises due to O=E conversion becomepredominant. Figure 5 shows the plots of variousnoises (power consumed through a 1Ω resistor inunit bandwidth Hz) versus optical power in thePD, with R ¼ 0:9A=W, id ¼ 4nA, T ¼ 290K, f e ¼10GHz, FA ¼ 2:5, Vπ ¼ 4V, and the central wave-length of the optical filter is 1550nm. Figure 5(a)is drawn with a fixed 1nm optical bandwidth. Wecan see that when the optical power is below−21dBm, the dominant noise source is electricalnoise, while above, which is the more common situa-tion, the dominant source is beat noise. In our experi-ments, the optical PSD is about −2dBm=nm. Underthis condition, the various noises versus different op-tical bandwidths, and thus different power levels, areshown in Fig. 5(b). The optical power ranges from −9to þ8dBm, and the beat noise is 44dB larger thanshot noise, 29–48dB larger than electrical noise.The total noise is primarily limited by beat noise,and we can see that the two plots completely overlapin Fig. 5(b).By considering only the contribution of beat noise,

the system noise factor can be written as

F ¼ 1þ i2bnRL

GkT≈

2V2π

π2kTRLðBo − f eÞsinc2ðϕ=2Þ: ð16Þ

And the system output CNR is given by

CNR ¼ Ps

Pbnðf eÞ¼ m2ðBo − f eÞsinc2

�ϕ2

�: ð17Þ

From Eqs. (16) and (17) we find that F and CNR de-pend on the optical filter bandwidth, when the otherparameters are specified. There is an optimal valueBo;opt to maximize CNR and minimize F. This can beexplained physically. The power of the signal first in-creases with the optical bandwidth Bo when Bo > f e,and then decreases due to the cancellation effect re-sulting from the summation of δis, which has differ-ent phases because of dispersion [see Eq. (7)].However, the beat noise power increases monotoni-cally withBo [see Eq. (10)]. When the increased speedof the signal meets the speed of the noise for the firsttime, the CNR gets the maximum value and F theminimum. The optimal values are given by

Bo;opt ¼1:1656πf eDν

þ f e; ð18Þ

Fig. 5. Various noises versus optical power in PD: (a) fixed opticalbandwidth 1nm with the optical PSD ranging from −40 to10dBm=nm; (b) fixed optical PSD −2dBm=nm with the opticalbandwidth ranging from 0.2 to 10nm.

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Fmin ¼ 2:8V2πf eDν

πkTRL; ð19Þ

CNRmax ¼0:18πV2

s

f eDνV2π: ð20Þ

For f e ¼ 10GHz, Dν ¼ 4:088 × 10−22 s=Hz(51:1ps=nm around 1550nm), T ¼ 290K, RL ¼50Ω, Vπ ¼ 4V, and Vs ¼ 0:5V, the optimal valuesare Bo;opt ¼ 100:8GHz (about 0:8nm around1550nm), Fmin ¼ 2:9 × 108 ð84:6dBÞ, and CNRmax ¼2:16 × 109 ð93:3dBÞ. This CNR value of 93:3dB ismuch lower than that of systems employing lasersfor which one typical value is 128dB [6]. Similar ana-lysis can be applied to filters with different shapes.

3. Experimental Results and Discussion

Our experimental setup is shown in Fig. 2. First, weuse two different optical filters, whose full band-widths of −3, −10, and −20 dB to validate the methodthat we used to derive the CNR above.Filter1: W3dB ¼ 1:37nm, W10dB ¼ 2:03nm,

W20dB ¼ 2:74nm.Filter2: W3dB ¼ 0:48nm, W10dB ¼ 0:68nm,

W20dB ¼ 0:86nm.The output power of the broadband optical source

is 19dBm, and the spectrum width is about 20nm.The system insertion loss is about 18dB. The opticalpower into the photodetector is about −15dBm afterfilter2, and −12dBm after filter1. At this power level,the shot noise is at least 16dB lower than the elec-trical noise, and the beat noise is about 13dB largerthan the electrical noise for filter1 and 17 dB largerfor filter2, according to our analysis above. So thesystem is beat noise dominated.The output power of the microwave source is chan-

ged and a series of the system output CNR are mea-sured. The experimental results are shown in Fig. 6and the theoretical results according to Eqs. (4) and(10) are shown simultaneously for comparison. Thefilter shapes for theory calculation are approximat-ing Gaussian functions, which have the same −10dBbandwidths as the actual filters. The modulation in-dex can be calculated from the central to sidebandpower ratio (CSR) measured by replacing the BSby a laser. The dispersion of CFG is 51:1ps=nm.Good agreement is obtained between the measure-

ment and the calculation, and the average deviationis about 0:7dB for filter1 and 0:2dB for filter2, whichis mainly attributed to the difference in filter shapesbetween calculation and experiment. It is worth not-ing that the CNR with filter1 is about 4dB lowerthan that with filter2, although filter1 has a largerbandwidth and a larger optical power. This resultagrees with our prediction because the equivalentbandwidth of filter2 is closer to the optimal valuein our system.We also explored the relationship between CNR

and optical bandwidth experimentally using a band-width-variable tunable optical filter from Alnair

Labs. The filter shapes are shown in Fig. 7, whichare very close to ideal rectangles. The transition be-tween the −3 and −20dB point is about 0:1nm foreach roll-off side, and the dispersion of the filter with-in the passband is negligible. To ensure the opticalpower, when the optical bandwidth is very small,above the level at which the beat noise begins to dom-inate, we add an EDFA right before the optical filter.Since the optical source is broadband, the EDFA willnot induce considerable additional noise, such asspontaneous–spontaneous beat noise, comparedwith the source beat noise itself. The total opticalpower before the EDFA is about 0dBm, and theEDFA has a gain of 11dB. So the optical PSD afterthe filter is about −2dBm=nm.

The experimental results are shown in Fig. 8. Thehorizontal axis is the filter full bandwidth measuredat −20dB minus 0:1nm, which implies the equiva-lent rectangular bandwidth W ¼ W20dB − 0:1. InFig. 8(a), the RF frequency is 10GHz. When thesystem dispersion is zero, the CNR increases mono-tonically with the optical bandwidthW. For example,if W is doubled, the CNR will increase by 3dB. So we

Fig. 6. Measured CNR versus modulation index using two differ-ent optical filters.

Fig. 7. Bandwidth-variable filter.

662 APPLIED OPTICS / Vol. 48, No. 4 / 1 February 2009

can simply increase the optical bandwidth to meetthe system demand. However, when the dispersion isnonzero, the CNR will change periodically with W,and the peak CNR at each period decreases withthe period number. Theoretically the peak CNR ofthe second period is about 4:8dB lower than thatof the first period. At some points, the CNR falls nearzero. The optical bandwidth needs to be carefully de-signed to maximize CNR. For example, in Fig. 8(a),when the system dispersion is 51:1ps=nm, the opti-cal bandwidth should be chosen at 0:8nm, and themaximum CNR might be 93:3dB. This result agreeswith the measured values very well which are93:4dB at 0:7nm and 93:3dB at 0:9nm (the experi-mental bandwidth increment is 0:2nm). The plot ofCNR versus optical bandwidth when f e ¼ 5GHz isalso shown in Fig. 8(b), and good agreement betweenexperiment and theory is obtained again. The chan-ging period of CNR versus W when f e ¼ 5GHz istwice the value when f e ¼ 10GHz, which yields low-er accuracy demand when designing the opticalbandwidth.In our experiments, we used an EDFA with a gain

of 11dB, which does not induce considerable noise, tokeep the optical power when the optical bandwidth isvery small. However, when the system insertion lossgets very large, as is the case for large-scale phasearray antennas, the EDFA will be essential. And

when the optical power before the EDFA is signifi-cantly low, the noise induced by the EDFA will benot negligible. This still needs further investigation.

4. Conclusion

We have investigated the noise in BS- and disper-sion-based optical TTD systems. We found that thebeat noise induced by the optical source begins todominate and grows far larger than other noiseterms quickly as long as the optical power level isabove the electrical noise. The system noise figureand output CNR were calculated and found to de-pend on the bandwidth of the optical filter, whichcan be properly designed to maximize the CNR. Ex-periments were carried out and good agreement be-tween experiment and calculation was obtained.

This work is supported in part by National NaturalScience Foundation of China (NSFC) under grants6052130298 and 60432020, Project 863 undergrant 2006AA01Z261, Project 973 under grant2006CB302805, and by the Italian Ministry of For-eign Affairs Project iCHIP .

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Fig. 8. CNR versus optical bandwidth. (a) f e ¼ 10GHz;(b) f e ¼ 5GHz.

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