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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 22, NOVEMBER 15, 2012 3475

RF Nonlinearities in an Analog Optical Link andTheir Effect on Radars Carrying Linear and Nonlinear

Frequency Modulated PulsesLior Yaron, Student Member, IEEE, and Moshe Tur, Fellow, IEEE, Fellow, OSA

Abstract—Optical links are afflicted with RF nonlinearities,originating from various opto-electronic devices, like intensitymodulators. The effects of such nonlinearities on nonlinear fre-quency modulated (NLFM) pulses are studied in detail for bothclutter and discrete returns and compared to the linear frequencymodulated case. While for discrete returns, NLFM coding gener-ates weaker ghosts, this advantage is lost for clutter, where bothcodings exhibit similar performance. When such an optical linkin a photonic beamformer processes the signal returned from ahigh contrast scene, third and higher order nonlinearities severelylimit the obtainable contrast of the imaging radar. For third-ordernonlinearities and high RF input powers, an inverse fourth-orderrelationship exists between the contrast and the input voltage.For lower inputs, the contrast is initially limited by system noiseand then by the skirts of the compressed impulse response. Theseresults for the two frequency codings are experimentally con-firmed in a high frequency (10 GHz), wideband (1 GHz) photoniclink, where the nonlinearities originate from the behavior of aMach–Zehnder intensity modulator. Specifically, to achieve acontrast better than 25 dB, the RF input to the Mach–Zehndermodulator should not exceed 1/8 of the link input third-orderintercept voltage.

Index Terms—Clutter, microwave communication, nonlineari-ties, optical communication, optical devices, radar.

I. INTRODUCTION

A NALOG optical links can benefit a variety of microwaveapplications due to their well-known advantages of

extremely wide bandwidth, multibeam capabilities, low trans-mission loss, immunity to electromagnetic interference, andthe ability to realize compact and lightweight systems [1]–[3].Whether for carrying RF over fiber for the cellular industryor in a sophisticated photonic beamformer, the basic opticallink (see Fig. 1) comprises an optical source, usually a laser,a means to impress the RF information on the optical carrier,either through direct modulation of the source itself, or usingan external modulator, an optical fiber, an optional optical am-plifier, and an optical receiver to convert the optical modulationback to RF. Pre- and post-electronic amplification are alsoquite common. Dispersion effects in the link may be minimizedby employing single longitudinal-mode lasers, working at the

Manuscript received February 09, 2012; revised July 24, 2012, August 24,2012; accepted September 11, 2012. Date of publication September 21, 2012;date of current version October 31, 2012.The authors are with the Faculty of Engineering, Tel-Aviv University, Tel-

Aviv 69978, Israel (email: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JLT.2012.2220336

Fig. 1. Schematic analog-optical link. E/O: electrical-to-optical converter;O/E: optical-to-electrical converter. The optical and electronic paths aredenoted by solid and dashed lines, respectively.

zero-dispersion wavelength of the fiber, together with externalmodulation to avoid laser chirp. Well-designed optical linkshave been demonstrated with a flat response and linear phaseover many gigahertz [4], [5]. However, these links are notexempt from dynamic range limitations. In fact, the commonlyused electro-optic modulators, such as the Mach–Zehndermodulator (MZM) and electro-absorption modulator, and toa somewhat lesser extent, nowadays, also photodetectors andtheir associated amplifiers, may significantly add to the linknonlinear budget and limit the achievable dynamic range[3], [6]–[8]. Specifically, the transfer function of such a link,employing the more commonly used MZM, is indeed quitenonlinear

(1)

Here, is the input average optical power, is the optical gain(loss) of the link, is the detector responsivity (in Volts/Watts),

is a dc bias voltage and characterizes the sensitivity ofthe modulator to input voltage changes.RF nonlinearities in a suboctave microwave link (i.e., when

harmonics fall outside the receiver bandwidth) are usually an-alyzed by considering the two-tone third-order intercept point(at the input: IIP3) [3]. This calculated power value denotesthe input RF power of each of the two continuous wave tonesfor which the power of the output third-order intermodulationproducts is equal to the output power of either tone. The ac-tual effect of these intermodulation products on a particularsystem also depends on the type of involved signals. In radars,achieving their range resolution through the use of very narrowpulses, two received pulses, which are temporally disjoint,will not generate any intermodulation products. This is not thecase, however, in pulse compression radars, employing thecommonly used quite long linear frequency modulated (LFM)pulsed waveforms [9]–[12]. Here, in a typical scenario whentwo or more such LFM pulses overlap in time, the nonlinear

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3476 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 22, NOVEMBER 15, 2012

characteristics of the receiver tend to produce, after matchedfilter compression, ghost returns that cannot be distinguishedfrom the true ones [13]–[15]. Optical microwave links, em-ploying MZMs, are especially vulnerable due to the relativelylow IIP3 of this kind of modulator [7]. Electronic linearizationof the MZM response, while possible [16], is quite a difficulttask in multigigahertz wideband links, where photonic solutionsexcel.In this paper and for the first time to our knowledge, the tol-

erance of a different modulation format: nonlinear frequencymodulation (NLFM) [10], to RF nonlinearities in radar receiversis investigated both theoretically and experimentally for a pho-tonic link. While the linear performance of these more complexwaveforms is quite comparable to that of their LFM counter-parts, they are shown to have a different nonlinear behavior.Two typical receiving scenarios are studied: 1) The receiver col-lects overlapping echoes from a number of discrete, spatiallyresolvable targets; and 2) an imaging radar receives many over-lapping echoes from spatially unresolvable clutter. It is shownthat while in the first case NLFM offers higher immunity to RFnonlinearities, it has practically no advantage over LFM in thesecond scenario.Following a brief review of NLFM and LFM signals

in Section II, and a definition of our nonlinear model inSection III, the effect of intermodulation products on echoesfrom two discrete targets is studied in Section IV. Using simu-lations, Section V compares the contrast performance of NLFMand LFM in an imaging scenario of clutter. The results ofSection V are experimentally examined in Section VI, followedby a discussion in Section VII and conclusions in Section VIII.

II. LFM AND NLFM SIGNALS

The LFM signal has been the workhorse of radar signals dueits ability to provide, upon compression, a very narrow im-pulse response, without taxing the transmitter output amplifier[9]–[12]. An ideal LFM square pulse is defined by [9]

(2)

where and are the pulse’s center in time and frequency,respectively. , where is the total bandwidth ofthe signal ( gigahertz), and is the pulse length ( tens ofmicroseconds). During the pulse duration ,the instantaneous frequencylinearly scans the frequency range[see Fig. 2(a)]. Compression of a received LFM pulse is donethrough correlation with the undistorted original pulse shape(i.e., a matched filter) [9]. The squared magnitude of the re-sulting narrow impulse response, to be denoted by iscentered around of the LFM pulse, (2), with a width, whichis inversely proportional to , and a skirt of sidelobes, whose

peak level (PSL) is approximately 13 dB below .These unacceptably high sidelobes can be suppressed by usingan apodizing weighting filter [9], at the expense of a slightlywider main lobe. Normally implemented at the receiver, thisapodization also leads to a reduction of the postprocessingsignal-to-noise ratio (SNR) [9]. Fig. 2(b) and (c) plots theimpulse responses for a 1 GHz wide LFM pulse, apodized witheither a Hamming window or a low PSL Taylor one.An alternative approach to achieve a narrow impulse re-

sponse with low sidelobes level is to use nonlinear frequencymodulation [10], where is a nonlinear function of the time. Out of the quite few already suggested forms for[9], [10], we study the NLFM function suggested by [17],which was chosen for its good performance in the context ofthis paper, and was also employed in optical beamformingnetworks [18].Analytically, its power spectrum is given by (3), shown at the

bottom of the page.With the further requirement that the derivedNLFM signal is also of temporal amplitude

(4)

the method of stationary phase [19] is numerically invoked toproduce the NLFM shape of Fig. 2(a) (we also as-sumed that ). Contrary to LFM, the NLFM signalspends more time near its center frequencies. After matched fil-tering but with no need for apodization, the resulting isvery similar to that of a weighted LFM in terms of sidelobeslevel and temporal resolution [see Fig. 2(d)]. The absence ofwindow weighting results in an improved postprocessing SNRof 1–2 dB. If proven advantageous, NLFM modulation is quiteeasy to implement today with the availability of high-speed ar-bitrary waveform generators, as done here in Section VI.Of importance in many radar applications is the integrated

sidelobe ratio (ISLR) [9], defined here by ( is the full widthat half maximum of ):

(5)which represents the ratio between the energy in the sidelobeskirt and that of the main lobe (it should be noted that slightlydifferent choices of the integration limits appear in the litera-ture [11]). The width of , the associated PSL and ISLRvalues, and the apodizing window SNR losses are summarizedin Table I for an LFM signal with a Hamming and a low PSLTaylor windows, and an NLFM waveform, all with s

(3)

YARON AND TUR: RF NONLINEARITIES IN AN ANALOG OPTICAL LINK 3477

Fig. 2. (a) Instantaneous frequency of LFM, (2) and NLFM, (3), pulses around their center frequency. (b)–(d) Normalized impulse responses for these two fre-quency modulations with s and GHz. The LFM pulse is processed with either (b) a Hamming window or (c) a high-order Taylor one (c). (d)NLFM is processed with an ideal rectangular window.

TABLE IPROPERTIES OF THE IMPULSE RESPONSE FOR NLFM AND LFM WAVEFORMS

WITH S AND GHZ

and GHz. Clearly, lower PSL and ISLR are achieved atthe expense of a wider . In practice, the nonideal RF fre-quency response of the link, together with the unavoidable linknoise, result in worse values for both the PSL and ISLR.

III. NONLINEAR MODEL

Expanding the overall transfer function of the photonic link(prior to compression), , in a Taylor series,and keeping only the ac terms up to the third order, we get

(6)

where is the input third-order interceptvoltage of the link [3], and (also

). Motivated by radar beamforming and other sub-octave applications, where the information is centered around ahigh-frequency RF carrier, the second-order term was omittedsince it produces only high harmonics, usually outside the RFbandwidth of the link.For a photonic link, where RF nonlinearities are governed

by the MZM of (1) at a positive-slope quadrature, whereis an even integer, (1) becomes

---------------------------------------------------------

(7)

Again but here [7].This third-order nonlinear transfer function of the photonic

link will be now used for the various analyses. The experimentalresults will justify the truncation of the Taylor expansion of thelink transfer function at the third-order level.

IV. EFFECT OF INTERMODULATION PRODUCTS IN A DISCRETERETURN SCENARIO

In order to analyze the intermodulation products arising fromseveral discrete returns of frequencymodulated signals, we con-sider the simple case of two echoes returning from two realpoint targets with delays and , with respectiveamplitudes and . For both LFM and NLFM, the received


voltage signal, which is also the input to the photonic link, willthen be

(8)

is taken from either (2) or (4).Substituting (8) in our nonlinear model of the link (6)

(9)

stands for terms of the order ) one findsthat the third-order term in (6), while only slightly modifying thecoefficient of , generates a bothersome intermodulationsignal , of which only two terms seem to fall within thesystem bandwidth

(10)

For the case of LFM coding, the time-dependent arguments inthe cosines of the two intermodulation terms in (10) are of theform , with , or , rep-resenting mathematically legitimate, though spurious, LFM sig-nals [13], [14], sharing the same and with the true echoes,but with time delays of and [see Fig. 3(a)].These false signals will be properly compressed by the matchedfilter, resulting in two ghost reflections at and

[see Fig. 3(b)], whose magnitude depends on thedegree of overlap between the two signals, as determined byand (see Fig. 5). These ghosts clearly limit the free dynamicrange of a photonic beamformer.It seems that for most coded schemes, other than LFM, the

intermodulation products may not produce ghost signals of thesame shape as the original. Specifically, for the case of two LFMechoes of (3), the intermodulation products obey a frequencylaw different from the one for which the matched filter was de-signed, resulting in much weaker ghosts (see Figs. 4 and 5).In summary, coded radar signals are much longer than the

simple, uncoded short ones. Therefore, their echoes from dis-crete targets tend to experience much more overlap, leadingto higher sensitivity to third-order, intermodulation nonlineari-ties in the transmitter–receiver transfer function. Interestingly,the NLFM waveform, while exhibiting similar linear perfor-mance to that of its widely used LFM counterpart, appears tobe more immune to intermodulation-originated spurious echoesfrom discrete scatterers.

Fig. 3. (a) Instantaneous frequency of two partially overlapping LFM echoesand their intermodulation products, created after passing in the nonlineartransfer function of (6) with s, GHz, s, and

. The dashed horizontal lines mark the original signals band-width. (b) Real and ghost echoes created after compressing the received signalwith a matched filter.

V. EFFECT OF INTERMODULATION PRODUCTS FOR THE CASEOF DENSELY DISTRIBUTED MULTIPLE SCATTERERS

A. Model

Consider the radar returns from a river and its two banks,where the river direction is perpendicularly oriented with re-spect to the grazing radar line of sight. Assuming zero radarcross section for the river water, we model the surface of theriver banks as a manifold of randomly distributed scatterers.Thus, each transmitted frequency modulated pulse is multiplyreflected from scatterers on the river banks, but not from theriver itself, resulting in an infinite contrast between the river andits two banks, practically bounded by the system noise floor. Buteven for a noise-free system, the river nevertheless appears to bereflecting. This reduction in observed contrast is brought aboutby two independent effects, one linear: the finite skirt of the im-pulse response (see Fig. 2); and the other nonlinear: the appear-ance within the river extent of third-order intermodulation prod-ucts of the type discussed in the previous section, originatingfrom scatterers on the river banks. Both effects limit the con-trast sensitivity of imaging radars, but while the first providesa pretty high contrast, being independent of the power receivedfrom the banks, the third-order effect ceaselessly degrades thecontrast as the square of the received power. Thus, to maintaina specified contrast, the input power must be restricted at the


Fig. 4. (a) Instantaneous frequency of two partially overlapping NLFMechoes and their intermodulation products, created after passing in the non-linear transfer function of (6) with s, GHz, s,and . The dashed horizontal lines mark the original signalsbandwidth. (b) Real and ghost echoes created after compressing the receivedsignal with a matched filter.

Fig. 5. Simulation of the ghost to real echo peak power ratio as a function offor two overlapping NLFM pulses (blue lines) and two LFM pulses (green

lines), after passing in the nonlinear transfer function of (6) with s,GHz, and several temporal shifts: . It can be

seen that for LFM pulses, the lines for and practicallyoverlap.

expense of the obtained SNR. These two contrast-limiting phe-nomena are studied in the following for the general case of LFMand NLFM radar returns from randomly distributed scatterers,i.e., from clutter, as in returns from rain, snow, and the surfaceof the earth.For a simplified, yet easily generalized analysis, a 1-D model

is assumed (see Fig. 6), where a large number of scatterers arerandomly distributed along the banks length with a density ofscatterers per unit time. Each scatterer returns the incident pulse

Fig. 6. Radar cross section of a zero-scattering narrow river sitting betweentwo wide, unity cross-section banks, normalized to unity at the river banks.

with a random delay , and an amplitude of . The overallnumber of scatterers is assumed to be large enough so that eachresolution cell of the system ( in our case) contains manyscatterers. Under such assumptions, the time events andthe corresponding number of scatterers in a given time segmentobey Poisson statistics [20], [21], the properties of which willbe used later to determine various properties of the receivedsignal. Once a frequency modulated pulse is transmitted, thesignal amplitude generated at the receiver input by the randomscatterers is given by

(11)

Here, for a given time , the sum extends over all scattererswhose yield a nonzero value for the “rect” function, de-pends on the transmitted RF amplitude, as well as on the geo-metric and absorption losses, assumed to be range-independent,as well as other system parameters. Without loss of generality,we assume that .

B. Linearly Processed Pulse Returns From Clutter

We have already shown [14] that for a linear receiver, i.e.,in (6), the average received RF power, ,

is given by

(12)

Thus, apart from the factor, is the mean returned en-ergy per pulse times the number of scatterers per unit time, and

is related to the bank’s normalized radar cross section[9]. In a homogeneous scenario, both and will notchange with range (or equivalently, with time). Since for eachtime point, gets contributions from a large number ofwide pulses, the river, being much narrower than the pulse width, cannot be identified in the versus time curve of Fig. 7(a).

Upon compressing with a matched filter, [seeFig. 7(b)–(d), the averaged power in the banks becomes[14], [20], [21]

(13)

which is basically the same as (12) for an NLFM pulse. Theapodization of the LFM spectrum leads to a small reduction of


Fig. 7. (a) Simulated instantaneous received power from the river. (b) Com-pressed signal for NLFM pulse. (c) Compressed signal for LFM pulse witha Hamming window. (d) Compressed signal for an LFM pulse with a Taylorwindow. Horizontal heavy lines measure the average power. Here, a linear RFtransfer function was assumed ( in (6)). The powers in curves (a)–(d) werenormalized to have unity average at the banks.

the energy of the impulse response from its unapodized valueof to (Hamming) or (Taylor). Whilethe nonscattering river is distinctly resolved, its time slot, [seeFig. 7(b)–(d)], displays a nonzero reflection, originating fromthe skirts of the impulse responses generated by scatterers on theriver banks. Indeed, for narrow enough rivers, where the skirtscontributions from the two banks overlap, the observed contrast,as defined by , is of the order ofthe relevant ISLR. For wider rivers, the individual skirts contri-butions may become visible and the contrast in the middle im-proves. Since the compressed signal at both the banks and riverlocations originates from the same banks’ signal, the observedcontrast is clearly power independent for a linear RF transferfunction ( in (6)).

C. Nonlinearly Processed Returns From Clutter

For a nonlinear transfer function, in (6), intermodula-tion ghost echoes, created by the scatterers on the banks, willalso raise the level of the river signal. Here, however, while thereflection from the banks scales linearly with , the riversignal due to third-order nonlinearities is expected to scale as

, resulting in a fast shrinking contrast with increasing. Thus, to maintain a required contrast, must be

limited, compromising the available SNR. To quantify this ef-fect, of (11) was simulated using a large number of scat-terers, uniformly scattered along the banks with a density of

scatterers/s and random locations . A 100 nswide river separated the two banks, each of width (no in-termodulation products originate from points, whose distancefrom the river is larger than ). Pulse width was sand with a bandwidth of GHz. The simulatedwas then fed into the transfer function given in (6). The output

Fig. 8. Results for a nonlinear RF transfer function, in (6). The com-pressed signals are shown for an NLFM pulse with s, GHz,and (a) (b) .

Fig. 9. Simulations of the banks-river contrast as a function of for NLFMpulse (blue solid thick line) and LFM pulse with a Hamming window (red solidthin line) and low PSL Taylor window (green solid line), all with sand GHz.

was digitally processed by a matched filter and the contrastwas calculated as a function of

the ratio (6). Typically, processed returnsof NLFM pulse appears in Fig. 8 for and

. For weak enough signals, the processed image looksidentical to the linear one [see Fig. 8(a)]. However, for strongreceived signals, a distinct reduction in contrast is observed [seeFig. 8(b)].Fig. 9 compares LFM with NLFM with respect to the de-

pendence of the resulting river contrast on the received signalstrength. As expected, for weak signals, the river level is con-trolled by the skirts of the impulse responses, exhibiting inde-pendence of signal strength. While qualitatively, the contrastof the various coding/windows in the linear regime follows thetrend of the ISLRs of Table I, the actual values depend on theexact shapes of the different skirts (see Fig. 2). For stronger sig-nals, the contrast quickly deteriorates at the expected rate of(since , while ), or40 dB/decade (only for the fifth-order nonlinear-ities will be manifested). Unlike their different behavior in thediscrete case (see Section IV), both LFM and NLFM signals ex-hibit similar contrasts in the nonlinear regime. Reexaminationof the results of Section IV reveals that while the NLFM wave-forms produce lower peak power discrete ghost echoes, the en-ergy of each ghost is the same as that of its LFM counterpart byless than a decibel. Therefore, in the clutter scenario, where itis the energy of the IR which determines the contrast (13), theNLFM coding has no advantage over the LFM one.


Fig. 10. Experimental setup. MZM: Mach–Zehnder modulator; EDFA:erbium-doped fiber amplifier; O/E: optical-to-electrical converter. A/D: analog-to-digital converter. Recorded data were then processed by a computer. Theoptical and electronic paths are denoted by solid and dashed lines, respectively.

TABLE IIEXPERIMENTAL SETUP PARAMETERS

VI. EXPERIMENT AND RESULTS

The experimental setup is shown in Fig. 10 and importantlink parameters are summarized in Table II. The setup has beenspecifically designed and implemented to make the MZM thedominant nonlinear element in the link; see the following fordetails. The digitally simulated signals of (11), with the param-eters of Section V, for both LFM and NLFM coding, were con-verted to an analog signal using a wideband arbitrary waveformgenerator and an X-band upconverter. After electronic amplifi-cation by a highly linear, low-noise multistage amplifier, havingan output IP3 of 48 dBm, these signals served as the modu-lating voltage to a high-frequency MZM with a of 9 V at

GHz, corresponding to an IIP3 of dBm. ThisIIP3 value is lower enough from the nonlinear specificationsof the driving amplifier to ensure that the MZM was the dom-inant nonlinear bottleneck of the link. This assertion has beenexperimentally verified also in the presence of the chosen O/Ephotoreceiver, postamplifier, downconverter, and A/D samplingcircuitry (see Fig. 10). The MZM was biased at quadrature,as in (7). After passing through the optical part of the system,the detected RF signal was down-converted to 1 GHz and thenrecorded, following digitization by a 20 GSamples/s, 8 bits A/Dconverter. The final river image was obtained after processingthe recorded signal with a matched filter, as described previ-ously. Following the photonic microwave usage, the modula-tion index was defined as

(14)

Fig. 11. Experimental results for the banks-river contrast as a function of themodulation index . Each marker represents an average of five measurements.The contrast for NLFM pulse results appears as blue pluses, while the contrastfor LFM pulse processed with a Taylor window appears as green asterisks. Forevery second value, the LFM signal was also processed with a Hammingwindow. These results appear as red circles on top of the green asterisks. Thesimulation results of Fig. 9 are also shown (solid lines).

(Note that for a sinusoidally modulated signal, reduces to itscommon definition [3]). was controlled by the variable RFattenuator over a wide range and the experimentally obtainedresults appear in Fig. 11 for the LFM signal processed with aTaylor/Hamming windows (green asterisks/red circles), as wellfor the NLFM signal (blue pluses). There is a fairly good agree-ment between the experimental and simulation results for highmodulation indices , and the dependence isclearly obeyed. While a Taylor expansion of the sine function in(7) also contains fifth and higher order powers of ,their presence is not seen in Fig. 11, indicating the sufficientaccuracy of the third-order approximation for the chosen rangeof .For lower values of the modulation index, however, system

noise causes the measured contrast to be -dependent, see inthe following.

VII. DISCUSSION

The observed deviation of the measured results from those ofthe simulation at low to moderate values of is a direct con-sequence of the presence of noise. Indeed, the measured riversignal has contributions not only from the skirts of the impulseresponses of the banks scatterers, and/or intermodulation prod-ucts of the banks signals, but also from system noise, i.e.,

(15)

represents a few sources of noise [22] which are presentin the setup, namely: 1) the laser relative intensity noise (RIN)of dB/Hz; 2) signal-ASE beat noise from the optical am-plifier ( dB), resulting in a RIN of dB/Hz; 3)detector thermal noise; and 4) shot noise. At a received optical


Fig. 12. Solid light blue line describes the experimental system SNR (prior tocompression). The initial slope is due to the -independent optical noisewith an effective RIN of dB/Hz. The finite SNR of the electronicallygenerated NLFM and LFM signals determines the saturation value of the systemSNR for values of above 0.1. The dashed light blue line and the dash-dottedpurple line plot the system SNR for an infinitely high electronic SNR and aneffective system RIN of dB/Hz and dB/Hz, respectively.

power of 2.6 dBm, the detector and shot noises may be inter-preted as having effective RIN values of and dB/Hz,respectively, leading to a total effective system RIN ofdB/Hz. Thus, for values of , the signal attributed tothe river signal is dominated by this -independent combinednoise, rather than by the much weaker actual river return. In thisregime, the contrast increases linearly with (see Fig. 11).Another electronic source of noise, which grows like the

signal as , is related to the finite ( dB) SNR of theelectronically generated LFM and NLFM signals. With ap-proximately equal contributions from the arbitrary waveformgenerator and the upconverter, this noise is initially weakerthan the -independent sources. However, for high enoughvalues of it clamps the SNR (before compression) to

dB, (see Fig. 12), and the processed contrast to a slightlyhigher value.Whenever the contrast is governed by noise, the improved

postprocessing SNR of the NLFM signal (see Section II) ex-plains the observed slightly higher performance of this codingover LFM (see Fig. 11). As for the LFM contrast measurements,the interplay between the better ISLR of the Taylor window andthe lower SNR loss of the Hamming window (see Table I) re-sults in an almost unnoticeable difference between the two win-dows for the relevant values . Finally, for still highervalues of , the nonlinear nature of the modulator becomesdominant, resulting in the behavior. The measured noiseparameters were used to estimate the achievable contrast forNLFM coding in a noisy system. The resulting thin dashed linein Fig. 13 shows a very good match with the measured data.For a powerful enough laser, where the optical amplifier

is no longer needed and for a high power detector, the pho-tonic-related noise is governed by the laser RIN [3]. Assumingalso infinitely high SNR of the RF signal at the MZM input,the expected SNR (for ) approximately increases as

(see Fig. 12). For a low RIN laser ofdB/Hz, the skirt-limited contrast is closely approached (upperpurple dash-dotted line in Fig. 13).Clearly, the achieved contrast cannot exceed the values set ei-

ther by the skirts of the LFM/NLFM impulse response (for low

Fig. 13. Banks-river contrast as a function of the modulation index underdifferent noise scenario for NLFM signal. The experimental results appear asblue pluses. The blue solid thick line shows noiseless simulation results, whilethe two dashed lines represent simulation with noise. The lower light blue thindashed line uses the actual system noise parameters of a RIN of dB/Hz,the somewhat noisy RF input to the MZM with 30 dB SNR. The upper purpledash-dotted line describes the contrast dependence on for a low RIN laser of

dB/Hz, and a noiseless input RF signal. Both curves represent the averageof 50 Monte Carlo simulations with different noise seeds.

values of ), or the intermodulation products of the Banks sig-nals (for higher values of ). In particular, to achieve a contrastof 25 dB, should not exceed 0.5.The spurious-free dynamic range (SFDR) of the optical link

can be roughly estimated from the experimental curve of Fig. 11as follows: an extrapolated contrast of unity determines the min-imum value of , while the intersection of a linear extrapola-tion of the low part of the curve with its one is used toestimate the maximum spurious-free . The obtained SFDR is

. This SFDR could be improved by using a laserwith a better RIN and higher power so that no optical amplifica-tion would be required (see Fig. 13), and also by operating theMZM modulator below quadrature [23].While the analysis has concentrated on the nonlinearities of

the MZM, it is equally applicable to any other optical link withdominant third-order nonlinearities. Thus, should not ex-ceed (i.e., if a contrast of 25 dB isrequired.

VIII. CONCLUSION

The transmission of LFM and NLFM pulses through an RFtransfer function with RF nonlinearities results in ghost echoes,unless the input level to the link is limited. In a discrete echoesscenario, the level of the spurious echoes depends not only onthe nonlinear coefficient in the system transfer function, but alsoon the temporal overlap between the received echoes and thespecific signal used. NLFM coding, while demonstrating linearperformance similar to that of its LFM counterpart (in termsof their ), exhibits a much lower peak power of the ghostechoes after the matched filter processing. In an imaging sce-nario, involving a narrow zero-cross-section area, surroundedby a scattering area, ghost echoes originating from the scat-tering area fall inside the zero-cross-section area and decreasethe obtained contrast. The degradation in the observed contrastdepends on the energy of the spurious echoes, rather than ontheir peak power and no improvement is achieved by using theNLFM coding, other than a slightly better postprocessing SNR.


In this imaging scenario, the input voltage to an optical link,afflicted with third-order RF nonlinearities, should not exceed0.125 to attain a contrast of 25 dB, posing a limitation onthe achievable dynamic range. While this behavior is shared byany link with third-order nonlinearities, it is of particular interestin microwave photonic systems, where Mach–Zehnder or othermodulators and components dictate a rather low .

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Lior Yaron (S’09) received the B.Sc. degree in physics and electrical en-gineering from Tel-Aviv University, Tel-Aviv, Israel, in 2008, where he iscurrently working toward the Ph.D. degree on microwave photonics, photonicanalog signal processing, and fiber-optic sensing at the Faculty of Engineering.

Moshe Tur (F’98) received the B.Sc. degree in mathematics and physics fromthe Hebrew University, Jerusalem, Israel, in 1969, the M.Sc. degree in appliedphysics from the Weizmann Institute of Science, Rehovot, Israel, in 1973, andthe Ph.D. degree from Tel-Aviv University, Tel-Aviv, Israel in 1981.He is currently the Gordon Professor of electrical engineering in the School

of Electrical Engineering, Tel-Aviv University, where he has establisheda fiber-optic sensing and communication laboratory. He has authored orcoauthored more than 350 journal and conference technical papers withemphasis on microwave photonics, fiber-optic sensing (with current emphasison structural health monitoring, using fiber Bragg gratings and the Brillouineffect), polarization mode dispersion, and advanced fiber-optic communicationsystems.Dr. Tur is a Fellow of the Optical Society of America.

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