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Ambiguity function analysis of pulse trainpropagation: applications to temporal Lau filtering

Jorge Ojeda-Castaeda,1,3 Jess Lancis,1,*Cristina M. Gmez-Sarabia,1 Vctor Torres-Company,1 and Pedro Andrs2

1Grup de Recerca dptica de Castell, Departament de Fsica, Universitat Jaume I, E12080 Castell, Spain2Departamento de ptica, Universitat de Valncia, E46100 Burjassot, Spain

3Permanent address, Instituto de Investigacin en Comunicacin ptica,Universidad Autnoma de San Lus Potos, Mxico

*Corresponding author: [email protected]

Received November 21, 2006; revised February 9, 2007; accepted March 13, 2007;posted March 28, 2007 (Doc. ID 77300); published July 11, 2007

We use the periodic-signal ambiguity function for visualizing the intensity-spectrum evolution through propa-gation in a first-order dispersive medium. We show that the degree of temporal coherence of the optical sourceplays the role of a low-pass filter on the signals ambiguity function. Based on this, we present a condition onthe temporal Lau effect for filtering harmonics at fractions of the Talbot length. This result allows one to in-crease the repetition rate of a pulse train obtained from a sinusoidally phase-modulated CW signal. 2007

Optical Society of AmericaOCIS codes: 070.6770, 320.5390.

1. INTRODUCTIONPhase-space representations provide useful tools for char-acterizing and analyzing the propagation of ultrashort op-tical pulses[17]. The temporal Talbot effect is useful forregenerating a periodic pulse train that propagates inguided dispersive media [8,9]. Alternatively, high-repetition-rate pulse trains can be achieved at fractions ofthe Talbot length [10,11], including tunable duty cycle[12]. In this way, the fractional Talbot effect can be inter-preted as a filter that removes harmonics of the input in-

tensity spectrum [13].Recently, a multiwavelength source was used to discuss

the temporal Lau effect [14]. From a practical point ofview, this type of source can be obtained with FabryPerot laser diodes[15] or by spectral slicing an amplifiedspontaneous emission radiation source [16], commonlyused in telecom. The periodic pulses are produced by anexternal amplitude modulator, which allows for the inde-pendent control of the pulse repetition rate. After propa-gation in a group delay dispersion (GDD) circuit satisfy-ing the Talbot condition, every line produces a temporallyshifted version of the input intensity. Since the globalsource is spectrally incoherent, the final intensity is an in-coherent superposition of every shifted replica. In this

way, one regenerates the original sequence with an ad-equate selection of both the spacing between channelsand the repetition rate of the external modulator.

Here, we use the ambiguity function[1719] of the pe-riodic signal for exploring the temporal Lau effect at frac-tions of the Talbot length. Our approach is depicted sche-matically in Fig.1. Going from top to bottom, in the firstline of Fig.1(a),we represent the evolution of a signal ina GDD circuit. In the second line, we depict the use of asquare-law detector for evaluating the temporal intensity.Here, we attempt to formulate a mapping (dotted arrow)between the input temporal intensity and the output tem-

poral intensity for periodic signals. To that goal, we showthat for a monochromatic optical source, the signal ambi-guity function represents the spectrum intensity evolu-tion in a GDD circuit as depicted at the bottom of Fig.1(a). Furthermore, for a spectrally incoherent source, thedegree of temporal coherence plays the role of a filter onthe signals ambiguity function as depicted Fig. 1(b).

Hence, our present aim is threefold. First, if the sourceis monochromatic, we show that the periodic signals am-biguity function contains all changes suffered by the spec-

trum intensity as the signal propagates in guided para-bolic dispersive media. Second, if the source is broadbandand spectrally incoherent, we show that the complex de-gree of coherence acts as a low-pass filter on the spectrumintensity evolution. This gives an extra degree of freedomto filter higher-order harmonics. Third, we present a mis-match of the temporal Lau condition for filtering harmon-ics at fractions of the Talbot length. Based on this, we ob-tain optical pulse trains with higher repetition rates thanthe original sequence.

To our end, in Section 2, we consider as input a periodicpulse train that propagates in a GDD circuit. We relatethe spectrum intensity with the ambiguity function. InSection 3, we discuss the use of the signals ambiguity

function as a polar display. In Section 4, we extend ourtreatment to broadband sources. Finally, in Section 5, weapply the temporal Lau effect for filtering the temporalintensity at fractions of the Talbot length.

2. SPECTRUM INTENSITY EVOLUTION:COHERENT CASE

As depicted in Fig. 2, we use first an optical monochro-matic source with carrier angular frequency 0. At the in-put of a GDD circuit (say z =0) an external modulator gen-

2268 J. Opt. Soc. Am. A/ Vol. 24, No. 8 / August 2007 Ojeda-Castaeda et al.

1084-7529/07/082268-6/$15.00 2007 Optical Society of America


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erates the signal, which is a periodic complex amplituderepresented by the Fourier series

gt = m=

amexp imt. 1

In Eq.(1)we denote as =2/Tthe fundamental angularfrequency of the slowly varying envelope. At the input,the temporal intensity is

It,z = 0 = m=

n=

am+nan * exp imt, 2where * stands for the complex conjugate. The intensityspectrum is obtained by taking the Fourier transform ofEq.(2) to give

I,0 = m=

n=

am+nan * m. 3For the sake of simplicity, we assume that the dispersivemedium is a single-mode fiber with a parabolic dispersionrelation. In this paper, we assume no fiber loss, althougha general nonfrequency dependent attenuation coefficientshould lead to the same results. Then at the output, sayat z0, the slowly varying envelope is

u,z = m=

amexpi2/2m2

2zexp im. 4

In Eq. (4) we denote as = t 1z the proper time. Thesymbols1 and 2denote, respectively, the inverse of the

group velocity and the first-order dispersion coefficient.We express the temporal intensity at z0 as

I,z = m=

n=

am+nan * expi2mn2z

expi2/2m2

2zexp im. 5

Hence, the intensity spectrum at z0 is

I,z = m=

n=

am+nan * expi2mn2z

expi2/2m2

2z m. 6

Now, we invoke the definition of the ambiguity function:

A,t =

gt + t/2g * t t/2expitdt

= 1/2

G + /2G * /2

exp it

d

, 7where G is the Fourier transform of the signal gt.The ambiguity function of the pulse train in Eq. (1) is

A,t = m=

n=

am+nan * exp intexp imt/2 m. 8

Next, we note that the ambiguity function of the pulsetrain in Eq.(8) contains as two particular cases the spec-trum intensity in Eqs.(3) and(6).That is, fort =0 Eq.(8)becomes Eq.(3), while for t = 2mz Eq. (8) becomes Eq.(6).

3. POLAR DISPLAY

From the above observations, we claim that for any valueofz the intensity spectrum is

I,z = m=

Am, 2zm m. 9

Hence, at the output of the GDD circuit, the temporal in-tensity is

I,z = m=

Am, 2zmexp im. 10

It is apparent from Eqs.(9) and(10)that the signals am-biguity function contains (in a single picture) the evolu-

tion of the intensity spectrum I,z for variable z. Thisresult is depicted schematically in Fig. 3, where we dis-play the modulus of the ambiguity function of a sinusoidalphase signal; this is further analyzed in Section 5.

In other words, we note that the values of the spectrumintensity are sampled along the straight line t = 2zm. Since the values along the horizontal axis are = m, then the values along the vertical axis are obtainedthrough the slope s = 2z. Consequently, the ambiguity

Fig. 1. Block diagram of the proposed approach: (a) monochro-matic case, (b) spectrally incoherent case.

Fig. 2. Schematic diagram of the optical setup.

Ojeda-Castaedaet al. Vol. 24, No. 8/ August 2007/ J. Opt. Soc. Am. A 2269


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function can be thought of as a polar display of the spec-trum intensity evolution, with variable slope 2z.

Within the celebrated spacetime analogy the above re-sult is equivalent to the polar display of the optical trans-fer function, of optical systems that suffer from focus er-rors [20]. This approach has been applied to extend thedepth of field of an optical system [21,22].

4. SPECTRALLY INCOHERENT SOURCEIf the spectral distribution of the optical source is takeninto account, the averaged temporal intensity at a dis-tancez along a fiber for a given input signal, gt, is[23]

It,z = 1/2

SRt,z,2d, 11

where S is the normalized spectral density function ofthe source peaked at the optical frequency 0 and theguided mode integral is

Rt,z, =

G expiz itd. 12

As before, G denotes the Fourier transform of the sig-nalgt. For parabolic dispersive media, and for a periodic

pulse train, it is straightforward to evaluate the squaremodulus of Eq.(12)to obtain

R,z,2 = m=

n=

am+nan * expi2mn2z

expi2/2m2

2z

exp im+ i2mz 0 , 13

where again we denote as the proper time. By using theresults in Eqs.(9) and (10),we can rewrite Eq.(13)as

R,z,2 = m=

Am, 2zm

exp im+ i2mz 0 . 14

Also, by substituting Eq.(14)in Eq.(11), we obtain thatfor a spectrally incoherent and broadband source the tem-poral intensity is

I,z

= m=

1/2

Sexpi2mz 0d 0Am, 2zmexp im. 15

Equivalently, if we recognize the definition of the complexdegree of coherence, t=S+0expitd, the tem-poral intensity becomes

I,z = m=

2mzAm, 2zmexp im.

16

This remarkably simple result makes apparent the fol-lowing. The complex degree of coherence, 2mz,plays the role of a low-pass filter on the ambiguity func-tion as depicted in Fig. 1(b). Of course, for a monochro-matic source, Eq. (16) reduces to Eq. (10). We illustratethe above results by filtering out harmonics of the tempo-ral intensity at fractions of the Talbot length.

5. TEMPORAL LAU FILTERING

Let us consider that an electro-optic phase modulatordriven by an RF sinusoidal signal modulates the beam ofan optical source. In this case,

gt = expisin2t/T . 17

In Eq.(17))we denote as the modulation index. FromEq.(17), it is straightforward to evaluate

A,t =

expi2sint/2costexpitdt

= n=

inJn2sint/2 n. 18

In Fig.3we display the modulus, A, t, of the above ex-pression for =/ 2 andT=50 ps. From Eq.(18)we havethat

Am, 2mz = imJm2sin2

2zm/2 .

19

By substituting Eq.(19)in Eq.(16),we obtain that for aspectrally incoherent source, the output temporal inten-sity is

Fig. 3. (Color online) Modulus of the ambiguity function of asinusoidal phase signal. The repetition rate is 20 GHz and themodulation index value is fixed to / 2 rad.

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