Page 1

Ambiguity function analysis of pulse train

propagation: applications to temporal Lau filtering

Jorge Ojeda-Castañeda,1,3Jesús Lancis,1,* Cristina M. Gómez-Sarabia,1Víctor Torres-Company,1and Pedro Andrés2

1Grup de Recerca d’Óptica de Castelló, Departament de Física, Universitat Jaume I, E12080 Castelló, Spain

2Departamento de Óptica, Universitat de València, E46100 Burjassot, Spain

3Permanent address, Instituto de Investigación en Comunicación Óptica,

Universidad Autónoma de San Luís Potosí, México

*Corresponding author: lancis@fca.uji.es

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 source

plays the role of a low-pass filter on the signal’s ambiguity function. Based on this, we present a condition on

the 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 America

OCIS codes: 070.6770, 320.5390.

1. INTRODUCTION

Phase-space representations provide useful tools for char-

acterizing and analyzing the propagation of ultrashort op-

tical pulses [1–7]. The temporal Talbot effect is useful for

regenerating a periodic pulse train that propagates in

guideddispersivemedia

repetition-rate pulse trains can be achieved at fractions of

the 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 of

view, this type of source can be obtained with Fabry–

Perot laser diodes [15] or by spectral slicing an amplified

spontaneous emission radiation source [16], commonly

used in telecom. The periodic pulses are produced by an

external 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 temporally

shifted version of the input intensity. Since the global

source 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 channels

and the repetition rate of the external modulator.

Here, we use the ambiguity function [17–19] 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 first

line of Fig. 1(a), we represent the evolution of a signal in

a GDD circuit. In the second line, we depict the use of a

square-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-

[8,9].Alternatively,high-

poral intensity for periodic signals. To that goal, we show

that 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, the

degree of temporal coherence plays the role of a filter on

the signal’s ambiguity function as depicted Fig. 1(b).

Hence, our present aim is threefold. First, if the source

is monochromatic, we show that the periodic signal’s 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 broadband

and spectrally incoherent, we show that the complex de-

gree of coherence acts as a low-pass filter on the spectrum

intensity evolution. This gives an extra degree of freedom

to 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 than

the original sequence.

To our end, in Section 2, we consider as input a periodic

pulse train that propagates in a GDD circuit. We relate

the spectrum intensity with the ambiguity function. In

Section 3, we discuss the use of the signal’s ambiguity

function as a polar display. In Section 4, we extend our

treatment to broadband sources. Finally, in Section 5, we

apply the temporal Lau effect for filtering the temporal

intensity 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 2007Ojeda-Castañeda et al.

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

Page 2

erates the signal, which is a periodic complex amplitude

represented by the Fourier series

g?t? =?

m=−?

?

amexp?− im?t?.

?1?

In Eq. (1) we denote as ?=2?/T the fundamental angular

frequency of the slowly varying envelope. At the input,

the temporal intensity is

?? ?

where * stands for the complex conjugate. The intensity

spectrum is obtained by taking the Fourier transform of

Eq. (2) to give

?? ?

For the sake of simplicity, we assume that the dispersive

medium is a single-mode fiber with a parabolic dispersion

relation. In this paper, we assume no fiber loss, although

a general nonfrequency dependent attenuation coefficient

should lead to the same results. Then at the output, say

at z?0, the slowly varying envelope is

I?t,z = 0? =?

m=−?

n=−?

?

am+nan*?exp?− im?t?,

?2?

I˜??,0? =?

m=−?

n=−?

?

am+nan*???? − m??.

?3?

u??,z? =?

m=−?

?

?amexp?i??2/2?m2?2z??exp?− im???.

?4?

In Eq. (4) we denote as ?=t−?1z the proper time. The

symbols ?1and ?2denote, respectively, the inverse of the

group velocity and the first-order dispersion coefficient.

We express the temporal intensity at z?0 as

am+nan* exp? ?i?2mn?2z???

?exp?i??2/2?m2?2z?exp?− im???.

I??,z? =?

m=−?

?? ?

n=−?

?

?5?

Hence, the intensity spectrum at z?0 is

I˜??,z? =?

m=−?

?? ?

?exp?i??2/2?m2?2z???? − m??.

n=−?

?

am+nan* exp?i ??2mn?2z???

?6?

Now, we invoke the definition of the ambiguity function:

A??,t? =?

−?

= ?1/2???

−?

?

g?t? + t/2?g * ?t? − t/2?exp?i?t??dt?

?

G??? + ?/2?G * ??? − ?/2?

?exp?− it???d??,

?7?

where G??? is the Fourier transform of the signal g?t?.

The ambiguity function of the pulse train in Eq. (1) is

am+nan* exp?− in?t??

?exp?− im?t/2???? − m??.

A??,t? =?

m=−?

?? ?

n=−?

?

?8?

Next, we note that the ambiguity function of the pulse

train in Eq. (8) contains as two particular cases the spec-

trum intensity in Eqs. (3) and (6). That is, for t=0 Eq. (8)

becomes Eq. (3), while for t=−?2m?z Eq. (8) becomes Eq.

(6).

3. POLAR DISPLAY

From the above observations, we claim that for any value

of z the intensity spectrum is

I??,z? =?

m=−?

?

A?m?,− ?2zm????? − m??.

?9?

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

tensity is

I??,z? =?

m=−?

?

A?m?,− ?2zm??exp?− im???.

?10?

It is apparent from Eqs. (9) and (10) that the signal’s am-

biguity function contains (in a single picture) the evolu-

tion of the intensity spectrum I˜??,z? for variable z. This

result is depicted schematically in Fig. 3, where we dis-

play the modulus of the ambiguity function of a sinusoidal

phase signal; this is further analyzed in Section 5.

In other words, we note that the values of the spectrum

intensity are sampled along the straight line t=−??2z?

??m??. Since the values along the horizontal axis are ?

=m?, then the values along the vertical axis are obtained

through the slope s=−??2z?. Consequently, the ambiguity

Fig. 1.

matic case, (b) spectrally incoherent case.

Block diagram of the proposed approach: (a) monochro-

Fig. 2.Schematic diagram of the optical setup.

Ojeda-Castañeda et al.

Vol. 24, No. 8/August 2007/J. Opt. Soc. Am. A2269

Page 3

function can be thought of as a polar display of the spec-

trum intensity evolution, with variable slope −?2z.

Within the celebrated space–time 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 the

depth of field of an optical system [21,22].

4. SPECTRALLY INCOHERENT SOURCE

If the spectral distribution of the optical source is taken

into account, the averaged temporal intensity at a dis-

tance z along a fiber for a given input signal, g?t?, is [23]

I?t,z? = ?1/2???

−?

?

S????R?t,z,???2d?,

?11?

where S??? is the normalized spectral density function of

the source peaked at the optical frequency ?0and the

guided mode integral is

R?t,z,?? =?

−?

?

G??? − ??exp?i?????z − i??t?d??. ?12?

As before, G??? denotes the Fourier transform of the sig-

nal g?t?. For parabolic dispersive media, and for a periodic

pulse train, it is straightforward to evaluate the square

modulus of Eq. (12) to obtain

?? ?

?exp?i??2/2?m2?2z?

?R??,z,???2=?

m=−?

n=−?

?

am+nan* exp?i??2mn?2z???

?exp?− im?? + i?2m?z?? − ?0??,

?13?

where again we denote as ? the proper time. By using the

results in Eqs. (9) and (10), we can rewrite Eq. (13) as

?R??,z,???2=?

m=−?

?

A?m?,− ?2zm??

?exp?− im?? + i?2m?z?? − ?0??.

?14?

Also, by substituting Eq. (14) in Eq. (11), we obtain that

for a spectrally incoherent and broadband source the tem-

poral intensity is

I??,z?

=?

m=−?

???1/2???

?A?m?,− ?2zm??exp?− im???.

−?

?

S???exp?i?2m?z?? − ?0??d?? − ?0??

?15?

Equivalently, if we recognize the definition of the complex

degree of coherence, ??t?=?S??+?0?exp?−i?t?d?, the tem-

poral intensity becomes

I??,z? =?

m=−?

?

??− ?2m?z?A?m?,− ?2zm??exp?− im???.

?16?

This remarkably simple result makes apparent the fol-

lowing. The complex degree of coherence, ??−?2m?z?,

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 illustrate

the 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 modulator

driven by an RF sinusoidal signal modulates the beam of

an optical source. In this case,

g?t? = exp?i?? sin?2?t/T??.

?17?

In Eq. (17)) we denote as ?? the modulation index. From

Eq. (17), it is straightforward to evaluate

A??,t? =?

−?

?

exp?i?2?? sin??t/2??cos??t???exp?i?t??dt?

=?

n=−?

?

?i?nJn?2?? sin??t/2????? − n??.

?18?

In Fig. 3 we display the modulus, ?A??,t??, of the above ex-

pression for ??=?/2 and T=50 ps. From Eq. (18) we have

that

A?m?,− ?2m?z? = ?− i?mJm?2?? sin??2?2zm/2??.

?19?

By substituting Eq. (19) in Eq. (16), we obtain that for a

spectrally incoherent source, the output temporal inten-

sity is

Fig. 3.

sinusoidal phase signal. The repetition rate is 20 GHz and the

modulation index value is fixed to ?/2 rad.

(Color online) Modulus of the ambiguity function of a

2270J. Opt. Soc. Am. A/Vol. 24, No. 8/August 2007Ojeda-Castañeda et al.

Page 4

I??,z? =?

m=−?

?

??− m?2?z?Jm?2?? sin??2?2zm/2??

?exp?− im??? + ?/2??.

?20?

By assuming that the normalized spectral density func-

tion is an even function, then the complex degree of coher-

ence is also an even function. Hence, the temporal inten-

sity becomes

I??,z? = 1 + 2?

m=1

?

??m?2?z?Jm?2?? sin??2?2zm/2??

?cos?m??? + ?/2??.

?21?

At fractions of the Talbot length, z=ZT/M with ZT

=4?/?2?2, Eq. (21) becomes

I??,ZT/M? = 1 + 2?

m=1

?

??2mT/M?Jm?2?? sin?2?m/M??

?cos?m??? + ?/2??.

?22?

From the argument of the Bessel functions, in Eq. (22) we

note that if sin?2?m/M?=0, then the cosinusoidal har-

monics ?m=pM/2 with p=1,2,3...) are filtered out. Con-

sequently, there is temporal filtering effect solely due to

the length of the dispersive media. For example, at 1/4 of

the Talbot length Eq. (22) becomes

I??,ZT/4? = 1 + 2?

m=1

?

??mT/2?Jm?2?? sin???/2?m??

?cos?m??? + ?/2??.

?23?

It is apparent from Eq. (23) that in the temporal intensity

one has filtered out the mth harmonic, if m is equal to an

even integer number. Note that, for a strictly monochro-

matic source ??t?=1, when ??=?/4 only the first and

third harmonic play a significant role, which produces the

well-known result of flat-top pulse generation [24,25].

Next, we discuss the influence of the multiwavelength

source that is used for discussing the temporal Lau effect.

The spectral density function is

S??? = ?1/?2Q + 1???

q=−Q

Q

??? − ?0− q?s?.

?24?

Here, we denote as ?san angular frequency shift from the

carrier frequency ?0. Such a source could be provided by a

multiwavelength Fabry–Perot laser diode with 2Q+1

taps. The spacing between adjacent channels is ?s. The

uniformity in the energy can be achieved by proper spec-

tral filtering, previous to the stage of the modulation with

a bandpass properly designed Bragg grating.Additionally,

as a first approximation, we neglect the spectral line-

width of the modes, since for Fabry–Perot laser diodes the

individual linewidth can be in the submegahertz range.

With the above assumptions, the complex degree of tem-

poral coherence is

???? = ?1/?2Q + 1???1 + 2?

q=1

Q

cos?q?s???.

?25?

At the sampling points ?=2mT/M, the values of the de-

gree of temporal coherence are

??2mT/M? = ?1/?2Q + 1???1 + 2?

q=1

Q

cos?2mq?sT/M??.

?26?

Next, we select the following mismatch of the Lau condi-

tion:

?s= ??M/2N?,

?27?

where N denotes any positive integer. With this condition

Eq. (26) becomes

??2mT/M? = ?1/?2Q + 1???1 + 2?

q=1

Q

cos?2?mq/N??.

?28a?

In Fig. 4 we depict the fact that as Q increases, the value

??2mT/M? is equal to unity if m=N,2N,3N.... Otherwise

??2mT/M? is practically zero. That is, the values of the

complex degree of coherence can be approximated by a

comb of Kronecker’s delta, with period N,

??2mT/M? = ?m,nN.

?28b?

In other words, the proposed mismatch generates values

of the mutual coherence that behaves as a highly peaked

window for filtering harmonics of the temporal intensity.

Specifically, by using the result in Eq. (28), we express Eq.

(22) as

Fig. 4.

wavelength source. For the plot we assume an ideal infinite num-

ber of spectral lines producing a comblike structure.

Plot of the complex degree of coherence of the multi-

Ojeda-Castañeda et al.

Vol. 24, No. 8/August 2007/J. Opt. Soc. Am. A2271

Page 5

I??,ZT/M? = 1 + 2?

m=1

?

JmN?2?? sin?2?m?N/M???

?cos?mN??? + ?/2??.

?29?

It is apparent from Eq. (29) that at z=ZT/M, the temporal

intensity contains only the Nmth harmonics. For ex-

ample, at 1/4 of the Talbot length, and by setting the mis-

matched temporal Lau condition as ?s=2?/3, the tempo-

ral intensity is

I??,ZT/4? = 1 + 2?

m=1

?

J3m?2?? sin??3?/2?m??

?cos?3m??? + ?/2??.

?30?

In other words, the temporal intensity contains only the

3mth harmonics. In order to illustrate the capabilities of

the source filtering procces, we plot in Fig. 5(a) the output

intensity obtained with monochromatic source, Eq. (23))

with ??t?=1, for ??=2.1 rad and 20 GHz repetition rate.

The achieved waveform results useless for telecommuni-

cations applications since there are variations from pulse

to pulse due to the contribution of different harmonics.

However, as represented in Fig. 5(b), by selecting a mul-

tiwavelength optical source satisfying the Lau mismatch

condition ?s=2?/3, we obtain a well-defined sequence

[see Eq. (30)].

Note the resulting wavelength spacing is physically

feasible for a commercial Fabry–Perot laser diode. Fur-

thermore, a repetition rate higher ?3?? than the driving

signal is achieved at the output. This is due to the fact

that the modulation index value is taken to select essen-

tially the third harmonic. The influence from the rest of

the harmonics is noted in the low dc-floor level. It should

be mentioned that it is not possible to completely remove

with the present technique. In order to estimate the oper-

ating bandwidth, we choose Q?10 so that ??2mT/M? is

equal to unity only for m=N,2N,3N..., but otherwise it is

practically zero. In other words, at least ten spectral lines

must be considered. In this way, it results a total band-

width of 200 GHz. This value ensures the validity of the

first-order approximation to the dispersion relation for a

single-mode fiber operating at 1.55 ?m.

Finally, we point out that at the Talbot length, M=1,

our discussed formulas predict (as expected) uniform tem-

poral intensity.

6. CONCLUSIONS

We proposed to use the signal’s ambiguity function for vi-

sualizing (in a single picture) the evolution of the spec-

trum intensity as the periodic pulse propagates in guiding

parabolic dispersive media. We indicated that there is a

temporal filtering effect solely due to the length of the dis-

persive media.

Furthermore, we showed that for spectrally broad

sources the degree of temporal coherence acts as a low-

pass filter on the signal’s ambiguity function.

We presented a mismatch (in the temporal Lau condi-

tion) for obtaining values of the complex degree of tempo-

ral coherence, which behave as highly peaked periodic

windows for filtering harmonics at fractions of the Talbot

length. We showed that one can increase the repetition

rate of the pulse trains achievable with the electro-optic

phase-modulation method.

ACKNOWLEDGMENTS

This research was funded by Dirección General de Inves-

tigación Cientifica y Técnica, Spain, project FIS2004-

02404. J. Ojeda-Castañeda gratefully acknowledges fi-

nancial support from “Convenio UJI-Bancaixa” (grant

06I005.27). V. Torres gratefully acknowledges financial

support from a Formación de Profesorado Universitario

grant of the Ministerio de Educación y Ciencia.

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