Article
Ambiguity function analysis of pulse train propagation: Applications to temporal Lau filtering
University of Valencia, Valenza, Valencia, Spain
Journal of the Optical Society of America A (Impact Factor: 1.56). 09/2007; 24(8):226873. DOI: 10.1364/JOSAA.24.002268 Source: PubMed
Fulltext
Available from: Jorge OjedaCastaneda, Apr 16, 2015Ambiguity function analysis of pulse train
propagation: applications to temporal Lau ﬁltering
Jorge OjedaCastañeda,
1,3
Jesús Lancis,
1,
*
Cristina M. GómezSarabia,
1
Víctor TorresCompany,
1
and Pedro Andrés
2
1
Grup de Recerca d’Óptica de Castelló, Departament de Física, Universitat Jaume I, E12080 Castelló, Spain
2
Departamento de Óptica, Universitat de València, E46100 Burjassot, Spain
3
Permanent 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 periodicsignal ambiguity function for visualizing the intensityspectrum evolution through propa
gation in a ﬁrstorder dispersive medium. We show that the degree of temporal coherence of the optical source
plays the role of a lowpass 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 phasemodulated CW signal. © 2007
Optical Society of America
OCIS codes: 070.6770, 320.5390
.
1. INTRODUCTION
Phasespace 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
guided dispersive media [8,9]. Alternatively, high
repetitionrate 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
squarelaw 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 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 lowpass ﬁlter on the spectrum
intensity evolution. This gives an extra degree of freedom
to ﬁlter higherorder 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 2007 OjedaCastañeda et al.
10847529/07/0822686/$15.00 © 2007 Optical Society of America
Page 1
erates the signal, which is a periodic complex amplitude
represented by the Fourier series
g共t兲 =
兺
m=−⬁
⬁
a
m
exp共− im⍀t兲. 共1兲
In Eq. (1) we denote as ⍀=2
/T the fundamental angular
frequency of the slowly varying envelope. At the input,
the temporal intensity is
I共t,z =0兲 =
兺
m=−⬁
⬁
再
兺
n=−⬁
⬁
a
m+ n
a
n
*
冎
exp共− im⍀t兲, 共2兲
where * stands for the complex conjugate. The intensity
spectrum is obtained by taking the Fourier transform of
Eq. (2) to give
I
˜
共
,0兲 =
兺
m=−⬁
⬁
再
兺
n=−⬁
⬁
a
m+n
a
n
*
冎
␦
共
− m⍀兲. 共3兲
For the sake of simplicity, we assume that the dispersive
medium is a singlemode 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
u共
,z兲 =
兺
m=−⬁
⬁
兵a
m
exp关i共

2
/2兲m
2
⍀
2
z兲兴exp共− im⍀
兲. 共4兲
In Eq. (4) we denote as
=t −

1
z the proper time. The
symbols

1
and

2
denote, respectively, the inverse of the
group velocity and the ﬁrstorder dispersion coefﬁcient.
We express the temporal intensity at z⬎0as
I共
,z兲 =
兺
m=−⬁
⬁
再
兺
n=−⬁
⬁
a
m+n
a
n
* exp关共i

2
mn⍀
2
z兲兴
冎
⫻exp关i共

2
/2兲m
2
⍀
2
z兲exp共− im⍀
兲. 共5兲
Hence, the intensity spectrum at z⬎ 0is
I
˜
共
,z兲 =
兺
m=−⬁
⬁
再
兺
n=−⬁
⬁
a
m+ n
a
n
* exp关i共

2
mn⍀
2
z兲兴
冎
⫻exp关i共

2
/2兲m
2
⍀
2
z兲
␦
共
− m⍀兲. 共6兲
Now, we invoke the deﬁnition of the ambiguity function:
A共
,t兲 =
冕
−⬁
⬁
g共t
⬘
+ t/2兲g * 共t
⬘
− t/2兲exp共i
t
⬘
兲dt
⬘
= 共1/2
兲
冕
−⬁
⬁
G共
⬘
+
/2兲G * 共
⬘
−
/2兲
⫻exp共− it
⬘
兲d
⬘
, 共7兲
where G共
兲 is the Fourier transform of the signal g共t兲.
The ambiguity function of the pulse train in Eq. (1) is
A共
,t兲 =
兺
m=−⬁
⬁
再
兺
n=−⬁
⬁
a
m+n
a
n
* exp共− in⍀t兲
冎
⫻exp共− im⍀t/2兲
␦
共
− m⍀兲. 共8兲
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=−

2
m⍀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共
,z兲 =
兺
m=−⬁
⬁
A共m⍀,−

2
zm⍀兲
␦
共
− m⍀兲. 共9兲
Hence, at the output of the GDD circuit, the temporal in
tensity is
I共
,z兲 =
兺
m=−⬁
⬁
A共m⍀,−

2
zm⍀兲exp共− im⍀
兲. 共10兲
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
˜
共
,z兲 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 =−共

2
z兲
⫻共m⍀兲. Since the values along the horizontal axis are
=m⍀, then the values along the vertical axis are obtained
through the slope s =−共

2
z兲. 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.
OjedaCastañeda et al. Vol. 24, No. 8 / August 2007/ J. Opt. Soc. Am. A 2269
Page 2
function can be thought of as a polar display of the spec
trum intensity evolution, with variable slope −

2
z.
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共t兲,is[23]
I共t,z 兲 = 共1/2
兲
冕
−⬁
⬁
S共
兲兩R共t,z ,
兲兩
2
d
, 共11兲
where S共
兲 is the normalized spectral density function of
the source peaked at the optical frequency
0
and the
guided mode integral is
R共t,z ,
兲 =
冕
−⬁
⬁
G共
⬘
−
兲exp关i

共
⬘
兲z − i
⬘
t兴d
⬘
. 共12兲
As before, G共
兲 denotes the Fourier transform of the sig
nal g共t兲. For parabolic dispersive media, and for a periodic
pulse train, it is straightforward to evaluate the square
modulus of Eq. (12) to obtain
兩R共
,z,
兲兩
2
=
兺
m=−⬁
⬁
再
兺
n=−⬁
⬁
a
m+ n
a
n
* exp关i共

2
mn⍀
2
z兲兴
冎
⫻exp关i共

2
/2兲m
2
⍀
2
z兲
⫻exp关− im⍀
+ i

2
m⍀z共
−
0
兲兴, 共13兲
where again we denote as
the proper time. By using the
results in Eqs. (9) and (10), we can rewrite Eq. (13) as
兩R共
,z,
兲兩
2
=
兺
m=−⬁
⬁
A共m⍀,−

2
zm⍀兲
⫻exp关− im⍀
+ i

2
m⍀z共
−
0
兲兴. 共14兲
Also, by substituting Eq. (14) in Eq. (11), we obtain that
for a spectrally incoherent and broadband source the tem
poral intensity is
I共
,z兲
=
兺
m=−⬁
⬁
冋
共1/2
兲
冕
−⬁
⬁
S共
兲exp关i

2
m⍀z共
−
0
兲兴d共
−
0
兲
册
⫻A共m⍀,−

2
zm⍀兲exp共− im⍀
兲. 共15兲
Equivalently, if we recognize the deﬁnition of the complex
degree of coherence,
␥
共t兲= 兰S共
+
0
兲exp共−i
t兲d
, the tem
poral intensity becomes
I共
,z兲 =
兺
m=−⬁
⬁
␥
共−

2
m⍀z兲A共m⍀ ,−

2
zm⍀兲exp共− im⍀
兲.
共16兲
This remarkably simple result makes apparent the fol
lowing. The complex degree of coherence,
␥
共−

2
m⍀z兲,
plays the role of a lowpass 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 electrooptic phase modulator
driven by an RF sinusoidal signal modulates the beam of
an optical source. In this case,
g共t兲 = exp关i⌬
sin共2
t/T兲兴. 共17兲
In Eq. (17)) we denote as ⌬
the modulation index. From
Eq. (17), it is straightforward to evaluate
A共
,t兲 =
冕
−⬁
⬁
exp兵i关2⌬
sin共⍀t/2兲兴cos共⍀t
⬘
兲其exp共i
t
⬘
兲dt
⬘
=
兺
n=−⬁
⬁
共i兲
n
J
n
关2⌬
sin共⍀t/2兲兴
␦
共
− n⍀兲. 共18兲
In Fig. 3 we display the modulus, 兩A共
,t兲兩, of the above ex
pression for ⌬
=
/2 and T = 50 ps. From Eq. (18) we have
that
A共m⍀,−

2
m⍀z兲 = 共− i兲
m
J
m
关2⌬
sin共

2
⍀
2
zm/2兲兴.
共19兲
By substituting Eq. (19) in Eq. (16), we obtain that for a
spectrally incoherent source, the output temporal inten
sity is
Fig. 3. (Color online) Modulus of the ambiguity function of a
sinusoidal phase signal. The repetition rate is 20 GHz and the
modulation index value is ﬁxed to
/2 rad.
2270 J. Opt. Soc. Am. A / Vol. 24, No. 8 / August 2007 OjedaCastañeda et al.
Page 3
I共
,z兲 =
兺
m=−⬁
⬁
␥
共− m

2
⍀z兲J
m
关2⌬
sin共

2
⍀
2
zm/2兲兴
⫻exp关− im共⍀
+
/2兲兴. 共20兲
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共
,z兲 =1+2
兺
m=1
⬁
␥
共m

2
⍀z兲J
m
关2⌬
sin共

2
⍀
2
zm/2兲兴
⫻cos关m共⍀
+
/2兲兴. 共21兲
At fractions of the Talbot length, z=Z
T
/M with Z
T
=4
/

2
⍀
2
,Eq.(21) becomes
I共
,Z
T
/M兲 =1+2
兺
m=1
⬁
␥
共2mT/M兲J
m
关2⌬
sin共2
m/M兲兴
⫻cos关m共⍀
+
/2兲兴. 共22兲
From the argument of the Bessel functions, in Eq. (22) we
note that if sin共2
m/M兲= 0, then the cosinusoidal har
monics 共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共
,Z
T
/4兲 =1+2
兺
m=1
⬁
␥
共mT/2兲J
m
兵2⌬
sin关共
/2兲m兴其
⫻cos关m共⍀
+
/2兲兴. 共23兲
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
␥
共t兲= 1, when ⌬
=
/4 only the ﬁrst and
third harmonic play a signiﬁcant role, which produces the
wellknown result of ﬂattop 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共
兲 = 关1/共2Q +1兲兴
兺
q=−Q
Q
␦
共
−
0
− q⍀
s
兲. 共24兲
Here, we denote as ⍀
s
an 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
␥
共
兲 = 关1/共2Q +1兲兴
冋
1+2
兺
q=1
Q
cos共q⍀
s
兲
册
. 共25兲
At the sampling points
=2mT/ M, the values of the de
gree of temporal coherence are
␥
共2mT/M兲 = 关1/共2Q +1兲兴
冋
1+2
兺
q=1
Q
cos共2mq⍀
s
T/M兲
册
.
共26兲
Next, we select the following mismatch of the Lau condi
tion:
⍀
s
= ⍀共M/2N兲, 共27兲
where N denotes any positive integer. With this condition
Eq. (26) becomes
␥
共2mT/M兲 = 关1/共2Q +1兲兴
冋
1+2
兺
q=1
Q
cos共2
mq/N兲
册
.
共28a兲
In Fig. 4 we depict the fact that as Q increases, the value
␥
共2mT/M兲 is equal to unity if m=N ,2N ,3N. ... Otherwise
␥
共2mT/M兲 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,
␥
共2mT/M兲 =
␦
m,nN
. 共28b兲
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. Plot of the complex degree of coherence of the multi
wavelength source. For the plot we assume an ideal inﬁnite num
ber of spectral lines producing a comblike structure.
OjedaCastañeda et al. Vol. 24, No. 8 / August 2007/ J. Opt. Soc. Am. A 2271
Page 4
I共
,Z
T
/M兲 =1+2
兺
m=1
⬁
J
mN
关2⌬
sin共2
m共N/M兲兲兴
⫻cos关mN共⍀
+
/2兲兴. 共29兲
It is apparent from Eq. (29) that at z = Z
T
/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共
,Z
T
/4兲 =1+2
兺
m=1
⬁
J
3m
关2⌬
sin共共3
/2兲m兲兴
⫻cos关3m共⍀
+
/2兲兴. 共30兲
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
␥
共t兲= 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 welldeﬁned 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 共3⫻ 兲 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 dcﬂoor 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
␥
共2mT/M兲 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
ﬁrstorder approximation to the dispersion relation for a
singlemode 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 electrooptic
phasemodulation method.
ACKNOWLEDGMENTS
This research was funded by Dirección General de Inves
tigación Cientiﬁca y Técnica, Spain, project FIS2004
02404. J. OjedaCastañeda gratefully acknowledges ﬁ
nancial support from “Convenio UJIBancaixa” (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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 "Further, several approaches have been recently advanced to synthesize novel partially coherent pulses from uncorrelated–or partially correlated–superpositions of elementary pulses in time [7] and frequency [8]. In addition, several phasespace approaches to partially coherent pulse representation were discussed in the literature [9, 10]. Lately, a general phasespace approach has been put forward to describe partially coherent pulse synthesis from complex Gaussian pulses [11] . "
[Show abstract] [Hide abstract] ABSTRACT: We introduce the concept of phasespace separability degree of statistical pulses and show how it can be determined using a biorthogonal decomposition of the pulse Wigner distribution. We present explicit analytical results for the case of chirped gaussian Schellmodel pulses. We also demonstrate that chirping of the pulsed source serves as a powerful tool to control coherence and phasespace separability of statistical pulses. 
Article: Temporal Filtering in Phasespace
[Show abstract] [Hide abstract] ABSTRACT: We use the ambiguity function, of the slowly varying complexamplitude envelope, for visualizing the intensityspectrum evolution through propagation in a firstorder dispersive medium. We comment on the use of this formalism for temporal filtering. 
Article: Intensity spectra after firstorder dispersion of composite models of scalar cyclostationary light
[Show abstract] [Hide abstract] ABSTRACT: The spectrum of the intensity of dispersed waves obeying cyclostationary statistics is studied. The formalism is based on an exact formula by Marshall and Yariv [IEEE Photon. Technol. Lett.12, 302 (2000)] relating the intensity spectrum after firstorder dispersion to the Fourier transform of a certain restriction of the timeaveraged fourthorder correlation of the optical wave e(t) before dispersion. The formalism permits a simple computation of the spectrum of composite models defined by the independent addition or multiplication of a stationary and a cyclostationary field. The computations are simplified by introducing the auxiliary field z(tau)(t)=e(t)*e(t+tau), whose power spectral density represents the basic building block for solving the spectrum of composite models. The results are illustrated by a number of examples, including the intensity spectrum after dispersion of analogmodulated, partially coherent carriers, or the complete spectrum of intensity fluctuations of multiwavelength dispersionbased microwave photonic filters.