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Optics Communications 233 (2004) 431–437
www.elsevier.com/locate/optcom
Estimation of higher order chirp in ultrashort laser pulsesusing modified spectrum auto-interferometric correlation
A.K. Sharma *, P.A. Naik, P.D. Gupta
Laser Plasma Division, Centre for Advanced Technology, Indore 452 013, India
Received 28 August 2003; received in revised form 20 January 2004; accepted 27 January 2004
Abstract
A simple method based on modified spectrum auto-interferometric correlation (MOSAIC) is presented to determine
various higher order chirps in ultrashort laser pulses. Dependence of peak amplitude and temporal shape of the locus of
the interference minima of the MOSAIC signals on different higher order chirps facilitates their determination using a
simple computer algorithm.
� 2004 Published by Elsevier B.V.
PACS: 42.65.Re
Keywords: Femtosecond laser pulse; Frequency chirp; Autocorrelation
1. Introduction
In recent years, there has been a great ad-
vancement in generation and use of ultrashort la-
ser pulses for a variety of scientific research and
applications e.g., laser matter interaction at ultra-
high intensities [1–3], high speed (terabit/s) optical
communication systems [4,5], etc. Temporal char-
acterization of these laser pulses is an important
aspect of such investigations. Simple diagnosticsystems such as single-shot second-order autocor-
relators have been reported for measurement of
* Corresponding author. Tel.: +91-7312488478; fax: +91-
7312488430.
E-mail address: [email protected] (A.K. Sharma).
0030-4018/$ - see front matter � 2004 Published by Elsevier B.V.
doi:10.1016/j.optcom.2004.01.065
pulse duration, angular chirp and resulting pulse-
front tilt of femtosecond laser beams [6–8]. De-tection and measurement of the frequency chirp in
ultrashort laser pulses is another important task as
various order chirp components produce temporal
broadening of the laser pulse. While a linear chirp
primarily increases the full width at half maximum
(FWHM) duration of the laser pulse, higher order
chirp enhances the intensity in the wings of the
laser pulse. The latter is thus a hindrance inachieving a high value of peak to pedestal intensity
contrast in the ultrashort laser pulses. Chirp may
be present in the laser pulse either due to imperfect
alignment in the compressor stage of chirped pulse
amplification (CPA) based laser systems [9] or
acquired during propagation through some dis-
persive optical components [10,11]. Diagnostics of
432 A.K. Sharma et al. / Optics Communications 233 (2004) 431–437
the higher order chirp is therefore important for
applying necessary corrections in a laser system to
achieve desired temporal characteristics.
A variety of diagnostic tools have been devel-
oped for chirp measurements of ultrashort laser
pulses. A detailed characterization of the linear andhigher order chirp components can be carried out
using sophisticated techniques like frequency re-
solved optical gating (FROG) [12], spectral phase
interferometry for direct electrical field recon-
struction (SPIDER) [13], etc. However, these re-
quire a rather bulky experimental set-up and
involve complex numerical algorithms. On the
other hand, a detailed characterization may not benecessary in many practical situations. For exam-
ple, gross detection of the presence of chirp in a
laser pulse can be helpful in a real time optimiza-
tion of the laser system to eliminate or minimize the
chirp. In principle, interferometric auto-correlation
(IAC) [14] contains information on the chirp
present in the laser pulse. However, it does not
constitute a sensitive method for detection of chirp,especially for small magnitude of chirp. Recently,
Hirayama and Sheik-Bahae [15] proposed an in-
teresting technique of modifying the spectral con-
tents of the standard IAC signal and demonstrated
its usefulness for real time monitoring of the chirp.
The modified spectrum auto-interferometric cor-
relation (MOSAIC) signals were directly displayed
on an oscilloscope, facilitating a visual recognitionof the presence of chirp. The peak amplitude of the
locus of interference minima of the MOSAIC sig-
nals was also calculated for different values of
various order chirps for a known sech2 (t=tp) pulseshape. However, since the same peak amplitude
can be generated by several combinations of vari-
ous order chirps, this cannot be used for their un-
ambiguous quantitative determination.In this paper, we present a detailed investigation
of the characteristics of the MOSAIC signals. It is
seen that not only the peak amplitude of the locus
of the interference minima of the MOSAIC signals
but its temporal shape also depend on the mag-
nitude of different orders of chirp present in the
pulse. For instance, the temporal location of the
peak, its full width at half maximum duration, andthe asymmetry in the locus of the interference
minima of the MOSAIC signals vary differently
for different orders of the chirp and their magni-
tude. These, in turn, can be used to derive quan-
titative information on various orders of chirp.
The results are illustrated from typical examples of
MOSAIC signal computed for laser pulse with
different combinations of chirps, and deriving theinformation back using simple iterative algorithm
for a known laser pulse shape.
2. Modified spectrum auto-interferometric correla-
tion technique
The modified spectrum auto-interferometric
correlation is essentially an interferometric auto-
correlation (IAC) in which the weight factors of
different spectral components are changed to en-
hance its sensitivity for the chirp. In a second-orderautocorrelator, an ultrashort pulse laser beam is
split into two beams of equal intensity, which are
overlapped in space and time in a suitable nonlinear
medium/detector. To record the second-order in-
terferometric autocorrelation, only one fringe
should fall on the two-photon based detector.
Output of the detector as a function of time delay
between the two pulses then represents the IACsignal.
We may first examine some characteristics of the
IAC signals by considering two identical laser pulses
with their electric field amplitude given by�EðtÞ ¼ Re½EðtÞ expð�ix0tÞ�, where EðtÞ ¼ ½IðtÞ�1=2exp½i/ðtÞ�, x0 is the central frequency, IðtÞ is the
pulse intensity envelope function, and /ðtÞ is the
phase function. The latter may be expressed as
/ðtÞ ¼ aðt=tpÞ2 þ bðt=tpÞ3 þ cðt=tpÞ4; ð1Þwhere a, b and c are the magnitudes of linear,
quadratic and cubic chirps, respectively, and tp is
the FWHM duration of the laser pulse. Theoverlap intensity Iðt; sÞ as a function of the time
delay s between the two pulses would be
½EðtÞ þ Eðt � sÞ�2 i.e.,
Iðt; sÞ ¼ E2ðtÞ þ E2ðt � sÞ þ 2EðtÞEðt � sÞ cosðx0sÞ:ð2Þ
The second-order autocorrelation function,
SIACðsÞ, is then represented byRI2ðt; sÞdt, along
with the normalizing conditionRI2ðtÞdt ¼ 1, as
A.K. Sharma et al. / Optics Communications 233 (2004) 431–437 433
SIACðsÞ ¼ 2þ 4
ZIðtÞIðt � sÞdt
þ 4
Z½IðtÞ þ Iðt � sÞ�I1=2ðtÞI1=2ðt � sÞ
� cosðx0sþ D/Þdt
þ 2
ZIðtÞIðt � sÞ cosð2x0sþ 2D/Þdt;
ð3Þ
where D/ ¼ /ðtÞ � /ðt � sÞ, is the phase differencebetween the two laser pulses.The above equation
may also be written in the form
SIACðsÞ ¼ 2þ 4G2ðsÞ þ 8F21ðsÞ cosx0s
þ 2F22ðsÞ cos 2x0s; ð4Þ
where
G2ðsÞ ¼Z
IðtÞIðt � sÞdt; ð5Þ
Normalised delay (τ/tp)-6 -4 -2 0 2 4 6
IAC
sign
al
0
4
8
12
16
Normalised delay (τ /tp)
-6 -4 -2 0 2 4 6
IAC
sign
al
0
4
8
12
16
Fig. 1. The IAC signals for no chirp (left-upper trace) and a linear c
signals are shown on the right-hand side. (A similar figure for differe
F21ðsÞ ¼ 1=2
Z½IðtÞ þ Iðt � sÞ�EðtÞE�ðt � sÞdt;
ð6Þand
F22ðsÞ ¼Z
E2ðtÞE�2ðt � sÞdt: ð7Þ
In Eq. (4), while the G2ðsÞ term, representing the
second-order intensity autocorrelation, containsinformation about the laser pulse duration, the
interference terms F21ðsÞ and F22ðsÞ are governed
by the laser pulse duration as well as the chirp
present in the pulse. However, the IAC signals are
not very sensitive to the magnitude and order of
the chirp. For instance, Fig. 1 shows the IAC
signals corresponding to no chirp condition and a
linear chirp of a ¼ 0:3 computed for a sech2 (t=tp)laser pulse. It is seen that the two signals are vi-
sually quite similar.
Normalised delay (τ/tp)-6 -4 -2 0 2 4 6
MO
SA
ICS
igna
l
2
4
6
8
10
Normalised delay (τ/tp)
-6 -4 -2 0 2 4 6
MO
SA
ICS
igna
l
2
4
6
8
10
hirp of a ¼ 0:3 (left-lower trace). The corresponding MOSAIC
nt values of the linear chirp is given by Hirayama et al. [15].)
434 A.K. Sharma et al. / Optics Communications 233 (2004) 431–437
It is seen from Eqs. (5) and (7) that for the no
chirp condition (i.e., EðtÞ ¼ E�ðtÞ), G2ðsÞ ¼ F22ðsÞ.This fact is exploited to enhance the chirp sensi-
tivity of the IAC signals. It may be realized from
Eq. (4) that the IAC signal has three frequency
components located at dc, x0 and 2x0. The chirpsensitivity can be increased [15] by eliminating the
x0 term, amplifying the 2x0 term by a factor of 2,
and retaining the dc terms in the IAC signal. This
spectrally modified auto-interferometric correla-
tion (MOSAIC) signal is then represented as
SMOSAICðsÞ ¼ 2þ 4G2ðsÞ þ 4F22ðsÞ cos 2x0s: ð8ÞIt is easily seen from Eq. (8) that for no chirp
condition the locus of the minima of the oscil-
lating interference function SMOSAICðsÞ is a
straight line at a dc level of 2. Presence of any
chirp in the laser pulse would result in a change
in the locus of the interference minima from this
straight line. Fig. 1 shows the MOSAIC signalscomputed for the no chirp condition and for a
linear chirp of a ¼ 0:3, for a sech2 (t=tp) pulse.
This is similar to Fig. 1 shown in work of Hi-
rayama et al. [15] for a linear chirp of 0.15 and
0.25 to illustrate the MOSAIC technique. A clear
difference in the locus of interference minima is
observed for the chirp case, which shows two
peaks symmetrically located about the zero timedelay. Occurrence of such peaks, henceforth re-
ferred to as MOSAIC peaks, facilitates visual
detection of the chirp.
Normalized mean chirp
0.0 0.2 0.4 0.6 0.8 1.0
Nor
mal
ised
ampl
itude
ofM
OS
AIC
peak
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Linear chirp
Cubic chirp
Quadratic chirp
Fig. 2. Dependence of the normalized amplitude of MOSAIC
peak on the normalized mean chirp for linear, quadratic and
cubic chirps. (This figure has been adapted from the work of
Hirayama et al. [15].)
3. Dependence of the MOSAIC signal on the chirp
parameters
Eqs. (4)–(7) reveal that the peak amplitudes of
the loci of interference maxima and minima of the
MOSAIC signals will depend on the magnitude
and order of the chirp. Moreover, the two loci
have a common dc background of 2. We therefore
define normalized peak amplitude of the MOSAIC
signal as the ratio of peak amplitude of the locus
of minima to that of maxima, after subtracting thedc background. In general, this normalized peak
amplitude increases with increasing magnitude of
the chirp. Further, in order to facilitate compari-
son of the results for different orders of the chirp,
we use normalized mean chirp (NMC) parameter
[15] (weighted over the laser pulse shape) as
NMC ¼Z
IðtÞ t0d/dt
��������dt
ZIðtÞdt
�: ð9Þ
The normalized peak amplitude of the MOSAIC
signal was computed for different values of NMCfor linear, quadratic and cubic chirps for a sech2
(t=tp) pulse. Fig. 2 shows the dependence of the
normalized peak amplitude on the NMC in the
range from 0 to 1 rad. This figure has been adapted
from the work of Hirayama et al. [15] to facilitate
extension of MOSAIC technique to determine
various order chirps. It is seen that the MOSAIC
peak varies differently for different orders of chirp.It may be observed that a small amplitude of the
MOSAIC peak does not necessarily mean a neg-
ligible chirp because higher order chirps with large
values of NMC may be present. Further, it is no-
ted from Fig. 2 that a given peak amplitude of the
MOSAIC signal can occur for several different
combinations of various order chirps. For illus-
tration, Fig. 3 shows the loci of interference min-ima of the MOSAIC signal for two different chirp
conditions viz. (a) quadratic chirp of b � 0:17(NMC¼ 0.43) and (b) cubic chirp of c � 0:05(NMC¼ 0.28). The normalized peak amplitudes of
the MOSAIC signals in the two cases are same
(equal to 0.05). The same situation also occurs for
Normalised mean chirp0.2 0.4 0.6 0.8 1.0
Tem
pora
lasy
met
rypa
ram
eter
(∆t
2/∆t
1)
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Cubic chirp
Linear chirp
Quadratic chirp
Fig. 5. The temporal asymmetry parameter of the MOSAIC
signal for different values of the normalized mean chirp for
linear, quadratic, and cubic chirps.
Temporal delay (τ/tp)
-5 -4 -3 -2 -1 0 1 2 3 4 5Nor
mal
ised
ampl
itude
ofin
terf
eren
cem
inim
a
0.00
0.02
0.04
0.06
0.08
Quadratic chirp[β γ~ 0.17 (NMC ~ 0.43)]
Cubic chirp [ ~ 0.05 (NMC ~ 0.28)]
∆t1 ∆t2
Fig. 3. Loci of the interference minima of the MOSAIC signal
for two different chirp conditions are shown: (a) quadratic chirp
b � 0:17, NMC¼ 0.43 (solid line); (b) cubic chirp c � 0:05,
NMC¼ 0.28 (dotted line). The amplitudes of the MOSAIC
peaks are equal in the two cases.
Normalised mean chirp0.0 0.2 0.4 0.6 0.8 1.0
Tem
pora
lloc
atio
nof
MO
SA
ICP
eak
[ τ/t p]
1.0
1.5
2.0
2.5
3.0
Linear chirp
Quadratic chirp
Cubic chirp
Fig. 4. Variation of the temporal location of the MOSAIC
peak with normalized mean chirp for linear, quadratic, and
cubic chirps.
A.K. Sharma et al. / Optics Communications 233 (2004) 431–437 435
a linear chirp of a � 0:17 (NMC¼ 0.24). Thus a
knowledge of the normalized peak amplitude of
the MOSAIC signal alone can not predict the
magnitude and order of the different chirps.
Next, we examine the dependence of temporalshape of the MOSAIC peak on the chirp param-
eters. For instance, one may note from Fig. 3 that
the temporal location of the MOSAIC peak is
different for the two chirp conditions shown. It is
also observed from Fig. 3 that the MOSAIC sig-
nals are asymmetric with respect to the temporal
location of the peak. This asymmetry is more ev-
ident for the case of quadratic chirp. The temporalfeatures of the MOSAIC peak may be broadly
expressed in terms of: (1) the location of the peak
(2) the FWHM duration and (3) the asymmetry
parameter. The last one may be defined as ratio of
the two half widths at half maximum i.e., Dt2=Dt1as depicted in Fig. 3. Dependence of the temporal
location of the peak on the normalized mean chirp
computed for linear, quadratic and cubic chirpsfor sech2 (t=tp) pulse is shown in Fig. 4. It is seen
that, for all the three cases, the peak time decreases
with increase in magnitude of the chirp. Further, it
is observed from Fig. 4 that for cubic chirp the
peak of MOSAIC signal occurs at a much larger
time delay as compared to that for linear and
quadratic chirps.
Figs. 5 and 6 show the temporal asymmetry
parameter (Dt2=Dt1) and the FWHM duration of
the MOSAIC signal, respectively, as a function of
the normalized mean chirp for linear, quadratic
and cubic chirp. It is interesting to note that in
presence of cubic chirp, the MOSAIC signal re-mains almost symmetric with respect to its peak in
contrast to the asymmetric profiles for linear and
quadratic chirps. This may be useful as a visual
indicator of the presence of cubic chirp. Next, the
FWHM duration of the MOSAIC signal (Fig. 6)
decreases with increasing magnitude of the chirp.
This decrease is more steep for linear chirp as
Normalised mean chirp0.0 0.2 0.4 0.6 0.8 1.0F
WH
Mdu
ratio
n[( ∆
t 1+∆t
2)/t
p]of
MO
SA
ICpe
ak
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
Quadratic chirp
Linear chirpCubic chirp
Fig. 6. FWHM duration of the MOSAIC signal for different
values of the normalized mean chirp for linear, quadratic and
cubic chirps.
436 A.K. Sharma et al. / Optics Communications 233 (2004) 431–437
compared to that for the quadratic and cubic
chirps.
4. Determination of the chirp parameters from the
MOSAIC signals
Different dependences of the peak amplitude
and temporal characteristics of the MOSAIC sig-
nal on the magnitude and order of the chirp pre-
sented above may be used for quantitativeestimation of the chirp parameters. In many
practical situations such as chirp pulse amplifica-
tion based laser systems, the linear chirp can be
easily eliminated/minimized. Detection of any
quadratic and cubic chirp from the MOSAIC sig-
nals in this case becomes much easier because for
an observed amplitude of the MOSAIC peak, the
temporal location (Fig. 4) and the asymmetry pa-rameter (Fig. 5) are quite different for the two
cases. In general, various order chirp components
may be simultaneously present in the laser pulse so
that the amplitude and temporal characteristic of
the MOSAIC peak may not conform to any par-
ticular order. Here the chirp parameters can be
determined by matching the observed MOSAIC
signals with the computed ones through an itera-tive procedure for various combinations of differ-
ent order phase terms involving a, b and c.The task of iterative fitting of MOSAIC signals
gets much simplified by putting conditions of
minimum and maximum on the values of NMC.
The latter are identified from the observed values
of peak amplitude and temporal location of the
MOSAIC peak. From Fig. 2, it is clear that the
amplitude of the MOSAIC peak increases with
increase in the NMC for all chirp orders. This putsa lower limiting value on the normalized mean
chirp, NMCmin, which is the smallest of the NMC
values corresponding to the observed value of the
normalized peak amplitude for different chirp or-
ders. Similarly, Fig. 4 shows a decreasing behavior
for the time of occurrence of the MOSAIC peak
with increasing NMC for all chirp orders. This, in
turn, puts an upper limiting value on the normal-ized mean chirp, NMCmax, which is the largest of
the NMC values corresponding to the observed
value of the time of occurrence of the MOSAIC
peak for different chirp orders. The actual NMC
for the given MOSAIC signal should lie between
NMCmin and NMCmax.
A simple computer program has been written to
find out the values of the chirp parameters a, b andc from the given IAC signal. Firstly, the observed
IAC signal is converted into MOSAIC signal by
carrying out the following steps: (a) the IAC signal
is decomposed into its Fourier components using
fast Fourier transform algorithm, (b) the 2x0
component is multiplied by a factor of 2 and the
same is added to the dc term, and finally (c) by
using inverse fast Fourier transform, the modifiedsignal in frequency domain is converted into time
domain to get the MOSAIC signal i.e., SMOSAICðsÞ.Next, the program generates various possible
combinations of a, b and c which have NMC in the
range NMCmin to NMCmax. In the present exercise,
the step size was taken to be Da ¼ 0:05, Db ¼ 0:02and Dc ¼ 0:01. The program then computes MO-
SAIC signals for a sech2 (t=tp) laser pulse andcompares them with the given MOSAIC signal.
The computed MOSAIC signal with minimum
value of rms error (with respect to the reference
MOSAIC signal) was identified, and the corre-
sponding values of a, b and c give the chirp pa-
rameters. This procedure was tested on a number
of pre-calculated MOSAIC signals for different
combinations of a, b and c. For example a MO-SAIC signal was computed for a ¼ 0:2(NMC� 0.28), b ¼ 0:06 (NMC� 0.15), and
A.K. Sharma et al. / Optics Communications 233 (2004) 431–437 437
c ¼ 0:04 (NMC� 0.22). It had a normalized peak
amplitude of 0.10, temporal location of the peak
s=tp ¼ �1:6, and asymmetry parameter Dt2=Dt1 ¼1:19 (Dt2 ¼ 1:00tp and Dt1 ¼ 0:84tp). The lower andupper limiting values of NMC for the above am-
plitude and temporal location of the peak wereNMCmin ¼ 0.39 and NMCmax ¼ 0.91. It was possi-
ble to get back the original values of a, b and cwithin the step accuracy. Moreover, this determi-
nation was not sensitive to a random noise of �5%
imposed on the MOSAIC signal.
In conclusion, we have presented a simple
method of quantitative estimation of linear and
higher order chirps based on analysis of themodified spectrum auto-interferometric correla-
tion (MOSAIC) signals. It is shown that the peak
amplitude, the temporal location of the peak, the
FWHM duration, and the asymmetry of the locus
of the interference minima exhibit different de-
pendences on magnitude of the chirp and its order.
These, in turn, facilitate diagnostic of the chirp
parameters of ultrashort laser pulses through aFourier transform based computer algorithm. The
method is particularly suited to determine small
values of higher order chirps in ultrashort laser
pulses.
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