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Complete characterization of optical pulses by real-time
spectral interferometry
Naum K. Berger, Boris Levit, Vladimir Smulakovsky, and Baruch Fischer
We demonstrate a simple method for complete characterization (of amplitudes and phases) of short optical
pulses, using only a dispersive delay line and an oscilloscope. The technique is based on using a dispersive
delay line to stretch the pulses and recording the temporal interference of two delayed replicas of the pulse
train. Then, by transforming the time domain interference measurements to spectral interferometry, the
spectral intensity and phase of the input pulses are reconstructed, using a Fourier-transform algorithm. In
the experimental demonstration, mode-locked fiber laser pulses with durations of ⬃1 ps were characterized
with a conventional fast photodetector and an oscilloscope. © 2005 Optical Society of America
OCIS codes: 320.7100, 320.5550, 140.3510, 070.4790, 120.3180, 100.5070.
1. Introduction
As optical pulses are becoming shorter and have wide
use for many basic and applied purposes, there is a
strong need for simple and quick measurement tech-
niques. Indeed, a variety of methods were developed
throughout the years for what is called complete
characterization, for finding the amplitudes and the
phases of optical pulses. A widely used pulse measur-
ing method is frequency-resolved optical gating (see,
for instance, the reviews in Refs. 1 and 2), which
belongs to the class of nonlinear methods. Other
techniques that include linear interferometric mea-
surements in the spectral,
3–5
spectral–temporal,
6,7
time,
8,9
and spatial–spectral
10,11
domains or spectral
filtering
12
provide simple and direct (i.e., nonitera-
tive) processing of the results and much higher sen-
sitivity. Nonetheless, most of those linear techniques
have nonlinear ingredients, such as cross-correlation
recording,
6,8
nonlinear frequency shear in spectral
phase interferometry for direct electric-field recon-
struction,
5
or frequency-resolved optical gating for
characterizing a reference pulse.
4
Additionally, they
require special equipment, for instance, for tunable
filtering of the pulse harmonics.
12
In the present paper we propose a novel method for
complete characterization of optical pulses that is
entirely linear and simple to implement. This method
is similar to spectral shearing interferometry,
5
but
the interference is formed in the time domain and
then translated to the spectral domain, owing to the
linear relation between the patterns in the temporal
and the frequency domains. We call this method real-
time spectral interferometry. Because the pulse char-
acterization is performed in the spectral domain,
there is no need of high temporal resolution, and thus
we were able to characterize fiber laser pulses of
⬃1 ps by using a conventional fast photodetector and
an oscilloscope.
2. Description of the Method
We start with stretching of the optical pulses to be
characterized by a dispersive delay line. First, let us
assume that this dispersive element has a quadratic
phase response and choose line length L to meet the
condition L ⬎⬎
2
兾共8
2
兲, where is the original pulse
width and 
2
is the group-velocity dispersion. Then
the output pulse’s shape is the temporal analog of the
spatial Fraunhofer diffraction
13
:
E
out
共
t
兲
⬀ exp
关
it
2
兾
共
2
2
L
兲
兴
F
共
t兾
2
L
兲
, (1)
where F共兲 ⫽ |F共兲|exp关i共兲兴 is the complex spec-
trum of the pulse to be measured and |F共兲|
2
and
共兲 are the spectral intensity and the spectral phase
of the input pulse, respectively.
According to relation (1), measured intensity |E
out
共t兲|
2
of the stretched pulse gives spectral intensity I共兲
The authors are with the Department of Electrical Engineering,
Technion—Israel Institute of Technology, Haifa 32000, Israel. B.
Fischer’s e-mail address is fischer@ee.technion.ac.il.
Received 5 May 2005; revised 17 August 2005; accepted 22 Au-
gust 2005.
0003-6935/05/367862-05$15.00/0
© 2005 Optical Society of America
7862 APPLIED OPTICS 兾 Vol. 44, No. 36 兾 20 December 2005
⫽ |F共兲|
2
of the input pulse after a substitution:
t ⫽
2
L, (2)
where is measured relative to the center of the
spectrum. Such real-time spectral analysis of optical
pulses was performed with an optical fiber
14
or a
chirped fiber Bragg grating
15,16
used as a dispersive
delay line. However, relation (1) also contains infor-
mation on the pulse’s spectral phase 共兲 that can be
extracted by conventional interferometric measure-
ments. We use for that extraction the method of
shearing interferometry, next described.
The stretched pulses are reflected from two sides of
a glass plate (see Fig. 1), and the interference be-
tween the two reflected replicas is measured by a
photodiode and a sampling oscilloscope. It is signifi-
cant here that the interferometry is performed in the
time domain and then transformed to spectral inter-
ferometry, according to time-to-frequency conversion
[Eq. (2)]. Then the nonlinear frequency shear of
Ref. 5 is replaced in our method by linear operation of
the temporal delay between the two reflected replicas
of the pulses.
The interference intensity is given by
I
int
共
t
兲
⫽ I
共
兲
⫹ I
共
⫹⌬
兲
⫹ 2
关
I
共
兲
I
共
⫹⌬
兲
兴
1兾2
cos
关
⌬t ⫹⌬
共
兲
兴
, (3)
where
⌬
共
兲
⫽
共
⫹⌬
兲
⫺
共
兲
⬇
共
d兾d
兲
⌬ (4)
and and frequency shear ⌬ are related to t and to
time delay ⌬t in the glass plate by Eq. (2). The spec-
tral phase can be obtained from relation (4):
共
兲
⫽
共
1兾⌬
兲
冕
⌬
共
兲
d. (5)
Spectral intensity I共兲 and spectral phase 共兲 can be
obtained from the measured spectral interferogram
by use of relations (3)–(5), as shown below. Then by a
Fourier transform we obtain the reconstructed time-
dependent quantities, I共t兲 and 共t兲, of the original
pulse.
It is important to emphasize the fundamental dif-
ference between our method and that presented in Ref.
9. There too the pulse stretching in a dispersive delay
line is used. However, the stretched pulses are first
completely characterized there in the time domain,
whereas in our method the original pulses are com-
pletely characterized in the spectral domain.
5
To char-
acterize the pulse completely in the time domain, one
should sample it, according to the sampling theorem,
with temporal resolution ␦t
t
⬇ 1兾f
p
, where f
p
is the
spectral interval of the nonzero pulse energy. If the
temporal resolution of an oscilloscope is insufficient
for the measurement of the original pulses, stretch-
ing of the original pulses cannot improve this situa-
tion, because the original and stretched pulses have
the same energy spectrum and, therefore, the same
temporal resolution is required for their character-
ization. According to the sampling theorem, spectral
resolution ␦f, required for the pulse characterization
in the spectral domain (in our method), is equal to
␦f ⬇ 1兾
p
, where
p
is the temporal interval during
which the pulse has nonzero energy. Taking into ac-
count Eq. (2), we obtain the temporal resolution re-
quired for the real-time spectral interferometry:
␦t
s
⬇ 2
2
L兾
p
. (6)
It is shown below that the needed temporal resolution
is readily provided by a conventional fast oscilloscope.
3. Experimental Results
We used an erbium-doped fiber ring laser with pas-
sive mode locking for the optical pulse source. The
laser generated optical pulses with a repetition rate
of 10 MHz at a wavelength of 1530.2 nm. The disper-
sive delay line was a fiber with high dispersion, com-
monly used for dispersion compensation. Figure 2
shows the experimental temporal interferogram,
Fig. 1. Schematic representation of the measurement setup.
EDFA, erbium-doped fiber amplifier.
Fig. 2. Oscilloscope trace of the interference between two replicas
of the stretched pulse reflected from the two surfaces of a glass
plate. The time and frequency scales are related by Eq. (2).
20 December 2005 兾 Vol. 44, No. 36 兾 APPLIED OPTICS 7863
measured with a photodiode and an oscilloscope (both
with a bandwidth of 50 GHz. The frequency scale,
calculated according to Eq. (2), is also shown in Fig. 2.
We used a phase retrieval procedure similar to that
described in Ref. 17. First, the Fourier transform of the
interference pattern was calculated, as shown in Fig. 3.
Then the pattern of the Fourier transform was shifted
to the left by an amount ⌬兾2. This corresponds to
eliminating the linear component ⌬t of the phase
difference in Eq. (3). The central and left sidebands
were filtered out, and the remaining sideband was
inverse Fourier transformed. The absolute value and
argument of the signal obtained give, respectively,
spectral intensity I共兲 [we neglect ⌬ in I共⫹⌬兲]
and phase difference ⌬共兲. Spectral phase 共兲 was
calculated by integration, according to Eq. (5).
In reality, the spectral phase response of a disper-
sive delay line such as an optical fiber is not exactly
quadratic. However, we took this deviation into ac-
count in our calculation. We considered that the real-
time spectral analysis is accomplished in this case
only by the quadratic component of the phase re-
sponse for the pulse that has the so-called distorted
complex spectrum |F共兲|exp关i共兲 ⫹ i
nq
共兲兴, where
nq
共兲 is the nonquadratic contribution of phase re-
sponse 共兲 (the linear part can be neglected). For
this case the temporal Fraunhofer condition has to be
met for this distorted pulse. Component
nq
共兲 was
measured and subtracted from the spectral phase
obtained by the procedure described above.
The measurement of phase response 共兲 of the
dispersive delay line was done in the same manner as
for the pulse’s spectral phase. For that purpose, we
placed an additional dispersive delay line before the
former line. Then 共兲 is given by the difference of the
spectral phases obtained in the two measurements,
one with both dispersive elements and the second
with the additional line only. From the fitting of the
experimental data we obtained 共兲⬇⫺4.60 ⫻
10
⫺23
2
⫹ 7.16 ⫻ 10
⫺38
3
⫺ 2.066 ⫻ 10
⫺51
4
. Note
that dispersion and dispersion slopes were measured
by a similar method by Dorrer.
18
The calibration of time delay ⌬t in the glass plate
was made by use of spectral interferometry with a
broadband light source (amplified spontaneous emis-
sion from an erbium-doped fiber amplifier). The light
reflected from two sides of the glass plate was ana-
lyzed by an optical spectrum analyzer with a resolu-
tion of 0.015 nm. The Fourier transform of the
spectral interferogram was calculated as a function of
frequency. The position of the sideband peak on the
time axis in the Fourier transform corresponds to
time delay ⌬t. The measured value of ⌬t was
12.51 ps.
The measured spectral intensity I共兲 (solid curve)
and spectral phase 共兲 (dashed curve) of the original
laser pulse are shown in Fig. 4. The relation between
the frequency and the time scales in this figure is
given by Eq. (2). Figure 5 shows the intensity (solid
curve) and the phase (dashed curve) of the fiber laser
pulse calculated by the Fourier transform of the pulse
spectrum shown in Fig. 4. The pulse width is 1.3 ps.
To test our method we measured the spectral phase
response of a 9 m long standard single-mode fiber and
compared with a direct group-delay measurement
that needs a much longer fiber, for which we used
Fig. 3. Absolute value of the Fourier transform of the interference
pattern shown in Fig. 2.
Fig. 4. Spectral intensity I() (solid curve) and spectral phase
() (dashed curve) of the input pulse, reconstructed from Fig. 3.
The time and frequency scales are related by Eq. (2).
Fig. 5. Reconstructed temporal intensity I(t) (solid curve) and
phase (t) (dashed curve) of the fiber laser pulse.
7864 APPLIED OPTICS 兾 Vol. 44, No. 36 兾 20 December 2005
20 km of the same fiber. The results of the two mea-
surements are shown in Fig. 6. The average deviation
between them was 0.25 rad. We also measured the
autocorrelation of the laser pulse and compared it
with that calculated for the reconstructed pulse in-
tensity shown in Fig. 5. The results of this compari-
son are presented in Fig. 7. It can be seen that the
agreement between the two curves is excellent.
To estimate the required temporal resolution we use
Eq. (6). It should be taken into account that
p
in relation
(6) is the duration of the original pulse, distorted by
the nonquadratic component of the dispersion. We
estimate that in our experiments
p
⬇ 10 ps and the
spectral resolution is ␦f ⬇ 100 GHz, which corre-
sponds to a required temporal resolution of 58 ps. To
use the Fourier-transform algorithm we chose a spec-
tral shear of 21.6 GHz 共⬃3% of the pulse’s band-
width), which corresponds to a temporal resolution of
12.5 ps. Such accuracy is provided by a 50 GHz os-
cilloscope (the accuracy of the time base of our oscil-
loscope was 7 ps). For comparison, to characterize a
pulse of ⬃1 ps in the time domain, the temporal res-
olution of an oscilloscope should be less than 0.5 ps
(for Gaussian pulses).
It is clear that in our method the stretched pulses
should not overlap. This imposes a certain limitation
on the maximal repetition rate of the measured pulse
trains. Thus the pulse characterization is suitable for
fiber lasers with pulse repetition rates of tens or hun-
dreds of megahertz. However, it was shown in Ref. 19
that the interference between stretched overlapping
pulses can also be used for pulse characterization. In
this way the method in the present paper can be
expanded for characterization of high-repetition-rate
pulse trains.
4. Conclusions
We have demonstrated a novel method for optical
pulse characterization in which spectral interferom-
etry is performed in the time domain. The method is
simple and requires only the use of a dispersive delay
line and a conventional oscilloscope. The advantage
of the method is that the frequency shear that is used
in conventional spectral interferometry is replaced in
our method by a simple operation of the temporal
delay between two replicas of the stretched pulse
train. Our method does not require high resolution
for temporal measurements and still permits charac-
terization of ⬃1 ps pulses with a conventional fast
photodetector and an oscilloscope.
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