All-optical temporal differentiator based on a single
phase-shifted fiber Bragg grating
Naum K. Berger, Boris Levit and Baruch Fischer
Department of Electrical Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel.
Mykola Kulishov and David V. Plant
Department of Electrical and Computer Engineering, McGill University, Montréal, Québec H3A 2A7, Canada
Institut National de la Recherche Scientifique (INRS), Montréal, Québec H5A 1K6, Canada
Abstract: We propose and demonstrate an all-optical temporal differentiator based on a single π-phase-
shifted fiber Bragg grating operated in reflection. This device can efficiently process arbitrary optical
temporal waveforms with bandwidths up to a few gigahertz.
All-optical temporal differentiators are of general interest as basic building blocks in ultrahigh-speed optical signal
processing circuits in exact analogy with their electronic counterparts. More specifically, all-optical differentiators could be
used for optical pulse shaping, and optical control and sensing. As they can alleviate the bandwidth limitations typical of
electronic solutions, they are also attractive for realizing specific signal processing operations over ultra-wideband
microwave signals (by first transferring the microwave signal into the optical domain using e.g. electro-optic modulation).
Very recently, it has been shown theoretically  that a single uniform long-period fiber grating (LPG) can operate as a
temporal differentiator over optical signals with sub-picosecond temporal features. Based on this finding, LPG-based
optical differentiators capable of processing signals with temporal features as fast as ≈ 180 fs have been fabricated and
experimentally tested . The LPG-based solution is of specific interest for differentiating optical waveforms with
bandwidths > 1THz. However this previous solution is not suitable (e.g. it is very energetically inefficient) for processing
optical signals with bandwidths in the GHz regime. In this letter, we propose and experimentally demonstrate a new all-
optical differentiation approach, which is specially suited for operation on arbitrary optical signals with bandwidths up to a
few GHz. Specifically, we demonstrate that a single π phase shifted fiber Bragg grating (FBG) operated in reflection can
accurately calculate the first time derivative of the complex field (amplitude and phase) of a GHz-bandwidth input optical
The temporal operation of a first-order optical differentiator can be mathematically described as
and v(t) are the temporal optical waveforms (complex envelopes) at the input and at the output of the system, respectively,
and t is the time variable. In the spectral domain,
and v(t), respectively and ω is the base-band frequency (ω = ωopt - ω0, where ωopt is the optical frequency and ω0 is the
carrier frequency). Thus, first-order optical differentiation can be implemented using an optical filter with a transfer
function that varies linearly on frequency,
. A key feature of a first-order optical differentiator is that it must
introduce a π phase shift exactly at the signal’s central frequency, ω0. We have found out that these required spectral
characteristics are provided by a phase-shifted FBG consisting of two concatenated identical uniform FBGs with a π phase
shift between them. Specifically, this FBG device provides the required spectral features for optical differentiation within a
narrow bandwidth (up to a few GHz) around the transmitted resonance wavelength (reflection dip). In principle, in order to
achieve the required exact π phase shift at the signal’s central frequency the FBG structure must exhibit an ideal π phase
shift and the two concatenated uniform gratings must be identical (these features would ensure an exact null at the FBG
transmission resonance frequency). In practice, our numerical simulations show that a resonance deep > 30dB is sufficient
to ensure a deviation error < 2% in the differentiation process of an optical signal with a sufficiently narrow bandwidth).
3. FBG fabrication and characterization
A π phase-shifted Bragg grating was fabricated using the UV radiation of a cw frequency doubled Ar-ion laser with power
of about 160 mW. The UV beam was focused by a cylindrical lens trough a 1 mm slit and a phase mask (period of 1065
nm) onto the core of a hydrogen-loaded germano-silicate fiber. The written phase-shifted grating consists of two
consecutive uniform FBGs, each 1-mm long. A π-phase shift between these two uniform gratings was produced by
accurately shifting the mask relative to the fiber using a piezoelectric actuator following the photoinscription of the first
, where u(t)
, where U(ω) and V(ω) are the Fourier transform of u(t)
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grating. The spectral reflectivity of the fabricated FBG structure was measured by a tunable semiconductor laser (Ando Download full-text
AQ4321D) and an optical spectrum analyzer (Ando AQ6317B) with a wavelength resolution of 1 pm. The measured
spectrum around the grating reflection dip is shown in the inset of Fig.2(a). The measured depth of this reflection dip is -
34.5 dB, which should be sufficient for optical differentiation, according to our simulations. The 3-dB bandwidth of the dip
is 0.17 nm and we estimate that optical signals with bandwidths up to ≈ 5 GHz can be accurately differentiated with this
device (this is the bandwidth over which the reflection amplitude in the dip exhibits a linear variation). The grating spectral
phase was measured using the method reported in Ref.  and the result of this measurement is shown in the inset of Fig.
2(b). As anticipated, the measured π phase jump is abrupt enough to accurately implement optical differentiation.
4. Experimental results
The operation of the fabricated phase-shifted fiber Bragg grating as an optical differentiator was tested using the
experimental setup shown in Fig. 1. As an optical source, we used the same tunable semiconductor laser as for the grating
spectrum measurement. The CW radiation of the laser was sinusoidally modulated by a LiNbO3 electro-optic (EO) phase
modulator driven by an RF sinusoid from an RF synthesizer and RF amplifier. In the experiment presented here, the input
optical pulses were created by propagating the sinusoidally phase modulated light through a span of dispersion
compensating fiber (DCF) with a total dispersion of −4074 ps/nm. These pulses were reflected from the FBG-based optical
differentiator and directed by a circulator to a photodiode and an oscilloscope with a bandwidth of 50 GHz. Fig. 2(a) (solid
curve) shows the pulses formed after propagation through DCF dispersion for a RF modulation frequency fm = 1 GHz and a
modulation index in the EO modulator of A = 2.405 rad. For comparison, we also show in fig. 2(a) (dotted curve) the
numerically calculated pulses assuming the same parameters as in the experiment. The repetition rate of the generated
optical pulse train is fixed by the modulation frequency to 1GHz. The results of optical differentiation of the input pulses in
Fig. 2(a) after reflection in the phase-shifted FBG structure are shown in Fig. 2(b) (solid curves for experimental results and
dotted curves for an ideal differentiation process). The agreement between theory and experiments is remarkable (average
deviation between theoretical and experimental curves < 7%).
It should be noted that the demonstrated optical differentiator provides the derivative of the complex pulse field (including
amplitude and phase). In another set of experiments, the optical differentiator was used for phase-to-intensity conversion of
a sinusoidally phase modulated signal, demonstrating again an excellent agreement between theory and experiments.
A very simple and efficient all-optical (all-fiber) first-order temporal differentiation technique based on the use of a single
phase-shifted FBG has been proposed and experimentally demonstrated. This device calculates the first temporal derivative
of the complex envelope (amplitude and phase) of arbitrary optical waveforms with bandwidths up to a few gigahertz. This
development should prove very useful for all-optical information processing or for implementing specific signal processing
tasks over ultra-wideband microwave waveforms.
 M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators”, Opt. Lett. 30, 2700-2702 (2005).
 R. Slavik, M. Kulishov, Y. Park, J. Azaña, and R. Morandotti, “Temporal differentiation of sub-picosecond optical pulses using a single long-period
fiber gratings”, OSA - CLEO 2006, May 21-26, 2006, Long Beach, CA, USA, Paper CtuBB5, pp. 1-3.
 N. K. Berger, B. Levit, and B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251, 315-321
Fig. 1. Experimental setup
Intensity (arb. units)
T im e (ns)
Fig. 2. Experimental results (solid curves) and theoretical predictions (dotted
curves): Input (a) and output (b) signals in the FBG differentiator. The insets
show the measured FBG spectral characteristics.
T im e (n s )
1542.10 1542.15 1542.20 1542.25