We show that a reduction in the pulse distortion caused by chromatic dispersion can be achieved through pulse shaping. We argue that a simple binary phase mask in the Fourier plane of the laser spectrum can improve the transmission of short pulses in a dispersive channel through reduced broadening. The argument was tested experimentally, and a good agreement was found with the theory

Al A. Salvatore

Amnon Yariv

Danny Eliyahu

Jean-jacques Drolet

Joseph Rosen

Randal A. Salvatore

English

lity of pulses that do not disperse by virtue of their unique shaping was introduced. Based on the time -- space analogy Rosen et al.  9  proposed   usT,0d-u Cs T,0d-    2ipb 2 f    a      3 exp    i  Tt 0  b 2 f    dt 0 (2) as the initial pulse envelope to reduce the effect of GVD. a, b, f , and p . 2 are real positive parameters of the pulse. This pulse su - u C , Dv - d propagates along a dispersive channel at a distance of Dz . 2 p 2pb 2 f    pf4s11pdg  s22pd/p  yb    while maintaining a nearconstant peak value of    C s0, zdj.  The spectrum of uC is  U sv,0d-U Cs v,0d-cosfsT b vd  4  2 sT a vd    g , (3) where T b - b 2 fyb, T a - b 2 fya, and the power p - 4 has been chosen as an example.  In what follows we describe an experimental investigation of the possibility of creating dispersion-resistant pulses through pulse shaping and show that it can be accomplished by use of a simple binary phase mask in the spectral plane. We find that using a binary approximation of the initial p

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Retardation and Reduction of Pulse Distortion by Group-Velocity Dispersion through Pulse Shaping

Eliyahu, Danny

Salvatore, Randal A.

Rosen, Joseph

Yariv, Amnon

Drolet, Jean-Jacques

Caltech Authors - Main

1412 OPTICS LETTERS / Vol. 20, No. 12 / June 15, 1995Retardation and reduction of pulse distortion by
group-velocity dispersion through pulse shaping
Danny Eliyahu, Randal A. Salvatore, Joseph Rosen, Amnon Yariv, and Jean-Jacques Drolet
California Institute of Technology, M/S 128-95, Pasadena, California 91125
Received February 15, 1995
We show that a reduction in the pulse distortion caused by chromatic dispersion can be achieved through pulse
shaping. We argue that a simple binary phase mask in the Fourier plane of the laser spectrum can improve
the transmission of short pulses in a dispersive channel through reduced broadening. The argument was
tested experimentally, and a good agreement was found with the theory.Optically transmitted pulses in a single-mode channel
can be severely distorted and broadened as the result
of chromatic dispersion. Several methods have been
used to overcome the distortion of the pulse envelope
caused by group-velocity dispersion (GVD): a grat-
ing pair,1 a Fabry–Perot interferometer,2 a phase con-
jugator,3 a laser modulated with a periodic injection
current sweep,4,5 and time lenses for Gaussian-shaped
pulses.6 In soliton propagation self-phase modulation
compensates the GVD of a hyperbolic secant pulse
shape.7
With z denoted as the position along the dispersive
channel, yg as the group velocity, and T ­ t 2 zyyg as
the delayed time, the inf luence of GVD on the complex
temporal pulse electric field usT , zd envelope is given
by8
usT , zd ­
Z Dv/2
2Dv/2
U sv, z ­ 0dexp
µ
ib2zv2
2
2 ivT
¶
dv, (1)
where U sv, 0d is the Fourier transform of usT , 0d, v
is the angular frequency (relative to the laser’s center
frequency v0), Dv is the spectral width corresponding
to the pulse, and b2 ­ ›2by›v2jv0 is the GVD per unit
length.
Recently9,10 the possibility of pulses that do not dis-
perse by virtue of their unique shaping was introduced.
Based on the time–space analogy Rosen et al.9 pro-
posed
usT , 0d ­ uC sT , 0d ­
1p
2ipb2f
Z ‘
2‘
cos
•
t0
b
p
2
µ
t0
a
¶2‚
3 exp
µ
i
Tt0
b2f
¶
dt0 (2)
as the initial pulse envelope to reduce the effect of
GVD. a, b, f , and p . 2 are real positive parameters
of the pulse. This pulse su ­ uC , Dv ­ ‘d propagates
along a dispersive channel at a distance of Dz .
2
p
2p b2f 2pf4s1 1 pdgs22pd/pyb2 while maintaining a near-
constant peak value of juC s0, zdj.
The spectrum of uC is
U sv, 0d ­ UC sv, 0d ­ cosfsTbvd4 2 sTavd2g , (3)
where Tb ­ b2fyb, Ta ­ b2fya, and the power p ­ 4 has
been chosen as an example.0146-9592/95/121412-03$6.00/0In what follows we describe an experimental investi-
gation of the possibility of creating dispersion-resistant
pulses through pulse shaping and show that it can be
accomplished by use of a simple binary phase mask
in the spectral plane. We find that using a binary
approximation of the initial pulse spectrum [Eq. (3)],
UB sv, 0d ­ UCsv, 0dyjUC sv, 0dj, still leads to a consid-
erable reduction in pulse distortion. This approxima-
tion enables us to use a binary (0 or p) phase mask
instead of a complicated amplitude and phase filter.
The pulse field UB and its capability of maintaining
its form without distortion when traveling through a
dispersive channel depend on the ratio TbyTa. We find
numerically that TbyTa ­
p
8y5 is sufficient in that the
intensity IB sT ­ 0, zd ­ juB s0, zdj2 of the center of the
symmetric pulse and its shape remains at a nearly
constant height over the interval Dz . 8T 2b yb2 near
z ­ 0. One may increase this interval by increasing
the ratio TbyTa, yet increasing the interval spoils
the constancy of IB s0, zd in the region of interest.
At z ­ 0 the pulse has a main lobe width around
T ­ 0 of order Tb and a tail length that depends on
TbDv. Although reducing TbDv reduces the pulse tail
length, it increases the tail intensity compared with
IB s0, 0d and introduces small oscillations in IB s0, zd as
a function of z.
In Fig. 1 the pulse intensity shape I sT , zd nor-
malized by I s0, 0d is shown as a function of the
delayed time T for different values of GVD, i.e.,
the normalized distance b2zy2T2b . Figures 1(a)–
1(d) describe IB sT , zdyIB s0, 0d with a binary phase
mask, U sv, 0d ­ UBsv, 0d and TbDv ­ 4.044, using
the first nine lobes of the cosine spectrum function
of Eq. (3). The pulse maintains its basic form and
height even for b2zy2T2b ­ 1.6, as can be seen from
comparison among Figs. 1(a)–1(c). In these cases
the main lobe of the pulse envelope as well as IB s0, zd
is essentially invariant. The changes that are due
to chromatic dispersion are mostly in the tails. The
reduction of IB s0, zd for b2zy2T2b ­ 0.8 [Fig. 1(b)] is a
result of finite TbDv, as mentioned above. Note that
this result is independent of the sign of b2 because
IB s0, 2zd ­ IB s0, zd. Therefore, in practice one can
exploit this fact and shape the pulse uB sT , , 24T 2b yb2d
at the beginning of the dispersive channel and use the
full interval of 8T 2b yb2. 1995 Optical Society of America
June 15, 1995 / Vol. 20, No. 12 / OPTICS LETTERS 1413Fig. 1. Pulse intensity I sT , zd normalized by I s0, 0d
versus TyTb for different values of b2zy2T2b . (a)–(d)
U sv, 0d ­ UB sv, 0d and TbDv ­ 4.044. (e)–(h) The solid
curves result from USsv, 0d ­ 1 and TbDv ­ 2.156 having
the same pulse main lobe width at z ­ 0; the dotted curves
represent a Gaussian pulse with the same pulse width at
z ­ 0. The GVD is equal to zero in (a) and (e), whereas
for (b) and (f ) b2zy2T2b ­ 0.8, for (c) and (g) b2zy2T
2
b ­ 1.6,
and for (d) and (h) b2zy2T2b ­ 2.2.
These results are compared with those from a rect-
angular spectrum without the phase mask. For this
comparison [the solid curves in Figs. 1(e)–1(h)] we
choose U sv, 0d ­ USsv, 0d ­ 1 and spectral width Dvs ­
2.156yTb to generate the same pulse main lobe width as
with the mask. The pulse shape is IS sT , 0dyIS s0, 0d ­
sinc2sDvST d, where sincsxd ­ sinsxdyx. The primary
difference between the pulse’s UB [Eq. (2)] and US
is in the tails. The second pulse has shorter tails
and therefore suffers more distortion and reduction of
I s0, zd compared with I s0, 0d along its propagation in
the dispersive channel.
Results for the familiar Gaussian pulse [the dotted
curves in Figs. 1(e)–1(h)] are also compared. This
pulse has just one lobe and suffers a reduction of
IG s0, zd even at distances shorter than those of US . Its
complete dependence on T and z can be written in
closed form and is given by
IG sT , zd
IG s0, 0d
­
exp
‰
2
T 2
T 2G f1 1 sb2zyT
2
G d2g
¾
f1 1 sb2zyT 2Gd2g1/2
. (4)
We chose TG ­ 1.44Tb so that the main lobe width
is identical to that of the two cases described above
at z ­ 0. From Figs. 1(c) and 1(g) one can see that
although the Gaussian pulse decays to 54% of its initial
intensity [Fig. 1(g)], the pulse with the UB spectrum
maintains its value.The above theoretical results were tested experi-
mentally for the cases of UB and the rectangular
spectrum US . The experimental setup is shown in
Fig. 2. The laser is a passively mode-locked quad-
ruple quantum-well two-section, GaAs buried het-
rostructure as described previously.11,12 Its spectrum
is between 839.9 and 845.2 nm. The pulse spectrum
is approximated by a rectangular function with a
quadratic phase.11,12 The laser pulses were sent to
a telescoped dual-grating compressor (1yd ­ 2000
linesymm) to achieve pulse shaping,13 laser chirp
cancellation, and GVD addition. Different optical
frequency components of the input laser pulse are
spatially dispersed by the grating at the left and then
resolved spatially in the Fourier plane of the lens. At
this plane the spectrum is filtered through either the
spatial binary phase mask or a window to reshape
the pulse. The spectrum is recombined spatially by a
second lens and grating. We easily achieve laser chirp
cancellation and GVD addition by moving the grating
at the right in Fig. 2. We cancel the laser chirp by
simply moving the grating to minimize the pulse
width (without the phase mask or the window) and
specifying the position of the grating as z ­ 0. Then
the phase mask is introduced in the Fourier plane and
autocorrelation measurements are taken for different
positions z of the right-hand grating. The GVD is
given by14
b2z
2T2b
­
l30z
4pT 2b c2d2 cos2sud
, (5)
where z is the distance of the right-hand grating loca-
tion along the central axis of the dual-grating compres-
sion system. Also, u ­ 51.1– is the angle between this
axis and the normal to the grating surface, l0 ­ 2pcyv0
is the optical carrier wavelength, and c is the velocity
of light.
The reshaped pulses are sent into a Michelson
interferometer and a second-harmonic generation
crystal to measure their temporal autocorrelation.
Autocorrelations were taken for the case when
U sv, 0d ­ UB sv, 0d and TbDv ­ 4.044. We find nu-
merically that the pulse shape is insensitive to small
variations in the amplitude distribution of the spec-
trum. Therefore the assumption of a rectangular
function between 2Dvy2 and Dvy2 for the laser
amplitude spectrum is justified, and the need for
Fig. 2. Fourier-transform pulse shaping, compression,
and fiber GVD simulated apparatus. Optical frequency
components of the input laser pulse are spatially dispersed
and filtered through the spatial binary phase mask to
reshape the pulse and then spatially recombined. The
left-hand grating is placed in the focal point of the lens,
whereas the right-hand grating is movable to achieve
compression and different chirp values for GVD.
1414 OPTICS LETTERS / Vol. 20, No. 12 / June 15, 1995Fig. 3. Comparison between experimental (dotted curves)
and theoretical (solid curves) autocorrelation results.
(a)–(c) U sv, 0d ­ UB sv, 0d and TbDv ­ 4.044; (d)–(f )
US sv, 0d ­ 1 and TbDv ­ 2.156. The measurements were
taken for different GVD’s: (a), (d) b2zy2T2b ­ 0; (b), (e)
b2zy2T2b ­ 1.1; (c), (f) b2zy2T
2
b ­ 2.2.
a complicated mask is reduced to that for a simple
binary phase mask. The mask, made by photo-
lithography,15 has a width of 3.4 mm (the spatial
width of the optical spectrum in the Fourier plane),
which is equivalent to Dv ­ 13.3 3 1012 radys. The
mask was encircled by a window of the same width
to eliminate undesirable inf luences from the tails
of the spectrum. The index of refraction of the
glass mask is n ­ 1.5; therefore the depth of the
etching is l0. The autocorrelations measured while
the phase mask is used are given in Figs. 3(a)–
3(c). Two parameters (Tb and b2) describe the fit
between the theoretical (solid curves) and experi-
mental (dotted curves) autocorrelation results. First,
for z ­ 0 [Fig. 3(a)] the parameter Tb was found
to be equal to Tb ­ 370 fs, whereas the theoretical
calculation gives Tb ­ 4.044yDv . 340 fs). A second
measurement [Fig. 3(b)] was taken at b2zy2T 2b ­ 1.1
(z ­ 2.5 cm) to specify the second parameter b2
(theoretical calculation gives b2zy2T2b ­ 1.16 at this
location). A third measurement [Fig. 3(c)] was taken
for b2zy2T2b ­ 2.2. A good fit is indicated between
theory and experimental results.
A similar series of autocorrelations was measured
without the phase mask, in which US sv, 0d ­ 1 and
TbDv ­ 2.156, to check the rectangular spectrum case.
In this case a window was located in the Fourier
plane, allowing the center of the laser spectrum to pass
and blocking the rest of the spectrum. Its width was
adjusted to give the same autocorrelation half-width,
which is nearly equivalent to getting the same main
lobe pulse width as with the phase mask. The allowed
spectrum had Dv . 6.62 3 1012 radys. The results ofthis experiment are shown in Figs. 3(d)–3(f). The
fit was done without any free parameters. The same
values of Tb and b2zy2T2b as before were used. A
good agreement between theory and the experimental
results was found again.
Two differences are prominent from the different
pulses used to generate Figs. 3(a)–3(f). First, the
autocorrelation tails in Figs. 3(a)–3(c) for the pseudo-
nondispersive pulse sUB d decay slowly, reaching the
background value of 1 only for very large t. This
is a direct result of the pulse tail, as can be seen
from Figs. 1(a)–1(d). The US pulse, which has a much
shorter tail, has autocorrelations that reach the value
of 1 [Figs. 3(d)–3(f)] for smaller t. Second, for higher
values of b2zy2T2b the autocorrelation has the form of
a cascade. This occurs when the main lobe of the
pulse [Fig. 1(d)] becomes comparable in magnitude
with the secondary lobes on each side. The width of
the autocorrelation for small tyTb indicates the width
of the main peak of the pulse.
In conclusion, the possibility of pulse shaping to
reduce the distortion caused by GVD was theoretically
considered and checked experimentally. It was found
that a simple phase mask in the spectral plane can
add tails with many lobes, which considerably alleviate
distortion to the main lobe of the pulse for longer
propagation distances in dispersive channels than for
other pulses.
This research was supported by the National Sci-
ence Foundation and the Advanced Research Projects
Agency.
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Retardation and reduction of pulse distortion by group-velocity dispersion through pulse shaping

One-dimensional beam shaping,’’

Optical Electronics,

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Retardation and reduction of pulse distortion by group-velocity dispersion through pulse shaping

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