It is proposed that the process of nonlinear optical phase conjugation can be utilized to compensate for channel dispersion and hence to correct for temporal pulse broadening. Specifically, a four-wave nonlinear interaction is shown to achieve pulse renarrowing. Spectral bandwidth constraints of the input pulse are presented for typical phase-conjugate interaction parameters

Fekete, Dan

Pepper, David M.

Yariv, Amnon

English

Caltech Authors - Main

52 OPTICS LETTERS / Vol. 4, No. 2 / February 1979
Compensation for channel dispersion by nonlinear
optical phase conjugation
Amnon Yariv, Dan Fekete, and David M. Pepper
California Institute of Technology, Pasadena, California 91125
Received October 2, 1978
It is proposed that the process of nonlinear optical phase conjugation can be utilized to compensate for channel dis-
persion and hence to correct for temporal pulse broadening. Specifically, a four-wave nonlinear interaction is
shown to achieve pulse renarrowing. Spectral bandwidth
phase-conjugate interaction parameters.
Recent theoretically and experimental"5'7-l0 studies
indicate that the generation of phase-conjugate replicas
of incident optical waves has certain formal aspects of
time reversal and that this property can be used to
correct for spatial-propagation distortion. In addition,
chirp compensation using four-wave mixing has been
analyzed."1
In this Letter we explore theoretically what happens
following conjugation to a short electromagnetic pulse
that has traversed a dispersive channel. We find,
reassuringly, that the group delay of the pulses is not
time reversed, i.e., pulses retain their relative temporal
order. The effect of group velocity dispersion, dvg/dw,
however, is time reversed to first order. This implies
that pulses broadened in propagation can be renarrowed
following conjugation by merely propagating through
a second channel. We remark here that the notion of
equalization or estimation and chirp techniques as ap-
plied to phase compensation due to pulse propagation
is a well-known concept in data communication systems
and in radar applications.12 The application of non-
linear optical techniques with respect to this problem
has not, to our knowledge, been analyzed and is the
topic of this Letter.
The model analyzed is shown in Fig. 1. An input
pulse,
fl(t) = g(t) exp(iwot),
is incident on a dispersive channel. wo is the optical
carrier frequency; the pulse envelope is g(t). The
Fourier transform of g(t) is F(Q)
g(t) = F(Q) exp(i2tdQ), (1)
so that
fi(t) = f F(Q) exp[i(wo + Q)tJdQ. (2).
Since the width of g(t) is very large compared with the
optical period 2,r/wo, it follows that Q << wo over the re-
gion where F(Q) is appreciable.
The propagation constant through the channel is
0,Bc), so that the output pulse f2(t) is
constraints of the input pulse are presented for typical
fc., F(Q) exp fi[(wo + Q)t - l(wo + f)Lfdf
where L, is the length of the channel.
Expanding I3(w) in a Taylor series near coo,
fl(Wo + 9) = f(Wo) + d:+ 2d1 2 +.
aw 2 aw2
leads to
f2 (t) = expfi(wot - 3oLfl)] F(M)
Xexp iQ (t- L d 22ll, dQ,
aw ' 2d2 1 ]J
(3)
(4)
where fOo - (wo). The term (aol/do)Ll can be written
as Ll/vg, where vg = wl/aO is the group velocity. The
term a 2 l/dcW 2 , which can be written as [- (I/vg 2 ) (dOgi
do)], is the group velocity dispersion term and causes
f2(t) to be broader than fI(t) when f,(t) is a trans-
form-limited pulse.
The pulse f2 (t) undergoes phase conjugation-the
result being the pulse f3 (t). To be specific, we will as-
sume that the phase conjugation is achieved by four-
wave mixing in a nondispersive medium with two (es-
sentially cw) pump waves, Al and A2, at wo. We further
assume that the response time of the nonlinear inter-
action is faster than the pulse duration. This causes a
Fourier component at wo + Q to be reflected'3 at a fre-
quency wo - Q, so that the sum 2wo of the pump
frequencies is equal to the sum of the incident and re-
flected frequencies. In general, an incident Fourier
component A exp[i(wo + Q)t] is reflected from the
conjugator as r(Q) exp[ik0(9)] A* exp[i(wo - Q)tJ, where
r(Q) exp[iq(9)] is the complex reflection coefficient of
the conjugator'3 at wo + Q (r is the magnitude of the
reflectivity). The total reflected wave is
f 3 (t) = exp[i(wot + floL1)J
X r(Q)F*(Q) exp - t - )
+ 21 ad: Q2L± + 0(Q)|| dQ. (6)
0146-9592/79/020052-03$0.50/ 0 © 1979, Optical Society of America
February 1979 / Vol. 4, No. 2 / OPTICS LETTERS 53
f2 (t)
1 BS Lfi
1i3 --(2
f3 (t)
Fig. 1. Schematic diagram of pulse renarrowing scheme.
The phase delays of channels 1 and 2 are given by #(co) and
Q'(w), respectively. The input pulse spectral envelope of f[(t)
is assumed to be transform limited. BS is a beam splitter.
The wave f3(t) is incident on a medium of length L2
whose propagation constant at wo - Q can be expanded
as
fl'(wo - Q) = do' - ' Q
Ow
1 23' +
+ -2Q 2 + .. .
2 9w2
The output f4(t) is given by
f4(t) = exp[i(coot + OL1 - fo'L2 )J r(t)F*(Q)
__ a2$ 
X ep i l-r ( L4Q2 +4((QidQ. (8)
2(2 aWdx2 ( | ()
The total group delay, (Ofi/Oco) L1 + (OfS'/Ow) L2, is the
sum of the individual channel delays. The group ve-
locity dispersion term, which involves the second-order
derivatives in Eq. (8), disappears, to first order, if
8
-L -8 L2 - 9
OW2 'a&(9
This requires that the sign of 62 $/8W2 be the same in
both channels. At the output of channel 2, the pulse is
renarrowed. For maximum renarrowing, it is necessary
that r(Q) be a constant over the spectral range of the
envelope function F(Q) and that 0(Q) = a + bQ, where
a and b are two real constants. Under these conditions
the output from channel 2 is
f4(t) = exp[i(wot + OoLi - 3o'L2 + a)] F*(Q)
X expPi - Q (t
Thus,
(o L O -b)l dQ. (10)
Hence, in addition to conjugation and a reflection of
the frequency spectrum from wo + 9 to coo - Q, the
four-wave mixing process introduces a group delay of
b. Under these conditions and neglecting higher orders
in expansion (4), the pulse at the output of channel 2
regains its original width.
To check the assumptions mentioned above con-
cerning the phase conjugator, we plot in Fig. 2 the cal-
culated complex reflection coefficient r(&) exp[iq(Q)]
of a typical conjugator.1 3 We find that, for Q < 4 GHz,
the deviation of 0(Q) from a straight line is very small
compared with ir, so that pulses with width of >2 X
10-10 sec can be renarrowed by conjugation. Under
these conditions, the constants a and b in the expansion
of O(M) are op - v/2 and nL/c, respectively, where L and
n are the interaction length and linear refractive index,
respectively, of the phase conjugator and ,p is the phase
of the pump waves. Thus the additional group delay
due to the four-wave interaction is given by tg' =
nL/c.
On the philosophical side, it is interesting to note that,
according to Eq. (11), the effect of conjugation plus the
flipping of the frequency spectrum is to time-invert the
effect of group velocity dispersion and all additional
terms in Eq. (4) that are even in powers of Q, but not the
group delay (along with other terms odd in powers of Q).
The group delays (0//) L1 and (ao'/I&)L2 of the two
legs are additive. The application of the term "time-
inversion" to the conjugation process should thus be
highly qualified.
We note that other nonlinear interactions that in-
volve phase conjugation and frequency-flipping effects,
such as three-wave mixing 2' 7 [i.e., wout = 2w -(w + Q)],
can also give rise to pulse renarrowing, as described
above. An analysis of the complex frequency-depen-
dent transmission coefficient due to a three-wave mix-
ing process process yields a constant phase factor similar
to that given above. However, because of the copro-
pagating nature of the input and output fields in a
three-wave interaction, the acceptable input frequency
spectrum [F(Q)] depends primarily on material dis-
F_ a: 3
CL 112& A
< I
O 2
C C OLD _
0 LI
ft [ = 37r /8 cm 1
L 1.0 cm
no 1.5
(11)
where
* = fOL1- f3o'L2 + a
-30 -20 -10 0 10
FREQUENCY OFFSET tGHz)
20
2
0
U2
2 X
aL
_ - 23J2
30
tg = "LI+ a#L2+b,
dw C o
which corresponds to the total group delay.
Fig. 2. Magnitude and phase of the complex reflection
coefficient versus frequency offset for typical four-wave phase
conjugator parameters (after Ref. 13).
CHANNEL 2 6-i(w)z
| CHANNEL 2 e iG'W)z |
1. L2 ,
4(t) = eit[eiwotg*(t -tg)]
and
54 OPTICS LETTERS / Vol. 4, No. 2 / February 1979
persion, and, to first order in Q, the medium does not
introduce an additional overall group delay. Thus, the
acceptable bandwidth for pulse renarrowing may be
greater in three-wave versus four-wave interactions.
Interactions that yield phase-conjugate fields without
frequency flipping (e.g., stimulated Brillouin scattering
and stimulated Raman scattering) may also be useful
in pulse renarrowing.
In conclusion, we have shown that nonlinear optical
conjugation of a dispersion-broadened pulse can be used
to obtain a renarrowed pulse by subsequent traversal
through a second channel. The input-frequency
bandwidth is essentially limited by the phase-matching
requirement of the specific nonlinear mixing scheme.
This research was supported by the Air Force Office
of Scientific Research. D. Fekete is a Weizmann In-
stitute Postdoctoral Fellow and wishes to acknowledge
the support of the Weizmann Institute. D. M. Pepper
is a Hughes Research Laboratories Doctoral Fellow and
is grateful and thankful for the support granted by the
Hughes Aircraft Company.
References
1. B. Ya. Zel'dovich, V. I. Popovichev, V. V. Ragul'skii, and
F. S. Faisullov, JETP Lett. 15,109 (1972).
2. A. Yariv, Appl. Phys. Lett. 28, 88 (1976).
3. R. W. Hellwarth, J. Opt. Soc. Am. 67,1 (1977).
4. A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 (1977).
5. D. M. Bloom and G. C. Bjorklund, Appl. Phys. Lett. 31,
592 (1977).
6. R. L. Abrams and R. C. Lind, Opt. Lett. 2, 94 (1978).
7. P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs,
A. Tomita, and K. H. Womack, Appl. Phys. Lett. 31, 435
(1977).
8. P. F. Liao and D. M. Bloom, Opt. Lett. 3, 4 (1978), and
references therein.
9. S. M. Jensen and R. W. Hellwarth, Appl. Phys. Lett. 32,
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10. D. M. Pepper, D. Fekete, and A. Yariv, Appl. Phys. Lett.
33, 41 (1978).
11. J. H. Marburger, Appl. Phys. Lett. 32, 372 (1978).
12-. See, for example, H. Bode, Network Analysis and Feed-
back Amplifier Design (Van Nostrand, New York, 1945)
or F. G. Stremler, Introduction to Communication Sys-
tems (Addison-Wesley, Reading, Mass., 1977).
13. D. M. Pepper and R. L. Abrams, Opt. Lett. 3, 212
(1978).


Compensation for channel dispersion by nonlinear optical phase conjugation

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Network Analysis and Feedback Amplifier Design

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Compensation for channel dispersion by nonlinear optical phase conjugation

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