A preview of this full-text is provided by Optica Publishing Group.
Content available from Optics Letters
This content is subject to copyright. Terms and conditions apply.
Nonresonant cascaded acousto-optic mode
coupling
Jianhui Zhao, Ren Miao, and Xiaoming Liu
Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
Received June 1, 2006; revised July 12, 2006; accepted July 15, 2006;
posted July 19, 2006 (Doc. ID 71582); published September 11, 2006
Nonresonant cascaded acousto-optic mode coupling under an overall phase-matching condition is presented,
where phase matching is not satisfied for each of the coupling stages but the sum of the two acoustic mo-
menta matches the optical phase detuning between the source mode and the destination mode. As a special
nonresonant cascaded case, conversion from LP01 to LP21 and LP02 driven by a single acoustic frequency is
predicted theoretically and demonstrated experimentally. © 2006 Optical Society of America
OCIS codes: 230.1040, 060.2340, 050.2770.
Acousto-optic (AO) mode coupling in regular stan-
dard single-mode fiber (SMF) is an attractive topic
and has been studied widely for more than a decade.
According to the systemic theory in Ref. 1, the input
LP01 mode can be coupled only to the antisymmetric
mode LP1n共n=1,2,3...兲at the phase-matching con-
dition, because the refractive-index perturbation in-
duced by the traveling flexural acoustic wave is anti-
symmetric. Recently, we pointed out theoretically
and demonstrated experimentally that AO mode cou-
pling could happen between any two modes with ad-
jacent azimuthal numbers, for instance, from LP11 to
LP21 or from LP11 to LP02.2This was demonstrated by
a cascaded AO mode coupling process, in which mode
LP01 was coupled first to LP11 and then from LP11 to
LP21 or LP02, with phase-matching conditions satis-
fied for each of the coupling stages.
In this Letter we present a more general cascaded
AO mode coupling in which phase matching may be
not satisfied individually for each of the coupling
stages. By solving cascaded coupled mode equations,
it is pointed out that complete conversion between
two optical modes with nonadjacent azimuthal num-
bers is realizable as long as an overall phase-
matching condition, which is called nonresonant, is
satisfied. Experimentally, coupling from LP01 to LP02
and LP21 is demonstrated by applying a single acous-
tic wave, where the two cascaded coupling stages are
driven by the same frequency. The measured far-field
patterns and data of coupling peak wavelength ver-
sus acoustic frequency all agree well with numerical
calculations.
Taking a cascaded coupling process from LP01 to
LP11 and then sequentially to LP21 as an example, a
diagram of the momenta is shown in Fig. 1, where

01,

11, and

21 are the photon momenta for the op-
tical modes and K1,K2are the momenta of the acous-
tic waves that are involved. The coupled mode equa-
tions can be written as1,2
dE01/dz=−i
1ei2
␦
1zE11,
dE11/dz=−i
1e−i2
␦
1zE01 −i
2ei2
␦
2zE21,
dE21/dz=i
2e−i2
␦
2zE11,共1兲
where
␦
1=共

01 −

11 −K1兲/2 and
␦
2=共

11 −

21 −K2兲/2
are the detuning parameters for coupling from LP01
to LP11 and from LP11 to LP21, respectively;
1,2 are
the coupling coefficients for each of the stages. As dis-
cussed in our previous work,2when the two acoustic
waves satisfy the phase-matching conditions for each
coupling stage 共
␦
1=
␦
2=0兲, as shown by the solid ar-
rows in Fig. 1, conversion from LP01 to LP21 can be
completed. This process can be called resonant cas-
caded coupling.
Now we consider a more general case in which
phase matching may not be satisfied for the indi-
vidual coupling stage but the sum of the two acoustic
momenta matches the momentum difference between
LP01 and LP21, i.e.,

01 −

21 =K1+K2, as shown by the
dashed arrows in Fig. 1. Then we have
␦
=
␦
1=−
␦
2,
and Eqs. (1) have a simple solution for the initial con-
ditions of E01共0兲=1, E11共0兲=E21共0兲=0:
E01共z兲=
1
2
1
2+
2
2exp共i
␦
z兲
冋
cos共sz兲−i
␦
ssin共sz兲
册
+
2
2
1
2+
2
2,
E11共z兲=−i
1
sexp共−i
␦
z兲sin共sz兲,
Fig. 1. Diagram of the photon momenta

and involved
acoustic momenta K. Solid arrows, resonant case, dashed
arrows, nonresonant case.
October 1, 2006 / Vol. 31, No. 19 / OPTICS LETTERS 2909
0146-9592/06/192909-3/$15.00 © 2006 Optical Society of America
