December 1, 1998 / Vol. 23, No. 23 / OPTICS LETTERS 1835
Frustration of Bragg reflection by cooperative dual-mode
interference: a new mode of optical propagation
Amnon Yariv
California Institute of Technology, Pasadena, California 91125-0001
Received July 20, 1998
A new optical mode of propagation is described, which is the natural eigenmode (supermode) of a fiber (or any
optical waveguide) with two cospatial periodic gratings. The mode frustrates the backward Bragg scattering
from the grating by destructive interference of its two constituent submodes (which are eigenmodes of a uniform
waveguide). It can be used in a new type of spatial mode conversion in optical guides. 1998 Optical Society
of America
OCIS codes: 050.2770, 060.2310, 030.4070.
Consider an optical mode ja. propagating in a fiber
that is incident upon a grating with a period L
1
and
satisfies the Bragg condition
b
a
1b
b
2p
L
1
,
(1)
which is necessary for strong coherent scattering from
right-traveling mode ja. into left-traveling mode jb..
b
a
and 2b
b
are the propagation constants of the two
guided modes. This situation is depicted in Fig. 1(a).
Also shown is a momentum diagram in which the val-
ues of b (momenta) of the various uniform fiber modes
involved in the scattering as well as the grating mo-
mentum are shown. The scattering causes the inten-
sity of mode ja. to decay evanescently in the grating
region of the waveguide. Typical fiber gratings can
yield an ja. !jb.reflection that exceeds 99% in less
than 1 cm of grating length.
1
A similar fate awaits a higher transverse mode, jc.,
propagating in the fiber when the mode is incident
upon a second grating, 2, with a period L
2
that satisfies
b
b
1b
c
2p
L
2
.
(2)
This mode will also undergo scattering into the same
backward mode, jb., as shown in Fig. 1(b).
Now consider what happens when gratings 1 and 2
are cospatial, i.e., occupy the same z stretch of the fiber.
If the forward-propagating fields of ja. and jc. are
temporally coherent, then the total scattering into jb.
is obtained by an algebraic addition of the ja. !jb.
and the jc. !jb.scattering amplitudes, as shown in
Fig. 1(c). There exists, obviously, a ratio of the modal
amplitudes AyC of the two forward modes for which
the net backward scattering S
ab
1 S
cb
into jb. is zero.
Under these conditions the backward mode jb.,ifitis
not present at z 0, the beginning of the dual-grating
region, will not be excited. Modes ja. and jc. under
this condition lose no power, thus defeating the Bragg
scattering. It follows, self-consistently, that the above
superposition of uniform fiber modes ja. and jc.
constitutes a supermode of the dual-grating fiber.
This new mode of optical propagation propagates
Fig. 1. (a) A fiber (or optical waveguide) modulated spatially by a grating with a period L
1
, which Bragg scatters
resonantly between forward mode ja. and backward mode jb.. A momentum diagram shows the initial and the final
photon momenta, b, involved as well as the lattice momentum 2pyL
1
. (b) Fiber with a second grating sL
2
d, which Bragg
scatters a higher-order forward mode jc. into backward mode jb.. (c) Dual-grating fiber, which can scatter resonantly
and simultaneously between modes ja. and jc. and backward mode jb. but not between modes ja. and jc..
0146-9592/98/231835-02$15.00/0 1998 Optical Society of America
1836 OPTICS LETTERS / Vol. 23, No. 23 / December 1, 1998
without loss through the dual-grating region that is
essentially impenetrable by either ja. or jc. alone.
To formalize the above description we consider the
coupled-mode equations that describe the interaction
among the uniform fiber modes ja., jb., and jc., all
different, that is due to the dual grating as in Fig. 1(c).
The complex normalized modal amplitudes are A, B,
and C such that the total field at z is given by
Esx, y, zd AszdE
1
sx, ydexps2ib
a
zd 1 BszdE
3
sx, yd
3 expsib
b
zd 1 CszdE
2
sx, ydexps2ib
c
zd . (3)
Representing grating 1 by its coupling constant K
1
and grating 2 by K
2
, we can describe the interaction by
the coupled-mode equations
2
dA
dz
iK
1
B expfisb
a
1b
b
2G
1
dzg, (4)
dB
dz
2 iK
1
A expf2isb
a
1b
b
2G
1
dzg2iK
2
C
3 expf2isb
b
1b
c
2G
2
dzg, (5)
dC
dz
iK
2
B expfisb
c
1b
b
2G
2
dzg,
G
1, 2
;
2p
L
1, 2
.
(6)
Modes ja. and jc. are not directly coupled. For such
coupling to occur we require a grating with a long
period L, where L 2pysb
a
2b
c
d. There exists,
however, indirect coupling between ja. and jc., which
is mediated by jb. and is expressed by Eqs. (4)–(6).
Nonlinear interactions in the writing of the grating,
which could lead to grating momenta ms2pyL
1
d 6
ns2pyL
2
d, m and n integers, are assumed negligible.
From Eq. (5) it follows that when
A
C
2
K
2
K
1
,
(7)
D ;sb
c
2b
a
1G
1
2G
2
d0, (8)
dB
dz
0.
If Bs0d 0, then Bszd 0 and Aszd and Cszd are
constant; i.e., there is no loss of forward-propagating
power. We can obtain the supermode, i.e., the stable
ratio of A, B, and C in Eq. (3), by diagonalizing interac-
tion equations (4)–(6). The solution that corresponds
to our mode can be represented symbolically by a col-
umn vector (unnormalized):
˜
E
1
8
>
>
>
>
<
>
>
>
>
:
2K
2
K
1
expf2isG
1
2 G
2
dzg
0
exp
∑
2
i
2
sG
2
2 G
1
dz
∏
9
†
†
†
†
=
†
†
†
†
;
expf2isb
a
1b
c
dzg
,
(9)
where we have chosen both individual Bragg mis-
matches as zero; i.e., b
a
1b
b
G
1
and b
b
1b
c
G
2
.
The remaining two solutions involve B fi 0 and are not
of interest in the present context.
Strictly speaking, this special mode in which the
reflected component is dark sB 0d exists only when
resonance condition (8) is satisfied. The consequences
of deviations from Eq. (8) resonance are discussed
separately.
The existence of this new mode of propagation leads
to some interesting applications and phenomena. If
we allow K
1
and K
2
to depend on z adiabatically,
3,4
then
according to Eq. (9) we can cause
˜
E
1
to vary between
0
B
B
@
1
0
0
1
C
C
A
,
when
K
1
szd
K
2
szd
0
and
0
B
B
@
0
0
1
1
C
C
A
,
when
K
2
szd
K
1
szd
0;
i.e., we can bring about ja. $jc.mode conversion by
mere adiabatic control of the gratings’ profile. Note
that mode jb. is never excited. This mode conversion
is expected to have the high wavelength selectivity
that is characteristic of backward Bragg scattering;
i.e., Dlyl ø ly2nL, where L is the grating length.
This new form of mode conversion by adiabatic grating
evolution is the spatial equivalent of coherent temporal
population control in atomic physics.
5
This subject
will be considered in detail in the future.
The author is indebted to Phil Willems and to Eva
Peral for many stimulating discussions. This research
was supported by the Defense Advanced Research
Projects Agency, the U.S. Office of Naval Research, and
the U.S. Air Force Office of Scientific Research. This
support is gratefully acknowledged.
References
1. T. Erdogan, J. Lightwave Technol. 15, 8 (1997).
2. A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
3. D. Marcuse, Theory of Dielectric Optical Waveguides
(Academic, New York, 1974), p. 106.
4. A. W. Snyder and J. D. Love, Optical Waveguide Theory
(Chapman & Hall, London, 1983).
5. M. P. Fewel, B. W. Shore, and K. Bergmann, Aust. J.
Phys. 50, 281 (1977).
