Lossless Propagation with Bragg Waveguide
Pawan Kumar1, Sudhanshu Kumar2,Vivek Srivastava3
NIET, GR.NOIDA (UP), INDIA
pawan.kumar_ec2006@yahoomail.com
ABSTRACT: The properties of nonlinear semiconductor finite
Bragg reflector waveguides are analyzed. It is shown that in
the nonlinear case, guided modes may propagate in structures
which have no bound modes in the linear regime
KEYWORD: Optical fiber, Reflection, Air Core, Refractive
index, Transverse Field.
Optical dielectric waveguides having a slab configuration that
can be used for supporting lossless confined modes having
higher refractive index of the inner layer than the outer layer
of the two bounding media. This condition provides an
imaginary transverse propagation constant which corresponds
to an evanescent decay of the mode field.
There are several practical examples where it is necessary or
significant to guide power in a layer having refractive index
than that in the two bounding media. An example of such
situation is the waveguide laser in which the inner layer is air
whose refractive index is n0=1, which can result in lossy
modes whose loss constant rises as the third power of the
reciprocal thickness of the inner layer.
We will show that in principle, if a low index slab is provided
with the bounding media having refractive index larger than
that of the inner slab, a periodic lossless propagation is
possible. The use of stratified periodic media in dielectric
waveguide has been proposed by Ash[1] who,did not
consider, the confined propagation in the low index
media.Referring Fig 1, we consider the case where,
na<ng<n1,n2. In TE modes, the only field components are
Ey,Hx,Hz. Each of these field components satisfies wave
equation such as for Ey,
Fig.1A Bragg reflection (slab) dielectric waveguide.
𝑑 2 𝐸𝑦
𝑑𝑧 2
+
𝑑 2 𝐸𝑦 𝑤2 2
+ n (x)Ey=0
𝑑𝑥 2 𝑐 2
(1)
If we take Ey(x,y,z) = E(x) exp (iβz) the wave equation
𝑑 2 𝐸(𝑥)
𝑑𝑥 2
𝑤2
+ ( 2 n2(x) - β2) E(x)=0
𝑐
(2)
We take a solution in the form
exp[𝑞𝑎 (𝑥 + 𝑡)],
𝑥 < −𝑡
(3𝑖)
E(x) = { 𝑐1 cos(𝑘𝑔 𝑥) + 𝑐2 𝑠𝑖𝑛(𝑘𝑔 𝑥), −𝑡 ≤ 𝑥 < 0 (3𝑖𝑖)
𝐸𝑘 (𝑥) exp(𝑖𝑘𝑥) ,
0≤𝑥
(3𝑖𝑖𝑖)
2
𝑤
2
𝑤
qa = √β2 − ( 𝑛𝑎 ) , kg = √( 𝑛𝑔 ) − β2
𝑐
𝑐
(4)
The characteristics of Bloch wave in the semi-finite stratified
medium when x>0, can be determined by the elements A, B,
C, D which depends on the propagation constant K1x, k2x,
alternatively. According to (9), where [
correspond to real k and thus allow propagation of wave that
𝑚𝜋
is Bloch waves. when [ 1 (A+D)]2>1, then k =
+iki
∧
2
It is assumed that solution in region (i) and (ii) are similar to
that of slab dielectric configuration.The new parameters which
exist in this region are Ek(x)exp(ikx) in the stratified periodic
medium (iii).Ek(x) is periodic as the wave have the Bloch
form(3) according to Floquet’s theorem and have period
denoted by∧,∧= a+b
EK(x+∧) = EK(x)
(5)
The field solution Ek(x)exp(ikx) at k≥0 is obtained by diagonal
zing the unit cell translation matrix(A,B,C,D) which further
gives a relation between complex amplitude of the incident
plane wave an and the reflected wave bn.
1
(A+D)]2<1
2
imaginary part Ki, in this case Bloch wave is evanescent
which creates gaps in the periodic medium, this gaps are
known as “forbidden gaps” of the medium.
To obtain the mode of waveguide of Fig 1, we match the
fields and their x derivatives at x = 0 and x = -t. The
dispersion relation can be determine by using (3),(7) & (8).
kg
𝑞𝑎 𝑐𝑜𝑠𝑘𝑔 𝑡−𝑘𝑔 𝑠𝑖𝑛𝑘𝑔 𝑡
𝑞𝑎 𝑠𝑖𝑛𝑘𝑔 𝑡+𝑘𝑔 𝑐𝑜𝑠𝑘𝑔 𝑡
= - ik1x
𝑒 −𝑖𝑘∧ −𝐴−𝐵
(10)
𝑒 −𝑖𝑘∧ −𝐴+𝐵
In the above equation the left side contains only the
parameters of the guiding (ng) and substrate (na) layers. For
confined propagation qa,kg and β are real so that the left side is
real number.The right side is real only if [ 1 (A+D)]2>1, which
2
A=𝑒
−𝑖𝑘1𝑥𝑎
𝑖
[cos k2xh - (
𝑘2𝑥
2 𝑘1𝑥
𝑖
B = 𝑒 𝑖𝑘1𝑥𝑎 [- (
𝑘2𝑥
2 𝑘1𝑥
−
𝑘1𝑥
𝑘2𝑥
−
𝑘1𝑥
𝑘2𝑥
)sink2xb],
)sink2xb],
(6)
C = B* , D = A*,
𝑤
2
Kix = √( 𝑛𝑖 ) − β2 , i = 1,2..
𝑐
Note that the transition matrix is unimodular i.e. AD-BC=1.
The resulting expression for the Bloch field at x ≥ 0 in the
region with index n1 is
E(x) = Ek(x) eikx
= {[ao𝑒
𝑖𝑘
1𝑥(𝑥−𝑛∧) +
b0𝑒
−𝑖𝑘
1𝑥(𝑥−𝑛∧)
] 𝑒 −𝑖𝑘(𝑥−𝑛∧) }×eikx
(7)
where a0 and b0 are the components of the eigen vector
𝐵
For(𝑎𝑏𝑜 ) = (𝑒 −𝑖𝑘∧−𝐴
)
(8)
𝑜
&
𝑒 −𝑘∧ = (
𝐴+𝐷
2
)± √(
𝐴+𝐷
2
2
) −1
(9)
means that the propagation condition in the periodic medium
fall within one of the forbidden gap.It follows that confined
lossless modes of waveguides, we can determine the eigen
mode by starting with some value of β<(  )ng, this determine
c
Kg,qa,k1x&k2x.
For lossless and confined mode propagation complete
reflection must take place at the boundary between the layered
medium and guiding layer. This situation exists only when the
wave having zigzag function incident on the interface under
the condition of forbidden gap.A field distribution of such a
waveguide as shown in Fig 2. It should be noted that in the
periodic medium the field correspond to a Bloch wave in a
forbidden gap and a periodic under an evanescent envelope
exp(-ikx).
A waveguide consists of a lower refractive index slab, say air
separated by two semi- infinite periodic media which is
possible.Such a waveguide can be designed by replacing the
structure to the left of plane a (where dE/dx=0) the structure to
the right. The distribution of field remains symmetric about
plane a as resulted in Fig 3.
The Bragg waveguide described above represents differently
against high order transverse mode i.e modes with large
number of nodes in the central guiding region. We can show
that if the waveguide is constructed for the fundamental
transverse mode (s=0), then in order that sth modeexist as well,
the condition
∧2
𝑡2
𝑙
= 𝑙 = 1,2..
𝑠
It is assumed that ∆=
(𝑛1 −𝑛2 )
𝑛1
<< 1.
CONCLUSION
It is proved that the Bragg reflection at the boundaries in a
waveguide can allow confined and lossless propagating modes
in regions having lower refractive index.
REFERENCES
[1]. P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. 68,1196 (1978).
Fig.2. Transverse field distribution of the fundamental mode of a typical
Bragg reflection (slab) waveguide with na = 1.0,ng= 3.24,n2 = 3.43, n1 = 3.55,
 =1.15µ, a = b = 0.5, t= 1.3738µ, = 0.2659µ.
[2]. M. Lbanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D.Joannopoulos,
Science 289, 415 (2000).
[3] Y. Xu, R. K. Lee, and A. Yariv, Opt. Lett. 25, 1756(2000).
[4] G. Ouyang, Y. Xu, and A. Yariv, Opt. Express9,733(2001),
http://www.opticsexpress.org.
[5] Y. Xu, G. Luoyang, R. K. Lee, and A. Yariv, J. LightwaveTechnol. 20,
428 (2002).
[6]
K.
Tetsuya
and
M.
Izutsu,
(2000),http://www.opticsexpress.org.
Opt.
Express
7,
10
[7] J. P. Meunier, J. Pigeon, and J. N. Massot, Opt. QuantumElectron. 13, 71
(1981).
[8] S. Guo, “Study of dispersion compensation single modeoptical fiber,”
M.S. thesis (Northern Jiaotong University, Beijing, China, 1996).
[9] P. K. Mishra, I. C. Goyal, A. K. Ghatak, and E. K.Sharma, Opt. Acta 31,
1041 (1984).
[10]. H. Etzkom and T. Heun, Opt. Quantum Electron. 18,1 (1986).
[11] A. Sharma and S. Banarjee, J. Lightwave Technol. 7,1919 (1989).
Fig. 3 Transverse field distribution of the fundamental.Mode of a typical
Bragg reflection (slag) waveguide with na = 1.0,n2 = 3.38, n1 = 2.89,
 =1.15µ, a = b = 0.5, t= 1.53µ, ∧ = 1.53µ.
Such a waveguide can be used for gaseous
laser.Mathematically,the mode condition for TE modes of the
symmetric Bragg waveguide can be shown as.
- ik1x
𝑒 −𝑖𝑘∧ −𝐴−𝐵
𝑒 −𝑖𝑘∧ −𝐴+𝐵
𝑤
={
𝑘𝑎 tan (
𝑘𝑎 cot (
) 𝑒𝑣𝑒𝑛𝑇𝐸𝑚𝑜𝑑𝑒𝑠
2
𝑘𝑎 𝑡
2
) 𝑜𝑑𝑑𝑇𝐸𝑚𝑜𝑑𝑒𝑠
(11)
2
Ka = √( 𝑛𝑎 ) − β2
𝑐
𝑘𝑎 𝑡
(12)
[12] W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman & Hall,
New York, 1983).
[13] G. Arf ken, Mathematical Methods for Physicists (Academic,Orlando,
Fla., 1985).