NEAR FIELD COMPUTATION IN 1D PHOTONIC CRYSTAL WAVEGUIDES FOR TE POLARIZATION
R. S. DUBEY*, NEMALA SIVA SANKAR
Department of Nanotechnology, Swarnandhra College of Engineering and Technology, Seetharampuram, Narsapur (A.P.), INDIA
Instead of conventional waveguides, photonic crystal waveguides are preferred due to better confinement of light within air core medium and zero radiation losses at the sharp bends. In this paper, we have designed one-dimensional (1D) photonic crystal waveguides for different parametrical values. It is observed that as the number of cladding stacks increased the oscillation of light waves is suppressed which is helpful to provide a better confinement of light in air guiding medium. It is also noted that in omnidirectional based waveguide, the guided mode can lie and well confined in the active region with zero propagation losses irrespective to the incident angle.
(Received March 19, 2012; Accepted June 20, 2012)
Keywords: Optical Waveguides, Photonic Crystals, Omnidirectional Reflector, Polarization
1. Introduction
Earlier optical waveguides were the natural replacement over the metallic waveguides due its losses occurred at optical frequencies. In dielectric waveguides, the reflection is restricted to small incidence angles with respect to the waveguide surface and light is guided by total internal reflection at the boundary of the waveguide. But in optical waveguides, the radiation losses occurred at the bending cannot be ignored. In order to suppress these losses, the radius of curvature of waveguides needed to be large with respect to the wavelength. To achieve better confinement of light in a waveguide, it is desirable to move away from the common total internal reflection occurs in the conventional waveguides. In such circumstances, the discovery of photonic crystals [1-2] has put a new alteration on light guiding. Recently, the study of group velocity in photonic crystals has opened a door to realize new photonic devices [3]. One major advantage of photonic crystals is the possibility of designing electromagnetic modes. The ability to modify the dispersion diagram of a guided mode in a photonic crystal waveguide is very useful for practical applications. In photonic crystal waveguides, the transverse guiding is accomplished by distribution reflection within the cladding layers [4-5]. document In addition, the active layer can be made of lower refractive index than cladding layers due to which the transverse light propagation lies in the forbidden band of one-dimensional (1D) photonic crystals (clads).
Presently, for the realization of optical integrated circuits the planer photonic crystal waveguides have became an essential building block due to its high transmission efficiency of light at sharp corners. Recently, much attention has been paid towards the study of omnidirectional reflectors [6- 9]. If omnidirectional reflectors are used as a clad, better confinement of light can be expected. In addition there is no limitation of incidence angle of light as well. This mechanism is helpful for
redirecting the scattered light in any direction completely within the guiding layer.
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*Corresponding author: [email protected]
In this paper, we have first designed the 1D photonic crystal waveguides for different parametrical values. To observe better field distribution in active region, we have designed omnidirectional reflector which is used as cladding layer for the waveguides. In section two, the
mathematical approach has been presented; the results and discussion are summarized in section third. Finally, section fourth concludes the paper.
2. Design approach
To study the guiding mechanism of 1D photonic crystal waveguide we have assumed that the air guiding layer is sandwiched between two 1D photonic crystals. One-dimensional photonic crystals are composed of high and low refractive index layers respectively. The considered structure of one-dimensional photonic crystal waveguide is shown in figure 1.
Fig. 1. Schematic of 1D photonic crystal waveguide
The cladding consists of two alternate layers of refractive index n1 and n2 and thickness d1 and d2. The refractive index and thickness of core region are nc and lc respectively. By employing transfer matrix method, the dispersion relation between the angular frequency ‘ω’ and the tangential component ‘β’ as well as Bloch wavevector ‘K’ is expressed as [3]
A D
K 2
cos 1
, 1 1
(1)
where in case of transverse electric polarization, A and D is defined as
2 2
2 1 1 2 2
2 sin
2 cos 1
1 k d
k k k i k d k e
A ika (2)
2 2
2 1 1 2 2
2 sin
2 cos 1
1 k d
k k k i k d k e
D ika
(3)
The solutions of Bloch wave vector ‘K’ gives forbidden and pass bands. After the solution of ‘K’ we have obtained the omnidirectional reflection bands by substituting it in equation (4). The expression for reflectivity is expressed as
2 2 2 2
Im sinh
Im
sinh
K N C K
rN C (4)
for larg in 1D p
coeffic
unders parame Fig. 2(
shown either t electric sides o parame lc=0.29
It is impor ge ‘N’ (numb
By ignorin photonic cry
Here, lc a cients define 3. Resul We have d stand the fie eters such a (a) and Fig. 2 n in figure 2
the sides of c c field profil of core regio etrical value 91µm, N=10
rtant to note ber of bilaye ng the interm ystal wavegui
1
2 2
C C C x E
TE TE TE
TE
and Λ are th d in equation lts and dis designed 1D eld distributi s core thickn 2(b) shows th 2(a), the fiel
core region t le but for inc on can be o es used for 0 and n1=2.3,
(a)
(b)
that the seco ers or stacks) mediate mathe
ide for TE po
cos sin , 2 cos
1 1 2 1 1 2
k k k k k k
x l x k
n E
n E
c c E
he thickness n (5) can be f scussion
photonic cry ion within t ness, refracti he near field d distributio the oscillatio creased num observed as figure 2(a)
n2=2.0, nc=1
)
)
ond term in t ).
ematical step olarization is
2 2 ,
a l n x
n l n x
c c
of active la found in ref.
ystal symmet the air core ive index, n d distribution on is well co on of light ca mber of bilaye
the number and figure 1, a=b=2µm,
the denomina ps, the equat s expressed a
, 2 2
x l a n
l n x
c c
ayer and latt [10].
tric wavegui region we number of bi in air core p onfined with an be observe ers. The redu of bilayers 2(b) are n1 , lc=0.291µm
ator exponen tion of electr as
1 n a
tice constant
ide for TE po have varied layers of cla photonic crys hin the core ed. Figure 2 uced wave o increased fr
=2.3, n2=2.0 m, N=15 resp
ntially tends ric field distr
nt. However
olarization c d various de ad and wave stal wavegui
medium ho 2(b) shows th oscillation eit from 10 to 1 0, nc=1, a=b pectively.
to zero ribution
(5)
r, other
ase. To signing elength.
des. As owever;
he same ther the 15. The b=2µm,
light w change n1=2.3 lc=0.35 observ confin require
confin scatter omnid wave w
Fig 2. Elect a=b=2µm, If the oscil will be confin
ed parametri 3, n2=1.5, n 5µm, N=5 re ved in figur
ement of lig ed.
Fig. 3. Ele nc=1, a=b=
The use o ement and z red light in a irectional re which is dep
tric field Inten lc=0.291µm, llation of ligh ne within co
cal values. T nc=1, a=b=2
espectively.
re 3(a). How ght in core r
ectric field In
=2µm, lc=0.29
of omnidirec zero losses a any direction flection ban picted in figu
(b) (a)
nsity versus pr , N=10 and ht either the re region. Th To plot figur 2µm, lc=0.29
An efficient wever, it is region the o
tensity versus 91 µm, N=6
l
ctional refle at the corner n completely
ds of one-di ure 4. The o
ropagation dir d figure (b) lc=0.291µm, side of activ herefore we re 3(a) and fi 91µm, N=6 t reduction i almost sup optimal value
s propagation and figure lc=0.35µm, N
ectors as a rs/bends. Thi within the g imensional p omnidirectio
rection figure n1=2.3, n2= N=15.
ve region is r have plotted igure 3(b), t 6 and n1=2.
is oscillation pressed in f es of structu
n direction fig (b) n1=2.5, n2
N=5.
clad is ben is mechanism guiding laye photonic crys
nal photonic
(a) n1=2.3, n 2.0, nc=1, reduced then d figure 3(a) the used para 5, n2=1.4, n of light wa
figure 3(b).
ural and opti
gure (a) n1=
2=1.4, nc=1, a
neficial in o m is helpful r. Hence, we stal for trans c bandgap fo
n2=2.0, nc=1, a=b=2µm, n maximum i
) and figure ametrical val nc=1, a=b=
aves can be . Hence, for ical paramet
2.3, n2=1.5, a=b=2 µm,
order to get for redirect e have obtain sverse electr or both TE a
ncident 3(b) at lues are
=2 µm, clearly r better ters are
t better ting the
ned the ic (TE) and TM
polarizations are defined by the lower bandedge at the normal incidence (θ=00) and the upper bandedge at the perpendicular incidence (θ=900). The normalized frequency ranges of omnidirectional reflection bands for TE wave are 0.31-0.39 and 0.72-0.73 (in the units of ωd/c). In figure 4, the white region, dark gray region and light gray strips are corresponding to the pass, forbidden and omni bands respectively. Once the range of omnidirectional reflection bands are known efficient guiding of light can be done in the waveguide.
Fig. 4. Projected band diagram of 1D photonic crystal showing omnidirectional bands for TE polarization.
If we used our designed omnidirectional reflectors for a waveguide then guided mode can lie and well confined in the active region with zero propagation losses. Therefore, we have plotted figure 5 in which the near field intensity of fundamental mode is depicted. Here, the omnidirectional reflectors designed for 25 bilayers of alternate layers of n1=1.3 and n2=2.4 is used as clad at 1.55µm wavelength.
Fig. 5. Electric field Intensity in 1D photonic crystal waveguide for TE polarization.
It is observed that the electric field is well guided in the guiding region where the refractive index of guiding layer is lower than the refractive indices of the cladding layers. It is also observable that the field strength decays rapidly on both side of the air guiding layer. The
assumed thicknesses of upper and lower cladding layers are ab2µm however, the thickness of air guiding medium is lc 0.5µm . In simple words, the wider omni band means wider frequency (wavelength of light) selection is permissible to forbid hundred percent of light after striking upon the omnidirectional reflectors. If we used these reflectors as a cladding 100 % light will be strongly reflected after made incident and enforced to forward in the guiding medium to give better confinement.
4.Conclusions
By employing transfer matrix method the designing of 1D photonic crystal waveguides have been carried out. Initially, we have designed and analyzed the near field in the active air region for different parametrical values. It is observed that the confinement of near field is better at optimized parametrical values. In addition, the oscillation of light waves either the sides of core region is suppressed for optimal values of structural and optical parameters. The suppression of oscillation of waves is useful to confine maximum part of incident light. Further, we have obtained the omni bands for TE polarization which provides 100% the reflection of light irrespective to the angle of incidence. In omnidirectional reflector based waveguide it is found that field is well confined within the air guiding medium where the refractive index of guiding layer is lower than the refractive indices of the cladding layers.
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