Modified step-theory for investigating mode coupling mechanism in photonic crystal waveguide taper

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Modified step-theory for investigating mode coupling mechanism in photonic crystal waveguide taper
E. H. Khoo, A. Q. Liu and J. H. Wu
School of Electrical and Electronic Engineering Nanyang Technological University, Nanyang Avenue, Singapore 639798 eaqliu@ntu.edu.sg
J. Li and D. Pinjala
Institute of Microelectronics, Singapore, 11 Science Park Road, Singapore Science Park II, Singapore 117685
Abstract:
In this paper, the mathematical model of the modified step-theory is derived based on the platform of two-dimensional photonic crystal structure that is infinitely long in third dimension. The mode coupling mechanism of photonic crystal tapers is theoretically studied using the modified step-theory. The model is verified by comparing the transmission spectrum obtained for the input/output defect coupler where it shows a good match of less than 5% discrepancy. The modified step-theory is applied to different taper structures to investigate the power loss during the transmission. The power loss at the relative position of the taper provides an explanation as to which taper designs give the highest coupling efficiency.
2006 Optical Society of America
OCIS codes:
(130.0130) Integrated Optics; (230.7380) Waveguides, channeled; (250.5300) Photonic integrated circuits
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#68765 - $15.00 USDReceived 7 March 2006; revised 19 May 2006; accepted 4 June 2006
(C) 2006 OSA26 June 2006 / Vol. 14, No. 13 / OPTICS EXPRESS 6035
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1. Introduction
Photonic crystals (PCs) are regular array of structures that are periodic in one or more dimensions. The periodic structures forbid electromagnetic waves propagation within a frequency range, which is known as photonic bandgap (PBG) in all directions. Since the last two decades, extensive research has been focused on photonic crystals because of its ability in controlling and molding the flow of electromagnetic lightwave, as an analogy to manipulate electrons flow in conventional semiconductor [1]. When defects are introduced to the crystal lattice, its applications widen, which results in the innovation of many highly compact integrated photonic devices such as photonic crystal waveguides (PCWGs), wide angle beam splitters, waveguide bends and photonic crystal cavity filters [2-6]. Photonic crystal waveguides is one of the key components because it forms a medium for light path propagation in photonic integrated circuit [1-2]. Line defects are introduced to the bulk crystal lattice by changing the geometry or simply removing a row of crystal cells to create a photonic crystal waveguide. The line defects give rise to a localized mode in the bandgap, where light is trapped and guided in the defects location of the photonic crystal. This special property allows PCWGs to guide light around sharp bend operating at very narrow width [7], which overcomes the restriction of classical conventional waveguide with high transmission power and low radiation loss. Nevertheless, for infrared or visible light wavelength–sized integrated devices, the size of the photonic crystal structure is in sub-mico or nano-sized range. Researchers and scientists
#68765 - $15.00 USDReceived 7 March 2006; revised 19 May 2006; accepted 4 June 2006
(C) 2006 OSA26 June 2006 / Vol. 14, No. 13 / OPTICS EXPRESS 6036
have to face many challenges in the design and analysis, fabrication process and experimental measurement techniques. These technical difficulties have also posed challenges for the commercialization of the photonic crystal devices. However, the most critical problem that arises from the small size of photonic crystal devices is to couple sufficient amount of light to the narrow waveguide. In designing photonic crystal devices, this is the most important step because poor coupling results in low transmission of power to the waveguide, which in turn affects the functionality and reliability of the optical devices. For this reason, photonic crystal tapers [8-14] have been designed to couple light source of larger cross-sectional size to the narrow PCWG. Light is tapered to the narrow guide by either structural means [8-12] or by varying the size of unit lattice to provide mode conversion [13-14]. The former uses the structural shape of the taper to slowly match the mode profile of the light source to PCWG while the latter uses progressively-variation geometry to achieve adiabatic mode conversion. Many different methods [15-20] are used to calculate the coupling efficiency and transmission loss in tapered waveguide transmission. Among all, the coupled mode theory, perturbation method, multipole method and beam propagation method are commonly used. The coupled mode theory and the perturbation theory [15-16] are able to converge fast for analysis of weak geometrical waveguide variation. However, stronger perturbation requires higher-order corrections, which may affect the convergence to exact solution. For high-index contrast waveguide, the multipole method [17-18] can be applied to analyze the eigenmodes with high numerical accuracy but it does not allow perturbative formulation. The beam propagation method [19-20] can give high accuracy but needs very fine resolution. This can be computationally demanding. The step-theory [21-24] is used to calculate transmission efficiency in conventional tapered waveguide by dividing it into vertical strips called steps. By matching the field components at the boundary between steps, the transmitted amplitude can be obtained. The theory can handle tapered waveguide with small or moderate variation in the cross sectional size with very high convergence [24]. For large geometrical variation, the theory can still converge moderately. Additionally, the method also demonstrates the analysis of intermodal coupling mechanism in the tapering section with respect to the relative position. Many steps are required for calculation of the transmission amplitude, which increases the computational time and resources. However, for applications in photonic crystal tapered structure, this problem is reduced as shown in the later part of the paper. This method also requires the propagation constants for all the guided modes in each step to be known, which is quite hectic for very large number of modes. The objective of this paper is to modify the step-theory and applied it to investigate the coupling mechanism in photonic crystal tapered waveguide. The paper will applied it to several taper designs in photonic crystal and also provides an explanation as of why some taper curvature has higher coupling efficiency than the other. The paper will be organized as followed. In the next section, the fundamental of photonic crystal waveguide is briefly introduced. Section three shows the detail derivation of the modified step-theory for its application in photonic crystal taper waveguide. In the subsequent section, the method is verified with the structure designs in [9]. Section five discusses the numerical results with special focus in abrupt step waveguide, step taper waveguide and smooth linear and nonuniform taper design. Lastly, the paper concludes the results obtained.
2. Fundamentals of light propagation in photonic crystal waveguide
The field profile for any uniform planar waveguide can be described in term of the eigenfields of the normal waveguide modes. The solution of the normal modes is given as [25] (1) where
E
0
is the real amplitude of the propagating
E
-field at a position
x
.
ξ
(
z
) is the z dependent field distribution of the guided mode.
ξ
is dependent on the width and the mode
( )
xi
e z E E
β
ξ
×=
0
#68765 - $15.00 USDReceived 7 March 2006; revised 19 May 2006; accepted 4 June 2006
(C) 2006 OSA26 June 2006 / Vol. 14, No. 13 / OPTICS EXPRESS 6037
number of the waveguide.
β
is defined as the propagation constant of the guided mode and the electromagnetic wave is propagating in the
x
direction. For light to be guided in photonic crystal, two conditions must be satisfied. First, the mode considered is in the bandgap of a photonic crystal. Second, the mode in operation must be truly guided and does not radiate out. For photonic crystal waveguide, the waveguide solution is more complex due to the presence of the periodicity profile in the crystal lattice. Assuming a two-dimensional crystal structure in the
x
and
z
direction, the field mode of a periodic function has the following relationship [2] (2) where
u
kx,kz
(x,z)
is a periodic function in
x
and
z
direction. The periodicity in
x
and
z
direction leads to a
xz
dependent for
E
-field, which is a product of plane wave with a
xz
-periodic function. This is known as Bloch’s theorem. Photonic crystal guided mode field resembles modes in conventional waveguide [26-27]. This is because the dispersion band diagram for the conventional waveguide is similar to the photonic crystal waveguide for rod structure. A good approximation will be a case of a metallic waveguide of width
b
, on which an artificial periodicity of
d
is imposed. The equation for the mode field of the guided modes is given as [26] (3) The modal field distribution is based on a sine function, which is dependent on the mode number,
m
and the width of the waveguide,
b
. For
m
= 1, this refers to the fundamental even mode where larger values imply higher order modes. When m =
1, 3, 5
, … , (2
υ
+1), … , it corresponds to even modes with a cosine mode profile that is symmetrical about the
x
direction. When
m
=
2, 4, 6
, … , (2
υ
), … , it corresponds to odd modes with a sine profile that is asymmetrical about the
x
direction. The constant
b
is dependent on the structure of the lattice arrangement. It corresponds to the width of the periodicity imposed waveguide, which is usually in multiples of the lattice constant.
ψ
is the phase coefficient, which is dependent on the artificial periodicity of
d
given as (4) where
β
is the wave vector of the wave and
η
is the propagation direction. For periodic structure, the modes are also characterized by the Bloch modes. The integer
η
is related to the folding of the bands at wave vector,
k = 2*
π
/d
, which is at the first Brillouin zone of the photonic crystal. The phase coefficient is conserved in the direction of propagation. The mode field for guided mode in Eq. (3) is used for the derivation of modified step-theory, which is discussed in the next section.
3. Mathematical model of Modified step-theory
The modified step-theory is an extension of the srcinal step-theory in calculating the amplitude of the transmitted modes in periodic photonic crystal waveguide. Eq. (2) is used to describe the mode field of the guided modes in the derivation of the srcinal step-theory. In the modified step-theory, Eq. (3) is used instead to account for the periodicity and discontinuous dielectric boundary in photonic crystal structures. The equations for calculating the transmission amplitudes and coupling constant are presented in this section. Subsequently, the modified step-theory is used to calculate the transmission and reflection loss in an iterative manner from one step to another until the end of the taper in the next section. Before deriving the modified step-theory, a few assumptions are made. Firstly, the coupling between reflected modes is negligible. This will simplify the derivation and for
( ) ( ) ( ) ( )
z xu zik xik r E
z x z x
k k z xk k
,expexp
,,
××∝
( )
[ ]
xib zm E E
ψ π
−
+=
exp21sin
0
( )
xd x
+=
πη β ψ
2
η
= 0,
±
1,
±
2,…,
#68765 - $15.00 USDReceived 7 March 2006; revised 19 May 2006; accepted 4 June 2006
(C) 2006 OSA26 June 2006 / Vol. 14, No. 13 / OPTICS EXPRESS 6038
gradual taper, reflection is very low with negligible coupling and adiabatic condition is achieved. Secondly, the mode coupling of reflected and transmitted radiation onto the guided modes is small and negligible. This is especially true for the reflected radiation in the case of gradual slow taper where there is only forward transmitted radiation, which is very small because near adiabatic or adiabatic condition is achieved. Thirdly, intermodal loss for short taper can be neglected because the reflection loss of the higher order mode is the main loss, as the taper becomes narrower. The higher order modes are cut-off and reflected backward. Consider a simple case where there are two guided modes,
i
and
j
traveling in a two-dimensional photonic crystal waveguide of width
b
and
artificial periodicity of
d
. Both modes travel to a narrower waveguide of width
b’
with the same periodicity. The wider waveguide is labeled as the
n
th step with subscript
n
, while the narrower waveguide is labeled as the (
n
+1)th step with subscript
n
+1. Using Eq. (3) for the description of the mode field for
i
and
j
, the incident, reflected and transmitted components for the
E
and
H
field at the boundary between steps are matched and the amplitude of transmitted mode
j
in the (
n
+1)th step is given by (5) where
A
in
refers to the ratio of the modal field amplitude of the
i
th mode at the
n
th step, in the presence of mode conversion to the initial incident amplitude that corresponds to the fundamental mode and has a unit power. Similar assignments are given for
A
jn+1
and
A
jn
. The coupling constant
c
ij
between mode
i
and
j
is given as (6a) where (6b) and (6c) where the purpose of
γ
is to account for the Bloch modes in periodic structure for convergence in high index contrast material. When
γ
is set to zero, Eq. (6b) is simplified to the case of the conventional waveguide. The field overlap integral
I
in,jn+1
is given by (7) The detail derivations of the equations for the coupling and transmission calculation are attached as in Appendix A. The field overlap integral account for the common field distribution between the incident and transmitted field profile at different steps. As the width of the taper waveguide varies along the direction of propagation, the values of
b
change. Therefore, the integral varies and is evaluated for every transmission between the steps. For the case of multiple modes propagating in the waveguide, the general expression for the transmitted
j
th mode is given as the sum of all the field profile of the incident mode which is given as (8) Based on the real and imaginary part of Eq. (7), the transmitted amplitude for the
j
th
mode is given by
dzb zmb zm I
jnin1 jnin,
+
+=
++
∫
21'sin21sin
1
π π
d
πη γ
4
=
2122
sincos
+
=
∑ ∑
+
iininijiininij1 jn
Ac Ac A
ψ ψ
( )
( )
ininiij jn1 jn
i Aci A
ψ ψ
−=−
∑
++
expexp
1
( )
( )
γ β β γ β β
γβ γβ γβ γ β β β β
++++
+++++
=
++++
11121
22
inin jn jn
in jnininin jnin
2G
1,1,
1,
+++
=
jn jninin
jninin jnij
I I I Gc
ψ ψ
)exp()exp(exp
1
jn jn jjininij jn1 jn
i Aci Aci A
ψ ψ ψ
−+−=−
++
#68765 - $15.00 USD
Received 7 March 2006; revised 19 May 2006; accepted 4 June 2006
(C) 2006 OSA26 June 2006 / Vol. 14, No. 13 / OPTICS EXPRESS 6039

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