| Summary: | 碩士 === 國立成功大學 === 機械工程學系 === 102 === SUMMARY
We numerically analyze photonic crystals with Archimedean tilings arrangement, which were consisted of dielectric materials and magneto-optical (MO) materials. In the beginning, the dispersion curves of three different type of photonic crystals were calculated by plane wave expansion method. Among them, the one which is composed of air holes in the MO background is suitable for applications. We futher remove six air holes of the photonic crystal to form a ring cavity, and the whispering-gallery-mode is observed in it. When the magnetic field is applied along the out-of-plane direction, the permittivity of MO materials will be changed and the resonant frequency of whispering-gallery-mode will split. Besides, we introduce different size of circular hole and ring-shape hole into the center of the ring cavity to optimize the efficacy. Finally, a common photonic crystal which is consisted of air holes in the silicon background at the same filling ratio is used to compare the performance.
Key words: Magneto-optical materials, whispering-gallery-mode, Archimedean tilings
INTRODUCTION
Over past decades, people pay a lot of attention to a kind of periodic dielectric structure named photonic crystal. In certain regions of the frequency, the optical wave can not pass the photonic crystals, and these regions are called photonic band gaps. When point defects or line defects are introduced into the perfect photonic crystals, it will cause highly localized resonance and guide modes relatively at the frequency within the band gap, and thus many novel applications such as sensors, filters and coupler are realized.
Archimedean tilings are constituted by regular convex polygons which are not identical necessarily. There are not any intervals or overlaps between these regular convex polygons, therefore, the vertices are all identical. Although there exists many kinds of Archimedean tilings, once the parameters of photonic crystal are determined, the wave propagation behaviors are fixed. Thus, how to tune the properties of photonic crystals becomes an important issue. Magneto-optical materials are one kind of tunable materials. When the external magnetic field is employed, the permittivity or the permeability will be changed. Based on this phenomenon, tunable photonic crystal devices can be designed. In this work, we combine photonic crystals with Archimedean tilings, and the wave propagation properties, i.e. width of band gaps and resonant frequencies of photonic crystal which is composed of magneto-optical materials with Archimedean tilings arrangement are studied. The influence of external magnetic field is analyzed, too. Computations are performed by using plane wave expansion and finite element software.
MATERIALS AND METHODS
In this section, we will illustrate the plane wave expansion method which is used to calculate the dispersion curves of photonic crystals.
The Maxwell equations are shown below:
(1)
(2)
Where E ⃑ is the intensity of electric field, H ⃑ is the intensity of magnetic field, ε _r is the relative permittivity, ε _0 is permittivity in vacuum, and μ _0 is permeability in vacuum.
Assume that the electric field and the magnetic field are both time harmonic functions and can be defined as:
(3)
(4)
When we introduce equations (3) and (4) into equation (1) and (2), the Helmholtz’s equation can be derived, as shown below:
(5)
According to Bloch’s theorem, the magnetic field in the periodic structure can be expressed as:
(6)
Where G ⃑ is the reciprocal lattice vector, k ⃑ is Bloch’s wavevector, and (e _λ ) ̂ are two unit vectors which are perpendicular to (k ⃑+G ⃑).
Since 1 /(ε _r ( r ⃑) ) is a periodic function, it can be expanded into Fourier series, as shown below:
(7)
(8)
Where Ω represents the unit cell, and V is the volume of the unit cell.
By substituting equation (7) and (8) into equation (5), we can obtain an eigenfunction defined as:
(9)
Equation (9) can be separated into two different eigenfunctions as shown as equation (10) and (11) relatively according to E-polarization(TE) and H-polarization(TM).
(10)
(11)
If the external magnetic field is applied along z direction, the permittivity of the MO materials can be defined as[72]:
ε ̂= [■(ε _xx ＆ε _xy ＆0 @ε _yx ＆ε _yy ＆0 @0 ＆0 ＆ε _zz )] (12)
And the equation (9) will be modified as shown below:
∑_(G ^' )▒|k ⃑+ G ⃑ ||k ⃑+ G ⃑^' |[■(e ̂_2 ∙κ( G ⃑-G ⃑^' ) e ̂_2 ^' ＆-e ̂_2 ∙κ( G ⃑-G ⃑^' ) e ̂_1 ^' @-e ̂_1 ∙κ( G ⃑-G ⃑^' ) e ̂_2 ^' ＆e ̂_1 ∙κ( G ⃑-G ⃑^' ) e ̂_1 ^' )]{■(h _(1, G ^' )@h _(2, G ^' ) )} = ω ^2 /c ^2 {■(h _(1, G ^' )@h _(2, G ^' ) )} (13)
Equation (13) can be separated into two different eigenfunctions as shown as equation (14) and (15) relatively according to E-polarization(TE) and H-polarization(TM).
∑_(G ^' )▒〖H ⃑_k ^TM (G ⃑, G ⃑^' ) h _(1, G ⃑^' ) = ω ^2 /c ^2 h _(1, G ⃑ ) 〗 (14)
H ⃑_k ^TM (G ⃑, G ⃑^' )=ε _xx ^-1 (G ⃑-G ⃑^' )(k _y + G _y )(k _y + G _y ^' )
+ ε _yy ^-1 (G ⃑-G ⃑^' )(k _x + G _x )(k _x + G _x ^' )
-ε _xy ^-1 (G ⃑-G ⃑^' )(k _y + G _y )(k _x + G _x ^' )
-ε _yx ^-1 (G ⃑-G ⃑^' )(k _x + G _x )(k _y + G _y ^' )
∑_(G ^' )▒〖H ⃑_k ^TE (G ⃑, G ⃑^' ) h _(2, G ⃑^' ) = ω ^2 /c ^2 h _(2, G ⃑ ) 〗 (15)
H ⃑_k ^TE (G ⃑, G ⃑^' )=ε _zz ^-1 ( G ⃑-G ⃑^' ) |k ⃑+ G ⃑ ||k ⃑+ G ⃑^' |
The two equations above indicate that the off-diagonal elements of the dielectric tensor will affect the dispersion relationship of photonic crystals only when the optical wave is TM polarized. Thus, we just discuss the behavior of TM polarized waves.
RESULTS AND DISCUSSION
The unit cell of the photonic crystal with Archimedean tilings arrangement is shown in figure 1. It is consisted of air holes arranged in MO background whose permittivity is 4.75. The largest width of band gap whose value is 0.025(a/λ) appears when the radius of air holes are 0.38a,as shown in figure 2, where a is the side of the regular polygon. We futher remove six air holes of the photonic crystal to creat a ring cavity, as shown in figure 3. The plane wave is incident from left, and there exists five eigenmode in the range of band gap. Moreover, the whispering-gallery-mode occurs when the normalized frequency is 0.351673(a/λ), and its quality factor is 3885. If the wavelength of this frequency is designed to locate at 1550 nm, then the value of side of polygon is 545.093 nm. When applying an external magnetic field to the photonic crystal, the resonant frequency of the whispering-gallery-mode will separate. Figure 4 shows the wavelength spectrum when g is 0.4. As long as the magnetic field is large enough, the two resonant peak can be identified, and the quality factor can maintain at a high level. By appropriately introducing circular hole or ring-shape hole into the center of the ring cavity, the performances of the cavity will be better. Finally, a common photonic crystal which is consisted of air holes arranged in silicon background is analyzed. Because the contrast of impedance between silicon and air is larger than that between MO materials and air. Thus, the value of quality factor and separation of the latter are more higher than the former.
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