Dispersion Characteristics of Fin Lines with Photonic Substrate

Humberto César Chaves Fernandes and Davi Bibiano Brito Department of Electrical Engineering - Federal University of Rio Grande do Norte, P. O. Box 1583,

Fax: + 55 84 2153767 , 59072-970-Natal/RN, Brazil humbeccf@ct.ufrn.br davibibiano@gmail.com

Abstract – Current and foreseen applications of photonic crystals especially the PBG (Photonic Band Gap) can be divided according to their principle of functioning. In this work an efficient method is presented for the accurate full-wave analyses of the unilateral and bilateral finlines with a 2D PBG material as substrate. To analyze its efficiency, it’s necessary to determine the phase, attenuation and effective dielectric constants. In order to analyze the behavior of the structure with this substrate, the full wave TTL (Transversal Transmission Line) method is used. Numerical results for the attenuation and effective dielectric constant of unilateral and bilateral finline with PBG substrate are presented. Index Terms – Finline; Photonic Band gap; Millimeter Waves; TTL-Transverse Transmission Line method; PBG materials; Unilateral and Bilateral Finlines. I. Introduction

Photonic Band Gap are periodic dielectric structures that forbid propagation of electromagnetic waves in a certain frequency range. This forbidden band of frequencies translates into a photonic band gap (PBG). The PBG structure was initially designed for optical applications [1]. The fact that the periodic structure is scalable makes PBG useful also in the microwave or millimeter-wave domain [2]. Application of PBG to microwave circuits has many advantages, one of which is the slow wave property that makes it possible to miniaturize the circuit size and reduce the circuit cost.

These PBG structures are made by a periodic arrangement of the index of refraction using two different materials with two different refraction indexes. In the same way that crystalline solids have bands for electrons due to the periodic atomic arrangement in the space, the periodic variation of the dielectric constant, or the index of refraction, in the photonic crystals produces a band structure for photons, with well-defined energy-momentum levels. These bands can be designed in a similar same way that bands for electrons can be engineered. The band engineering for photons allows to control photons inside these PBG materials, and obtain new properties that can be used for new optoelectronic devices [2]-[13]. The schematic illustration of this crystals geometry forms is show in Fig.1

The periodicity and the physical dimensions of the areas with different dielectric constant are related with the wavelength, or frequency, of the photons that can propagate inside the PBG material. Typical sizes and periods for these areas are inside the nanometers range for energies of

photons between the near infrared and the visible portion of the spectrum of the light.

(a) (b) (c)

Fig. 1 – PBG Structures, real and reciprocal spaces: (a) one-dimensional (b) two-dimensional (c) three-dimensional.

In this work we analyze the dispersion characteristics of

the unilateral and bilateral finlines, with a 2D PBG substrate, by the use of the TTL method. This PBG structures have acquired the widest diffusion due to the easy, albeit expensive, implementation of microelectronic technology techniques, such as photolithography [3]. As a reference example for 2D photonic crystal behavior we consider a structure based on a triangular lattice with air holes in a semiconductor material. This geometry, Fig.2 is of particular significance because it possess a complete band gap, in certain conditions, and because of its compatibility with several fabrication techniques.

Fig.2 – Top view of a 2D photonic crystal made of a triangular lattice of air holes of radius r, in a dielectric

medium with dielectric constant ε.

The 2D PBG crystal prevents photons from propagating within the plane of the active region, the only allowed modes for the light to couple into are the radiation modes. This translates to a higher flux of radiation being emitted perpendicular to the active region [5].

85-89748-04-9/06/$25.00 © 2006 IEEE ITS2006103

The proposed structures with PBG substrate are shown in Fig.3. The unilateral finline consist of two conductors fins on the sides of a dielectric substrate, adapted in the E-plane of a rectangular millimeter wave guide, as shown in Fig.3 (a). According to the figure, 2a and 2b are the height and width of the wave guide, respectly, s and g are the thickness of the regions 1 and 2 respectly, εr is the relative permittivity of the substrate material, w1 is the width of the slot and f is the infinitesimal thickness of the fin conductor. The bilateral finline has fins between the 1 and 2 regions Fig.3 (b).

Fig. 3 –Traverse section of: (a) an unilateral and (b) a bilateral finline. The TTL method is used to determine the attenuation and

effective dielectric constants of unilateral and bilateral finlines with PBG substrate, as the first time [2-9]. In this method the electromagnetic field components inside of the millimeter waveguide are obtained as functions of the fields in the “y” direction. By the use of the boundary conditions, the characteristic equation is determined, and its roots allow the obtention of the attenuation (α) and phase (ß) constants. II. Finlines Fields

A. TTL Method

The general equations of the fields in the TTL method are obtained after using the Maxwell’s equations, as [6-9]:

Where the index ‘T’ represents the transversal component directions (x, z):

zExEE zixiTi ˆˆrrr

+=

zHxHH zixiTi ˆˆrrr

+=

zz

xx

zx zxT ˆˆˆˆ∂∂

+∂∂

=∇+∇=∇

The equations are used for the analysis in the spectral

domain, in the "x" and "z" directions. Therefore it should be applied to the field, equations of Fourier transform, and the

general equations of the electric and magnetic fields in the TTL method, are obtained as:

2 21

xi n yi ji i

yiE j E Hyk∂α ωμ∂γ

−⎡ ⎤

= − Γ⎢ ⎥+ ⎣ ⎦

% % %

2 21

zi yi n yii i

E Eyk∂ ωμα∂γ

(4.1)

H⎡ ⎤

= −Γ −⎢ ⎥+ ⎣ ⎦

% % %

2 21

xi n yi ji i

H j Hyk∂α ωε∂γ

+

(4.2)

yiE⎡ ⎤

= − Γ⎢ ⎥+ ⎣ ⎦

% % %

2 21

zi yi n yii i

H Hyk∂ ωεα∂γ

+

(4.3)

E⎡ ⎤

= −Γ⎢ ⎥+ ⎣ ⎦

% % %

y

Where: , is the propagation constant in “y” direction; α

n is the spectral variable in “x” direction.

is the wave number of ith term of dielectric region; is the relative dielectric constant of the material with losses; is the dielectric constant of the ith region; is the complex propagation constant; and, is the complex angular frequency. B. The Admittance Matrix The equations below are applied, being calculated the Ey and Hy fields through the solution of the Helmoltz equations in the spectral domain [9]-[10], the equations for the regions of the unilateral finlines are: For the region 1:

1 1 1coshy eE A γ=%

1 1 1sinhy hH A yγ=% (5.1) For the region 2:

2 2 2 2 2sinh coshy e eE A y B yγ γ= +%

2 2 2 2 2sinh coshy h hH A y B yγ γ= +% (5.2) For the region 3:

( )3 3 3cosh 2y eE A aγ y= −%

( )3 3 3cosh 2y hH A a yγ= −% (5.3)

∗εμε riik

ε

=ω= 20

22 k

0rii ε⋅ε= ∗

0

iriri j

ωεε=ε ∗ σ

−

2 222ini k−Γ−α=γ

jβ+α=Γ

irjω+ω

=ω

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∇∂∂

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧−

×∇+

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

yi

yiT

yi

yiT

iiTi

Ti

H

E

yE

Hj

kH

Er

r

r

r

r

r

122

1ε

μω

γ

(3)

(2.2)

(2.1)

(1)

(4.4)

(a) (b)

104

Substituting these solutions in the TTL equations, and applying the boundary conditions, the eletromagnetic field components are determined for each region, K E~ x3 / y=t = E~ x2 / y=t = E~ xt (6.1) E~ z3 / y=t = E~ z2 / y=t = E~ zt (6.2) E~ x1 / y=s = E~ x2 / y=s (6.3) E~ z1 / y=s = E~ z2 / y=s (6.4) H~ x1 / y=s = H~ x2 / y=s (6.5) H~ z1 / y=s = H~ z2 / y=s (6.6) Determination of the propagation constants,

H~ x2 - H~ x3 / y=t = zJ% (7.1)

H~ z2 - H~ z3 / y=t = xJ− % (7.2) L

Where, and zJ% xJ− % are the current densities. A system of equations is then obtained, that in a matrix form is given as:

1 1 1 1

1 1 1 1

Yx x Yz x Exg JxEzg JzYz x Yz z

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢

⎢ ⎥ ⎣ ⎦ ⎣⎣ ⎦

% %

%

⎡ ⎤⎥⎦%

(8)

Where for the bilateral structure:

( ) ( )1 1 2 2 2 2 322 3

2 3

cothtanhn n

fgjYx x γγα γ α γωμ γ γ

⎡ ⎤−= − + −⎢ ⎥

⎣ ⎦ (8.1)

1 1 32

2 3

cothtanhn fgYx z α γγωμ γ γ

⎡− Γ= +⎢

⎣ ⎦

⎤⎥ (8.2)

1 1 32

2 3

cothtanhn fgYz x α γωμ γ γ

⎡ ⎤− Γ= +⎢ ⎥

⎣ ⎦

γ (8.3)

( ) ( )1 1 2 2 2 2 322 3

2 3

cothtanhn n

fgjYz z k k γγα αωμ γ γ

⎡ ⎤= − + −⎢ ⎥

⎣ ⎦ (8.4)

The "Y "matrix is the dyadic Green admittance function of

the bilateral finline. To eliminate the components of the current densities, the moment method was applied with the expansion of E~ xg and E~ zg in terms of known base functions, as:

(1 1 nExg Ax fx )α= %% (9.1)

(1 1 nEzg Az fz )α= %% (9.2) As a result, the equation is transformed in an homogeneous matricial equation, whose non-trivial solution corresponds to the characteristic equation, and its roots supply the phase and attenuation constants.

1 1 1 11

1 1 1 11

0

AxKx x Kx zAzKz x Kz z

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦ (10)

Where:

( ) (1 1 1 11 1n nKx x fx Yx x fx )α α

∞

−∞

=∑ % % (10.1)

( ) ( )1 1 1 11 1n nKx z fx Yx z fzα α

∞

−∞

=∑ % % (10.2)

( ) ( )1 1 1 11 1n nKz x fz Yz x fxα α

∞

−∞

=∑ % % (10.3)

( ) (1 1 1 1 11 n nKz z fz Yz z fz )α α

∞

−∞

=∑ % % (10.4)

The characteristic equation for determining the complex

propagation constant Γ, is obtained by setting the determinant of the system matrix equal to zero. The effective dielectric constant is determined after numerical solutions of the matrix determinant by the relation between the phase constant and the wave number of the free space:

εef = (β/ko)2 (11)

The characteristic impedance is then determined

2

2CV xZ

P= (12)

where the voltage in slot and the transmitted power are given by the expressions shown below :

1/ 2

1/ 2

w

X xw

V E+

−

= ∫ (13) gdx

(2

* *

0

1 Re8

a

x y y xP E H E H+∞

−∞)dy

⎡ ⎤= −⎢ ⎥

⎣ ⎦∑ ∫ % % % % (14)

III. PBG Structure

For a non-homogeneous structure submited, the incident sign goes at the process of multiple spread. A solution can be obtained through a numerical process called homogenization [13-18]. The process is based in the theory related to the diffraction of an incident electromagnetic plane wave imposed by the presence of a air immerged cylinders in a homogeneous material.

The process is based in the theory related to the diffraction of the incident electromagnetic plane wave imposed by the presence of air cylinders immerged in an homogeneous material [13].

105

In the Cartesian coordinates system of axes (O, x, y, z), are shown in the Fig. 3. A cylinder is considered with relative permittivity ε1, with a traverse section in the plane xy, embedded in a medium of permittivity ε2. For this process the two-dimensional structure is sliced in layers whose thickness is equal at the cylinder diameter. In each slice is realized the homogenization process.

Fig. 3 – Homogenized bidimensional crystal.

According to homogenization theory the effective permittivity depends on the polarization [13]. For the s and p polarization, respectively, we have:

( ) 221 εεεβε +−=eq

( )⎪⎭⎪⎬⎫

⎪⎩

⎪⎨⎧

+−+−= 3/143/10

211

3111βββ

βεε OAAeq

Where:

21

211 /1/1

/1/2εεεε

−+

=A

( )

21

212 /13/4

/1/1εεεεα

+−

=A

And β is defined as the ratio between the area of the cylinders and the area of the cells, α is an independent parameter whose value s equal to 0.523. The A1 and A2 variables in (17) and (18) were included only for simplify (16) equation. IV. Numerical Results

The computational program used to calculate the effective

dielectric constant and the attenuation constant for unilateral and bilateral finline, with PBG substrate, was developed in Fortran PowerStation and Matlab for Windows.

The structures are analyzed in a WR-28 millimeter wave guide with dimensions 2a = 7.112 mm and 2b = 3.556 mm. Fig. 4 and Fig.5 shows the attenuation constant (α) as functions of the frequency for s and p polarizations,

respectively, for two different slot widths in a bilateral finline with PBG substrate.

Fig. 4 – Attenuation constant as function of the frequency, in a bilateral finline.

(15)

(16)

Fig. 5 – Attenuation constant as function of the frequency, in a unilateral

finline.

The effective dielectric constant as function of the frequency is shown in the Fig. 6 and Fig. 7, for different slot widths of a bilateral and unilateral finline, respectively. In these figures the effective dielectric constant increase when the frequency increase.

(17)

(18)

106

Fig. 6 – Effective dielectric constant as a function of the frequency, in a bilateral finline.

Fig. 7 – Effective dielectric constant as a function of the frequency, in a unilateral finline.

Fig. 8 and Fig.9 shows the characteristic impedance as function of the frequency.

Fig. 8 - Characteristic impedance as function of the frequency, in a bilateral

finline.

Fig. 9 - Characteristic impedance as function of the frequency, in a unilateral finline.

V. Conclusions The full wave transverse transmission line (TTL) method was used to the characterization of the unilateral and bilateral finlines, considering a 2D Photonic Band Gap (PBG) substrate, at applications in millimeter waves. Numerical results for the attenuation and effective dielectric constant of unilateral and bilateral finlines with PBG substrate were presented. This work was partially supported by CNPQ.

References [1] E. Yablonovitch, “Inhibited spontaneous emission in solid-

state physics and electronics,” Phys. Rev. Lett., vol. 58, pp. 2059–2062, May 1987.

[2] Y. Qian and T. Itoh, “Microwave applications of photonic bandgap (PBG) structures,” in Proc. 1999 Asia Pacific Microwave Conf., vol. 2, Singapore, 1999, pp. 315–318.

[3] C. López, “Materials aspects of photonic crystals”, Advanced aterials, Vol.15, No. 20, 16. pp. 1679 -1704, Oct. 2003.

[4] J. P. Silva, S. A. P Silva and H. C. C. Fernandes, “Coupling analysis at the coupler and unilateral edge-coupled fin line”, International Conference on Millimeter and Submillimeter Waves and Applications II, SPIE's 1998 International Symposium on Optical Science, Engineering and Instrumentation, San Diego, CA, USA. Conf. Proc. pp. 53-54, Jul. 1998.

[5] Kuo-Yi Lim, “Design and Fabrication of One-dimensional and Two-dimensional Photonic Bandgap Devices,” Ph. D., November, 1998, now at Boston Consulting Group.

[6] O. S. D. Costa, S. A.P. Silva and H. C. C. Fernandes, "3D complex propagation of coupled unilateral and antipodal arbitrary finlines", Brazilian CBMAG'96-Congress of Electromagnetism, Ouro Preto-MG, pp. 159-162, Nov. 1996.

[7] P. J. Meier, “Integrated fin-line millimeter components”, IEEE Trans. on Microwave Theory and Tech., Vol. MTT-22, pp. 1209-1216, Dec. 1974.

[8] B. Baht and S. K. Koul, “Analysis, design and applications of finlines”, Artech House, 1987.

[9] H.C.C. Fernandes and M. C. Silva, “Dynamic TTL method applied to the fin-line resonators”, Journal of Microwaves and Optoelectronics, Vol.2,N.3, pp.57-66, Jul.2001.

[10] H.C.C. Fernandes and M. C. Silva, “Fin-Line resonators by TTL method “, Special Issue on Signals, Systems and Electronics Technology, IEICE-Transactions, pp.599-601, Tokyo -Japan, Mar. 2002.

[11] H.C.C. Fernandes, “TTL method applied to the fin-line resonators”, Journal of Electromagnetic Waves and Applications, USA, Vol. 15, 11pp. 2001.

[12] H.C.C. Fernandes, E. A. M. Souza and I. de S. Queiroz Jr., “Metallization thickness in bilateral and unilateral finlines”, International Journal of Infrared and Millimeter Waves”, Vol. 15 º.6, pp. 1001-1014, Plenum Press, USA, Jun.1994.

[13] E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals”, J. Optical Soc. American, Vol. 17, No.2, pp.320-327, Feb. 2000.

[14] D. S. Wiersma, P. Bartolini, Ad Lagendijk, and R. Righini, “Localization of light in a dimensional medium”, Letters to Nature - Vol.390 18/25, Dec. 1997.

[15] U. Gruning, V. Lehmann, S. Ottow, and K. Bush, “Macroporus silicon with a complete two dimensional photonic band gab centered at 5 μm”, Appl. Phys. Lett. 68 Feb. 1998.

107

[16] V. Radisic, Y. Qian, R. Coccioli, and T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines”, IEEE Microwave and Guided Wave Letters, Vol. 8 No. 2 Feb. 1998.

[17] K. Guillouard, M. F. Wong, V. Fouad Hanna, J. Citerne, “Diakoptics using finite element analysis”, MWSYM 96 Vol. 1, p. 363-366.

[18] J. Tan, G. Pan, “A general functional analysis to dispersive structures”, MWSYM 96 Vol. 2, p. 1027-1030.

[19] A. R. B. Rocha and H. C. C. Fernandes, “Analysis of antennas with PBG substrate”, International Journal of Infrared and Millimeter Waves, Vol. 24, pp. 1171-1176, USA, Jul. 2003.

[20] H. C. C. Fernandes and S. P. Santos, " Change in the directivity of planar array with PBG substrate”, WSEAS Trans. on Communications, Athens, Greece, pp. 433-437, Jul. 2004.

[21] H. C. C. Fernandes, E. A. D. Leal, D. M. Alvarenga and D. B. Brito, “Applications of photonic substrate at millimeter wave frequencies”, SBPMat2005, Proc. CD, Recife-PE, Brazil, Oct. 2005.

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