Simple Relations Between a Uniaxial Medium and an Isotropic Medium

(1)

Simple Relations between a Uniaxial Medium

and an Isotropic Medium

Saffet G. S¸en*

Abstract—In this article, in a simple way, simple relations are derived between the electric field components of an electrically uniaxial medium and those of an isotropic medium. The permittivity of the isotropic medium is the same as the permittivity of the uniaxial medium that is common to the axes transverse to the optic axis. Using the spectral representation, the vector wave equation for the electric field intensity vector of the uniaxial medium is solved for thexdirected,y directed andzdirected point sources. For thex directed andydirected point sources, the electric field components transverse to the optic axis are written in terms of the corresponding components of the isotropic medium plus some other terms. Part of these terms are closed forms expressions, and the rest are Sommerfeld type integrals. Elements of each group are related to each other by coordinate transformations. The electric field components parallel to the optic axis are shown to be obtained from the isotropic medium components using coordinate transformations. The relations between the uniaxial medium and isotropic medium field components are verified by comparing the results of a previous study in the literature to the results obtained using the relations in this study. Good agreement is achieved between these results.

1. INTRODUCTION

Anisotropic materials are materials whose electromagnetic properties depend on the direction. An electrically uniaxial material is a special case of anisotropic materials. An electrically uniaxial material has a permittivity along its distinguished or optic axis which is different from the common permittivity in the directions of the other two axes.

Anisotropic materials have a wide application range in technology. Microwave integrated circuits, resonators, microstrip antennas are some of the application areas. A great amount of literature on anisotropic materials exists. Some of the studies on the propagation of electromagnetic waves in homogeneous uniaxial media are given in [1–4]. In [1], using some coordinate variable transformations in Maxwell’s equations, equivalence is established between TMz electromagnetic waves in electrically uniaxial media andTMzwaves in isotropic media. A similar equivalence is built forTEzelectromagnetic waves. In [2], electrically uniaxial medium is studied in a separate chapter. The relations between the electrically uniaxial medium fields and isotropic medium fields are described by using scalar potential functions. The scalar potentials for the electrically uniaxial medium are expressed in terms of scalar potentials for the isotropic medium. In [3], the dyadic Green’s functions for an electrically uniaxial medium are determined in closed forms using the 3-dimensional Fourier transform. In [4], the dyadic Green’s functions are derived for a general uniaxial medium, i.e., a medium which is magnetically and electrically uniaxial medium, by solving a dyadic differential equation. Apart from these articles, the free-space Green functions for several types of anisotropic media are studied in the references [8–28]. The basics of uniaxial media are given in the books in [2, 3, 5–7].

In this article, the Green’s functions for the components of the electric field for an electrically uniaxial medium are found for three kinds of point sources, an x directed point source, a y directed

Received 13 April 2014, Accepted 28 May 2014, Scheduled 6 June 2014

* Corresponding author: Saffet Gokcen Sen ([email protected]).

(2)

point source and az directed point source. They are related to the Green’s functions of the isotropic medium with a permittivity the same as the permittivity of the uniaxial medium that is common to the axes transverse to the optic axis. The solution is made using the spectral representation method given in [5]. Each of the Green’s functions contains a term which is a scaled version of the corresponding isotropic medium Green’s function. For the components transverse to the optic axis, there is another term which is a Sommerfeld type integral, and the rest of the terms are in closed form. Sommerfeld integral type terms are convertible to each other by using coordinate scalings. The same conversion is possible for the closed form terms.

In Section 2, spectral domain wave equations are derived for the electric field components of the uniaxial medium. In Section 3, these spectral domain equations are solved for three basic point source configurations. In Section 4, the relations obtained are verified by comparing their results to those obtained from the closed form expressions given in [4]. Good agreement is obtained except for the field points with zenith angleθ close to 0.

2. SPECTRAL DOMAIN WAVE EQUATIONS

For the electrically uniaxial medium with its distinguished axis in the ˆz direction, first two of the Maxwell’s equations can be written in the following form assuming the time harmonic dependence of

e−jωt_:

~

∇ ×E~ =jωµ ~H (1)

~

∇ ×H~ =J~−jω¯¯²·E~ (2)

In these equations,E~ is the electric field intensity vector, H~ the magnetic field intensity vector, and J~

the electric current density vector. The scalarµ is the permeability of the medium, and the tensor ¯¯²is the permittivity tensor of the medium given by

¯¯

²= ˆxx²ˆ + ˆyy²ˆ + ˆzz²ˆ z (3)

ω is the angular frequency andj the imaginary unit. Taking the curl of both sides of Equation (1) and using the Gauss law given by

~

∇ · ³

¯¯

²·E~

´

= 1

jω∇ ·~ J,~ (4)

the following wave equation is obtained forE~:

∇2E~ +ω2µ²¯¯·E~ +∇~∇~.

·

ˆ

z(²z−²) ² Ez

¸

=−jωµ ~J+∇~ Ã

~

∇ ·J~

jω²

!

(5)

The spectral representation or 3-dimensional inverse Fourier transform of a scalar F is defined as follows [5]:

F(x, y, z) = 1 (2π)3

Z Z Z _∞

−∞

˜

F(k_x, k_y, k_z)ej~k·~rd~k (6)

where ˜F is the spectral domain form ofF and

~k= ˆxkx+ ˆyky+ ˆzkz, ~r= ˆxx+ ˆyy+ ˆzz, d~k=dkxdkydkz (7) If the spectral expansion is applied to Equation (5), then the following three equations are obtained:

¡

−k2+ω2µ²¢E˜x+(²z_²−²)(−kxkz) ˜Ez=

µ

−jωµ− k2x

jω²

¶

˜

Jx−k_jω²xkyJ˜y −k_jω²xkzJ˜z (8)

¡

−k2+ω2µ²¢E˜y+(²z−_² ²)(−kykz) ˜Ez=

Ã

−jωµ− k

2 y

jω²

!

˜

Jy− k_jω²xkyJ˜x− k_jω²ykzJ˜z (9)

¡

−k2+ω2µ²¢E˜_z+ (²z−²)

²

¡

−k_z2¢E˜_z=

µ

−jωµ− k2z

jω²

¶

˜

J_z−kxkz

jω²J˜x− kykz

jω² J˜y (10)

where

(3)

3. BASIC POINT SOURCE CONFIGURATIONS

In this section, the spectral domain wave equations are to be solved for basic point source configurations. They are the ˆx directed, ˆy directed and ˆzdirected point source configurations.

3.1. ˆx Directed Point Source

The ˆxdirected point source is given as follows:

~

J = ˆxIlδ(x)δ(y)δ(z) (12)

where

Il= 1A·m (13)

The spectral domain representation for this source is ˜

Jx= 1, J˜y = 0, J˜z= 0 (14)

If the spectral domain equations are solved for this point source, then the following expressions are obtained for spectral E~ components:

˜

Exxuni =

jωµ

(k_z−k_z₁) (k_z−k_z₂) −

jωµ

(k_z−k_z₃) (k_z−k_z₄)

−

jωµ kx2 k2

x+ky2 (k_z−k_z₁) (k_z−k_z₂) +

jωµ kx2 k2

x+ky2 (k_z−k_z₃) (k_z−k_z₄) +

jωµ+ k2x

jω²z

(k_z−k_z₃) (k_z−k_z₄) (15)

˜

Eyxuni =

−jωµ kxky

k2 x+ky2 (kz−kz1) (kz−kz2)

+

jωµ kxky k2

x+k2y (kz−kz3) (kz−kz4)

+

1

jω²_zkxky

(kz−kz3) (kz−kz4)

(16)

˜

Ezxuni =

1

jω²zkxkz (kz−kz3) (kz−kz4)

(17)

where

kz1 = q

ω2_µ²₋_k2

x−ky2, kz2 =− q

ω2_µ²₋_k2

x−k2y (18)

k_z₃ =

r

ω2_µ²₋ ²

²z

k2 x−

² ²z

k2

y, kz4 =− r

ω2_µ²₋ ²

²z

k2_x− ²

²z

k2_y (19)

Thekz integration in the spectral representation is performed by the residue integration indicated in [5] to obtain the following integrals:

Exxuni =Ixx1 −Ixx2−Ixx3 +Ixx4+Ixx5 (20)

Ixx1 = ³

−ωµ

8π2

´ Z Z ∞

−∞

ejkz1|z|ej~kt·~rt

kz1

d~kt=jωµe jkr

4πr (21)

Ixx2 = ³

−ωµ

8π2

´ Z Z ∞

−∞

ejkz₃|z|_ej~kt·~rt

kz3

d~kt (22)

I_xx₃ =

³

−ωµ

8π2

´ Z Z ∞

−∞ k2

x

k2 x+ky2

kz1

ejkz₁|z|_ej~kt·~rt_d~k

t (23)

Ixx4 = ³

−ωµ

8π2

´ Z Z ∞

−∞ k2

x

k2 x+ky2

k_z₃ e

jkz₃|z|_ej~kt·~rt_d~k

(4)

I_xx₅ = j 8π2

Z Z _∞

−∞

jωµ+ k 2 x

jω²z

kz3

ejkz3|z|ej~kt·~rt_d~k

t (25)

Eyxuni =Iyx1 +Iyx2 +Iyx3 (26)

Iyx1 =

ωµ

8π2

Z Z _∞

−∞ kxky

k2 x+ky2

k_z₁ e

jkz₁|z|_ej~kt·~rt_d~k

t (27)

Iyx2 = ³

−ωµ

8π2

´ Z Z ∞

−∞ kxky

k2 x+ky2

kz3

t (28)

I_yx₃ = 1 8π2_ω²_z

Z Z _∞

−∞ kxky

kz3

ejkz₃|z|_ej~kt·~rt_d~k

t (29)

Ezxuni =Izx1 =

1 8π2_ω²_z

Z Z _∞

−∞

|z|

z kxe

t (30)

In the above integrals, the following definitions are used:

~kt= ˆxkx+ ˆyky, ~rt= ˆxx+ ˆyy, d~kt=dkxdky (31) Each of the integral groups contains some integrals which are not totally independent of each other. Using the variable transformations given by

k_x=

r

² ²z

k0_x, k_y =

r

² ²z

k_y0, (32)

the following relations can be derived:

Ixx2(x, y, z) =

³_²

z

²

´

Ixx1 µr

²z

²x,

r

²z

²y, z

¶

(33)

Ixx4(x, y, z) =

³_²

z

²

´

Ixx3 µr

²z

²x,

r

²z

²y, z

¶

(34)

I_yx₂(x, y, z) =−

³_²

z

²

´

I_yx₁

µr

²_z ²x,

r

²_z ²y, z

¶

(35)

In addition to these symmetry relations, some integrals are related with the spectral field components for the isotropic medium with permittivity ². If the spectral domain scalar wave equations are solved for the isotropic medium, then the following integrals are obtained for the electric field components:

Exxiso =

j

8π2

Z Z _∞

−∞

jωµ+ kx2

jω² kz1

t (36)

E_yx_iso = 1 8π2_ω²

Z Z _∞

−∞ kxky

kz1

t (37)

Ezxiso =

1 8π2_ω²

Z Z _∞

−∞

|z|

z kxe

jkz1|z|ej~kt·~rt_d~k

t (38)

Utilizing the variable transformation in (32), the following relations can be written:

Ixx5 =

³_²

z

²

´

Exxiso µr

²z

²x,

r

²z

²y, z

¶

(39)

Iyx3 =

³_²

z

²

´

Eyxiso µr

²z

²x,

r

²z

²y, z

¶

(40)

Ezxuni = r

²z

²Ezxiso µr

²z

² x,

r

²z

²y, z

¶

(5)

Sommerfeld integral forms are to be derived by applying a series of variable transformations followed by the application of an integral identity for Bessel functions of the first kind as indicated in [5]. The variable transformations and related expressions are as follows:

kx=kρcos (α), ky =kρsin (α) (42)

~kt·~rt=kρρcos (α−φ), dkxdky =kρdkρdα (43)

x=ρcos (φ), y=ρsin (φ) (44)

The identity for the Bessel functions of the first kind is given by

Jn(kρρ)ejnφ= ₂1_π

Z _2π

0

ejkρρcos(α−φ)+jnα−jnπ₂_dα ₍₄₅₎

In the Sommerfeld integrals, the integration path denoted byP is the Sommerfeld integration path, and the branch of the square root relation used for the calculations ofkz1 is the Im{kz1}= 0 branch [5].

Ixx3 =

jωµ

8π ejkr

r +

·

ωµcos (2φ) 16π

¸ Z

P

kρ

k_z₁H

(1)

2 (kρρ)ejkz1|z|dkρ (46)

I_yx₁ =

·

−ωµsin (2φ)

16π

¸ Z

P

kρ

kz1

H₂(1)(k_ρρ)ejkz₁|z|_dk

ρ (47)

Hence,xcomponent of the uniaxial medium electric field can be related to that of the isotropic medium electric field as follows:

Exxuni =Ixx1 −Ixx2−Ixx3 +Ixx4+Ixx5 (48)

where

I_xx₁(x, y, z) =jωµejkr

4πr (49)

Ixx2(x, y, z) =

³_²

z

²

´

Ixx1 µr

²z

²x,

r

²z

²y, z

¶

(50)

Ixx3 =

jωµ

8π ejkr

r +

·

ωµcos (2φ) 16π

¸ Z

P

k_ρ kz1

H₂(1)(kρρ)ejkz1|z|dkρ (51)

I_xx₄(x, y, z) =

³_²

z

²

´

I_xx₃

µr

²_z ²x,

r

²_z ²y, z

¶

(52)

Ixx5(x, y, z) =

³_²

z

²

´

Exxiso µr

²z

²x,

r

²z

²y, z

¶

(53)

As toy component of the uniaxial medium electric field, the following equations are valid:

E_yx_uni =I_yx₁+I_yx₂+I_yx₃ (54)

where

Iyx1 = ·

−ωµsin (2φ)

16π

¸ Z

P

k_ρ kz1

I_yx₂(x, y, z) =−

³_²

z

²

´

I_yx₁

µr

²z

²x,

r

²z

²y, z

¶

(56)

Iyx3(x, y, z) =

³_²

z

²

´

Eyxiso µr

²z

²x,

r

²z

²y, z

¶

(57)

3.2. ˆy Directed Point Source

The ˆy directed point source is given as follows:

~

(6)

where

Il= 1A·m (59)

Jx= 0, J˜y = 1, J˜z= 0 (60)

If the spectral domain equations are solved for this point source, then the following expressions are obtained for spectral E~ components:

˜

Exyuni =

−jωµ kxky

k2 x+k2y (kz−kz1) (kz−kz2)

+

jωµ kxky k2

x+ky2 (kz−kz3) (kz−kz4)

+

1

jω²zkxky (kz−kz3) (kz−kz4)

(61)

˜

Eyyuni =

jωµ

(kz−kz1) (kz−kz2)

− jωµ

(kz−kz3) (kz−kz4)

−

jωµ k

2 y

k2 x+ky2 (kz−kz1) (kz−kz2)

+

jωµ k

2 y

k2 x+k2y (kz−kz3) (kz−kz4)

+

jωµ+ k 2 y

jω²z (kz−kz3) (kz−kz4)

(62)

˜

Ezyuni =

1

jω²z

k_yk_z

(k_z−k_z₃) (k_z−k_z₄) (63)

Ifkz integration in the spectral representation is performed, the following expressions are obtained:

E_xy_uni =I_xy₁+I_xy₂ +I_xy₃ (64)

Ixy1 =

ωµ

8π2

Z Z _∞

−∞ kxky

k2 x+ky2

kz1

t (65)

Ixy2 = ³

−ωµ

8π2

´ Z Z ∞

−∞ kxky

k2 x+k2y

kz3

t (66)

Ixy3 =

1 8π2_ω²_z

Z Z _∞

−∞ kxky

kz3

t (67)

E_yy_uni = = I_yy₁−I_yy₂−I_yy₃ +I_yy₄ +I_yy₅ (68)

Iyy1 = ³

−ωµ

8π2

´ Z Z ∞

−∞

ejkz1|z|ej~kt·~rt

kz1

d~kt=jωµe jkr

4πr (69)

Iyy2 = ³

−ωµ

8π2

´ Z Z ∞

−∞

ejkz₃|z|_ej~kt·~rt

kz3

d~kt (70)

Iyy3 = ³

−ωµ

8π2

´ Z Z ∞

−∞ k2

y

k2 x+k2y

k_z₁ e

t (71)

Iyy4 = ³

−ωµ

8π2

´ Z Z ∞

−∞ k2

y

k2 x+k2y

kz3

t (72)

Iyy5 =

j

8π2

Z Z _∞

−∞

jωµ+ k 2 y

jω²z

kz3

t (73)

Ezyuni =Izy1 =

1 8π2_ω²

z

Z Z _∞

−∞

|z|

z kye

(7)

Similar to the case for the ˆx directed source, the following relations can be written:

Ixy2(x, y, z) =−

³_²

z

²

´

Ixy1 µr

²z

²x,

r

²z

²y, z

¶

(75)

Iyy2(x, y, z) =

³_²

z

²

´

Iyy1 µr

²z

²x,

r

²z

²y, z

¶

(76)

I_yy₄(x, y, z) =

³_²

z

²

´

I_yy₃

µr

²_z ²x,

r

²_z ²y, z

¶

(77)

If the spectral domain equations are solved for the isotropic medium with permittivity ² for the y

directed point source, then the following integrals can be derived:

Exyiso =

1 8π2_ω²

Z Z _∞

−∞ kxky

kz1

t (78)

E_yy_iso = j 8π2

Z Z _∞

−∞

jωµ+ k 2 y

jω² kz1

t (79)

Ezyiso =

1 8π2_ω²

Z Z _∞

−∞

|z|

z kye

jkz1|z|ej~kt·~rt_d~k

t (80)

Comparing the isotropic medium solutions with the uniaxial medium solutions yields the following equations:

I_xy₃ =

³_²

z

²

´

E_xy_iso

µr

²_z ²x,

r

²_z ²y, z

¶

(81)

Iyy5 =

³_²

z

²

´

Eyyiso µr

²z

²x,

r

²z

²y, z

¶

(82)

Ezyuni = r

²z

²Ezyiso µr

²z

²x,

r

²z

²y, z

¶

(83)

After the Sommerfeld integral representations of integrals I_xy₁ and I_yy₃ are derived, the following equations can be written:

Ixy1 = ·

−ωµsin (2φ)

16π

¸ Z

P

kρ

kz1

Iyy3 =

jωµ

8π ejkr

r −

·

ωµcos (2φ) 16π

¸ Z

P

k_ρ kz1

Hence,x and y components of the electric field for the uniaxial medium can be written as follows:

Exyuni =Ixy1+Ixy2 +Ixy3 (86)

where

Ixy1 = ·

−ωµsin (2φ)

16π

¸ Z

P

kρ

kz1

H₂(1)(kρρ)ejkz1|z|dk_ρ (87)

Ixy2(x, y, z) =−

³_²

z

²

´

Ixy1 µr

²z

²x,

r

²z

²y, z

¶

(88)

Ixy3(x, y, z) =

³_²

z

²

´

Exyiso µr

²_z ²x,

r

²_z ²y, z

¶

(89)

(8)

where

Iyy1(x, y, z) =jωµ

ejkr

4πr (91)

Iyy2(x, y, z) =

³_²

z

²

´

Iyy1 µr

²z

²x,

r

²z

²y, z

¶

(92)

I_yy₃ =jωµ 8π

ejkr

r −

·

ωµcos (2φ) 16π

¸ Z

P

k_ρ kz1

H₂(1)(k_ρρ)ejkz1|z|dk_ρ (93)

Iyy4(x, y, z) =

³_²

z

²

´

Iyy3 µr

²z

²x,

r

²z

²y, z

¶

(94)

Iyy5(x, y, z) =

³_²

z

²

´

Eyyiso µr

²z

²x,

r

²z

²y, z

¶

(95)

3.3. ˆz Directed Point Source

The ˆzdirected point source is given as follows:

~

J = ˆzIlδ(x)δ(y)δ(z) (96)

where

Il= 1A·m (97)

Jx= 0, J˜y = 0, J˜z= 1 (98)

The spectral domain field components can be written as follows:

˜

E_xz_uni =

² ²z

kxkz

jω²

(kz−kz3) (kz−kz4)

(99)

˜

Eyzuni =

² ²_z

kykz

jω²

(kz−kz3) (kz−kz4)

(100)

˜

Ezzuni =

² ²z

µ

jωµ+ kz2

jω²

¶

(kz−kz3) (kz−kz4)

(101)

After performing thekzintegration in the spectral representation, the following two-dimensional integral forms can be obtained for the electric field components:

Exzuni =

1 8π2_ω²_z

Z Z _∞

−∞

|z|

z kxe

t (102)

E_yz_uni = 1 8π2_ω²_z

Z Z _∞

−∞

|z|

z kye

t (103)

Ezzuni = µ

− 1

8π2_ω² z

¶ Z Z _∞

−∞

ω2_µ²₋_k2 z3

k_z₃ e

t (104)

The counterparts of these expressions in the isotropic medium are

Exziso =

1 8π2_ω²

Z Z _∞

−∞

|z|

z kxe

t (105)

E_yz_iso = 1 8π2_ω²

Z Z _∞

−∞

|z|

z kye

t (106)

Ezziso = µ

− 1

8π2_ω²

¶ Z Z _∞

−∞

ω2_µ²₋_k2 z1

k_z₁ e

(9)

If these two groups of expressions are compared to each other, then the following relations can be derived:

Exzuni(x, y, z) = r

²z

²Exziso µr

²z

²x,

r

²z

²y, z

¶

(108)

E_yz_uni(x, y, z) =

r

²_z ²Eyziso

µr

²_z ²x,

r

²_z ²y, z

¶

(109)

Ezzuni(x, y, z) =Ezziso µr

²z

²x,

r

²z

²y, z

¶

(110)

In addition, the Sommerfeld integral forms for electric field components can be found as follows:

Exzuni =

jcos (φ) 8πω²z

Z

P

|z|

z k

2

ρH1(1)(kρρ)ejkz3|z|dkρ (111)

Eyzuni =

jsin (φ) 8πω²z

Z

P

|z|

z k

2

ρH1(1)(kρρ)ejkz3|z|dkρ (112)

Ezzuni = µ

− ²

8πω²2 z

¶ Z

P

k3 ρ

kz1

H₀(1)(kρρ)ejkz3|z|dk_ρ (113)

4. RESULTS

In this section, the validity of the formulas derived for the electric field components in an electrically uniaxial medium are to be shown using numerical results. The reference to be used is the article by Weiglhofer [4]. The sources are assumed at the origin of the coordinate system.

Forx directed point source, the electric field is obtained in [4] as follows:

~ Euni =

µ

− 1

jω²

¶ ½

~

∇ ·

∂ ∂xge(~r)

¸

+ ˆxω2µ²²z ²ge(~r)

¾

−jωµ

½h_²

z

²ge(~r)−gm(~r)

i_xy_ˆ 2₋_yxy_ˆ

x2₊_y2

+

µ

ˆ

x−2xyˆ 2−yxyˆ

x2₊_y2

¶

rege(~r)−rmgm(~r)

jω√µ²(x2₊_y2₎

¾

(114)

0 50 100 150 200 250 300 350 -500

-400 -300 -200 -100

φ (Degrees)

Re{

E

}

(V/m)

Weiglhofer Article

0 50 100 150 200 250 300 350 -350

-300 -250 -200 -150 -100

Weiglhofer Article

0 50 100 150 200 250 300 350 0.00

0.05 0.10 0.15

x

Im{

E

}

(V/m)

x

φ (Degrees)

% Error in |

E

|

(10)

Fory directed point source, the electric field is obtained in [4] as follows:

~ Euni =

µ

− 1

jω²

¶ ½

~

∇ ·

∂ ∂yge(~r)

¸

+ ˆyω2µ²²z ²ge(~r)

¾

−jωµ

½h_²

z

²ge(~r)−gm(~r)

i₋_xxy_ˆ _{+ ˆ}_yx2

x2₊_y2

+

µ

ˆ

y−2−xxyˆ + ˆyx 2

x2₊_y2

¶

r_eg_e(~r)−r_mg_m(~r)

jω√µ²(x2₊_y2₎

¾

(115)

Forz directed point source, the electric field is obtained in [4] as follows:

~ E_uni =

µ

− 1

jω²

¶ ½

~

∇ ·

∂ ∂zge(~r)

¸

+ ˆzω2µ²g_e(~r)

¾

(116)

0 50 100 150 200 250 300 350 -200

-100 0 100 200

Weiglhofer Article

0 50 100 150 200 250 300 350 -100

-50 0 50

100 Weiglhofer

Article

0 50 100 150 200 250 300 350 0.0858914

0.0858914 0.0858914 0.0858914 0.0858914

φ (Degrees)

Re{

E

}

(V/m)

y

Im{

E

}

(V/m)

y

φ (Degrees)

% Error in |

E

|

Figure 2. x directed point source,Ey versus φfor²rz = 2, R= 1 m.

0 50 100 150 200 250 300 350 -100

-50 0 50 100

Weiglhofer Article

0 50 100 150 200 250 300 350 -200

-100 0 100 200

Weiglhofer Article

0 50 100 150 200 250 300 350 5.×10 -15

1.×10 -14 1.5×10-14

2.×10 -14 φ (Degrees)

Re{

E

}

(V/m)

% Error in |

E

|_z

z

Im{

E

}

(V/m)

z

φ (Degrees)

φ (Degrees) 0

(11)

In the above expressions, the following definitions are valid:

r_e=

s

²_z

µ

x2

² + y2

² + z2

²z

¶

(117)

rm=

p

x2₊_y2₊_z2 ₍₁₁₈₎

ge(~r) =

ejω√µ²re

4πre (119)

gm(~r) = e

jω√µ²rm

4πrm (120)

~r= ˆxx+ ˆyy+ ˆzz (121)

0 20 40 60 80

-200 -100 0 100 200 300

Weiglhofer Article

0 20 40 60 80

-200 -100 0 100

200 _WeiglhoferArticle

θ (Degrees)

Re{

E

}

(V/m)

x

Im{

E

}

(V/m)

x

θ (Degrees)

20 40 60 80

0 10 20 30 40 50 60 70

x

θ (Degrees)

% Error in |

E

|

Figure 4. x directed point source,Ex versusθ for²rz = 2,R = 5 m.

0 20 40 60 80

-40 -30 -20 -10 0

10 _WeiglhoferArticle

0 20 40 60 80

-10 0 10

20 Weiglhofer

Article

20 40 60 80

0 20 40 60 80 100

θ (Degrees)

Re{

E

}

(V/m)

% Error in |

E

|_y

y

Im{

E

}

(V/m)

y

θ (Degrees)

(12)

The results are obtained for x directed point source and z directed point source. No calculations are made for y directed point source since they are similar to the x directed point source results. For example, x component of the electrical field for y directed point source is the same as y component of the electric field forx directed point source. The operation frequency is 1 GHz. The Sommerfeld type integrals are calculated by the steepest descent method (SDM).

The permittivity tensor of the medium is ¯¯

²=²0(ˆxxˆ 1 + ˆyyˆ1 + ˆzzˆ2) (122) The distance coordinate of the field point in spherical coordinate system is chosenR = 1 m for results which are versus varying azimuth angleφof the field point. The zenith angleθis equal to 60◦ _{for these}

0 20 40 60 80

-50 0

50 Weiglhofer

Article

0 20 40 60 80

-50 0

50 Weiglhofer

Article

0 20 40 60 80

0 5.×10 -13

1.×10 -12

1.5×10 -12

2.×10 -12

2.5×10 -12

θ (Degrees)

Re{

E

}

(V/m)

% Error in |

E

|

z

Im{

E

}

(V/m)

z

θ (Degrees)

Figure 6. x directed point source,Ez versusθ for²rz = 2, R= 5 m.

0 20 40 60 80

-400 -300 -200 -100 0 100 200 300

Weiglhofer Article

0 20 40 60 80

-300 -200 -100 0 100 200 300

Weiglhofer Article

0 20 40 60 80

0

2.×10 -14

4.×10 -14

6.×10 -14

8.×10 -14

φ (Degrees)

Re{

E

}

(V/m)

x

Im{

E

}

(V/m)

x

φ (Degrees)

% Error in |

E

|

(13)

0 20 40 60 80 -30

-20 -10 0 10 20

Weiglhofer Article

0 20 40 60 80

-30 -20 -10 0 10 20 30

Weiglhofer Article

0 20 40 60 80

0 2.×10-14 4.×10-14 6.×10-14 φ (Degrees)

Re{

E

}

(V/m)

y

Im{

E

}

(V/m)

y

φ (Degrees)

% Error in |

E

|

Figure 8. z directed point source,Ey versus θfor²rz = 2,R= 1 m.

0 20 40 60 80

400 -200 0 200 400

Weiglhofer Article

0 20 40 60 80

-400 -300 -200 -100 0 100 200 300

Weiglhofer Article

0 20 40 60 80

0 5.×10-14 1.×10-13

1.5×10-13

2.×10-13 2.5×10-13 3.×10-13

3.5×10-13 φ (Degrees)

Re{

E

}

(V/m)

% Error in |

E

|

z

Im{

E

}

(V/m)

z

φ (Degrees)

Figure 9. z directed point source,Ez versusθ for²rz = 2, R= 1 m.

results, which are displayed in Fig. 1 to Fig. 3. If the results obtained with respect to the azimuth angle

φare examined, it can be observed that they have got high accuracy forR = 1 m. The accuracy of the results increases as ²rz is increased, or the radial distance is increased.

For the results which are given versus the zenith angle θ of the field point, R is chosen 5 m. The reason of choosing minimumR as 5 m is that the results have low accuracy forθ closed to 0 when the field points are close to the source. The azimuth angle φis 5◦ _{for these results. The results which are}

(14)

components of the electric field for x directed and y directed point sources. The same case is valid for the electric field components produced by the z directed point source. The reason is that they are related to the isotropic field components by simple coordinate transformations.

5. CONCLUSION

In this article, the electric field components of an electrically uniaxial medium are related to those of an isotropic medium with a permittivity the same as the permittivity of the uniaxial medium that is common to the axes transverse to the optic axis using a simple method, i.e., the spectral representation method. The obtained relations are also simple. They contain terms which are convertible to each other by simple coordinate transformations. For the zenith angle θ not close to 0, the relations are verified with high accuracy using the SDM method for Sommerfeld integral type terms.

REFERENCES

1. Clemmow, P. C., “The theory of electromagnetic waves in a simple anisotropic medium,”

Proceedings IEE, Vol. 110, 101–106, Jan. 1963.

2. Felsen, L. B. and N. Marcuwitz,Radiation and Scattering of Waves, Wiley-IEEE Press, New Jersey, 2003.

3. Chen, H. C., Theory of Electromagnetic Waves, McGraw-Hill, New York, 1983.

4. Weiglhofer, W. S., “Dyadic Green’s functions for general uniaxial media,”Proceedings IEE, Vol. 137, 5–10, Feb. 1990.

5. Chew, W. C., Waves and Fields in Inhomogeneous Media, IEEE Press, New York, 1995. 6. Kong, J. A.,Electromagnetic Wave Theory, John Wiley & Sons, 1986.

7. Born, M. and E. Wolf,Principles of Optics, Cambridge University Press, 1999.

8. Lakhtakia, A., V. K. Varadan, and V. V. Varadan, “Radiation and canonical sources in uniaxial dielectric media,”International Journal of Electronics, Vol. 65, No. 6, 1171–1175, 1988.

9. Lakhtakia, A., V. V. Varadan, and V. K. Varadan, “Time-harmonic and time-dependent dyadic Green’s functions for some uniaxial gyro-electromagnetic media,” Applied Optics, Vol. 28, No. 6, 1049–1052, 1989.

10. Weiglhofer, W., “Analytic methods and free-space dyadic Green-functions,”Radio Science, Vol. 28, No. 5, 847–857, 1993.

11. Weiglhofer, W. S. and I. V. Lindell, “Analytic solution for the dyadic Green-function of a non-reciprocal uniaxial bianisotropic medium,” International Journal of Electronics and Communications, Vol. 48, No. 2, 116–119, Mar. 1994.

12. Weiglhofer, W. S., “Dyadic Green-function for unbounded general uniaxial bianisotropic medium,”

International Journal of Electronics, Vol. 77, No. 1, 105–115, Jul. 1994.

13. Weiglhofer, W. S. and A. Lakhtakia, “New expressions for depolarization dyadics in uniaxial dielectric-magnetic media,”International Journal of Infrared and Millimeter Waves, Vol. 17, No. 8, 1365–1376, Aug. 1996.

14. Olyslager, F. and B. Jakoby, “Time-harmonic two- and three-dimensional Green dyadics for a special class of gyrotropic bianisotropic media,” IEE Proceedings — Microwaves Antennas and Propagation, Vol. 143, No. 5, 413–416, Oct. 1996.

15. Marklein, R., K. J. Langenberg, and T. Kaczorowski, “Electromagnetic and elastodynamic point source excitation of unbounded homogeneous anisotropic media,” Radio Science, Vol. 31, No. 6, 1919–1930, 1996.

16. Lindell, I. V., “Decomposition of electromagnetic fields in bi-anisotropic media,” Journal of Electromagnetic Waves and Applications, Vol. 11, No. 5, 645–657, 1997.

(15)

18. Olyslager, F. and I. V. Lindell, “Green’s dyadics for a class of bi-anisotropic media with nonsymmetric bi-anisotropic dyadics,” AEU — International Journal of Electronics and Communications, Vol. 52, No. 1, 32–36, 1998.

19. Weiglhofer, W. S., “New expressions for depolarization dyadics in an axially uniaxial bianisotropic medium,” International Journal of Infrared and Millimeter Waves, Vol. 19, No. 7, 993–1005, Jul. 1998.

20. Weiglhofer, W. S., “Scalar Hertz potentials for nonhomogeneous uniaxial dielectric-magnetic mediums,” International Journal of Applied Electromagnetics and Mechanics, Vol. 11, No. 3, 131– 140, 2000.

21. Liu, S. H., L. W. Li, M. S. Leong, and T. S. Yeo, “Field representations in general rotationally uniaxial anisotropic media using spherical vector wave functions,” Microwave and Optical Technology Letters, Vol. 25, No. 3, 159–162, 2000.

22. Weiglhofer, W. S., “The connection between factorization properties and closed-form solutions of certain linear dyadic differential operators,” Journal of Physics A — Mathematical and General, Vol. 33, No. 35, 6253–6261, 2000.

23. Li, L. W., N. H. Lim, M. S. Leong, T. S. Yeo, and J. A. Kong, “Cylindrical vector wave function representation of Green’s dyadics for uniaxial bianisotropic media,” Radio Science, Vol. 36, No. 4, 517–523, 2001.

24. Li, K., S. O. Park, H. J. Lee, and W. Y. Pan, “Dyadic Green’s function for an unbounded gyroelectric chiral medium in cylindrical coordinates,” 3rd International Symposium on Electromagnetic Compatibility, 668–671, Beijing, 2002.

25. Tan, E. L., “Vector wave function expansions of dyadic Green’s functions for bianisotropic media,”

IEE Proceedings — Microwaves Antennas and Propagation, Vol. 149, No. 1, 57–63, Feb. 2002. 26. Olyslager, F. and I. V. Lindell, “Electromagnetics and exotic media: A quest for the holy grail,”

IEEE Antennas and Propagation Magazine, Vol. 44, No. 2, 48–58, 2002.

27. Lakhtakia, A. and T. G. Mackay, “Simple derivation of dyadic green functions of a simply moving, isotropic, dielectric-magnetic medium,”Microwave and Optical Technology Letters, Vol. 48, No. 6, 1073–1074, 2006.