Simple Relations between a Uniaxial Medium and an Isotropic Medium

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1 Progress In Electromagnetics Research B, Vol. 6, 79 93, 214 Simple Relations between a Uniaial Meium an an Isotropic Meium Saffet G. Şen * Abstract In this article, in a simple wa, simple relations are erive between the electric fiel components of an electricall uniaial meium an those of an isotropic meium. The permittivit of the isotropic meium is the same as the permittivit of the uniaial meium that is common to the aes transverse to the optic ais. Using the spectral representation, the vector wave equation for the electric fiel intensit vector of the uniaial meium is solve for the irecte, irecte an irecte point sources. For the irecte an irecte point sources, the electric fiel components transverse to the optic ais are written in terms of the corresponing components of the isotropic meium plus some other terms. Part of these terms are close forms epressions, an the rest are Sommerfel tpe integrals. Elements of each group are relate to each other b coorinate transformations. The electric fiel components parallel to the optic ais are shown to be obtaine from the isotropic meium components using coorinate transformations. The relations between the uniaial meium an isotropic meium fiel components are verifie b comparing the results of a previous stu in the literature to the results obtaine using the relations in this stu. Goo agreement is achieve between these results. 1. INTRODUCTION Anisotropic materials are materials whose electromagnetic properties epen on the irection. An electricall uniaial material is a special case of anisotropic materials. An electricall uniaial material has a permittivit along its istinguishe or optic ais which is ifferent from the common permittivit in the irections of the other two aes. Anisotropic materials have a wie application range in technolog. Microwave integrate circuits, resonators, microstrip antennas are some of the application areas. A great amount of literature on anisotropic materials eists. Some of the stuies on the propagation of electromagnetic waves in homogeneous uniaial meia are given in [1 4]. In [1], using some coorinate variable transformations in Mawell s equations, equivalence is establishe between TM electromagnetic waves in electricall uniaial meia an TM waves in isotropic meia. A similar equivalence is built for TE electromagnetic waves. In [2], electricall uniaial meium is stuie in a separate chapter. The relations between the electricall uniaial meium fiels an isotropic meium fiels are escribe b using scalar potential functions. The scalar potentials for the electricall uniaial meium are epresse in terms of scalar potentials for the isotropic meium. In [3], the aic Green s functions for an electricall uniaial meium are etermine in close forms using the 3-imensional Fourier transform. In [4], the aic Green s functions are erive for a general uniaial meium, i.e., a meium which is magneticall an electricall uniaial meium, b solving a aic ifferential equation. Apart from these articles, the free-space Green functions for several tpes of anisotropic meia are stuie in the references [8 28]. The basics of uniaial meia are given in the books in [2, 3, 5 7]. In this article, the Green s functions for the components of the electric fiel for an electricall uniaial meium are foun for three kins of point sources, an irecte point source, a irecte Receive 13 April 214, Accepte 28 Ma 214, Scheule 6 June 214 * Corresponing author: Saffet Gokcen Sen saffetgokcensen@hotmail.com). The author is with the Department of Electrical an Electronics Engineering, Atatürk Universit, Erurum 2524, Turke.

2 8 Şen point source an a irecte point source. The are relate to the Green s functions of the isotropic meium with a permittivit the same as the permittivit of the uniaial meium that is common to the aes transverse to the optic ais. The solution is mae using the spectral representation metho given in [5]. Each of the Green s functions contains a term which is a scale version of the corresponing isotropic meium Green s function. For the components transverse to the optic ais, there is another term which is a Sommerfel tpe integral, an the rest of the terms are in close form. Sommerfel integral tpe terms are convertible to each other b using coorinate scalings. The same conversion is possible for the close form terms. In Section 2, spectral omain wave equations are erive for the electric fiel components of the uniaial meium. In Section 3, these spectral omain equations are solve for three basic point source configurations. In Section 4, the relations obtaine are verifie b comparing their results to those obtaine from the close form epressions given in [4]. Goo agreement is obtaine ecept for the fiel points with enith angle θ close to. 2. SPECTRAL DOMAIN WAVE EQUATIONS For the electricall uniaial meium with its istinguishe ais in the ẑ irection, first two of the Mawell s equations can be written in the following form assuming the time harmonic epenence of e jωt : E = jωµ H 1) H = J jω E 2) In these equations, E is the electric fiel intensit vector, H the magnetic fiel intensit vector, an J the electric current ensit vector. The scalar µ is the permeabilit of the meium, an the tensor is the permittivit tensor of the meium given b = ˆˆ + ŷŷ + ẑẑ 3) ω is the angular frequenc an j the imaginar unit. Taking the curl of both sies of Equation 1) an using the Gauss law given b E ) = 1 jω J, 4) the following wave equation is obtaine for E: [ 2 E + ω 2 µ E +. ẑ ] ) E = jωµ J + ) J jω The spectral representation or 3-imensional inverse Fourier transform of a scalar F is efine as follows [5]: F,, ) = 1 F 2π) 3 k, k, k ) e j k r k 6) F is the spectral omain form of F an k = ˆk + ŷk + ẑk, r = ˆ + ŷ + ẑ, k = k 7) If the spectral epansion is applie to Equation 5), then the following three equations are obtaine: k 2 + ω 2 µ ) Ẽ + ) k 2 + ω 2 µ ) Ẽ + ) k 2 + ω 2 µ ) Ẽ + ) ) k k ) Ẽ = jωµ k2 jω ) k k ) Ẽ = jωµ k2 jω ) ) k 2 Ẽ = jωµ k2 jω 5) J k k jω J k k jω J 8) J k k jω J k k jω J 9) J k k jω J k k jω J 1) k 2 = k 2 + k 2 + k 2 11)

3 Progress In Electromagnetics Research B, Vol. 6, BASIC POINT SOURCE CONFIGURATIONS In this section, the spectral omain wave equations are to be solve for basic point source configurations. The are the ˆ irecte, ŷ irecte an ẑ irecte point source configurations ˆ Directe Point Source The ˆ irecte point source is given as follows: The spectral omain representation for this source is J = ˆIlδ ) δ ) δ ) 12) Il = 1A m 13) J = 1, J =, J = 14) If the spectral omain equations are solve for this point source, then the following epressions are obtaine for spectral E components: Ẽ uni = Ẽ uni = Ẽ uni = jωµ k k 1 ) k k 2 ) jωµ k k 3 ) k k 4 ) k 2 jωµ k 2 + k 2 jωµ k k 1 ) k k 2 ) + k 2 + k 2 jωµ + k k 3 ) k k 4 ) + jω k k 3 ) k k 4 ) jωµ k k k 2 + k 2 k k 1 ) k k 2 ) + 1 jω k k k k 3 ) k k 4 ) k 1 = ω 2 µ k 2 k, 2 k 3 = ω 2 µ k 2 k 2, k 2 jωµ k k k 2 + k 2 k k 3 ) k k 4 ) + k 2 = k 2 1 jω k k 3 ) k k 4 ) 15) 16) 17) ω 2 µ k 2 k 2 18) k 4 = ω 2 µ k 2 k 2 19) The k integration in the spectral representation is performe b the resiue integration inicate in [5] to obtain the following integrals: E uni = I 1 I 2 I 3 + I 4 + I 5 2) I 1 = ωµ ) e jk 1 e j k t r t 8π 2 k k t = jωµ ejkr 1 4πr 21) I 2 = ωµ ) e jk 3 e j k t r t 8π 2 k k t 3 22) I 3 = I 4 = ωµ 8π 2 ) ωµ 8π 2 ) k 2 k 2 + k 2 k 1 e jk 1 e j k t r t k t 23) k 2 k 2 + k 2 k 3 e jk 3 e j k t r t k t 24)

4 82 Şen I 5 = j jωµ + jω 8π 2 e jk 3 e j k t r t k k t 3 25) E uni = I 1 + I 2 + I 3 26) I 1 = ωµ k 2 + k 2 8π 2 e jk 1 e j k t r t k k t 27) 1 I 2 = ωµ ) k 2 + k 2 8π 2 e jk 3 e j k t r t k k t 28) 3 1 I 3 = 8π 2 e jk 3 e j k t r t ω k k t 29) 3 1 E uni = I 1 = 8π 2 ω k e jk 3 e j k t r t k t 3) In the above integrals, the following efinitions are use: kt = ˆk + ŷk, r t = ˆ + ŷ, k t = 31) Each of the integral groups contains some integrals which are not totall inepenent of each other. Using the variable transformations given b k = k, k = k, 32) the following relations can be erive: ) ) I 2,, ) = I 1,, 33) ) ) I 4,, ) = I 3,, 34) ) ) I 2,, ) = I 1,, 35) In aition to these smmetr relations, some integrals are relate with the spectral fiel components for the isotropic meium with permittivit. If the spectral omain scalar wave equations are solve for the isotropic meium, then the following integrals are obtaine for the electric fiel components: E iso = j 8π 2 E iso = 1 8π 2 ω E iso = 1 8π 2 ω k2 jωµ + k2 jω e jk 1 e j k t r t k k t 36) 1 e jk 1 e j k t r t k k t 1 37) k e jk 1 e j k t r t k t 38) Utiliing the variable transformation in 32), the following relations can be written: ) ) I 5 = E iso,, ) ) I 3 = E iso,, ) E uni = E iso,, 39) 4) 41)

5 Progress In Electromagnetics Research B, Vol. 6, Sommerfel integral forms are to be erive b appling a series of variable transformations followe b the application of an integral ientit for Bessel functions of the first kin as inicate in [5]. The variable transformations an relate epressions are as follows: k = k ρ cos α), k = k ρ sin α) 42) kt r t = k ρ ρ cos α φ), = k ρ k ρ α 43) = ρ cos φ), = ρ sin φ) 44) The ientit for the Bessel functions of the first kin is given b J n k ρ ρ) e jnφ = 1 2π 2π e jk ρρ cosα φ)+jnα jn π 2 α 45) In the Sommerfel integrals, the integration path enote b P is the Sommerfel integration path, an the branch of the square root relation use for the calculations of k 1 is the Im {k 1 } = branch [5]. I 3 = jωµ e jkr [ ] ωµ cos 2φ) k ρ + H 1) 2 k ρ ρ) e jk 1 k ρ 46) 8π r 16π P k 1 [ ] ωµ sin 2φ) k ρ I 1 = H 1) 2 k ρ ρ) e jk 1 k ρ 47) 16π k 1 P Hence, component of the uniaial meium electric fiel can be relate to that of the isotropic meium electric fiel as follows: E uni = I 1 I 2 I 3 + I 4 + I 5 48) I 1,, ) = jωµ ejkr 49) 4πr ) ) I 2,, ) = I 1,, 5) I 3 = jωµ e jkr [ ] ωµ cos 2φ) k ρ + H 1) 2 k ρ ρ) e jk 1 k ρ 51) 8π r 16π P k 1 ) ) I 4,, ) = I 3,, 52) ) ) I 5,, ) = E iso,, 53) As to component of the uniaial meium electric fiel, the following equations are vali: E uni = I 1 + I 2 + I 3 54) [ ] ωµ sin 2φ) k ρ I 1 = H 1) 2 k ρ ρ) e jk 1 k ρ 55) 16π P k 1 ) ) I 2,, ) = I 1,, 56) ) ) I 3,, ) = E iso,, 57) 3.2. ŷ Directe Point Source The ŷ irecte point source is given as follows: J = ŷilδ ) δ ) δ ) 58)

6 84 Şen Il = 1A m 59) The spectral omain representation for this source is J =, J = 1, J = 6) If the spectral omain equations are solve for this point source, then the following epressions are obtaine for spectral E components: Ẽ uni = Ẽ uni = jωµ k k k 2 + k 2 k k 1 ) k k 2 ) + jωµ k k k 2 + k 2 k k 3 ) k k 4 ) + jωµ k k 1 ) k k 2 ) jωµ k k 3 ) k k 4 ) k 2 k 2 1 jω k k 3 ) k k 4 ) jωµ k 2 + k 2 jωµ k k 1 ) k k 2 ) + k 2 + k 2 jωµ + k k 3 ) k k 4 ) + jω 62) k k 3 ) k k 4 ) 1 k k jω Ẽ uni = 63) k k 3 ) k k 4 ) If k integration in the spectral representation is performe, the following epressions are obtaine: E uni = I 1 + I 2 + I 3 64) I 1 = ωµ k 2 + k 2 8π 2 e jk 1 e j k t r t k k t 65) 1 I 2 = ωµ ) k 2 + k 2 8π 2 e jk 3 e j k t r t k k t 66) 3 1 I 3 = 8π 2 e jk 3 e j k t r t ω k k t 67) 3 E uni = = I 1 I 2 I 3 + I 4 + I 5 68) I 1 = ωµ ) e jk 1 e j k t r t 8π 2 k k t = jωµ ejkr 69) 1 4πr I 2 = ωµ ) e jk 3 e j k t r t 8π 2 k k t 7) 3 k 2 61) I 3 = I 4 = ωµ 8π 2 ) ωµ 8π 2 ) k 2 k 2 + k 2 k 1 e jk 1 e j k t r t k t 71) k 2 k 2 + k 2 k 3 e jk 3 e j k t r t k t 72) I 5 = j jωµ + 8π 2 1 E uni = I 1 = 8π 2 ω k2 jω e jk 3 e j k t r t k k t 73) 3 k e jk 3 e j k t r t k t 74)

7 Progress In Electromagnetics Research B, Vol. 6, Similar to the case for the ˆ irecte source, the following relations can be written: ) ) I 2,, ) = I 1,, ) ) I 2,, ) = I 1,, ) ) I 4,, ) = I 3,, 75) 76) 77) If the spectral omain equations are solve for the isotropic meium with permittivit for the irecte point source, then the following integrals can be erive: E iso = 1 8π 2 e jk 1 e j k t r t ω k k t 78) 1 E iso = j 8π 2 E iso = 1 8π 2 ω jωµ + k2 jω e jk 1 e j k t r t k k t 79) 1 k e jk 1 e j k t r t k t 8) Comparing the isotropic meium solutions with the uniaial meium solutions iels the following equations: ) ) I 3 = E iso,, 81) ) ) I 5 = E iso,, 82) ) E uni = E iso,, 83) After the Sommerfel integral representations of integrals I 1 an I 3 are erive, the following equations can be written: [ ] ωµ sin 2φ) k ρ I 1 = H 1) 2 k ρ ρ) e jk 1 k ρ 84) 16π P k 1 I 3 = jωµ e jkr [ ] ωµ cos 2φ) k ρ H 1) 2 k ρ ρ) e jk 1 k ρ 85) 8π r 16π k 1 Hence, an components of the electric fiel for the uniaial meium can be written as follows: P E uni = I 1 + I 2 + I 3 86) [ ] ωµ sin 2φ) k ρ I 1 = H 1) 2 k ρ ρ) e jk 1 k ρ 87) 16π P k 1 ) ) I 2,, ) = I 1,, 88) ) ) I 3,, ) = E iso,, 89) E uni = I 1 I 2 I 3 + I 4 + I 5 9)

8 86 Şen I 1,, ) = jωµ ejkr 91) 4πr ) ) I 2,, ) = I 1,, 92) I 3 = jωµ e jkr [ ] ωµ cos 2φ) k ρ H 1) 2 k ρ ρ) e jk 1 k ρ 93) 8π r 16π P k 1 ) ) I 4,, ) = I 3,, 94) ) ) I 5,, ) = E iso,, 95) 3.3. ẑ Directe Point Source The ẑ irecte point source is given as follows: J = ẑilδ ) δ ) δ ) 96) Il = 1A m 97) The spectral omain representation for this source is J =, J =, J = 1 98) The spectral omain fiel components can be written as follows: k k Ẽ uni = jω 99) k k 3 ) k k 4 ) k k Ẽ uni = jω 1) k k 3 ) k k 4 ) ) jωµ + k2 jω Ẽ uni = 11) k k 3 ) k k 4 ) After performing the k integration in the spectral representation, the following two-imensional integral forms can be obtaine for the electric fiel components: 1 E uni = 8π 2 ω k e jk 3 e j k t r t k t 12) 1 E uni = 8π 2 ω k e jk 3 e j k t r t k t 13) E uni = 1 ) ω 2 µ k 2 3 8π 2 e jk 3 e j k t r t ω k k t 14) 3 The counterparts of these epressions in the isotropic meium are E iso = 1 8π 2 ω k e jk 1 e j k t r t k t 15) E iso = 1 E iso = 8π 2 ω 1 8π 2 ω k e jk 1 e j k t r t k t 16) ) ω 2 µ k 2 1 e jk 1 e j k t r t k k t 17) 1

9 Progress In Electromagnetics Research B, Vol. 6, If these two groups of epressions are compare to each other, then the following relations can be erive: ) E uni,, ) = E iso,, 18) ) E uni,, ) = E iso,, 19) ) E uni,, ) = E iso,, 11) In aition, the Sommerfel integral forms for electric fiel components can be foun as follows: j cos φ) E uni = 8πω P k2 ρh 1) 1 k ρ ρ) e jk 3 k ρ 111) j sin φ) E uni = 8πω P k2 ρh 1) 1 k ρ ρ) e jk 3 k ρ 112) E uni = ) kρ 3 8πω 2 H 1) k ρ ρ) e jk 3 k ρ 113) P k 1 4. RESULTS In this section, the valiit of the formulas erive for the electric fiel components in an electricall uniaial meium are to be shown using numerical results. The reference to be use is the article b [4]. The sources are assume at the origin of the coorinate sstem. For irecte point source, the electric fiel is obtaine in [4] as follows: E uni = 1 jω + ) { ˆ 2 ˆ2 ŷ [ ] g e r) + ˆω 2 µ } { [ ] ˆ 2 g e r) jωµ g ŷ e r) g m r) ) } re g e r) r m g m r) jω µ ) 114) Re{E } V/m) φ Degrees) Im{E } V/m) φ Degrees) φ Degrees) Figure 1. irecte point source, E versus φ for r = 2, R = 1 m.

10 88 Şen For irecte point source, the electric fiel is obtaine in [4] as follows: E uni = 1 ) { [ ] jω g e r) + ŷω 2 µ } { [ ] ˆ + ŷ 2 g e r) jωµ g e r) g m r) ) } ˆ + ŷ2 re g e r) r m g m r) + ŷ jω µ ) For irecte point source, the electric fiel is obtaine in [4] as follows: E uni = 1 ) { [ ] } jω g e r) + ẑω 2 µg e r) 115) 116) Re{E } V/m) φ Degrees) Im{E } V/m) φ Degrees) φ Degrees) Figure 2. irecte point source, E versus φ for r = 2, R = 1 m. Re{E } V/m) Im{E } V/m) φ Degrees) φ Degrees) φ Degrees) Figure 3. irecte point source, E versus φ for r = 2, R = 1 m.

11 Progress In Electromagnetics Research B, Vol. 6, In the above epressions, the following efinitions are vali: ) 2 r e = ) r m = ) µre g e r) = ejω 119) 4πr e µrm g m r) = ejω 4πr m 12) r = ˆ + ŷ + ẑ 121) Re{E } V/m) θ Degrees) Im{E } V/m) θ Degrees) θ Degrees) Figure 4. irecte point source, E versus θ for r = 2, R = 5 m. Re{E } V/m) θ Degrees) Im{E } V/m) θ Degrees) θ Degrees) Figure 5. irecte point source, E versus θ for r = 2, R = 5 m.

12 9 Şen The results are obtaine for irecte point source an irecte point source. No calculations are mae for irecte point source since the are similar to the irecte point source results. For eample, component of the electrical fiel for irecte point source is the same as component of the electric fiel for irecte point source. The operation frequenc is 1 GH. The Sommerfel tpe integrals are calculate b the steepest escent metho SDM). The permittivit tensor of the meium is = ˆˆ 1 + ŷŷ 1 + ẑẑ 2) 122) The istance coorinate of the fiel point in spherical coorinate sstem is chosen R = 1 m for results which are versus varing aimuth angle φ of the fiel point. The enith angle θ is equal to 6 for these 5 5 Re{E } V/m) -5 Im{E } V/m) -5 θ Degrees) θ Degrees) θ Degrees) Figure 6. irecte point source, E versus θ for r = 2, R = 5 m. Re{E } V/m) φ Degrees) Im{E } V/m) φ Degrees) φ Degrees) Figure 7. irecte point source, E versus θ for r = 2, R = 1 m.

13 Progress In Electromagnetics Research B, Vol. 6, Re{E } V/m) Im{E } V/m) φ Degrees) φ Degrees) φ Degrees) Figure 8. irecte point source, E versus θ for r = 2, R = 1 m. Re{E } V/m) 4 2 φ Degrees) Im{E } V/m) φ Degrees) φ Degrees) Figure 9. irecte point source, E versus θ for r = 2, R = 1 m. results, which are isplae in Fig. 1 to Fig. 3. If the results obtaine with respect to the aimuth angle φ are eamine, it can be observe that the have got high accurac for R = 1 m. The accurac of the results increases as r is increase, or the raial istance is increase. For the results which are given versus the enith angle θ of the fiel point, R is chosen 5 m. The reason of choosing minimum R as 5 m is that the results have low accurac for θ close to when the fiel points are close to the source. The aimuth angle φ is 5 for these results. The results which are versus the enith angle θ are given in Fig. 4 to Fig. 9. The results obtaine with respect to the varing enith angle θ inicate that the SDM oes not perform well for θ close to. The accurac of the results increases as r is increase, or the raial istance is increase. There is no accurac problem for

14 92 Şen components of the electric fiel for irecte an irecte point sources. The same case is vali for the electric fiel components prouce b the irecte point source. The reason is that the are relate to the isotropic fiel components b simple coorinate transformations. 5. CONCLUSION In this article, the electric fiel components of an electricall uniaial meium are relate to those of an isotropic meium with a permittivit the same as the permittivit of the uniaial meium that is common to the aes transverse to the optic ais using a simple metho, i.e., the spectral representation metho. The obtaine relations are also simple. The contain terms which are convertible to each other b simple coorinate transformations. For the enith angle θ not close to, the relations are verifie with high accurac using the SDM metho for Sommerfel integral tpe terms. REFERENCES 1. Clemmow, P. C., The theor of electromagnetic waves in a simple anisotropic meium, Proceeings IEE, Vol. 11, 11 16, Jan Felsen, L. B. an N. Marcuwit, Raiation an Scattering of Waves, Wile-IEEE Press, New Jerse, Chen, H. C., Theor of Electromagnetic Waves, McGraw-Hill, New York, , W. S., Daic Green s functions for general uniaial meia, Proceeings IEE, Vol. 137, 5 1, Feb Chew, W. C., Waves an Fiels in Inhomogeneous Meia, IEEE Press, New York, Kong, J. A., Electromagnetic Wave Theor, John Wile & Sons, Born, M. an E. Wolf, Principles of Optics, Cambrige Universit Press, Lakhtakia, A., V. K. Varaan, an V. V. Varaan, Raiation an canonical sources in uniaial ielectric meia, International Journal of Electronics, Vol. 65, No. 6, , Lakhtakia, A., V. V. Varaan, an V. K. Varaan, Time-harmonic an time-epenent aic Green s functions for some uniaial gro-electromagnetic meia, Applie Optics, Vol. 28, No. 6, , , W., Analtic methos an free-space aic Green-functions, Raio Science, Vol. 28, No. 5, , , W. S. an I. V. Linell, Analtic solution for the aic Green-function of a non-reciprocal uniaial bianisotropic meium, International Journal of Electronics an Communications, Vol. 48, No. 2, , Mar , W. S., Daic Green-function for unboune general uniaial bianisotropic meium, International Journal of Electronics, Vol. 77, No. 1, , Jul , W. S. an A. Lakhtakia, New epressions for epolariation aics in uniaial ielectric-magnetic meia, International Journal of Infrare an Millimeter Waves, Vol. 17, No. 8, , Aug Olslager, F. an B. Jakob, Time-harmonic two- an three-imensional Green aics for a special class of grotropic bianisotropic meia, IEE Proceeings Microwaves Antennas an Propagation, Vol. 143, No. 5, , Oct Marklein, R., K. J. Langenberg, an T. Kacorowski, Electromagnetic an elastonamic point source ecitation of unboune homogeneous anisotropic meia, Raio Science, Vol. 31, No. 6, , Linell, I. V., Decomposition of electromagnetic fiels in bi-anisotropic meia, Journal of Electromagnetic Waves an Applications, Vol. 11, No. 5, , Cheng, D. J, W. Ren, an Y. Q. Jin, Green aics in uniaial bianisotropic-ferrite meium b clinrical vector wavefunctions, Journal of Phsics A Mathematical an General, Vol. 3, No. 2, , 1997.

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