Fields of a Dipole Near a Layered Substrate

Transcription

1 Appendix C Fields of a Dipole Near a Layered Substrate µ z θ µ 1 ε 1 µ 2 ε 2 µ 3 ε 3 d z o x,y Figure C.1: An electric dipole with moment µ is located at r o = (,, z o ) near a layered substrate. The fields in each medium are expressed in cylindrical coordinates r = (ρ, ϕ, z). Vertical Electric Dipole The cylindrical field components of a vertically oriented dipole µ=(,, µ z ) read as µ z e ik1 3 E 1ρ = ρ (z z o ) i µ z R 3 o 3ik 1 k 2 1 dk ρ J 1 (k ρ ρ)a 1 k ρ k 1z e ik1z(z+zo) 1 (C.1)

2 2 APPENDIX C. FIELDS OF A DIPOLE NEAR A LAYERED SUBSTRATE E 2ρ = i µ z dk ρ J 1 (k ρ ρ) A 2 e ik2zz A 3 e ik2zz k ρ k 2z e ik1zzo (C.2) E 3ρ = i µ z dk ρ J 1 (k ρ ρ)a 4 k ρ k 3z e i(k1zzo k3zz) (C.3) E 1ϕ = E 2ϕ = E 3ϕ = E 1z = µ z e ik1 3 (z z o ) 2 + µ z R 4 o 3ik 1 (z z o ) 2 R 3 o dk ρ J (k ρ ρ)a 1 k 2 ρ eik1z(z+zo) 1 + k2 1 (z z o) 2 + ik 1 + k 2 1 (C.4) (C.5) E 2z = µ z dk ρ J (k ρ ρ) A 2 e ik2zz + A 3 e ik2zz k 2 ρ eik1zzo (C.6) E 3z = µ z dk ρ J (k ρ ρ)a 4 k 2 ρ ei(k1zzo k3zz) (C.7) H 1ρ = H 2ρ = H 3ρ = H 1ϕ = i ω µ z ρ eik1 i ω µ z 2 1 ik 1 dk ρ J 1 (k ρ ρ)a 1 k ρ e ik1z(z+zo) (C.8) (C.9) H 2ϕ = i ω ε 2 µ z ε 1 H 3ϕ = i ω ε 3 µ z ε 1 dk ρ J 1 (k ρ ρ) A 2 e ik2zz + A 3 e ik2zz k ρ e ik1zzo dk ρ J 1 (k ρ ρ)a 4 k ρ e i(k1zzo k3zz) (C.1) (C.11) H 1z = H 2z = H 3z = (C.12)

3 3 Horizontal Electric Dipole The cylindrical field components of a horizontally oriented dipole µ=(,, ) read as E 1ρ = cosϕ e ik1 k ik 1 1 cosϕ + ρ 2 R 3 o 3 ρ J 1(k ρ ρ) k ρ B 1 ik 1z C 1 3ik 1 k i k 1z J (k ρ ρ) ik 1z B 1 k ρ C 1 } (C.13) E 2ρ = cosϕ { 1 kρ dk ρ e ik1z zo ρ J 1(k ρ ρ) B 2 + ik 2z C 2 e + k ρ B 3 ik 2z C 3 e ik2z i k 2z J (k ρ ρ) B 2 + k ρ C 2 e + } ik 2z B 3 k ρ C 3 e (C.14) E 3ρ = cosϕ dk ρ e i(k1z zo k3z z) { 1 ρ J 1(k ρ ρ) k ρ B 4 + ik 3z C 4 i k 3z J (k ρ ρ) ik 3z B 4 + k ρ C 4 } (C.15) E 1ϕ = sin ϕ e ik1 1 sin ϕ ik 1 k (C.16) ρ J 1(k ρ ρ) k ρ B 1 ik 1z C 1 k 2 1 J (k ρ ρ)b 1 } E 2ϕ = sin ϕ { 1 kρ dk ρ e ik1z zo ρ J 1(k ρ ρ) B 2 + ik 2z C 2 e + k ρ B 3 ik 2z C 3 e k 2 2 J (k ρ ρ) B 2 + B 3 } (C.17)

4 4 APPENDIX C. FIELDS OF A DIPOLE NEAR A LAYERED SUBSTRATE E 3ϕ = sin ϕ dk ρ e i(k1z zo k3z z) { 1 ρ J 1(k ρ ρ) k ρ B 4 + ik 3z C 4 k 2 3 J (k ρ ρ)b 4 } (C.18) E 1z = cosϕ ρ (z z o ) eik cosϕ 3ik 1 k 2 1 (C.19) dk ρ e ik1z (z+zo) k ρ J 1 (k ρ ρ) ik 1z B 1 k ρ C 1 E 2z = cosϕ ik2z dk ρ e {k ik1z zo ρ J 1 (k ρ ρ) B 2 + k ρ C 2 e } ik 2z B 3 k ρ C 3 e (C.2) E 3z = cosϕ dk ρ e i(k1z zo k3z z) k ρ J 1 (k ρ ρ) ik 3z B 4 + k ρ C 4 (C.21) H 1ρ = sin ϕ i ω (z z 1 o) eik1 2 ik 1 + sin ϕ i ω ρ J 1(k ρ ρ)c 1 ik 1z J (k ρ ρ)b 1 } (C.22) H 2ρ = sin ϕ i ω ε 2 ε 1 dk ρ e ik1z zo { 1 ρ J 1(k ρ ρ) C 2 e ik2zz + C 3 e ik2zz ik 2z J (k ρ ρ) B 2 e ik2zz } ik2zz B 3 e (C.23) H 3ρ = sin ϕ i ω ε 3 ε 1 dk ρ e i(k1z zo k3z z) { 1 ρ J 1(k ρ ρ)c 4 + ik 3z J (k ρ ρ)b 4 } (C.24) H 1ϕ = cosϕ i ω (z z 1 o) eik1 2 ik 1 (C.25)

5 5 cosϕ i ω ρ J 1(k ρ ρ)c 1 + J (k ρ ρ) ik 1z B 1 k ρ C 1 } H 2ϕ = cosϕ i ω ε 2 ε 1 { 1 dk ρ e ik1z zo ρ J 1(k ρ ρ) C 2 e ik2zz + C 3 e ik2zz ik2z J (k ρ ρ) B 2 + k ρ C 2 e } ik 2z B 3 k ρ C 3 e (C.26) H 3ϕ = cosϕ i ω ε 3 ε 1 { dk ρ e i(k1z zo k3z z) 1 ρ J 1(k ρ ρ)c 4 J (k ρ ρ) } ik 3z B 4 + k ρ C 4 (C.27) H 1z = sinϕ i ω ρ 1 eik1 2 ik 1 sin ϕ i ω dk ρ e ik1z (z+zo) k ρ J 1 (k ρ ρ)b 1 (C.28) H 2z = sinϕ i ω ε 2 ε 1 H 3z = sinϕ i ω ε 3 ε 1 dk ρ e ik1z zo k ρ J 1 (k ρ ρ) B 2 e ik2zz + B 3 e ik2zz dk ρ e i(k1z zo k3z z) k ρ J 1 (k ρ ρ)b 4 (C.29) (C.3) Definition of the coefficients A j, B j, and C j The coefficients A j, B j, C j are determined by the boundary conditions on the interfaces. Using the abbreviations f 1 = ε 2 k 1z ε 1 k 2z g 1 = µ 2 k 1z µ 1 k 2z f 2 = ε 2 k 1z + ε 1 k 2z g 2 = µ 2 k 1z + µ 1 k 2z f 3 = ε 3 k 2z ε 2 k 3z g 3 = µ 3 k 2z µ 2 k 3z f 4 = ε 3 k 2z + ε 2 k 3z g 4 = µ 3 k 2z + µ 2 k 3z, (C.31) the coefficients read as:

6 6 APPENDIX C. FIELDS OF A DIPOLE NEAR A LAYERED SUBSTRATE A 1 (k ρ ) = i k ρ (f 1 f 4 + f 2 f 3 e 2ik2zd ) k 1z (f 2 f 4 + f 1 f 3 e 2ik2zd ) (C.32) A 2 (k ρ ) = A 3 (k ρ ) = A 4 (k ρ ) = i i 2 ε 1 k ρ f 4 (f 2 f 4 + f 1 f 3 e 2ik2zd ) 2 ε 1 k ρ f 3 e 2ik2zd (f 2 f 4 + f 1 f 3 e 2ik2zd ) i 4 ε 1ε 2 k ρ k 2z e i(k2z k3z) d (f 2 f 4 + f 1 f 3 e 2ik2zd ) (C.33) (C.34) (C.35) B 1 (k ρ ) = i k ρ (g 1 g 4 + g 2 g 3 e 2ik2zd ) k 1z (g 2 g 4 + g 1 g 3 e 2ik2zd ) B 2 (k ρ ) = i ε 1 ε 2 2 µ 1 k ρ g 4 (g 2 g 4 + g 1 g 3 e 2ik2zd ) B 3 (k ρ ) = i ε 1 ε 2 2 µ 1 k ρ g 3 e 2ik2zd (g 2 g 4 + g 1 g 3 e 2ik2zd ) B 4 (k ρ ) = i ε 1 4 µ 1 µ 2 k ρ k 2z e i(k2z k3z) d ε 3 (g 2 g 4 + g 1 g 3 e 2ik2zd ) (C.36) (C.37) (C.38) (C.39) C 1 (k ρ ) = 2 kρ 2 (f4 + f 3 e 2ik2zd )(g 4 + g 3 e 2ik2zd )(ε 1 µ 1 ε 2 µ 2 ) (C.4) + 4 ε 1 µ 1 k2z 2 (ε 2 µ 2 ε 3 µ 3 )e 2ik2zd / (g2 g 4 + g 1 g 3 e 2ik2zd )(f 2 f 4 + f 1 f 3 e 2ik2zd ) ε 1 C 2 (k ρ ) = 2 kρ 2 f4 (g 4 + g 3 e 2ik2zd )(ε 1 µ 1 ε 2 µ 2 ) ε 2 2 µ 1 k 2z f 1 (ε 2 µ 2 ε 3 µ 3 )e 2ik2zd / (g2 g 4 + g 1 g 3 e 2ik2zd )(f 2 f 4 + f 1 f 3 e 2ik2zd ) (C.41) ε 1 C 3 (k ρ ) = 2 kρ 2 f3 (g 4 + g 3 e 2ik2zd )(ε 1 µ 1 ε 2 µ 2 )e 2ik2zd ε µ 1 k 2z f 2 (ε 2 µ 2 ε 3 µ 3 )e 2ik2zd / (g2 g 4 + g 1 g 3 e 2ik2zd )(f 2 f 4 + f 1 f 3 e 2ik2zd ) (C.42) C 4 (k ρ ) = 4 kρ 2 ε 1 k 2z e i(k2z k 3z) d ε 3 (g 4 + g 3 e 2ik2zd )(ε 1 µ 1 ε 2 µ 2 ) ε 3 + µ 1 (f 2 f 1 e 2ik2zd )(ε 2 µ 2 ε 3 µ 3 ) / (g2 g 4 + g 1 g 3 e 2ik2zd )(f 2 f 4 + f 1 f 3 e 2ik2zd ) (C.43)

7 7 In order to stay on the proper Riemann sheet, all square roots k jz = kj 2 k2 ρ j {1,2,3 } (C.44) have to be chosen such that Im{k jz } >. The integrals have to be evaluated numerically. The integration routine has to account for both oscillatory behavior and singularities. It is recommended that the integration range is split into sub-intervals and that the integration path is extended into the complex k ρ -plane. For some applications it is advantageous to express the Bessel functions J n in terms of Hankel functions since they converge rapidly for arguments with an imaginary part. An integration routine that proved very reliable is the so-called Gauss-Kronrod routine.