SI Units and Gaussian Units

Home , Admittance, Gaussian units, Gauss (unit), Hertz vector, Maxwell (unit), Statvolt

Appendix A

A.1 Conversion of Amounts All factors of 3 (apart from exponents) should, for accurate work, be replaced by 2.99792456, arising from the numerical value of the velocity of light. [43]

Physical quantity Symbol SI(MKSA) Gaussian

Length l 1 meter (m) 102 centimeters (cm) Mass m 1 kilogram (kg) 103 grams (gm or g) Time t 1 second (sec or s) 1 second (sec or s) Frequency f 1 hertz (Hz) 1 hertz (Hz) Force F 1 newton (N) 105 dynes Work, Energy W, U 1 joule (J) 107 ergs Power P 1 watt (W) 107 ergs/s Charge q 1 coulomb (C) 3 × 109 statcoulombs Charge density % 1 C/m3 3 × 103 statcoul/cm3 Current I 1 ampere (A) 3 × 109 statamperes Current density J 1 A/m2 3 × 105 statamp/cm2 Potential ϕ 1 volt (V) 10−2/3 statvolt Electric field E 1 V/m 10−4/3 statvolt/cm Electric induction D 1 C/m2 12π × 105 statvolt/cm Polarization P 1 C/m2 3 × 105 moment/cm3 Magnetic flux Φ 1 weber (Wb) 108 maxwell (Mx) Magnetic induction B 1 tesla (T) 104 gauss (Gs) Magnetic field H 1 A/m 4π × 10−3 oersted (Oe) Magnetization M 1 A/m 10−3 moment/cm3 Conductance G 1 siemens (S) 9 × 1011 cm/s Conductivity σ 1 S/m 9 × 109 1/s Resistance R 1 ohm (Ω) 10−11/9 s/cm Capacitance C 1 farad (F) 9 × 1011 cm Inductance L 1 henry (H) 109 cm 674 Appendix A

A.2 Formulas in SI (MKSA) Units and Gaussian Units

Name of formula SI (MKSA) Gaussian

∇ × E = −∂ B ∇ × E = −1 ∂ B ∂ t c ∂ t

Maxwell ∇ × H = ∂ D + J ∇ × H = 1 ∂ D + 4π J ∂ t c ∂ t c equations ∇ · D = % ∇ · D = 4π% ∇ · B = 0 ∇ · B = 0

1 Lorentz force F = q(E + v × B) F = q(E + c v × B)

Constitutional D = ²0E + P = ²ED = E + 4πP = ²E equations B = µ0(H + M) = µHB = H + 4πM = µH J = γE J = γE

Constitutional ² = ²0(1 + χe) = ²0²r ² = 1 + 4πχe = ²r parameters µ = µ0(1 + χm) = µ0µr µ = 1 + 4πχm = µr

n × (E2 − E1) = 0 n × (E2 − E1) = 0 4π Boundary n × (H2 − H1) = J s n × (H2 − H1) = c J s equations n · (D2 − D1) = %s n · (D2 − D1) = 4π%s n · (B2 − B1) = 0 n · (B2 − B1) = 0 Z Z 1 % 0 1 % 0 Coulomb’s law E = 4π² 2 rˆdV E = ² 2 rˆdV V r V r Z Z 1 % 0 1 % 0 ϕ = 4π² r dV ϕ = ² r dV V V Z Z µ J × rˆ 0 µ J × rˆ 0 Biot-Savart B = 4π 2 dV B = c 2 dV V r V r law Z Z µ J 0 µ J 0 A = 4π r dV A = c r dV V V

2 % 2 % Poison ∇ ϕ = − ² ∇ ϕ = −4π ² equations 2 2 4π ∇ A = −µJ ∇ A = − c µJ SI Units and Gaussian Units 675

Name of formula SI (MKSA) Gaussian

2 ²µ 2 Wave ∇2E − ²µ∂ E = 0 ∇2E − ∂ E = 0 ∂ t2 c2 ∂ t2 equations 2 ²µ 2 ∇2H − ²µ∂ H = 0 ∇2H − ∂ H = 0 ∂ t2 c2 ∂ t2

Dynamic B = ∇ × A B = ∇ × A potentials E = −∇ϕ − ∂ A E = −∇ϕ − 1 ∂ A ∂ t c ∂ t ∂ ϕ ²µ ∂ ϕ Lorentz gauge ∇ · A + ²µ = 0 ∇ · A + = 0 ∂ t c ∂ t

∂2ϕ % ²µ ∂2ϕ % D’Alembert ∇2ϕ − ²µ = − ∇2ϕ − = −4π ∂ t2 ² c2 ∂ t2 ² equations 2 ²µ 2 ∇2A−²µ∂ A =−µJ ∇2A− ∂ A =−4π µJ ∂ t2 c2 ∂ t2 c Z Z 1 %(t − r/c) 0 1 %(t − r/c) 0 Retarding ϕ= 4π² r dV ϕ= ² r dV V V potentials Z Z µ J(t − r/c) 0 µ J(t − r/c) 0 A= 4π r dV A= c r dV V V 1 1 Energy density w = 2(E · D + H · B) w = 8π (E · D + H · B) c Poynting vector P = E × H P = 4π E × H 676 Appendix A

A.3 Preﬁxes and Symbols for Multiples

Multiple Preﬁx Symbol

10−18 atto a 10−15 femto f 10−12 pico p 10−9 nano n 10−6 micro µ 10−3 milli m 10−2 centi c 10−1 deci d 10 deka da 102 hecto h 103 kilo k 106 mega M 109 giga G 1012 tera T 1015 peta P 1018 exa E Appendix B

Vector Analysis

B.1 Vector Diﬀerential Operations

B.1.1 General Orthogonal Coordinates

r ³ ∂ x ´2 ³ ∂ y ´2 ³ ∂ z ´2 u1, u2, u3, h1, h2, h3, hi = + + , i = 1, 2, 3 ∂ ui ∂ ui ∂ ui

A = uˆ1A1 + uˆ2A2 + uˆ3A3

X3 1 ∂ ϕ 1 ∂ ϕ 1 ∂ ϕ 1 ∂ ϕ ∇ϕ = uˆ = uˆ + uˆ + uˆ (B.1) i h ∂ u 1 h ∂ u 2 h ∂ u 3 h ∂ u i=1 i i 1 1 2 2 3 3

1 X3 ∂ ∇ · A = (hjhkAi) h1h2h3 ∂ ui · i=1 ¸ 1 ∂ ∂ ∂ = (h2h3A1) + (h3h1A2) + (h1h2A3) (B.2) h1h2h3 ∂ u1 ∂ u2 ∂ u3

· ¸ X3 1 ∂ ∂ ∇ × A = uˆi (hkAk) − (hjAj) hjhk ∂ uj ∂ uk i=1 · ¸ 1 ∂ ∂ = uˆ (h A ) − (h A ) 1 h h ∂ u 3 3 ∂ u 2 2 2 3 · 2 3 ¸ 1 ∂ ∂ + uˆ (h A ) − (h A ) 2 h h ∂ u 1 1 ∂ u 3 3 3 1 · 3 1 ¸ 1 ∂ ∂ + uˆ3 (h2A2) − (h1A1) (B.3) h1h2 ∂ u1 ∂ u2 678 Appendix B

1 X3 ∂ ³h h ∂ ϕ ´ ∇2ϕ = j k h1h2h3 ∂ ui hi ∂ ui · i=1 ¸ 1 ∂ ³h h ∂ ϕ ´ ∂ ³h h ∂ ϕ ´ ∂ ³h h ∂ ϕ ´ = 2 3 + 3 1 + 1 2 (B.4) h1h2h3 ∂ u1 h1 ∂ u1 ∂ u2 h2 ∂ u2 ∂ u3 h3 ∂ u3

∇2A = ∇(∇ · A) − ∇ × ∇ × A · ³ ´¸ 1 ∂ F0 1 ∂ F3 ∂ F2 = uˆ1 − − h1 ∂ u1 h2h3 ∂ u2 ∂ u3 · ³ ´¸ 1 ∂ F0 1 ∂ F1 ∂ F3 + uˆ2 − − h2 ∂ u2 h3h1 ∂ u3 ∂ u1 · ³ ´¸ 1 ∂ F0 1 ∂ F2 ∂ F1 + uˆ3 − − (B.5) h3 ∂ u3 h1h2 ∂ u1 ∂ u3 where F0 = ∇ · A · ¸ h1 ∂ ∂ F1 = h1(∇ × A)1 = (h3A3) − (h2A2) h2h3 ∂ u2 ∂ u3 · ¸ h2 ∂ ∂ F2 = h2(∇ × A)2 = (h1A1) − (h3A3) h3h1 ∂ u3 ∂ u1 · ¸ h3 ∂ ∂ F3 = h3(∇ × A)3 = (h2A2) − (h1A1) h1h2 ∂ u1 ∂ u2

B.1.2 General Cylindrical Coordinates

∂ h ∂ h u , u , z, h = 1, 1 = 0, 2 = 0 1 2 3 ∂ z ∂ z 1 ∂ ϕ 1 ∂ ϕ ∂ ϕ ∇ϕ = uˆ1 + uˆ2 + zˆ (B.6) h1 ∂ u1 h2 ∂ u2 ∂ z · ¸ 1 ∂ ∂ ∂ Az ∇ · A = (h2A1) + (h1A2) + (B.7) h1h2 ∂ u1 ∂ u2 ∂ z · ¸ 1 ∂ A ∂ ∇ × A = uˆ z − (h A ) 1 h ∂ u ∂ z 2 2 2 · 2 ¸ 1 ∂ ∂ A + uˆ (h A ) − z 2 h ∂ z 1 1 ∂ u 1 · 1 ¸ 1 ∂ ∂ + uˆ3 (h2A2) − (h1A1) (B.8) h1h2 ∂ u1 ∂ u2

· ³ ´ ³ ´¸ 2 2 1 ∂ h2 ∂ ϕ ∂ h1 ∂ ϕ ∂ ϕ ∇ ϕ = + + 2 (B.9) h1h2 ∂ u1 h1 ∂ u1 ∂ u2 h2 ∂ u2 ∂ z Vector Analysis 679

2 2 2 ∇ A = ∇ AT + zˆ∇ Az (B.10) where Az is the longitudinal component and AT is the transverse 2- dimensional vector of A

A = AT + zˆAz, AT = uˆ1A1 + uˆ2A2

· ³ ´ ³ ´¸ 2 2 1 ∂ h2 ∂ Az ∂ h1 ∂ Az ∂ Az ∇ Az = + + 2 h1h2 ∂ u1 h1 ∂ u1 ∂ u2 h2 ∂ u2 ∂ z µ ¶ 1 ∂ F 1 ∂ F ∂2A 2 ˆ 0 z 1 ∇ AT = u1 − + 2 h1 ∂ u1 h2 ∂ u2 ∂ z µ ¶ 1 ∂ F 1 ∂ F ∂2A ˆ 0 z 2 + u2 + + 2 (B.11) h2 ∂ u2 h1 ∂ u1 ∂ z where · ¸ 1 ∂ ∂ F0 = ∇ · AT = (h2A1) + (h1A2) h1h2 ∂ u1 ∂ u2 · ¸ 1 ∂ ∂ Fz = |∇ × AT | = (h2A2) − (h1A1) h1h2 ∂ u1 ∂ u2

B.1.3 Rectangular Coordinates

x, y, z, h1 = 1, h2 = 1, h3 = 1 ∂ ϕ ∂ ϕ ∂ ϕ ∇ϕ = xˆ + yˆ + zˆ (B.12) ∂ x ∂ y ∂ z ∂ A ∂ A ∂ A ∇ · A = x + y + z (B.13) ∂ x ∂ y ∂ z µ ¶ µ ¶ µ ¶ ∂ A ∂ A ∂ A ∂ A ∂ A ∂ A ∇×A = xˆ z − y +yˆ x − z +zˆ y − x (B.14) ∂ y ∂ z ∂ z ∂ x ∂ x ∂ y ∂2ϕ ∂2ϕ ∂2ϕ ∇2ϕ = + + (B.15) ∂ x2 ∂ y2 ∂ z2 2 2 2 2 ∇ A = xˆ∇ Ax + yˆ∇ Ay + zˆ∇ Az (B.16)

B.1.4 Circular Cylindrical Coordinates

ρ, φ, z, h1 = 1, h2 = r, h3 = 1 ∂ ϕ 1 ∂ ϕ ∂ ϕ ∇ϕ = ρˆ + φˆ + zˆ (B.17) ∂ ρ ρ ∂ φ ∂ z 1 ∂ 1 ∂ A ∂ A ∇ · A = (ρA ) + φ + z (B.18) ρ ∂ ρ ρ ρ ∂ φ ∂ z 680 Appendix B · ¸ · ¸ · ¸ 1 ∂ A ∂ A ∂ A ∂ A 1 ∂ ∂ A ∇ × A=ρˆ z − φ + φˆ ρ − z + zˆ (ρA ) − ρ (B.19) ρ ∂ φ ∂ z ∂ z ∂ ρ ρ ∂ ρ φ ∂ φ µ ¶ 1 ∂ ∂ ϕ 1 ∂2ϕ ∂2ϕ ∇2ϕ = ρ + + (B.20) ρ ∂ ρ ∂ ρ ρ2 ∂ φ2 ∂ z2 µ ¶ µ ¶ 2 ∂ A A 2 ∂ A A ∇2A=ρˆ ∇2A − φ − ρ +φˆ ∇2A + ρ − φ +zˆ∇2A (B.21) ρ ρ2 ∂ φ ρ2 φ ρ2 ∂ φ ρ2 z

B.1.5 Spherical Coordinates

r, θ, φ, h1 = 1, h2 = r, h3 = r sin θ ∂ ϕ 1 ∂ ϕ 1 ∂ ϕ ∇ϕ = rˆ + θˆ + φˆ (B.22) ∂ r r ∂ θ r sin θ ∂ φ 1 ∂ 1 ∂ 1 ∂ A ∇ · A = (r2A ) + (sin θA ) + φ (B.23) r2 ∂ r r r sin θ ∂ θ θ r sin θ ∂ φ · ¸ 1 ∂ ∂ A ∇ × A = rˆ (sin θA ) − θ r sin θ ∂ θ φ ∂ φ · ¸ · ¸ 1 1 ∂ A ∂ 1 ∂ ∂ A + θˆ r − (rA ) + φˆ (rA ) − r (B.24) r sin θ ∂ φ ∂ r φ r ∂ r θ ∂ θ µ ¶ 1 ∂ ∂ ϕ 1 ∂ ³ ∂ ϕ´ 1 ∂2ϕ ∇2ϕ = r2 + sin θ + (B.25) r2 ∂ r ∂ r r2 sin θ ∂ θ ∂ θ r2 sin2 θ ∂ φ2 · ¸ 2 ³ ∂ A ∂ A ´ ∇2A = rˆ ∇2A − A + cot θA + csc θ φ + θ r r2 r θ ∂ φ ∂ θ · ¸ 1 ³ ∂ A ∂ A ´ + θˆ ∇2A − csc2 θA −2 r +2 cot θ csc θ φ θ r2 θ ∂ θ ∂ φ · ¸ 1 ³ ∂ A ∂ A ´ + φˆ ∇2A − csc2 θA −2 csc θ r −2 cot θ csc θ θ (B.26) φ r2 φ ∂ φ ∂ φ

B.2 Vector Formulas B.2.1 Vector Algebraic Formulas

A · B = B · A (B.27) A × B = −B × A (B.28) A · (B × C) = B · (C × A) = C · (A × B) (B.29) A × (B × C) = (A · C)B − (A · B)C (B.30) (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C) (B.31) (A × B) × (C × D) = (A × B · D)C − (A × B · C)D (B.32) Vector Analysis 681

B.2.2 Vector Diﬀerential Formulas

∇(ϕ + ψ) = ∇ϕ + ∇ψ (B.33) ∇(ϕψ) = ϕ∇ψ + ψ∇ϕ (B.34) ∇(A · B) = (A · ∇)B + (B · ∇)A + A × (∇ × B) + B × (∇ × A) (B.35) ∇ · (A + B) = ∇ · A + ∇ · B (B.36) ∇ · (ϕA) = A · ∇ϕ + ϕ∇ · A (B.37) ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) (B.38) ∇ × (A + B) = ∇ × A + ∇ × B (B.39) ∇ × (ϕA) = ∇ϕ × A + ϕ∇ × A (B.40) ∇ × (A × B) = A(∇ · B) − B(∇ · A) + (B · ∇)A − (A · ∇)B (B.41) ∇ · ∇ϕ = ∇2ϕ (B.42) ∇ × ∇ϕ = 0 (B.43) ∇ · ∇ × A = 0 (B.44) ∇ × ∇ × A = ∇(∇ · A) − ∇2A (B.45)

B.2.3 Vector Integral Formulas Volume V is bounded by closed surface S. The unit vector n is normal to S and directed positively outwards. Z I ∇ϕdV = ϕndS (B.46) V S Z I ∇ · AdV = A · ndS (Gauss0s theorem) (B.47) V Z S I ∇ × AdV = n × AdS (B.48) Z V I S (ϕ∇2ψ − ∇ϕ∇ψ)dV = ϕ∇ψ · ndS (Green0s ﬁrst identity) (B.49) Z V I S (ψ∇2ϕ−ϕ∇2ψ)dV = (ψ∇ϕ−ϕ∇ψ)·ndS (Green0s second identity) V S (B.50) Open surface S is bounded by closed line or contour l. Z I n × ∇ϕdS = ϕdl (B.51) S l Z I ∇ × A · ndS = A · dl (Stokes0s theorem) (B.52) S l 682 Appendix B

B.2.4 Diﬀerential Formulas for the Position Vector

x = xˆx + yˆy + zˆz x0 = xˆx0 + yˆy0 + zˆz0 p r = x − x0 = rˆr r = |r| = |x − x0| = (x − x0)2 + (y − y0)2 + (z − z0)2 ∂ ∂ ∂ ∂ ∂ ∂ ∇ = xˆ + yˆ + zˆ ∇0 = xˆ + yˆ + zˆ ∂ x ∂ y ∂ z ∂ x0 ∂ y0 ∂ z0 r ∇r = −∇0r = = rˆ (B.53) r 1 1 r rˆ ∇ = −∇0 = − = − (B.54) r r r3 r2 rˆ rˆ ∇ · = −∇ · = 4πδ(r) = 4πδ(x − x0) (B.55) r2 r2 1 1 ∇2 = ∇02 = −4πδ(r) = −4πδ(x − x0) (B.56) r r Appendix C

Bessel Functions

C.1 Power Series Representations

Bessel functions of the ﬁrst kind: X∞ (−1)m ³x´2m+n J (x) = (C.1) n m!(m + n)! 2 m=0 Bessel functions of the second kind or Neumann functions:

2 γx 1 nX−1 (n − m − 1)!³x´2m−n N (x) = ln J (x) − n π 2 n π m! 2 m=0 1 X∞ (−1)m ³x´2m+n³ 1 1 1 1 ´ − 1+ +· · ·+ +1+ +· · ·+ (C.2) π m!(m + n)! 2 2 m 2 m + n m=0 Modified Bessel functions of the first kind: X∞ 1 ³x´2m+n I (x) = (C.3) n m!(m + n)! 2 m=0 Modified Bessel functions of the second kind:

γx 1 nX−1 (n − m − 1)!³x´2m−n K (x) = (−1)n+1 ln I (x) − (−1)m n 2 n 2 m! 2 m=0 (−1)n X∞ 1 ³x´2m+n³ 1 1 1 1 ´ − 1+ +· · ·+ +1+ +· · ·+ (C.4) 2 m!(m + n)! 2 2 m 2 m + n m=0 where γ is the Euler’s constant

³ Xn 1 ´ ln γ = lim − ln n , ln γ = 0.5772, γ = 1.781 n→∞ m m=1 684 Appendix C

C.2 Integral Representations

−n Z 2π Z 2π j j(x cos α−nα) 1 jx cos α Jn(x)= e dα, J0(x)= e dα (C.5) 2π 0 2π 0 C.3 Approximate Expressions C.3.1 Leading Terms of Power Series (Small Argument)

x2 x x3 1 ³xń J (x) x−→→0 1 − , J (x) x−→→0 − , J (x) x−→→0 (C.6) 0 4 1 2 16 n n! 2 2 γx 2 (n − 1)!³ 2 ń N (x) x−→→0 ln , N (x) x−→→0 , N (x) x−→→0 , (n 6= 0) (C.7) 0 π 2 1 πx n π x x2 x x3 1 ³xń I (x) x−→→0 1 + , I (x) x−→→0 + , I (x) x−→→0 (C.8) 0 4 1 2 16 n n! 2 2 1 (n − 1)!³ 2 ń K (x) x−→→0 ln , K (x) x−→→0 , K (x) x−→→0 , (n 6= 0) (C.9) 0 γx 1 x n 2 x

C.3.2 Leading Terms of Asymptotic Series (Large Argument) r r 2 ³ π nπ ´ 2 ³ π nπ ´ J (x) x−→→∞ cos x− − , N (x) x−→→∞ sin x− − (C.10) n πx 4 2 n πx 4 2 r r (1) x→∞ 2 j x− π − nπ (2) x→∞ 2 −j x− π − nπ H (x) −→ e ( 4 2 ), H (x) −→ e ( 4 2 ) (C.11) n πx n πx r x→∞ 1 x x→∞ π −x In(x) −→ √ e , Kn(x) −→ e (C.12) 2πx 2x C.4 Formulas for Bessel Functions

(1) (2) Zn(x) represents Jn(x), Nn(x), Hn (x), and Hn (x).

C.4.1 Recurrence Formulas

2n Z (x) + Z (x) = Z (x) (C.13) n−1 n+1 x n 2n I (x) − I (x) = I (x) (C.14) n−1 n+1 x n 2n K (x) − K (x) = − K (x) (C.15) n−1 n+1 x n Bessel Functions 685

C.4.2 Derivatives

1 n n Z0 (x)= [Z (x) − Z (x)]=Z (x) − Z (x)= Z (x) − Z (x) n 2 n−1 n+1 n−1 x n x n n+1 (C.16) 1 n n I0 (x)= [I (x) + I (x)]=I (x) − I (x)= I (x) + I (x) (C.17) n 2 n−1 n+1 n−1 x n x n n+1 1 n n K0 (x)=− [K (x) + K (x)]=−K (x) − K (x)= K (x) − K (x) n 2 n−1 n+1 n−1 x n x n n+1 (C.18) 0 0 0 Z0(x) = −Z1(x), I0(x) = I1(x), K0(x) = −K1(x) (C.19) 1 1 1 Z0 (x) = Z (x)− Z (x), I0 (x) = I (x)− I (x), K0 (x) = −K (x)− K (x) 1 0 x 1 1 0 x 1 0 0 x 1 (C.20)

C.4.3 Integrals Z n+1 n+1 x Zn(x)dx = x Zn+1(x) (C.21) Z −(n−1) −(n−1) x Zn(x)dx = −x Zn−1(x) (C.22) Z x2 xZ2 (kx)dx = [Z2 (kx) − Z (kx)Z (kx)] (C.23) n 2 n n−1 n+1

C.4.4 Wronskian dF (x) dF (x) Deﬁne W [F (x),F (x)] = F (x) 2 − F (x) 1 as the Wronskian. 1 2 1 dx 2 dx 2 W [J (x), N (x)] = (C.24) n n πx 2 W [J (x), H(1)(x)] = j (C.25) n n πx 2 W [J (x), H(2)(x)] = −j (C.26) n n πx 4 W [H(1)(x), H(2)(x)] = j (C.27) n n πx 1 W [I (x), K (x)] = − (C.28) n n x Consequently 2 J (x)N (x) − N (x)J (x) = − (C.29) n n+1 n n+1 πx 1 I (x)K (x) − K (x)I (x) = (C.30) n n+1 n n+1 x 686 Appendix C

C.5 Spherical Bessel Functions C.5.1 Bessel Functions of Order n + 1/2 r 2 ³ 1 ´n dn sin x J (x) = xn+1/2 − (C.31) n+1/2 π x dxn x r 2 ³ 1 ´n dn cos x N (x) = (−1)n+1 xn+1/2 (C.32) n+1/2 π x dxn x r 2 Xn (n + k)! 1 H(1) (x) = e−jn(n+1)/2e jx (−1)k (C.33) n+1/2 πx k!(n − k)! (2jx)k k=0 r 2 Xn (n + k)! 1 H(2) (x) = e jn(n+1)/2e−jx (C.34) n+1/2 πx k!(n − k)! (2jx)k k=0 r r 2 2 ³sin x ´ J (x) = sin x, J (x) = − cos x (C.35) 1/2 πx 3/2 πx x r r 2 2 ³ cos x ´ N (x) = − cos x, N (x) = − sin x + − (C.36) 1/2 πx 3/2 πx x r r 2 e jx 2 ³e jx ´ H(1) (x) = , H(1) (x) = − e jx (C.37) 1/2 πx j 3/2 πx jx r r 2 e−jx 2 ³e−jx ´ H(2) (x) = , H(2) (x) = − e−jx (C.38) 1/2 πx −j 3/2 πx −jx

C.5.2 Spherical Bessel Functions r r π π j (x) = J (x), n (x) = N (x) (C.39) n 2x n+1/2 n 2x n+1/2 r r π π h(1)(x) = H(1) (x), h(2)(x) = H(2) (x) (C.40) n 2x n+1/2 n 2x n+1/2

C.5.3 Spherical Bessel Functions by S.A.Schelkunoﬀ r r πx πx Jˆ (x) = J (x), Nˆ (x) = N (x) (C.41) n 2 n+1/2 n 2 n+1/2 r r πx πx Hˆ (1)(x) = H(1) (x), Hˆ (2)(x) = H(2) (x) (C.42) n 2 n+1/2 n 2 n+1/2 Appendix D

Legendre Functions

D.1 Legendre Polynomials

n 1 d ¡ ¢n P (x) = x2 − 1 , (D.1) n 2n n! dxn 1 1 + x Xn 1 Q (x) = P (x) ln − P (x)Pn − 1(x). (D.2) n 2 n 1 − x l l−1 l=1 1 1 + x P (x) = 1, Q (x)= ln (D.3) 0 0 2 1 − x P1(x) = x, Q1(x)=xQ0(x) − 1 (D.4) 1 3 P (x) = (3x2−1), Q (x)=P (x)Q (x)− x (D.5) 2 2 2 2 0 2 1 5 3 P (x) = (5x3−3x), Q (x)=P (x)Q (x)− x2+ (D.6) 3 2 3 3 0 2 2 1 35 55 P (x) = (35x4−30x2+3), Q (x)=P (x)Q (x)− x3+ x (D.7) 4 8 4 4 0 8 24 D.2 Associate Legendre Polynomials

m m ¡ ¢m/2 d ¡ ¢m/2 d Pm(x)= x2 − 1 P (x) Qm(x) = x2 − 1 Q (x). (D.8) n dxm n n dxm n

1 2 1/2 P1(x) = (1 − x ) (D.9) 1 2 1/2 P2(x) = 3(1 − x ) x (D.10) 3 P1(x) = (1 − x2)1/2(5x2 − 1) (D.11) 3 2 5 P1(x) = (1 − x2)1/2(7x3 − 3x) (D.12) 4 2 688 Appendix D

2 2 P2(x) = 3(1 − x ) (D.13) 2 2 P3(x) = 15(1 − x )x (D.14) 15 P2(x) = (1 − x2)(7x2 − 1) (D.15) 4 2

3 2 3/2 P3(x) = 15(1 − x ) (D.16) 3 2 3/2 P4(x) = 105(1 − x ) x (D.17)

4 2 2 P4(x) = 105(1 − x ) (D.18)

D.3 Formulas for Legendre Polynomials

m In the following formulas, Rn(x) represents Pn(x) and Qn(x), Rn (x) repre- m m sents Pn (x) and Qn (x) including m = 0.

D.3.1 Recurrence Formulas

m m m (2n + 1)xRn (x) = (n + m)Rn−1(x) + (n − m + 1)Rn+1(x) (D.19) x 2m√ Rm(x) = (n − m − 1)(n + m)Rm−1(x) + Rm+1(x) (D.20) 1 − x2 n n n

D.3.2 Derivatives

0 0 (2n + 1)Rn(x) = Rn+1(x) − Rn−1(x) (D.21) 2 m0 m m (x − 1)Rn (x) = (n − m + 1)Rn+1(x) − (n + 1)xRn (x) (D.22)

D.3.3 Integrals Z R (x) − R (x) R (x)dx = n+1 n−1 (D.23) n 2n + 1 Z +1 Pn(x)Pl(x)dx = 0, for n 6= l (D.24) −1 Z +1 2 2 [Pn(x)] dx = , (D.25) −1 2n + 1 Z +1 m m Pn (x)Pl (x)dx = 0, for n 6= l (D.26) −1 Z +1 m 2 2 (n + m)! [Pn (x)] dx = , (D.27) −1 (2n + 1) (n − m)! Appendix E

Matrices and Tensors

E.1 Matrix

1. A matrix (a) or a is a rectangular array of real or complex scalars aij which are called the elements of the matrix,   a11 a12 · · · a1j · · · a1n    a21 a22 · · · a2j · · · a2n     · · · · · · · · · · · · · · · · · ·  a = (a) =   = (aij)mn (E.1)  ai1 ai2 · · · aij · · · ain   · · · · · · · · · · · · · · · · · ·  am1 am2 · · · amj · · · amn

2. A square matrix is a matrix with m = n.

3. A row matrix is a matrix with m = 1 and a Column matrix is a matrix with n = 1.

4. The determinant of a square matrix is a determinant consisting of the elements of the matrix, and is denoted by |A| or |(A)|.

5. The complementary minor is the determinant of the sub-matrix obtained from the square matrix (A) by deleting the ith row and the jth column, and is denoted by Mij .

6. The algebraic complement Aij is deﬁned as

i+j Aij = (−1) Mij

7. A diagonal matrix is a square matrix whose oﬀ-diagonal elements are zero, i.e., aij = 0, i 6= j. 690 Appendix E

8. A unit matrix (I) is a diagonal matrix where aii = 1; consequently, |(I)| = 1.

9. A zero matrix (0) is the matrix where aij = 0, for all i, j.

E.2 Matrix Algebra E.2.1 Deﬁnitions 1. Addition. The sum of two matrices exists only if the two matrices have the same size, i.e., the same number of rows and columns,

(A) + (B) = (aij)mn + (bij)mn = (aij + bij)mn (E.2)

2. Multiplication. If α is a scalar

α(A) = (αaij)mn (E.3)

3. Product of matrices. The product of two matrices exists only if the number of columns in (A) equals the number of rows in (B), i.e., (A) = (aij)mp,(B) = (bij)pn,

(A)(B) = (cij)mn (E.4)

where i = 1, 2 ··· m, j = 1, 2 ··· n, and

Xp cij = aikbkj = ai1b1j + ai2b2j + ··· + aipbpj k=1

E.2.2 Matrix Algebraic Formulas

In the following expressions, (A) = (aij)mn,(B) = (bij)mn,(C) = (cij)mn are matrices and α and β are scalars.

(A) + (B) = (B) + (A) (E.5) [(A) + (B)] + (C) = (A) + [(B) + (C)] (E.6) α[(A) + (B)] = α(A) + α(B) (E.7) (α + β)(A) = α(A) + β(A) (E.8) (A)(B) 6= (B)(A) (E.9) α[(A)(B)] = [α(A)](B) = (A)[α(B)] (E.10) [(A)(B)](C) = (A)[(B)(C)] (E.11) [(A) + (B)](C) = (A)(C) + (B)(C) (E.12) Matrices and Tensors 691

(A)[(B) + (C)] = (A)(B) + (A)(C) (E.13) |(A)(B)| = |A||B| (E.14) |α(A)| = αn|A| (E.15) (A) + (0) = (A) (E.16) (0)(A) = (A)(0) = (0) (E.17) (I)(A) = (A)(I) = (A) (E.18)

E.3 Matrix Functions

1. The negative of a matrix (A) is the matrix obtained by taking the negative value of all elements of (A), and is denoted by −(A),

−(A) = (−aij)mn, and (A)+(−(A)) = 0, −(−(A)) = (A) (E.19)

2. The conjugate of a matrix (A) is the matrix obtained by taking the conjugate of all elements of (A), and is denoted by (A)∗,

∗ ∗ (A) = (aij)mn (E.20) ((A)∗)∗ = (A) (E.21) ((A)(B))∗ = (A)∗(B)∗ (E.22)

3. The transpose of a matrix (A) is the matrix obtained by interchanging the rows of (A) and the columns of (A) and vice versa, and is denoted by (A)T, T (A) = (aji)nm (E.23) ((A)T)T = (A) (E.24) ((A)(B))T = (B)T(A)T (E.25) (A)∗T = (A)T∗ (E.26)

4. The conjugate transpose of a matrix (A) is the matrix obtained by apply- ing the conjugate and transpose operations on (A) simultaneously, and is denoted by (A)†, (A)† = (A)∗T (E.27) ((A)(B))† = (B)†(A)† (E.28)

5. The adjoint of a square matrix (A) is a square matrix whose elements is equal to the elements of the algebraic complement Aij in the matrix (A), and is denoted by (A)a,

a (A) = (Aij)nn (E.29) (A)(A)a = |A|(I) (E.30) 692 Appendix E

6. The inverse of a matrix (A) is the matrix that the product of (A) and its inverse matrix is a unit matrix. The matrix (A) has a unique inverse only if it is square and nonsingular, and is denoted by (A)−1,

(A)(A)−1 = (A)−1(A) = (I) (E.31)

1 1 (A)−1 = (A)a = (A ) (E.32) |A| |A| ij nn ((A)−1)−1 = (A) (E.33) 1 (α(A))−1 = (A)−1 (E.34) α ((A)(B))−1 = (B)−1(A)−1 (E.35) ((A)∗)−1 = ((A)−1)∗, ((A)T)−1 = ((A)−1)T, ((A)†)−1 = ((A)−1)† (E.36)

E.4 Special Matrices

1. Real matrix, ∗ (A) = (A), all aij are real

2. Symmetric matrix, T (A) = (A), aij = aji

3. Skew-symmetric matrix,

T (A) = −(A), aij = 0, for i = j, aij = −aji, for i 6= j

4. Hermitian matrix,

(A)† = (A), (A)T = (A)∗

5. Skew-Hermitian matrix,

(A)† = −(A), (A)T = −(A)∗

6. Unitary matrix or U matrix,

(A)†(A) = (I), (A)† = (A)−1, (A) = ((A)†)−1

7. Orthogonal matrix is a real unitary matrix,

(A)T(A) = (I), (A)T = (A)−1, (A) = ((A)T)−1 Matrices and Tensors 693

E.5 Tensors and vectors

1. A vector can be expressed by a row matrix or column matrix with three elements, and is denoted by a italic bold-face letter A,   Ax A = [Ax Ay Az] =  Ay  (E.37) Az

2. A tensor of rank 2 can be expressed by a 3 × 3 square matrix and is denoted by a bold face letter a,   axx axy axz a =  ayx ayy ayz  (E.38) azx azy azz

3. Vector and tensor operations     axx axy axz Ax a · A =  ayx ayy ayz   Ay  (E.39) azx azy azz Az   axx axy axz A · a = [Ax Ay Az]  ayx ayy ayz  (E.40) azx azy azz     axx axy axz bxx bxy bxz a · b =  ayx ayy ayz   byx byy byz  (E.41) azx azy azz bzx bzy bzz

A · a · A∗ = A∗ · aT · A (E.42) A · a∗ · A∗ = A∗ · a† · A (E.43) A · a · B = B · aT · A (E.44) A · a∗ · B = B · a† · A (E.45) Physical Constants

Physical constant Symbol

Speed of light in vacuum c 299792458 m/s ≈ 3 × 108 m/s

Vacuum permittivity ² 1 × 107 F/m 0 4πc2 ≈ 8.85418782 × 10−12 F/m

−7 Vacuum permeability µ0 4π × 10 H/m ≈ 12.5663706 × 10−7 H/m

−7 Vacuum wave impedance η0 4πc × 10 Ω ≈ 120π Ω

Electron charge magnitude e 1.6021892 × 10−19 C

Electron rest mass m 9.109534 × 10−31 kg

Electron charge to mass ratio e/m 1.75883 × 1011 C/kg

Proton rest mass m 1.6726485 × 10−27 kg

Gyromagnetic ratio γ 1.75883 × 1011 rad/(s·T) Smith Chart Bibliography

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ABCD law, 589 Bi-anisotropic media, 11 ABCD matrix, 147 Biconical cavity, 291 Admittance, Biconical line, 291 characteristic, 120 Bi-isotropic media, 11 mutual, 139 Binomial transducer, 173 normalized, 125 Biot–savart law, 3 self, 139 Birefringence, 512 Admittance matrix, 139 Borgnis’ potentials (functions), 188 Ampère force, 4 Boundary conditions, 19 Ampère’s circuital law, 3 for Helmholtz equations, 198 Ampère’s law of force, 3 general, 19 Angular wave number, 57 impedance surface, 23 Angle of incidence, 85 open-circuit surface, 22 Angle of reflection, 85 perfect conductor surface, 21 Angle of refraction, 85 short-circuit surface, 21 Anisotropic media (materials), 10, 493 Boundary value problems, 179 electric, 494 Brewster angle, 96 magnetic, 494 Brillouin diagram, 417 nonreciprocal, 496 reciprocal, 496 Carter chart, 134 Anti-reflection (AR) coating, 109, 111, 113 Cavity (resonator), 243 Associate Legerdre functions (polynomials) biconical, 294 217 capacity-loaded coaxial line, 304 Asymmetrical planar dielectric waveguide, capacity-loaded radial line, 303 339 circular cylindrical, 279 Attenuation, 63 coaxial, 273 absorption, 154 insertion, 154 radial line, 287 of the metallic waveguide, 241 rectangular, 259 reflection, 154 reentrant, 295 Attenuation coefficient, 63 Q of, 244 in circular waveguide, 277 sectorial, 264 in metallic waveguide, 241 spherical, 288 in optical fiber, 384 Characteristic impedance, 120 in rectangular waveguide, 251 of metallic waveguide, 240 Backward wave (BW), 178, 403, 416 Bessel equations, 210 of transmission line, 120 Bessel functions, 211 Chebyshev polynomials, 168 modified, 212 Chebyshev transducer, 170, 175 of order n + 1/2, 218 Chiral media, 12 of the first kind, 212 Circular cylindrical cavity, 27 of the second kind, 212 Circular cylinder coordinates, 200, 209 roots of, 270, 275, 289 Circularly asymmetric modes, 277, 369 spherical, 219 Circularly polarized wave, 72 Index 705

Circular polarization, 72 Cylindrical (coordinate) systems, 193, 200 Circularly symmetric modes, 263, 278 Circular waveguide, 274 D’Alembert’s equations, 43 Clockwise polarized wave (CW), 68, 73 complex, 45 Coaxial cavity, 273 Damped waves, 63 Coaxial line, 268 Decaying field, 99 Co-directional coupling, 455 Decibel, 63 Completeness (relation), 222 Degree of polarization, 76 Complex index (of refraction), 479 Density modulation, 529, 531 for metals, 482 DFB structure, 462 Complex Maxwell equations, 13 DFB laser, 469 Complex permeability, 15 DFB resonator, 466, 469 Complex permittivity, 15, 478 quarter-wave shifted, 469 Complex susceptibility, 15, 477 transmission, 466 Complex vector, 13 Dielectric coated conducting cylinder, 385 Complex wave equations, 29 Dielectric coated conducting plate, 339 Conductivity, 6, 483 Dielectric crystals, 504 Constitutional tensors, 11, 494 Dielectric layer, 109 Constitutive equations (relations), 8 double, 164 for anisotropic media, 494 multiple, 111, 166 Constitutive matrix, 11, 494 single, 109, 161 Constitutive parameters, 8, 11 small reflection approach, 171 Contra-directional coupling, 456, 460 Dielectric resonator, 317, 387 Coordinate system, 185 cutoff-waveguide approach, 391 circular cylindrical, 200, 209 cutoff-waveguide, cutoff-radial-line ap- cylindrical, 193, 201 proach, 393 KDB, 500 in microwave integrated circuits, 395 orthogonal curvilinear, 185 perfect-magnetic-wall (open-circuit rectangular, 200, 205 boundary) approach, 387 spherical, 200, 214 Dielectric waveguide, 317 Corrugated conducting surface, 404 channel, 346 as periodic system, 423 circular, 356 as uniform system, 404 nonmagnetic, 368 bounded structure, 406 weakly guiding, 377 unbounded structure, 404 planar, 327, 339 Cotton–Mouton effect, 560 symmetrical, 327 Coulomb gauge, 42 asymmetrical, 339 Coulomb’s law, 3 rectangular, 346 Counterclockwise polarized wave (CCW), slab, 327 68, 73 strip, 346 Coupled-cavity chain, 411, 418 weakly guiding, 338, 377 Coupling coefficient, 451, 459 Diffusion equation, 25 Coupling impedance, 404 complex, 29 Coupling of modes, 450 Diffraction, 621 Critical angle, 97 Fraunhofer, 627, 629 Crystals, 504 Fresnel-Kirchhof, 623, 632 biaxial, 505 in anisotropic media, 640 isotropic, 504 of Gaussian beams, 634 reciprocal, 504 Rayleigh-Sommerfeld, 625 uniaxial, 504 Disk-loaded waveguide, 407, 426 Cutoff (angular) frequency, 204, 237 as periodic system, 426 Cutoff state, modes, 204 as uniform system, 407 Cutoff wavelength, 238 Dispersion, 9, 15, 476 Cutoff (angular) wave number, 204, 237 anomalous, 481 Cylindrical harmonics, 212 normal, 481 Cylindrical horn waveguide, 282 Classical theory of, 476 706 Index

Ideal gas model of, 476 in nonreciprocal media, 547 Dispersion characteristics (relations), 239 in plasma, 548 of asymmetrical planar dielectric in reciprocal media, 518 waveguide, 343 in simple media, 25, 55 of circular dielectric waveguides, 371 in uniaxial crystals, 505, 521 of metallic waveguides, 239 Electron beam, 526 of planar dielectric waveguides, 333 Elliptically polarized wave, 68 of slow waves, 402 Elliptic polarization, 68 Dispersion curves, 239, 334, 344, 355, 371, Elliptic Gaussian beam, 592 416 Energy density, 31, 34 Dispersive media, 9, 15, 476 in anisotropic media, 38 displacement current, 4, 7 in dispersive media, 35 Dissipation, 16 Equal ripple response, 170, 175 conducting, 31 Equation of continuity, 3 Distributed feedback (DFB) structure, 462 Equivalent (fictitious) magnetic charge, 18 Dominant mode, 252, 382 Equivalent (fictitious) magnetic current, 18 Double dielectric layer. 164 Equivalent transmission line, 134 Double refraction, 512 Evanescent modes, 204 Dual boundary conditions, 50 Expansion theorem, 221 Dual equations, 50 Extraordinary (e) wave, 499, 508, 511 Duality, 18, 50 Faraday rotation (effect), 550, 557 Effective-index surface, 522 Faraday’s law, 3 Effective permeability, 555 Fast wave (modes), 89, 100, 203 Effective permittivity, 549 Ferrite, 537 Eigenfunctions, 220 lossy, 542 vector, 223, 225 Field matching, 183, 230 Eigenvalues, 220, 222 approximate, 230 two-dimensional, 224 Flattest response, 173 variational principle of, 228 Floquet’s theorem, 412 Eigenvalue problems, 220 Forward wave (FW), 403, 416 Eigenwave (mode) expansions, 658 Fraunhofer diffraction, 627, 629, 634, 640, in anisotropic media, 665 645 in cylindrical coordinate system, 660 Frequency-domain Maxwell equations, 14 in inhomogeneous and anisotropic me- Frequency-domain wave equations, 29 dia, 666 Fresnel diffraction, 627, 632, 638, 649 in inhomogeneous media, 662 Fresnel–Kirchhoff diffraction formula, 623 in rectangular coordinate system, 658 Fresnel’s law, 91 Electric charge density, 3 Fundamental Gaussian beam, 577 bound, 5 free, 5 Gadolinium gallium garnet (GGG), 561 Electric current density, 3 Gaussian beams, 577 conductive, 5 beam radius of, 581 convection, 5 curvature radius of phase front of, 581 molecular, 6 electric and magnetic fields in, 583 polarization, 5 elliptic, 592 Electric field (strength), 4 energy density in, 584 Electric induction, 7 fundamental, 577 Electric susceptibility, 8 Hermite-, 596 Electric wall, 22 high-order modes, 595 Electromagnetic waves, 25 in quadratic index media, 603 in anisotropic media, 497 in anisotropic media, 614 in biaxial crystals, 505, 519 Laguerre-, 600 in dispersive media, 475 phase velocity of, 582 in electron beam, 556 power flow in, 584 in ferrite, 552 reflection and refraction of, 652, 668 Index 707

transformation of, 585 quarter-wavelength, 109, 161 Gauss’s law, 3 small reflection approach, 171 Good conductor, 65 Impedance transformation, 107, 126 Goos-Hänchen shift, 102 Incident wave, 77, 84 Group velocity, 203, 238, 487 Inclined-plate line, 283 in reciprocal crystals, 525 Index (of refraction), 86 Guided layer, 340 complex, 479 Guided modes, 203, 237, 333 Index ellipsoid, 513 Guided waves, 202 Insertion attenuation, 154 Guided-wavelength, 238 Insertion loss, 154 Gyromagnetic media, 537 Insertion phase shift, 155 Gyrotropic media, 497, 534 Insertion reflection coefficient, 153 Insertion VSWR, 153 Half far-field divergence angle, 581 Interaction impedance, 404 Hankel functions, 211 for periodic systems, 422 Harmonics, 206 Inward wave, 139 cylindrical, 212 Isotropic media, 8 rectangular, 206 spherical, 217 Joule’s law, 31 Helix, 431 Joule loss, 31, 35, sheath, 432 tape, 442 k–β diagram, 239, 415 Helmholtz’s equations, 29 KDB coordinate system, 500 approximate solution of, 228 Kirchhoff integral theorem, 621 solution of, 188 Kirchhoff’s diffraction theory, 621 HEM modes, 205, 347, 362 Klystron, 533 Hermite-Gaussian beam, 596 Kronig-Kramers relations, 479 Hertz vector (potential), 46 complex, 49 Laguerre-Gaussian beam, 600 electric, 47, 49 Lame coefficients, 186 instantaneous, 46, Larmor precession, 539 magnetic, 47, 49 Larmor frequency, 539 Method of, 194 Left-handed polarized wave, 68, 72 High-order mode Gaussian beam, 595 Legendre equation, 216 High-reflection (HR) coating, 111, 113 Legendre functions (polynomials), 216 associate, 217 Ideal gas model, 476 of the first kind, 217 Ideal transformer, 156 of the second kind, 217 Ideal waveguide, 237 Linearly polarized modes, 380 Impedance, 121 Linearly polarized wave, 68, 71 characteristic, 120 Liquid-phase epitaxy (LPE), 561 coupling, 404 Lorentz force, 4 interaction, 404 Lorentz gauge, 42 mutual, 137 Loss angle, 17 normalized, 125 Loss tangent, 17 self, 137 Lossless line, 121 wave, 28, 57 Lossless network, 144 Impedance matrix, 136 LSE mode, 209, 254 Impedance surface, 23 LSM mode, 209, 255 Impedance transducer, 109, 161 binomial, 173 Magnetic field (strength), 7 chebyshev, 170, 175 Magnetic flux density, 4 double-section, 164 Magnetic induction, 4 equal ripple response, 170, 175 Magnetic susceptibility, 8 flattest response, 173 Magnetic wall, 22 multi-section, 111, 166 Magnetization, 6 708 Index

Magnetization dissipation, 16 TEM, 26, 202 Magnetization damping loss, 35 TM, 87, 205, 363 Magnetization vector, 6 Modiﬁed Bessel functions, 212 Magnetoelectric material, 13 Monochromatic wave, 76 Magnetostatic waves (MSW), 560 Monopole, 3 backward volume (MSBVW), 567 Multiple-layer coating, 116, 166 forward volume (MSFVW), 572 as DFB transmission resonator, 470 surface (MSSW), 569 with an alternating index, 111 Matched line, 127 Multi-section impedance transducer, 111, Matrix, 136 166 admittance, 138 Mutual admittance, 139 impedance, 136 Mutual impedance, 137 network, 136 scattering, 139 Nabla operator, 187 transfer, 147 Natural (angular) frequency, 244 transmission, 148 Natural (angular) wave number, 243 Maxwell’s equations, 1 Neper, 63 Basic 2 Network, 136 for uniform simple media, 9 lossless, 144 frequency-domain (complex), 14 multi-port, 136 in anisotropic media, 17 reciprocal, 142 in derivative form, 2 source-free, 144 in integral form, 2 symmetrical, 151 in KDB system, 503 two port, 146 in material media, 6 Network matrix, 136 in vacuum, 5 Network parameters, 136 time-domain (instantaneous), 2 Neumann functions, 211 Media, Nonlinear media, 10 anisotropic, 10, 493 Non-polarized wave, 76 electric, 494 Nonreciprocal media, 496, 534 gyrotropic, 534 Non-thin lens, 591 gyromagnetic, 537 Normalized admittance, 126 magnetic, 494 Normalized impedance, 126, 135 nonreciprocal, 496, 534 Normal mode, 219 reciprocal, 496 Normal mode expansion, 225 bi-anisotropic, 11 Normal surface, 522 bi-isotropic, 11 n wave, 86, 91 chiral, 12 dispersive, 9, 15, 476 isotropic, 8 Ohm’s law, 6 nonlinear, 10 Open-circuit line, 127 simple, 8 Open-circuit surface, 22 Meridional ray (wave), 356, 361 Optical axis, 504 Method of Borgnis’ potentials, 188 Optical ﬁbers, 356 Method of Hertz vectors, 194 low-attenuation, 384 Method of longitudinal components, 195 nonmagnetic, 368 Mode, weakly guiding, 377 circularly asymmetric, 369 Optical resonators, 611 circularly symmetric, 363 Ordinary (o) wave, 499, 507, 510 EH, 369, 374 Orthogonal curvilinear coordinate systems, HE, 369, 374 185 HEM, 205 Orthogonal eigenfunction set, 221 linearly polarized, 380 Orthogonal expansion, 221 LSE, 209, 254 Orthogonality theorem, 221 LSM, 209, 255 Orthonormal eigenfunction set, 221 TE, 87, 205,363 Outward wave, 139 Index 709

Parallel-plate transmission line, 256 elliptically, 69 Paraxial approximation, 580 linearly, 68, 71 Partially polarized wave, 76 polarizing angle, 96 Penetration depth, 64 Potentials, Perfect conductor, 21, 67 scalar and vector, 41 Perfect electric conductor (PEC), 388 retarding, 41, 45 Perfect magnetic conductor (PMC), 388 Poynting’s theorem, 30 Periodic structures (systems), 411 for anisotropic media, 38 Permeability, 4, 8 for dispersive media, 35 complex, 15 frequency-domain (complex), 32 of vacuum, 4 the perturbation formulation of, 39 tensor, 10 time-domain (instantaneous), 30 for ferrite, 541, 544 Poynting vector, 32 Permittivity, 4, 8 complex, 33 complex, 15, 478 Principle axes (system), 496 of vacuum, 4 Principle mode, 250 tensor, 10 Principle of perturbation, 305 for electron beam, 529 Propagation coefficient (constant), 63 for plasma, 536 p wave, 86, 91 Perturbation, 305 cavity wall, 305 Q factor, 244 material, 308 of resonant cavity, 244 of cutoff frequency, 311 external, 245 of propagation constant, 312 loaded, 245 Phase coefficient (constant), 57 unloaded, 245 Phase matching, 453 of circular cylindrical cavity, 281 Phase synchronous, 453 q parameter, 585 Phase velocity, 27, 203, 238, 486 Quality factor, 244 in dispersive media, 486 Quarter-wave shifted DFB resonator, 469 in reciprocal crystals, 525 Quarter-wavelength impedance transducer, of plane waves, 27 109, 161 of Gaussian beam, 582 Quasi-polarized wave, 76 of guided waves, 238 Quasi-monochromatic wave, 76 Phasor, 13 Plane waves, 25, 55 Radial (transmission) line, 285 in biaxial crystals, 519 Radial line cavity, 287 in lossy media, 63 Radiation modes, 334, 346 in uniaxial crystals, 505, 521 Rayleigh–Sommerfeld diffraction formula, sinusoidal, 55 625 Plasma, 484, 534 Reciprocal crystal, 504 Plasma frequency, 484 Reciprocal network, 142 Poincarésphere, 74 Reciprocity (theorem), 51 Polarization, 6, 67, 76 in network theory, 142 circular, 72 Rectangular coordinates, 200, 205 degree of, 76 Rectangular dielectric waveguide, 346 elliptic, 69 Rectangular harmonics, 206 linear, 68, 71 Rectangular (resonant) cavity, 259 of plane waves, 67 Rectangular waveguide, 245 Polarization dissipation, 16 Reentrant cavity, 295 Polarization loss, 35, 63 approximate solution, 300 Polarization potentials, 46 exact solution, 297 Polarization vector, 6 Reflected wave, 77, 81, 91 Polarized wave, 67, 76 Reflection, 77, 81, 91, 121 circularly, 72 insertion, 153 clockwise, 68, 73 Reflection coefficient, 82, 121, 140, 153 counterclockwise, 68, 73 Refracted wave, 91 710 Index

Refraction, 91 Snell’s law, 84 of Gaussian beams in anisotropic me- Solution of Helmholtz’s equations, 188 dia, 652 approximate, 228 Refraction coefficient, 91 in circular cylindrical coordinates, 209 Resonant cavity, 243 in rectangular coordinates, 205 biconical, 294 in spherical coordinates, 214 capacity-loaded coaxial line, 304 Source-free network, 144 capacity-loaded radial line, 303 Space charge waves, 530 circular cylindrical, 279 Space harmonics, 412 coaxial, 273 Spherical Bessel functions, 219 radial line, 287 Spherical cavity, 288 rectangular, 259 Spherical coordinates, 214, 288 reentrant, 295 Spherical harmonics, 217 Q of, 244 Spin waves, 560 sectorial, 264 Standing wave, 77, 83 spherical, 288 Standing-wave ratio, 83, 121, 123 Resonator, 243, 387 Stationary formula, 229 dielectric, 387 Stokes parameters, 74 cutoff-waveguide approach, 391 Sturm–Liouville problem, 220 cutoff-waveguide, cutoff-radial-line Susceptibility, 8 approach, 393 complex, 477 in microwave integrated circuits, tensor, 10 395 Surface acoustic waves, 561 perfect-magnetic-wall approach, Surface admittance, 23 387 Surface impedance, 23 optical, 611 Surface loss, 241 Retarding Potentials, 41, 45 Surface wave, 100 Right-handed polarized wave, 68, 72 Symmetrical network, 151 Roots of the Bessel functions, 270, 275, 289 Symmetrical planar dielectric waveguide, 327 Scalar function, 2 Scalar potential, 41 Tape helix, 442 Scalar wave functions, 190 Telegraph equation, 119 Scale factors, 186 TEM wave (modes), 26, 202 Scattering matrix (S-matrix), 139 cylindrical, 284, 286 Scattering parameters (S-parameters), 140 spherical, 294 Sectorial cavity, 264 Tensor, 10 Sectorial waveguide, 267 permeability, 10, 494 Self admittance, 139 permittivity, 10, 494 Self impedance, 137 TE wave (modes), 87, 91, 204, 237 Separation of variables, 199 Thin lens, 590 Sheath helix, 432 Time-harmonic fields, 13 Shimdt chart, 133 TM wave (modes), 89, 93, 205, 237 Short-circuit line, 127 Total (internal) reflection, 97 Short-circuit surface, 21 Transfer matrix, 147 Signal velocity, 492 Transformation of impedance, 107, 125 Simple media, 8 Transmission coefficient, 93, 95, 141 Single dielectric layer, 109, 161 Transmission line, 117 Skew ray (wave), 357, 361 biconical, 291 Skin depth, 64 coaxial, 268 Slow wave (modes), 99, 204, 402 equivalent, 134 Slow-wave structures (systems), 402 parallel-plate, 256 dispersion characteristics of, 402 radial, 285 Small-amplitude analysis, 527 Transmission line chart, 130 Small-reflection approach, 171 Transmission matrix, 148 Smith chart, 136 Traveling wave, 27, 58 Index 711

persistent, 58 metallic, 235 Traveling-standing wave, 80 circular, 274 Two port (network), 146 cylindrical horn, 282 lossless, 149 general characteristics of, 236 reciprocal, 149 propagation characteristics of, 237 source-free, 149 rectangular, 245 symmetrical, 151 ridge, 311 sectorial, 267 Uniaxial crystals, 504 with different filling media, 319 Uniform plane waves, 25, 55 Waveguide coupler, 458 sinusoidal, 55 Waveguide switch, 458 Uniqueness theorem, 180 Wave impedance, 28, 57 with complicated boundaries, 182 in good conductors, 65 Unit vector, 2, 186 in lossy media, 64 of metallic waveguide, 217 Variational principle of eigenvalues, 228 of plane waves, 28, 57 Vector, 2 of TE and TM modes, normal, 93, 94 complex, 14 Wavelength, 62 Vector function, 2 Wave number, 62 Vector phasor, 14 angular, 62 Vector potential, 41 Waves, 27, 55 Velocity, circularly polarized, 72, 65 group, 203, 238, 487 clockwise (CW), 68, 73 of energy flow, 490 counterclockwise (CCW), 68, 73 phase, 27, 57, 203, 238, 486 damped, 63 signal, 492 elliptically polarized, 68 Velocity modulation, 528, 531 extraordinary (e), 499, 508, 511 Voltage reflection coefficient, 122 in electron beam, 526 Voltage-standing-wave ratio (VSWR), 121 inward, 139 insertion, 153 left-handed polarized, 68, 72 Volume loss, 241 linearly polarized, 68, 71 monochromatic, 76 non-polarized, 76 Wave equations, 24 ordinary (o), 499, 507, 510 frequency-domain (complex), 29 outward, 139 generalized, 24 partially-polarized, 76 homogeneous, 24 polarized, 67, 76 inhomogeneous, 24 quasi-polarized, 76 time-domain (instantaneous), 24 right-handed polarized, 68, 72 Wave impedance, 28 slow, 99, 204, 402 in good conductors, 65 TE, 87, 91, 204, 237 in lossy media, 64 TEM, 26, 202 of metallic waveguide, 240 TM, 89, 93, 205, 237 of plane waves, 28, 57 uniform plane, 25, 55 Waveguide, Wave vector, 59 dielectric, 317 Weakly guiding optical fiber, 377 asymmetrical planar, 339 circular, 356 Yttrium iron garnet (YIG), 561 circular, nonmagnetic, 368 channel, 346 ²-anisotropic media, 494 planar, 327, 339 µ-anisotropic media, 494 rectangular, 346 ω–β diagram, 239, 416 slab, 327 for coupled modes, 453 strip, 346 of DFB, 465 symmetrical planar, 327 of periodic system, 416 disk loaded, 407, 426 ideal, 237