Lecture 16 Waves in Isotropic Media: Dielectrics And

Lecture 16

Waves in Isotropic Media: Dielectrics and Conductors

In this lecture you will learn:

• Wave propagation in dielectric media

• Waves propagation in conductive media

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Review: Plane Waves in Free Space Faraday’s Law: r r r r ∇ × E()r = − j ω µo H ()r Ampere’s Law: r r r v r r ∇ × H()r = J()r + j ω εo E(r ) r Complex Wave Equation: Assume: J(rv) = ρ(rr) = 0

r r r r 2 r r ∇ × ∇ × E()r = − j ω µo ∇ × H(r ) = ω µo εo E(r ) 0 r r 2 r r 2 r r ⇒ ∇()∇ .E()r − ∇ E()r = ω µo εo E ()r 2 r r 2 r r ⇒ ∇ E()r = −ω µo εo E ()r

For a plane wave in free space we know the E-field and H-field phasors to be: r r r r ˆ − j k .r ω E r E()r = n Eo e k = ω µ ε = k o o c r E r r H()rr = kˆ × nˆ o e− j k .r µ ( ) η = o ≈ 377 Ω H ηo o εo

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

1 Waves in a Dielectric Medium – Wave Equation

Suppose we have a plane wave of the form, E r r r E()rr = nˆ E e− j k .r o ε traveling in an infinite dielectric medium with permittivity ε H What is different from wave propagation in free space?

Faraday’s Law: r r r r ∇ × E()r = − j ω µo H ()r Ampere’s Law: r r r ∇ × H()rr = J()rv + j ω ε E(rr) r Complex Wave Equation: Assume: J(rv) = ρ(rr) = 0

r r r r 2 r r compare with the ∇ × ∇ × E()r = − j ω µo ∇ × H(r ) = ω µo ε E(r ) 0 complex wave equation r r 2 r r 2 r r in free space ⇒ ∇()∇ .E()r − ∇ E()r = ω µo ε E ()r 2 r r 2 r r 2 r r 2 r r ⇒ ∇ E()r = −ω µo ε E ()r ∇ E(r )()= −ω µo εo E r

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Waves in a Dielectric Medium – Dispersion Relation

Substitute the plane wave solution: r E r r − j k .rr E()r = nˆ Eo e in the complex wave equation: ε 2 r r 2 r r H ∇ E()r = −ω µo ε E(r ) To get: 2 r r 2 r r ∇ E()r = −ω µo ε E(r ) r r r r 2 r r ⇒ − k .k E()r = −ω µo ε E ()r 2 2 ⇒ k = ω µo ε compare with k = ω µ ε ⇒ k = ω µ ε o o o for waves in free space

Refractive Index: ε Define refractive index “n” of a dielectric medium as: n = εo

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

2 Waves in a Dielectric Medium – Velocity

Plane wave: E r r r − j k .rr E()r = nˆ Eo e ε

Dispersion relation: H

k = ω µo ε ε ε n = ⇒ k = ω µo εo ε εo o n ω ⇒ k = ω compare with k = for waves in c free space c

The velocity of waves in a dielectric medium is reduced from the velocity of waves in free space by the refractive index

• Velocity of waves in free space: c c • Velocity of waves in dielectric medium of refractive index n: n

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Waves in a Dielectric Medium - Wavelength Plane wave in a dielectric medium: E r r r E()rr = nˆ E e− j k .r o ε n Dispersion relation: k = ω H c But the magnitude of the wavevector is 2π related to the wavelength by the relation: k = λ So for a dielectric medium we get: 2π c 2π c compare with λ = for waves λ = in free space ω ω n

The wavelength of plane waves in a dielectric medium is reduced from the wavelength of plane waves of the same frequency in free space by the refractive index

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

3 Waves in a Dielectric Medium – Magnetic Field

Plane wave: E r r r − j k .rr E()r = nˆ Eo e ε H Calculate the magnetic field:

r j r H()rr = ∇ × E()rr ω µo r r r ˆ k − j k .rr ⇒ H()r = ()k × nˆ Eo e ω µo r k r r ⇒ H()rr = ()kˆ × nˆ E e− j k .r ω µ o o The wave impedance also r E r r ⇒ H()rr = ()kˆ × nˆ o e− j k .r changes η µ η = o ε µ ε η = o o = o εo ε n

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Waves in a Conductive Medium – Complex Permittivity Suppose we have a plane wave of the form, σ E r r − j k z E()r = xˆ Eo e ε

traveling in an infinite medium with conductivity σ and H permittivity ε Faraday’s Law: r r r r ∇ × E()r = − j ω µo H ()r r Ampere’s Law: Now J()rv ≠ 0 r r r v r r ∇ × H()r = J()r + j ω ε E(r ) ⎛ σ ⎞ r r r r εeff ()ω = ε ⎜1− j ⎟ = σ E()r + j ω ε E ()r ⎝ ω ε ⎠ r r = j ω εeff ()ω E()r

Complex Wave Equation:

2 r r 2 r r The effect of conductivity has been ⇒ ∇ E()r = −ω µo εeff (ω)E(r ) absorbed in a complex frequency dependent effective permittivity

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

4 Waves in a Conductive Medium – Complex Refractive Index Plane wave: σ E r r − j k z E()r = xˆ Eo e ε H Dispersion relation:

k = ω µo εeff ()ω ε (ω ) εeff ()ω n ()ω = eff complex refractive ⇒ k = ω µo εo eff ε index εo o n ()ω ω ⇒ k = ω eff compare with k = for waves in c free space c

ε σ Since the refractive index is complex: n ()ω = 1− j eff ε ω ε the wavevector k is also complex o

Let: σ k = k'− j k'' = loss tangent ω ε Real part Imaginary part

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Waves in a Conductive Medium – Complex Wavevector Plane wave: σ E r r − j k z E()r = xˆ Eo e ε

Complex wavevector: k = k'− j k'' H

What are the implications of a complex wavevector? • Wave decays exponentially with distance as it propagates r r − j k z E()r = xˆ Eo e exponential decay r r − j k' z −k''z ⇒ E()r = xˆ Eo e e E σ e−k'' z ε • The wavelength is related to the real part of the wavevector: 2π k' = z λ λ

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

5 Waves in a Conductive Medium – Magnetic Field Plane wave: σ r r − j k z E E()r = xˆ Eo e ε Calculate the magnetic field: H

r j r H()rr = ∇ × E()rr ω µo r r k − j k z Don’t forget that the wavevector ⇒ H()r = yˆ Eo e ω µo “k” is complex now r r k − j k z ⇒ H()r = yˆ Eo e ω µo r E The impedance is now also ⇒ H()rr = yˆ o e− j k z complex ηeff ()ω µo ηeff ()ω = εeff ()ω

Note: The E-field and the H-field are no longer in phase since ηeff(ω ) is complex

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Waves in a Conductive Medium – Power Flow Plane wave: σ E r r − j k z r r Eo − j k z E()r = xˆ Eo e H()r = yˆ e ε ηeff ()ω k = k'− j k'' H 1 k k'− j k'' Note that: = = E σ η ()ω ω µ ω µ eff o o e−k'' z ε Poynting vector and time average power per unit area: r r 1 r r S()r ,t = Re[]S()r z 2 1 r r λ = Re[]E()rr × H * ()rr 2 1 ⎡ E2 ⎤ E2 ⎛ k' ⎞ = Re⎢zˆ o e−2k'' z ⎥ = zˆ o ⎜ ⎟ e−2k''z 2 * 2 ⎜ ω µ ⎟ ⎣⎢ ()ηeff ()ω ⎦⎥ ⎝ o ⎠ Time average power per unit area decays exponentially with distance because energy is dissipated in a conductive medium due to I2R (or J.E) type of losses and this energy dissipated is taken away from the plane wave

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

6 Loss Tangent and Dielectric Relaxation Time - I n (ω) The complex wavevector is: k = ω eff c ε σ The complex refractive index is: neff ()ω = 1− j εo ω ε

Loss tangent σ Loss tangent = ω ε ε But the dielectric relaxation time was: τ = d σ 1 ε 1 ⇒ Loss tangent = And: neff ()ω = 1− j ω τd εo ωτd

Loss tangent

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Loss Tangent and Dielectric Relaxation Time - II

There are two possible scenarios:

High frequency and/or low conductivity case (e.g. lossy dielectrics) ω ε ω τd >> 1 or >> 1 σ The frequency is much greater than the inverse dielectric relaxation time

⇒The conductive medium does not have enough time to react to the electromagnetic wave ⇒No appreciable currents flow in the conductive medium

Low frequency and/or high conductivity case (e.g. Imperfect metals) ω ε ω τd << 1 or << 1 σ The frequency is much smaller than the inverse dielectric relaxation time

⇒The conductive medium has enough time to react to the electromagnetic wave ⇒Appreciable currents flow in the conductive medium

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

7 Waves in a Conductive Medium – Lossy Dielectrics Plane waves: σ E r r − j k z E()r = xˆ Eo e ε Dispersion relation Refractive index H n ()ω ε σ k = ω eff neff ()ω = 1− j σ 1 c εo ω ε = = loss tangent ω ε ω τd Lossy dielectric approximation: σ ε ⎛ σ ⎞ If << 1 then: neff ()ω ≈ ⎜1− j ⎟ ω ε εo ⎝ 2ω ε ⎠ ω ε ⎛ σ ⎞ E σ ⇒ k = k'− j k'' = ⎜1− j ⎟ ⎜ ⎟ −k'' z ε c εo ⎝ 2 ω ε ⎠ e

ω ε σ µ 2π k' = k'' = o λ = c ε 2 ε o k' z r r − j k z E()r = xˆ Eo e λ r r − j k' z −k''z ⇒ E()r = xˆ Eo e e

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Waves in a Conductive Medium – Imperfect Metals

Now consider the case when: ω ε ω τ << 1 or << 1 d σ The frequency is much smaller than the inverse dielectric relaxation time

⇒The conductive medium has enough time to react to the electromagnetic wave ⇒Appreciable currents flow in the conductive medium c x r r E Current Density: J = σ E

z y H σ Plane wave ε

These currents try to screen out the magnetic field and, therefore, prevent the electromagnetic wave from going into the conductor

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

8 Waves in a Conductive Medium – Imperfect Metals Plane waves: σ E r r − j k z E()r = xˆ Eo e ε Dispersion relation Refractive index H ε σ neff ()ω k = ω neff ()ω = 1− j σ 1 c εo ω ε = = loss tangent ω ε ω τd

Imperfect metal approximation: σ ε σ σ Suppose >> 1 then: neff ()ω ≈ − j = ()1− j ω ε εo ω ε 2ω εo

σ ω µ 1 ⇒ k = k'− j k'' = o ()1− j = ()1− j 2 δ

ω µo σ 1 ω µo σ 1 2π k' = = k'' = = λ = = 2π δ 2 δ 2 δ k'

δ = penetration depth or skin-depth

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

Waves in a Conductive Medium – Imperfect Metals Due to current screening the wave decays within a few skin-depths: E σ r r − j k z ε E()r = xˆ Eo e r r − j k'z −k'' z ⇒ E()r = xˆ Eo e e z − z r r − j k'z δ ⇒ E()r = xˆ Eo e e z − δ δ = penetration depth or skin-depth e

Since the wavelength λ inside the medium is 2πδ , the wave hardly propagates one wavelength distance into the medium

r r The screening current density, given by J = σ E , is non-zero only in a layer of thickness equal to skin-depth δ near the surface

ECE 303 – Fall 2005 – Farhan Rana – Cornell University

9 Waves in a Conductive Medium

Lossy Dielectrics Imperfect Metals

ECE 303 – Fall 2005 – Farhan Rana – Cornell University