19-1 EM Lecture note W8.pdf

Jaesang Lee

Dept. of Electrical and Computer Engineering Seoul National University

(email: [email protected])

Electromagnetics

<Chap. 10> Waveguides and Cavity Resonators Section 10.5 ~ 10.6

(1st of week 8)

Textbook: Field and Wave Electromagnetics, 2E, Addison-Wesley

Chap. 10 | Contents for 1

^st

class of week 8

Sec 5. Circular waveguides

• Bessel’s differential equation and Bessel function

• Characteristics of TE and TM wave propagation

Chap. 10 | Introduction: Circular waveguide

• Circular waveguide

- Round metal pipe having a uniform circular cross-section - Inside filled with a dielectric (μ and ε)

• Wave equations for EM waves in a circular waveguide

∇

²

E + k

²

E = 0

∇

²

H + k

²

H = 0

⎧ ⎨

⎪

⎩⎪ → ( ∇

_r²_φ

+ ∇

_z²

) E + k

²

E = 0

∇

_r²_φ

+ ∇

_z²

( ) H + k

²

H = 0

⎧ ⎨

⎪

⎩⎪

where

∇

_r²_φ

:

∇

²_z

:

⎧ ⎨

⎪

⎩⎪

Laplacian for a transverse polar plane (r and φ) Laplacian for a longitudinal axis (z)

z r

φ

E = E

_T

+ a

_z

E

_z

H = H

_T

+ a

_z

H

_z

⎧ ⎨

⎩

• Longitudinal field components

- For TM mode

H

_z

= 0

∇

_r²_φ

+ ∇

²_z

( ) E

^z

+ k

²

E

^z

= 0

⎧ ⎨

⎪

⎩⎪

Here, where

E

_z

( r, φ , z ) = E

^z0

( ) r, φ e

^{−γ z}

H

_z

( r, φ , z ) = H

z

0

( ) r, φ e

^{−γ z}

⎧ ⎨

⎪

⎩⎪

- For TE mode

E

_z

= 0

∇

_r²_φ

+ ∇

²_z

( ) H

^z

+ k

²

H

^z

= 0

⎧ ⎨

⎪

⎩⎪

→ ∇

_r²_φ

E

_z⁰

+ ( γ

²

+ k

²

) E

^z0

= 0

→ ∇

_r²_φ

E

_z⁰

+ h

²

E

_z⁰

= 0

→ ∇

_r²_φ

H

_z⁰

+ ( γ

²

+ k

²

) H

^z0

= 0

→ ∇

_r²_φ

H

_z⁰

+ h

²

H

_z⁰

= 0

<Example of circular waveguide>

(By definition) (By definition)

Chap. 10 | Bessel’s differential equations and Bessel functions (1/3)

• Wave equation

- In cylindrical coordinate

Friedrich Wilhelm Bessel (Prussian (German))

(1784-1846)

∇

_r²_φ

E

_z⁰

+ h

²

E

_z⁰

= 0 → 1 r

∂

∂ r r ∂ E

_z⁰

∂ r

⎛

⎝⎜

⎞

⎠⎟ + 1 r

²

∂

²

E

_z⁰

∂ φ

²

+ h

2

E

_z⁰

= 0 ! (1)

- Separation of variables

E

_z⁰

( ) r, φ = R r ( ) Φ ( ) φ ! (2)

- By substituting (2) into (1),

1 r

∂

∂ r r ∂ R r ( )

∂ r

⎛

⎝⎜

⎞

⎠⎟ Φ ( ) φ + 1 r

²

∂

²

Φ ( ) φ

∂ φ

²

R r ( ) + h

²

R r ( ) Φ ( ) φ = 0

⎡

⎣ ⎢ ⎤

⎦ ⎥

→ r

R r ( ) dr d r dR r ( )

dr

⎛

⎝⎜

⎞

⎠⎟ + h

²

r

²

= − 1

Φ ( ) φ d

2

Φ ( ) φ

d

²

φ = n

²

only a function of r! only a function of φ!

: Both sides equal to the constant to be satisfied for all r and φ!

- Two ODEs

d

²

Φ ( ) φ

d

²

φ + n

2

Φ ( ) φ = 0 r

R r ( ) dr d r dR r ( )

dr

⎛

⎝⎜

⎞

⎠⎟ + h

²

r

²

= n

²

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

d

²

R r ( )

dr

²

+ 1 r

dR r ( )

dr + h

²

− n

²

r

²

⎛

⎝⎜

⎞

⎠⎟ R r ( ) = 0

Bessel’s Differential Equation

× r

²

R r ( ) Φ ( ) φ

(HW!)

Img src: Wikipedia

Chap. 10 | Bessel’s differential equations and Bessel functions (2/3)

• Bessel’s differential equation

- Second-order equation → Two linearly independent solutions exist!

J

_n

( ) hr = ( ) − 1

^m

( ) hr

^n+2m

m! ( n + m ) !2

^n+2m

m=0

∑

∞ where n is an integer value

- Jn(0) = 0 for all n, except for J0(0) = 1

- Jn(x): i) Alternating functions of decreasing amplitudes that ii) cross the zero level at iii) progressively shorter intervals. iv) As x becomes large, Jn(x) approach a sinusoidal form

<Bessel function of the 1st kind>

<Bessel function of the 2nd kind>

• Solution 2: Bessel function of the 2nd kind (of nth order)

Y

_n

( ) hr = ( cos n π ) J

n

( ) hr − J

−n

( ) hr

sin n π

where n is an integer value

• General solution

R r ( ) = C

n

J

_n

( ) hr + D

n

Y

_n

( ) hr

• Solution 1: Bessel function of the 1st kind (of nth order)

d

²

R r ( )

dr

²

+ 1 r

dR r ( )

dr + h

²

− n

²

r

²

⎛

⎝⎜

⎞

⎠⎟ R r ( ) = 0

Refer to Engineering Mathematics For derivation!

Chap. 10 | Bessel’s differential equations and Bessel functions (3/3)

• Bessel solution for a circular waveguide

- Characteristics of Bessel function of the 2nd kind

If hr → 0, Y

_n

( ) hr → ∞

- However, our region of interest should include the axis where r = 0

- ∴ A solution R(r) CANNOT have Yn(hr) that leads to unphysical situation!

∴ R r ( ) = C

n

J

_n

( ) hr

<Bessel function of the 1st kind>

• “Zeros” of Bessel function of the 1st kind

J

_n

( ) hr = ( ) − 1

^m

( ) hr

^n+2m

m! ( n + m ) !2

^n+2m

m=0

∑

∞

= 0

There are several hr values (zeros) that make Jn(hr) = 0!

p \ n 0 1 2 … 1 2.405 3.832 5.136 … 2 5.520 7.016 8.417 …

… … … … …

<Table 1>

Zeros of Jn(x) = xnp

<Table 2>

Zeros of Jn’(x) = x’np

p \ n 0 1 2 … 1 3.832 1.841 3.054 … 2 7.016 5.331 6.706 …

… … … … …

<Bessel function of the 2nd kind>

→Determine eigenvalues for TM mode! →Determine eigenvalues for TE mode!

Img src: Wolfram MathWorld

Chap. 10 | TM waves in circular waveguide (1/5)

• Circular waveguide

- Circular waveguide of radius “a”

- Dielectric medium (μ and ε) enclosed by metallic skin

• Longitudinal field components

z

- Solution components

R r ( ) = C

ⁿ

J

ⁿ

( ) hr

Φ ( ) φ

⎧ ⎨

⎪

⎩⎪

H

_z

= 0

E

_z

( r, φ , z ) = E

^z0

( ) r, φ e

^{−γ z}

⎧ ⎨

⎪

⎩⎪

r φ

a

where

∇

_r²_φ

E

_z⁰

+ h

²

E

_z⁰

= 0

^and

E

_z⁰

( ) r, φ = R r ( ) Φ ( ) φ

Solution of

d

²

Φ ( ) φ

d φ

²

+ n

²

Φ ( ) φ = 0

*

All the field components are periodic with respect to φ (period = 2π)

*

Φ(φ) should be in a sinusoidal form!

*

Because of the periodicity, n should be integer values

∴ E

_z⁰

( ) r, φ = C

n

J

_n

( ) hr cos n φ

^{(TM modes)}

*

sin(nφ) and cos(nφ) does not matter!

(only reference changes) (By definition)

Chap. 10 | TM waves in circular waveguide (2/5)

• Transverse field components

(Recall 10.2: General wave behaviors along uniform guides)

- Transverse E-field components expressed in terms of longitudinal E-field for TM modes

E

_T⁰

( )

_TM

= a

^x

E

^x0

+ a

^y

E

^y0

= − h γ

²

∇

^T

E

^{z0 where}

∇

^T

= a

^x

∂ ∂ x + a

^y

∂ ∂ y

E

_T⁰

( )

_TM

= a

^r

E

^r0

+ a

^φ

E

^φ0

= − h γ

²

∇

^T

E

^{z0 where}

∇

^T

= a

^r

∂ ∂ r + a

^φ

r ∂ ∂ φ

→ ( ) E

_T⁰ _TM

= a

^r

E

^r0

+ a

^φ

E

^φ0

= a

^r

⎛ ⎝⎜ − h γ

²

∂ ∂ E r

^z0

⎞ ⎠⎟ + a

^φ

⎛ ⎝⎜ − h γ

²

r ∂ ∂ E φ

^z0

⎞ ⎠⎟ ! (1)

- Transverse H-fields related to transverse E-fields via impedance ZTM

H

_T

( )

_TM

= 1

Z

_TM

⎡⎣ a

_z

× ( ) E

_{T TM}

⎤⎦

^where

Z

TM

= γ

j ωε ( ) Ω

H

_T

( )

_TM

= a

^r

H

^r0

+ a

φ

H

_φ⁰

= j ωε

γ a

^z

× ( a

^r

E

^r0

+ a

^φ

E

^φ0

)

= a

_r

− j ωε

γ E

^r

⎛

0

⎝⎜

⎞

⎠⎟ + a

_φ

j ωε

γ E

^φ

⎛

0

⎝⎜

⎞

⎠⎟ ! (2)

a

_r

× a

_φ

= a

_z

a

_φ

× a

_z

= a

_r

a

_z

× a

_r

= a

_φ

Right-hand rule

(Gradient in transverse plane) Cartesian:

Cylindrical:

Chap. 10 | TM waves in circular waveguide (3/5)

• Transverse field components

- From equations (1), (2), and (3)

a

_r

E

_r⁰

+ a

_φ

E

_φ⁰

= a

_r

− γ h

²

∂ E

_z⁰

∂ r

⎛

⎝⎜

⎞

⎠⎟ + a

_φ

− γ h

²

r

∂ E

_z⁰

∂ φ

⎛

⎝⎜

⎞

⎠⎟ ! (1) a

_r

H

_r⁰

+ a

_φ

H

_φ⁰

= a

_r

− j ωε

γ E

^φ

⎛

0

⎝⎜

⎞

⎠⎟ + a

_φ

j ωε

γ E

^r

⎛

0

⎝⎜

⎞

⎠⎟ ! (2)

E

_r⁰

= − j β h

²

∂ E

_z⁰

∂ r = − j β

h C

_n

J

_n

′ ( ) hr cos n φ E

_φ⁰

= − j β

h

²

r

∂ E

_z⁰

∂ φ =

j β n

h

²

r C

_n

J

_n

( ) hr sin n φ H

_r⁰

= − ωε

β E

^φ0

= −

j ωε n

h

²

r C

_n

J

_n

( ) hr sin n φ H

_φ⁰

= ωε

β E

^r

0

= − j ωε

h C

_n

J

_n

′ ( ) hr cos n φ

⎧

⎨

⎪ ⎪

⎪ ⎪⎪

⎩

⎪ ⎪

⎪ ⎪

⎪

E

_z⁰

( ) r, φ = C

ⁿ

J

ⁿ

( ) hr cos n φ ! (3)

• Eigenvalues h

- Eigenvalues provided by B.C. where tangential E-fields = 0 at r = a

Medium 1 (dielectric)

Medium 2 (Conductor)

E

_1t

= 0 E

_2t

= 0 H

_2t

= 0

a

_n2

× H

₁

= J

_S

a

_n2

⋅ D

₁

= ρ

_S

D

_2n

= 0 B

_2n

= 0 B

_1n

= 0

z

a

E

_φ⁰

E

_z⁰

E

_φ⁰

= E

_z⁰

= 0

for all φ at r = a

∴ J

_n

( ) ha = 0

From Chap. 7-5.1 - We can obtain transverse E and H-fields!

Chap. 10 | TM waves in circular waveguide (4/5)

• Lowest cutoff frequency for TM modes

- From <Table 1>, the lowest zero of Jn(x) is x01 = 2.405 - Thus, the smallest ha for J0(hr) = 0 → x01

p \ n 0 1 2 … 1 2.405 3.832 5.136 … 2 5.520 7.016 8.417 …

… … … … …

<Table 1>

Zeros of Jn(x) = xnp

<Bessel function of the 1st kind>

ha = 2.405 → h

_{TM 01}

= 2.405 a J

_n

( ) ha = 0

- Since cutoff frequency for TM modes is given by

f

_c

= h

2 π µε → ∴ ( ) f

^c _{TM 01}

= h

^{TM 01}

2 π µε =

0.383 a µε

Is it a dominant mode?

• Eigenvalue Notation for a circular waveguide

TM np

n: number of half-wave field variations in φ direction

p: number of half-wave field variations in r direction

← ∵ E

_z⁰

( ) r, φ = C

n

J

_n

( ) hr cos n φ

Chap. 10 | TM waves in circular waveguide (5/5)

• TM

01

mode

E

_r⁰

= − j β

h C

_n

J

_n

′ ( ) hr cos n φ = − j β

h C

₀

J

₀

′ ( ) hr (nonzero) E

_φ⁰

= j β n

h

²

r C

_n

J

_n

( ) hr sin n φ = 0

E

_z⁰

= C

_n

J

_n

( ) hr cos n φ = C

⁰

J

⁰

( ) hr (nonzero)

⎧

⎨

⎪ ⎪⎪

⎩

⎪ ⎪

⎪

<E & H-field patterns in a polar plane>

*

^{E-field ⊥ H-field}

*

E-field lines form the radial pattern

*

Density of H-field lines increases with “r”

(from 0 to a)

E-field H-field

H

_φ⁰

= − C

₀

j ωε

h J

₀

′ ( ) hr = C

0

j ωε

h J

₁

( ) hr

∵ J

₀

′ ( ) hr = − J

1

( ) hr

<Bessel function of the 1st kind>

ha

H

_r⁰

= − j ωε n

h

²

r C

_n

J

_n

( ) hr sin n φ = 0 H

_φ⁰

= − j ωε

h C

_n

J

_n

′ ( ) hr cos n φ = − j ωε

h C

₀

J

₀

′ ( ) hr (nonzero) H

_z⁰

= 0

⎧

⎨

⎪ ⎪⎪

⎩

⎪ ⎪

⎪

why?

E-field H-field

Img src: Wolfram MathWorld

Chap. 10 | TE waves in circular waveguide (1/2)

• Longitudinal field components

E

_z

= 0

H

_z

( r, φ , z ) = H

z

0

( ) r, φ e

^{−γ z}

⎧ ⎨

⎪

⎩⎪

^where

∇

_r²_φ

H

_z⁰

+ h

²

H

_z⁰

= 0

^and

H

_z⁰

( ) r , φ = R r ( ) Φ ( ) φ

- Similarly to the TM case,

∴ H

_z⁰

( ) r, φ = D

ⁿ

J

ⁿ

( ) hr cos n φ

^{(TE modes)}

• Transverse field components

- Transverse magnetic fields:

H

_T⁰

( )

_TE

= a

^r

H

^r0

+ a

^φ

H

^φ0

⎡⎣ ⎤⎦ = − γ

h

²

∇

_T

H

_z⁰

= − γ

h

²

a

_r

∂

∂ r + a

_φ

∂ r ∂ φ

⎛

⎝⎜

⎞

⎠⎟ H

_z⁰

⎡

⎣ ⎢ ⎤

⎦ ⎥

- Transverse electric fields:

E

_T⁰

( )

_TE

= a

^r

E

^r0

+ a

^φ

E

^φ0

⎡⎣ ⎤⎦ = ⎡ ⎣ ⎢ − Z

_TE

( a

_z

× ( ) H

_T⁰ _TE

) = − j ωµ γ ( a

^r

H

^r0

+ a

^φ

H

^φ0

) ⎤ ⎦ ⎥

H

_r⁰

= − j β h

²

∂ H

_z⁰

∂ r = − j β

h D

_n

J

_n

′ ( ) hr cos n φ H

_φ⁰

= − j β

h

²

r

∂ E

_z⁰

∂ φ =

j β n

h

²

r D

_n

J

_n

( ) hr sin n φ E

_r⁰

= − ωε

β H

^φ0

= −

j ωε n

h

²

r D

_n

J

_n

( ) hr sin n φ E

_φ⁰

= ωε

β H

^r0

= −

j ωε

h D

_n

J

_n

′ ( ) hr cos n φ

⎧

⎨

⎪ ⎪

⎪ ⎪⎪

⎩

⎪ ⎪

⎪ ⎪

⎪

Chap. 10 | TE waves in circular waveguide (2/2)

• Eigenvalues h

- Eigenvalues provided by B.C. where tangential E-fields = 0 at r = a

E

_φ⁰

= ωε

β H

^r

0

= − j ωε

h D

_n

J

_n

′ ( ) hr cos n φ

z

a

E

_φ⁰

E

_z⁰

= 0

E

_φ⁰

= 0

for all φ at r = a

∴ ′ J

_n

( ) ha = 0

by definition →

• Lowest cutoff frequency for TE modes

- From <Table 2>, the lowest zero of J’n(x) is x11 = 1.841 - Thus, the smallest ha for = J’1(ha) = 0 → x11

<Table 2>

Zeros of Jn’(x) = x’np

p \ n 0 1 2 … 1 3.832 1.841 3.054 … 2 7.016 5.331 6.706 …

… … … … …

ha = 1.841 → h

_TE11

= 1.841 a

- Since cutoff frequency for TE modes is given by

f

_c

= h

2 π µε → ∴ ( ) f

^c _TE11

= h

^TE11

2 π µε =

0.293 a µε

Is it a dominant mode?

- Comparison between TE and TM modes

∵ ( ) f

_{c TE11}

= 0.293

a µε < ( ) f

^c _{TM 01}

= 0.383

a µε ,

TE11 mode is a dominant mode of a circular waveguide!

Chap. 10 | TE waves in circular waveguide (Example)

Example 10-12

(a) A 10 GHz signal is transmitted inside a circular conducting pipe. Determine the inside diameter of the pipe such that its lowest fc is 20% below this signal frequency. (b) If the pipe is to operate at 15 (GHz), what

waveguide modes can propagate in the pipe?

f

_c

( )

_TE11

= 0.293

a µε =

0.293c

a = 0.293 × 3 × 10

⁸

a = 0.0879

a (GHz)

(a) Lowest fc = (fc)TE11

∴ 0.0879

a (GHz) = 10 × 0.8 (GHz) → d = 2 a = 2.2 (cm)

(b) Cutoff frequencies for various modes are given as

p \ n 0 1 2 … 1 2.405 3.832 5.136 … 2 5.520 7.016 8.417 …

… … … … …

<Table 1> (for TM!) Zeros of Jn(x) = xnp

<Table 2> (for TE!) Zeros of Jn’(x) = x’np

p \ n 0 1 2 … 1 3.832 1.841 3.054 … 2 7.016 5.331 6.706 …

… … … … …

f

_c

( )

_TE11

= 8 (GHz)

f

_c

( )

_TE21

= ( ) f

^c _TE11

3.054

1.841 = 13.27 (GHz) f

_c

( )

_TE01

= ( ) f

^c _TE11

3.832

1.841 = 16.25 (GHz) > 15 !

f

_c

( )

_{TM 01}

= ( ) f

^c _TE11

2.405

1.841 = 10.45 (GHz) f

_c

( )

_TM11

= ( ) f

^c _TE11

3.832

1.841 = 16.65 (GHz) > 15 !

∴ TE11, TE21, TM01 modes can propagate at a given

operating frequency 15 (GHz) and all other higher order modes should attenuate.

Jaesang Lee

Dept. of Electrical and Computer Engineering Seoul National University

(email: [email protected])

Electromagnetics

<Chap. 10> Waveguides and Cavity Resonators Section 10.5 ~ 10.6

(2nd of week 8)

Textbook: Field and Wave Electromagnetics, 2E, Addison-Wesley

Chap. 10 | Contents for 2

^nd

class of week 8

Sec 6. Dielectri-slab waveguide

• Odd and Even TE wave characteristics (try TM case at home!)

• Cutoff frequencies and possible modes

Chap. 10 | Introduction: Dielectric waveguide

• Dielectric-slab waveguide

- Thin dielectric slab (μ and ε) with thickness d situated in free space (μ0 and ε0) - Even without conducting walls, both TM and TE waves can be supported!

(shown later)

• Assumptions

- z: propagation direction

- x: Infinite in extent and no variation of the fields - Lossless dielectric (σd = 0)

• Wave equations

∇

²

E + k

²

E = 0

∇

²

H + k

²

H = 0

⎧ ⎨

⎪

⎩⎪

where

E

_z

( ) y, z = E

z

0

( ) y e

^{−γ z}

H

_z

( ) y, z = H

^z0

( ) y e

^{−γ z}

⎧ ⎨

⎪

⎩⎪

d

x z

y

→ ∂ E

∂ x = 0, ∂ H

∂ x = 0

∇

²_y

+ ∇

_z²

( ) E

^z

+ k

²

E

^z

= 0

∇

²_y

+ ∇

_z²

( ) H

^z

+ k

²

H

^z

= 0

⎧ ⎨

⎪

⎩⎪

In the z-direction,

∴ ∇

²_y

E

_z⁰

+ ( γ

²

+ k

²

) E

^z0

= 0 ! for TM modes with H

_z⁰

= 0

∇

²_y

H

_z⁰

+ ( γ

²

+ k

²

) H

^z0

= 0 ! for TE modes with E

_z⁰

= 0

⎧ ⎨

⎪

⎩⎪

<Dielectric-slab waveguide>

Wave equations for longitudinal fields

E = a

_x

E

_x

+ a

_y

E

_y

+ a

_z

E

_z

H = a

_x

H

_x

+ a

_y

H

_y

+ a

_z

H

_z

⎧ ⎨

⎪

⎩⎪

where

∵ ∇²_xE_z = 0

∇²_xH_z = 0

⎧⎨

⎪

⎩⎪

Chap. 10 | TE waves along a dielectric slab

• Longitudinal field components

- Ez = 0

- Hz satisfies wave equation:

• Transverse field components

d

x z

y

∇

²_y

H

_z⁰

+ ( γ

²

+ k

²

) H

^z0

= 0

<Dielectric-slab waveguide>

where

H

_z

( ) y, z = H

^z0

( ) y e

^{−γ z}

E

_x⁰

= − 1

h

²

γ ∂ E

^z0

∂ x + j ω µ ∂ H

^z0

∂ y

⎛

⎝⎜

⎞

⎠⎟

E

_y⁰

= − 1

h

²

γ ∂ E

^z0

∂ y − j ω µ ∂ H

^z0

∂ x

⎛

⎝⎜

⎞

⎠⎟

H

_x⁰

= − 1

h

²

γ ∂ H

^z0

∂ x − j ωε ∂ E

^z0

∂ y

⎛

⎝⎜

⎞

⎠⎟

H

_y⁰

= − 1

h

²

γ ∂ H

^z0

∂ y + j ωε ∂ E

^z0

∂ x

⎛

⎝⎜

⎞

⎠⎟

⎧

⎨

⎪ ⎪

⎪ ⎪

⎪

⎩

⎪ ⎪

⎪ ⎪

⎪

E

_x⁰

= − j ωµ h

²

∂ H

_z⁰

∂ y E

_y⁰

= 0

H

_x⁰

= 0 H

_y⁰

= − γ

h

²

∂ H

_z⁰

∂ y

⎧

⎨

⎪ ⎪

⎪⎪

⎩

⎪ ⎪

⎪ ⎪

y

z

^d

O

Free-space (μ0, ε0)

Dielectric (μd, εd)

y = d/2 y = -d/2

- Fields must be considered both in dielectric (core) & free-space (cladding) regions

- Field components should satisfy B.C. at y = d/2 and y = -d/2 (i.e. B.C. between two lossless dielectric)

Free-space (μ0, ε0)

Chap. 10 | TE waves along a dielectric slab (general solution)

• Solution for dielectric (y ≤ |d|/2)

- Modes propagating in z-direction without attenuation (γ = jβ) - A solution should be in a sinusoidal form

(i.e. a bounded standing wave)

h

_d²

= k

²

+ γ

²

= ω

²

µ

_d

ε

_d

− β

²

> 0.

∴ H

_z⁰

( ) y = H

o

sin h

_d

y + H

_e

cos h

_d

y

• Solution for free-space (y ≥ d/2 and y ≤ -d/2)

- Waves decay exponentially in the y-direction (“Evanescent wave”)

→ Waves bounded only within the guide (total internal reflected)

→ Waves not radiating away from it

∇

²_y

H

_z⁰

+ ( k

²

+ γ

²

) H

^z0

= 0 → ∇

^2y

H

^z0

+ h

^d2

H

^z0

= 0

Here,

→ “Odd” & “Even” functions

∇

²_y

H

_z⁰

+ ( γ

²

+ k

²

) H

^z0

= 0 → ∇

^2y

H

^z0

+ h

⁰²

H

^z0

= 0

h

₀²

= k

²

+ γ

²

= ω

²

µ

₀

ε

₀

− β

²

< 0.

Here, Thus,

h

₀²

! − α

²

∴ H

_z⁰

( ) y = C

^u

e

^{−α⎛⎝⎜y−d2⎞⎠⎟}

+ D

^u

e

^{α⎛⎝⎜ y−d2⎞⎠⎟}

where y ≥ d 2 H

_z⁰

( ) y = C

l

e

^{α y+}

d 2

⎛⎝⎜ ⎞

⎠⎟

+ D

_l

e

^{−α y+}

d 2

⎛⎝⎜ ⎞

⎠⎟

where y ≤ − d 2

⎧

⎨ ⎪

⎩ ⎪

→ Wavenumber for a bounded wave

Even Odd ^{μ0, ε0}

μ0, ε0

μd, εd

μd, εd

μ0, ε0

μ0, ε0

y

Img src: Cornell ECE 303 (Farhan Rana)

Chap. 10 | TE waves along a dielectric slab (Odd TE modes)

• Odd TE modes in the dielectric (|y| ≤ d/2)

- Longitudinal components

H

_z⁰

( ) y = H

^o

sin h

^d

y + H

^e

cos h

^d

y where y ≤ d 2 H

_z⁰

( ) y = C

u

e

^{−α y−}

d 2

⎛⎝⎜ ⎞

⎠⎟

where y ≥ d 2 H

_z⁰

( ) y = C

^l

e

^{α⎛⎝⎜ y+d2⎞⎠⎟}

where y ≤ − d 2

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

- Nonzero Transverse components

y=d/2

x z

y

y=-d/2

H

_z⁰

( ) y

H

_y⁰

( ) y E

_x⁰

( ) y

• Odd TE modes in the upper free-space (y ≥ d/2)

- Longitudinal components

where

- Nonzero Transverse components

E

_z⁰

= 0, H

_z⁰

( ) y = H

^o

sin h

^d

y

E

_x⁰

( ) y = − j ωµ

^d

h

_d²

∂ H

_z⁰

∂ y = − j ωµ

_d

h

_d

H

_o

cos h

_d

y H

_y⁰

( ) y = − γ

h

_d²

∂ H

_z⁰

∂ y = − j β

h

_d

H

_o

cos h

_d

y

⎧

⎨ ⎪⎪

⎩

⎪ ⎪

H

_z⁰

( ) y = C

u

e

^{−α y−}

d 2

⎛⎝⎜ ⎞

⎠⎟

H

_z⁰

d

2

⎛ ⎝⎜ ⎞

⎠⎟ = C

_u

= H

_o

sin h

_d

d 2

E

_x⁰

( ) y = − j ωµ

⁰

h

₀²

∂ H

_z⁰

∂ y = − j ωµ

₀

α C

^u

e

−α y−d 2

⎛⎝⎜ ⎞

⎠⎟

H

_y⁰

( ) y = − γ h

₀²

∂ H

_z⁰

∂ y = − j β

α C

^u

e

−α y−d 2

⎛⎝⎜ ⎞

⎠⎟

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

h

_d²

= ω

²

µ

_d

ε

_d

− β

²

α

²

= − h

₀²

= β

²

− ω

²

µ

₀

ε

₀

Here,

h

_d

:

Wavenumber for the bounded wave

α :

Attenuation coefficient for evanescent wave

(∵B.C. Continuous tangential H-fields)

Chap. 10 | TE waves along a dielectric slab (Odd TE modes)

• Relations between h

d

and α

- Provided by B.C. such that tangential E-fields (at y = ±d/2) should be continuous

E

_1t

= E

_2t

H

_1t

= H

_2t

⎡

⎣ ⎢

D

_1n

= D

_2n

B

_1n

= B

_2n

⎡

⎣ ⎢

B.C. between two lossless dielectrics

(tangential)

(normal)

E

_x⁰

d 2

⎛ ⎝⎜ ⎞

⎠⎟ → − j ωµ

_d

h

_d

H

_o

cos h

_d

d

2 = − j ωµ

₀

α H

^o

sin

h

_d

d 2

E

_x⁰

( ) y = − j ωµ

^d

h

_d

H

₀

cos h

_d

y ! for y ≤ d 2 E

_x⁰

( ) y = − j ωµ α

⁰

C

^u

e

^{−α⎛⎝⎜ y−d2⎞⎠⎟}

! for y ≥ d 2

⎧

⎨ ⎪⎪

⎩ ⎪

⎪

y=d/2

x z

y

y=-d/2

H

_z⁰

( ) y

H

_y⁰

( ) y E

_x⁰

( ) y

C

_u

= H

₀

sin h

_d

d

where

2

→ α

h

_d

= µ

₀

µ

_d

tan

h

_d

d 2

⎛ ⎝⎜ ⎞

⎠⎟ ! (1)

- By directly equating expressions for hd and α,

h

_d²

= ω

²

µ

_d

ε

_d

− β

²

α

²

= β

²

− ω

²

µ

₀

ε

₀

⎧ ⎨

⎪

⎩⎪

h

_d²

+ α

²

= ω

²

µ

_d

ε

_d

− ω

²

µ

₀

ε

₀

→ α = ω

²

( µ

_d

ε

_d

− µ

₀

ε

₀

) − h

^d2

! (2)

- By substituting equation (2) into (1), we get

ω

²

( µ

_d

ε

_d

− µ

₀

ε

₀

) − h

^d2

h

_d

= µ

₀

µ

_d

tan

h

_d

d 2

⎛ ⎝⎜ ⎞

⎠⎟

→ ∴ µ

_d

µ

₀

ω

²

( µ

_d

ε

_d

− µ

₀

ε

₀

) d

h

_d

d

( )

²

− 1 = tan ⎛ ⎝⎜ h 2

^d

d ⎞ ⎠⎟

:Transcendental equation for odd TE modes

:hd-α relation

for odd TE modes

H

_z⁰

( ) y = E

o

sin h

_d

y + E

_e

cos h

_d

y where y ≤ d 2 H

_z⁰

( ) y = C

u

e

^{−α y−}

d 2

⎛⎝⎜ ⎞

⎠⎟

where y ≥ d 2 H

_z⁰

( ) y = C

l

e

^{α y+}

d 2

⎛⎝⎜ ⎞

⎠⎟

where y ≤ − d 2

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

Chap. 10 | TE waves along a dielectric slab (Even TE modes)

• Even TE modes in the dielectric (|y| ≤ d/2)

- Longitudinal components

- Nonzero Transverse components

y=d/2

x z

y

y=-d/2

H

_z⁰

( ) y

H

_y⁰

( ) y E

_x⁰

( ) y

• Even TE modes in the upper free-space (y ≥ d/2)

- Longitudinal components

where

- Nonzero Transverse components

E

_z⁰

= 0, H

_z⁰

( ) y = H

^e

cos h

^d

y

E

_x⁰

( ) y = − j ωµ

^d

h

_d²

∂ H

_z⁰

∂ y = j ωµ

_d

h

_d

H

_e

sin h

_d

y H

_y⁰

( ) y = − γ

h

_d²

∂ H

_z⁰

∂ y = j β

h

_d

H

_e

sin h

_d

y

⎧

⎨ ⎪⎪

⎩

⎪ ⎪

H

_z⁰

( ) y = C

u

e

^{−α y−}

d 2

⎛⎝⎜ ⎞

⎠⎟

H

_z⁰

d

2

⎛ ⎝⎜ ⎞

⎠⎟ = C

_u

= H

_e

cos h

_d

d 2

E

_x⁰

( ) y = − j ωµ

⁰

h

₀²

∂ H

_z⁰

∂ y = − j ωµ

₀

α C

^u

e

−α y−d 2

⎛⎝⎜ ⎞

⎠⎟

H

_y⁰

( ) y = − γ h

₀²

∂ H

_z⁰

∂ y = − j β

α C

^u

e

−α y−d 2

⎛⎝⎜ ⎞

⎠⎟

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

h

_d²

= ω

²

µ

_d

ε

_d

− β

²

α

²

= − h

₀²

= β

²

− ω

²

µ

₀

ε

₀

Here,

h

_d

:

Wavenumber for the bounded wave

α :

Attenuation coefficient for evanescent wave

(∵B.C. Continuous tangential H-fields)

Chap. 10 | TE waves along a dielectric slab (Even TE modes)

• Relations between h

d

and α

- Provided by B.C. such that tangential E-fields (at y = ±d/2) should be continuous

E

_1t

= E

_2t

H

_1t

= H

_2t

⎡

⎣ ⎢

D

_1n

= D

_2n

B

_1n

= B

_2n

⎡

⎣ ⎢

B.C. between two lossless dielectrics

(tangential)

(normal)

E

_x⁰

d 2

⎛ ⎝⎜ ⎞

⎠⎟ → j ωµ

_d

h

_d

H

_e

sin h

_d

d

2 = − j ωµ

₀

α H

^e

cos

h

_d

d 2

E

_x⁰

( ) y = j ωµ

^d

h

_d

H

_e

sin h

_d

y ! for y ≤ d 2 E

_x⁰

( ) y = − j ωµ α

⁰

C

^u

e

^{−α⎛⎝⎜ y−d2⎞⎠⎟}

! for y ≥ d 2

⎧

⎨ ⎪⎪

⎩ ⎪

⎪

y=d/2

x z

y

y=-d/2

H

_z⁰

( ) y

H

_y⁰

( ) y E

_x⁰

( ) y

C

_u

= H

₀

cos h

_d

d

where

2

→ α

h

_d

= − µ

₀

µ

_d

cot

h

_d

d 2

⎛ ⎝⎜ ⎞

⎠⎟ ! (1)

- By directly equating expressions for hd and α,

h

_d²

= ω

²

µ

_d

ε

_d

− β

²

α

²

= β

²

− ω

²

µ

₀

ε

₀

⎧ ⎨

⎪

⎩⎪

h

_d²

+ α

²

= ω

²

µ

_d

ε

_d

− ω

²

µ

₀

ε

₀

→ α = ω

²

( µ

_d

ε

_d

− µ

₀

ε

₀

) − h

^d2

! (2)

- By substituting equation (2) into (1), we get

ω

²

( µ

_d

ε

_d

− µ

₀

ε

₀

) − h

^d2

h

_d

= − µ

₀

µ

_d

cot

h

_d

d 2

⎛ ⎝⎜ ⎞

⎠⎟

→ ∴ µ

_d

µ