Lecture 5 Plane Wave Reflection and Transmission · PDF fileReflection and Transmission ......

Lecture 5

Plane Wave Reflection and Transmission

1

1

1

ˆ( ) (0)1ˆ( ) (0)

zr r

zr r

z E e

z E e

E x

H y

Normal Incidence (Review)

Incident wave:1

1

1

ˆ( ) (0)1ˆ( ) (0)

zi i

zi i

z E e

z E e

E x

H y

Reflected wave:

Transmitted wave: 2

2

2

ˆ( ) (0)1ˆ( ) (0)

zt t

zt t

z E e

z E e

E x

H y

1 1 1( , , ) 2 2 2( , , )

x

y z

iE

rE

tEiH

rH

tH

iStS

rS

Nikolova 2012 2LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION

(0)(0)

r

iEE

reflection coefficient(0)(0)

t

iETE

transmission coefficient

note minus sign!

Nikolova 2012 3

Oblique Incidence – Parallel (or p) Polarization (Review)

plane of incidence – the plane defined by the interface normal ẑ) and the direction of incidence

x

zy

1 1( , ) 2 2( , )

ˆ ixˆ iy ||iE

||iH

ir t

ˆ ryˆ rx

||rE

||rH ˆ txˆ ty

||tE

||tH

PLANE OF INCIDENCE

angle of incidence

angle of reflection

angle of transmission

ˆ ˆ ˆsin cosˆ ˆ ˆcos sin

r r rr r r

x x zy x z


i i ii i i

x x zy x z


t t tt t t

x x zy x z

p-polarization – E-field is parallel to the plane of incidence


Oblique Incidence – p Polarization (Review) (2)

Incident wave:

1

1

( sin cos )|| ||0

||0 ( sin cos )||

1

ˆ ˆ( cos sin )

ˆ

i i

i i

i i x zi i

ii x z

E e

Ee

E x z

H y

Reflected wave:

1

||

||( sin cos )

|| ||0

ˆ ˆ( cos sin ) r r

r

rr r

i x z

E

E e

E x z

Transmitted wave:

2

||

||( sin cos )

|| ||0

ˆ ˆ( cos sin ) t t

t

tt t

i x z

E

T E e

E x z

1|| ||0 ( sin cos )

||1

ˆ r r

ir x z

Ee

H y

2

|| ||0 ( sin cos )||

2ˆ t t

it x z

T Ee

H y

reflection coefficient transmission coefficient

note minus sign!


Oblique Incidence – Perpendicular (or s) Polarization (Review)E-vector is perpendicular to the plane of incidence (tangential to the interface) x

zy

1 1( , ) 2 2( , )

ˆ ix

ˆ iy

iE

iH

ir t

ˆ ryˆ rx

rE r

H ˆ tx

ˆ ty

tE

tH


r r r

r r r

x x zy x zˆ ˆ ˆsin cosˆ ˆ ˆcos sin

i i i

i i i

x x zy x z


t t t

t t t

x x zy x z


Oblique Incidence – s Polarization (Review) (2)

Incident wave:1

1

( sin cos )0

0 ( sin cos )

1

ˆ

ˆ ˆ( cos sin )

i i

i i

i i x z

ii x z

i i

E e

E e

E y

H x z

Reflected wave:

10 ( sin cos )

1

ˆ ˆ( cos sin )

r r

rr r

ix zE e

H x z

Transmitted wave:

20 ( sin cos )

2

ˆ ˆ( cos sin )

t t

tt t

ix zT E e

H x z

1

0

( sin cos )0ˆ r r

r

ir x z

E

E e

E y

2

0

( sin cos )0ˆ t t

t

t i x z

E

T E e

E y

reflection coefficient transmission coefficient

• valid for all polarizations

• follows from the continuity of the tangential field components and the matching of their phases

• transmission into very good conductors


Snell’s Law (Partial Review)

1 1 1 1

2 2 2 2

sinsin

r i

t

i

nn

If medium 1 is a loss-free dielectric and medium 2 is a very good conductor (σ2 >> ωε2), show that θt is complex and tends to zero regardless of the angle of incidence. Derive the expression for the propagation factor in medium 2 when σ2 >> ωε2. 2 ( sin cos )t tx ze


Reflection and Transmission (Fresnel) Coefficients (Review)

• by definition the reflection coefficient Γ gives the ratio of the reflected to incident tangential E-field component at the interface

• the ratio of the reflected to incident tangential H-field component at the interface is −Γ

• by definition the transmission coefficient T gives the ratio of the transmitted to incident tangential E-field component at the interface


Fresnel Coefficients (Review) – 2

2 1||

2 1

2||

2 1

cos coscos cos

2 coscos cos

t i

t i

i

t iT

at normal incidence (θi = θt = 0), the above cases reduce to 2 1 2

2 1 2 1

2,

|| ||cos1cos

i

tT

for p-polarization:

for s-polarization:2 1

2 1

2

2 1

cos coscos cos

2 coscos cos

i t

i t

i

i tT

1T

1T


Oblique Incidence: Surface Impedance

• surface impedances (in the chosen coordinate system) for p and spolarizations are then obtained as

tan tan 0( )ˆs z

Z

E H z

||||

|| 0

xs

y z

EZ

H

0

ys

x z

EZ

H

• definition: the ratio of the values (complex in general) of the tangential E and H field vectors at the interface

note: in medium 1, these are total (incident + reflected) field values


Oblique Incidence: Surface Impedance – 2

2||0 || || 21

|| 1 2|| 2||0 1 ||

2(1 )cos 1

cos 1 sin1( /

c s)(1 )

oi

its i ii

EZ

E

20 1 22

0 1 21

2

(1 ) 1 =cos 1( / )(1 )cos co

ins

1 s

i

sii t

i

i

EZE

(1)|| 0 totalyH

(1)|| 0 totalxE (2) (2)

|| 0 || 0/x yE H

(1)0 totalyE

(1)0 totalxH

(2) (2)0 0/y xE H


Oblique Incidence: Surface Impedance of a Very Good Conductor

• the power-flux density penetrating into medium 2 (along z) is2

||, ||, ||, tan 00

1 1 | |ˆ2 2z s zz

S Z

E H z H

• this power-flux density is dissipated power per unit area

2 2||, s tan

1Re | (0) | , W/m2zp S R H

• we have already proven (sl. 7) that if medium 2 is a very good conductor, regardless of the angle of incidence, cosθt ≈ 1

2s 2

2 2 2

1Re2

R

2|| 2

2 (1 )

2s sZ Z j

sZ

2tan 0| | , ˆs s zz

J J H

Extreme Cases: Full Transmission, Full Reflection

full transmission occurs when the two media have the same constitutive parameters (no reflected wave)

1 2 and t r i 0, 1T

full reflection occurs when


• medium 2 is a PEC (short circuit)• medium 2 is a PMC (open circuit)• total internal reflection at a dielectric interface

• there is no reflected wave• the field in medium 1 is simply the traveling incident wave• the field in medium 2 is the same traveling wave

• the reflected wave is as strong as the incident one• there is no field in medium 2

Can total transmission happen if the 2 media are different? Explain.

Full Reflection


Prove that Γ = −1 and T = 0 if σ2 → ∞.

the conducting medium “shorts out” medium 1 by forcing a zero tangential E-field component (voltage between any 2 points on interface is 0); tangential H-field is doubled

• total field in medium 1 when Γ = ±1 for normal incidence

1 1

1 1

1

1

ˆ( ) (0)

ˆ( ) (0)

z zi

z zi

z E e e

z H e e

E x

H y

• full reflection at PEC

• full reflection at PMCPMC acts as an open circuit forcing a zero tangential H-field component and doubling the tangential E-fieldProve that Γ = 1 and T = 2 if µꞌꞌ→ ∞.

tan

2 2 2s tan s tan

2 (0)

0.5 | (0) | | (0) | , W/mi

i

H

p R R H H

Surface Current Density at the Surface of PEC

1

2

1, 0 reflected: (0) (0) transmitted: (0) 0

(0) (0) (0) 2 (0) (0) (0) 0

r i

t

i r i

t

T H HH

H H H HH H

• according to the boundary conditions:

2 1

1

ˆ ( ) , A/mˆ ˆ (0) , A/m

s

sH

z H H Jz y J

tan tan 1ˆ ˆ2 (0) 2 (0) /i is H E J x x

• the incident wave is “shorted” and thus induces current at the conductor’s surface

• this current is the source of the reflected wave

Nikolova 2012 15

• power loss in a very good conductor [see sl. 12]

Full Reflection at PEC – Animations

• E reverses phase at a perfectly conducting wall

• H does not reverse phase at a perfectly conducting wall


1(0) 0E

1 or 1(0) 2 (0)

HiH H

Standing Waves Due to Reflection from PEC (Γ = −1)

1 11 1ˆ ˆ( ) (0)( ) (0) 2 sin( )jk z jk zi iz E e e E j k z E x x

1 11 1

1

(0)ˆ ˆ( ) (0)( ) 2cos( )i

jk z jk zi Ez H e e k z

H y y

• assume medium 1 is loss-free (γ1 = jk1)

1

2

1 1 1 1 11 0.5sin(2 )

1 2 | (0) |ˆ sin( ) cos( )2

i

k z

j E k z k z

S E H z

• field forms a standing wave with clearly defined nulls and maxima along z (at PEC wall, z = 0, E has a null, H has a maximum)

• Poynting’s vector is purely imaginary – no power transferred along z

• where E has a null, H has a maximum, and vice versa


• for any z, E and H are in phase quadrature

Sinusoidal Standing Wave – Animation superposition of two sine waves traveling in opposite directions


position

Standing Wave – Animation standing H-field plane wave has a maximum at the shorted end (the E-field wave looks exactly the same only shifted by a quarter wavelength)


/ 2

Envelopes of Traveling, Standing, and Mixed Waves

the envelope of a standing wave has nulls and maxima spaced at λ/4[the distance between any two neighboring nulls (or maxima) is λ/2]

the envelope of a traveling wave is a constant line with no minima and maxima

between the extremes of a traveling and a standing wave lie infinite combinations of incident and reflected waves – mixed waves

mixed waves can be viewed as a superposition of a traveling wave and a standing wave where Γ can be any complex number


0 | | 1

http://wn.com/standing_wave_envelope

Locations of the Envelope Minima/Maxima• maxima occur where the standing wave and the traveling wave

interfere constructively• minima occur where the standing wave and the traveling wave

interfere destructively• maxima are (1 + |Γ|) times the magnitude of the incident wave• minima are (1 − |Γ|) times the magnitude of the incident wave

1 1( ) (0) (0) , | |jk z jk z ji iE z E e E e e 2

1| ( ) | ( ) ( ) | (0) | 1 | | 2 | | cos(2 )iE z E z E z E k z

max 1

min 1

| ( ) | | (0) | 1 | | for cos(2 ) 1| ( ) | | (0) | 1 | | for cos(2 ) 1

i

i

E z E k zE z E k z


envelope

Locations of the Envelope Minima/Maxima – 2

1 min,cos( 2 ) 1nk l

min,

1

(2 1) , 0,1, where 4 4

nl n n


• consider l = −z (the distance from the interface back into medium 1)

• the positions of the E-wave minima lmin,n (n = 0,1,2,…) are found from

Prove that the minima positions are given by

List the first 3 minima positions (n = 0,1,2) if medium 2 is PEC and the wavelength is 10 cm.

Hint: Note that lmin,n ≥ 0 must hold. To get the correct sign consider that1 1cos( 2 ) cos(2 )k l k l

Standing Wave Ratio

• SWR has a minimum value of 1 in the case of a traveling wave

• SWR is infinity in the case of a standing wave

• in microwave engineering, a SWR ≤ 2 is considered satisfactory for the purposes of matching and good power transfer

the SWR (Standing Wave Ratio) is the ratio of the maxima and the minima of the total wave in medium 1

1 | | 11 | |

SWR

max

min

| ( ) || ( ) |E zSWRE z

What is |Γ| if SWR = 2? What is the ratio of reflected to incident power density Sr/Si in this case? What is the ratio of transmitted to incident power density St/Si?

1| |1

SWRSWR


Standing Wave Ratio (2)

0.5

maxE

minE

max

min3ESWR

E


Summary• the reflection and transmission coefficients relate the respective

tangential E-field components at the interface

• at normal incidence full reflection, |Γ| = 1, is due to PEC or PMC terminations

• losses per unit area on very good conductors are calculated using

• the SWR gives the ratio of the wave envelope maximum and minimum (SWR ≥ 1)

• the distance between two neighbouring minima (or maxima) is λ/2

• at a PEC wall (short), the E-field has a null while the H-field has a maximum, which is double the value of the incident field magnitude

• at a PMC wall (open), the situation is reversed

2 2s s tan

1 1| | , W/m where and | | | |2 2s sp R R

J J H

Lecture 5 Plane Wave Reflection and Transmission · PDF fileReflection and Transmission ......

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