Fields and Waves I

Fields and Waves I

Lecture 23Reflection and Transmission at Normal Incidence

K. A. ConnorElectrical, Computer, and Systems Engineering Department

Rensselaer Polytechnic Institute, Troy, NY

April 22, 2023 Fields and Waves I 2

These Slides Were Prepared by Prof. Kenneth A. Connor Using Original Materials Written Mostly by the Following:

Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY

J. Darryl Michael – GE Global Research Center, Niskayuna, NY

Thomas P. Crowley – National Institute of Standards and Technology, Boulder, CO

Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY

Lale Ergene – ITU Informatics Institute, Istanbul, Turkey Jeffrey Braunstein – Chung-Ang University, Seoul, Korea

Materials from other sources are referenced where they are used. Those listed as Ulaby are figures from Ulaby’s textbook.



Figure References for Previous Slide

http://www.alanbauer.com/

http://contrapunctus.net/league/photo/pcd0077/003.php?fmt=base_jpeg&set=1

http://www.indelibleinc.com/kubrick/films/2001/images.html

http://www.widerange.org/images/large/palaReflection.jpg

http://z.about.com/d/goamsterdam/1/0/V/2/-/-/canal_reflection-1.jpg

http://www.alanbauer.com/

http://contrapunctus.net/league/photo/pcd0077/003.php?fmt=base_jpeg&set=1

http://www.indelibleinc.com/kubrick/films/2001/images.html


From H.G. Wells The Invisible ManSignet Classic, 2002


Overview

EM Waves in Lossless Media• Wave Equation & General Solution• Energy and Power

EM Waves in Lossy Media• Skin Depth• Approximate wave parameters

Low Loss Dielectrics Good Conductors

• Power and Power Deposition Wave Polarization

• Linear, circular & elliptical Reflection and Transmission at Normal

Incidence Plane Waves at Oblique Incidence


Normal incidence

• wave is normally incident on an infinite interface separating two different media

• impedance discontinuity

• similar to the transmission lines

• EM waves represented with rays or wavefronts

E

incident plane wave

transmitted plane wave

reflected plane wave

MEDIUM 1 η1 MEDIUM 2 η2


Step 1

As with transmission lines, the very first step is to write down the general solution for the electric and magnetic field waves in each of the media in the problem. Part of writing down the fields is to draw the little diagram with the three mutually perpendicular vectors for each component of the plane wave.

Maybe the first step should be to review the properties of transmission line waves since we will be doing identical analysis for uniform plane waves.


Lossless media

• two lossless, homogenous, dielectric media

E

z=0~ ( ) E z a E ei

xi jk z 0

1Incident wave

~ ( ) ~ ( ) H z a E z a

Eei

z

i

y

ijk z

1 1

0 1 1

1

1

k 1 1 1

.

.

.

MEDIUM 1 ε1, μ1 MEDIUM 2 ε2, μ2E i

H i a i

E r

H ra r

E t

H t a t.

.

x

zy


Lossless media

~ ( ) E z a E etx

t jk z 0

2

~ ( ) ( )~ ( ) H z a E z a

Eet

z

r

y

rjk z

1 1

0 1

2

2

2

k 2 2 2 transmitted wave

reflected wave~ ( ) E z a E er

xr jk z0

1

~ ( ) ~ ( ) H z a E z a

Eet

z

t

y

tjk z

2 2

0 2

1

1

1

k 1 1 1


Step 2

Next we write down the boundary conditions that obtain for the waves we are to analyze. Note from the diagram that both the electric and magnetic fields have only tangential components. For a conducting boundary the electric field must be zero if it is only tangential. This is possible by having the sign of the reflected wave be the negative of the incident wave while their magnitudes are the same. For a dielectric boundary, both the electric and magnetic fields must be continuous. That is, the field on one side of the boundary (incident + reflected) must equal the field on the other side of the boundary (transmitted).


Boundary conditions

E E Ei r t0 0 0 ~ ( ) ~ ( )E E1 20 0 or

Tangential component of the electric field should be continuous across the boundary

Tangential component of magnetic field should be continuous (no current source)

At the boundary (z=0):

~ ( ) ~ ( )H H1 20 0 orE E Ei r t

0 0 0

1 1 2

Note that inc + ref = trans not inc = ref + trans


Example 1

A 10 GHz plane wave has an electric field magnitude of 100 V/m and propagates in the az direction through a perfect dielectric with = 9. E is in the ax direction. a. What are the incident E and H phasors? b. At z = 0, the wave strikes a perfect conductor. What are the reflected E and H phasors? c. Use the boundary conditions to find the surface current density in the conductor. d. Draw the standing wave pattern for E and H (include numbers for amplitude and position). e. Simulate this case with sing_bnd.m by using a large imaginary dielectric for region 2.f. Calculate the total E and H. (phasor & time domain form).

r


Example 1

E tan 0


Example 1


Example 1


Standardize notation

We now use the standardized notation of reflection and transmission coefficients, just as we did with transmission lines.

Ratio of the reflected wave magnitude to the incident wave magnitude. (We use the electric field by convention.)

Ratio of the transmitted wave magnitude to the incident wave magnitude. (We use the electric field by convention.)


reflection and transmission coefficients

EE

r

i0

0

2 1

2 1

Normally incident

1

EE

t

i0

0

2

2 1

2Normally incident

Γ and τ are real for lossless dielectric media

Normally incident

For nonmagnetic media

EE

r

ir r

r r

0

0

1 2

1 2


Standing wave ratio

• One to one correspondence between the transmission line parameters and plane wave parameters

• incident and reflected waves create a standing wave pattern

transmission line analogy

~ , ~ , ,E H k

SE

E

~

~m ax

m in

1

1

11

if η1= η2 Γ=0 S=1

η2= 0 (perfect conductor) Γ=-1 S= ∞

~ , ~ , ,V I Z 0

• Smith chart can be used

• the max and min locations of the electric field intensity from the boundary are defined by the same expressions for the voltage max and min locations


Example 2

The same wave as in example 1 strikes a dielectric-air boundary at z=0 as shown below.a. Find the reflection and transmission coefficients.b. What are the reflected and transmitted electric field phasors?c. What are the reflected and transmitted H phasors? What is Ht/Hi? d. What is the standing wave ratio in the dielectric? Sketch the standing wave pattern for E and H. Run sing_bnd.m for this problem. e. What is the average power density of the incident, reflected, and transmitted waves?

Region 1

=9r

Region 2

=1r

z=0


Example 2


Example 2


Power flow in lossless media

The average power density flowing in medium 1

S S Sav avi

avr

1

S aE

avi

z

i

0

2

12S a

ESav

rz

i

avi 2 0

2

1

2

2

The average power density flowing in medium 2

S aE

avr

z

i

22 0

2

22

For lossless media

2

2

2

11 2

1

S Sav av


General Solution

Boundary

Ei & Er

Hi & Hr

Et

Ht

What quantities can be specified or calculated?Magnitude of incident E, dielectric properties of each region, intrinsic impedance of each region, frequency, wavelength, incident power, reflected power, transmitted power, transmission coefficient, reflection coefficient, standing wave ratio …


Lossless Transmission Lines

v z V e V ej z j z( )

i zV e V e

Z

j z j z

o

( )

Phase velocity:u

lc

1

Propagation constant:

lc

Characteristic/Intrinsic impedance:

Z lco

Uniform Plane Waves in Lossless Media

E z E e E exj z j z( )

H zE e E e

y

j z j z

( )

u

1

Total voltage and current: Total fields:


Lossless Transmission Lines Uniform Plane Waves in Lossless Media

zVV

e j z

2 zEE

e j z

2

Total again:

v z V e zj z( ) 1

i zVZ

e zo

j z( ) 1

E z E e zxj z( )

1

H zE

e zyj z( )

1

Generalized Reflection Coefficient:



Also:

Line or Wave Impedance:

Z zv zi z

Zzzo( )

11

Z zE zH z

zz

x

y

( )

11

( )zZ z ZZ z Z

o

o

( )zZ zZ z



Continuity of Z(z) at a boundary between two regions:

At another location z’:

( ' ) ( ) 'z z e j z z 2 ( ' ) ( ) 'z z e j z z 2

Z z Z z Z z Z z



Input impedance:

Zj dj dinpu t

2

3 2 2

2 3 2

tantanZ Z

Z jZ dZ jZ din o

L o

o L

tantan

Z in Z LZ o Z in

1 2 3

dd


Multiple Boundaries

z = 0 z = d

z

Region 1 Region 2 Region 3

A C

BG

K

F D E

J H I

If A is incident, each reflection produces the additional waves shown.


Multiple BoundariesThe waves take the usual general form in each region.


Radomes

http://www.cmmacs.ernet.in/nal/picts/

http://igscb.jpl.nasa.gov/network/site/areq.html

http://www.cmmacs.ernet.in/nal/picts/


Radomes

http://www.air-and-space.com/2003%20Miramar%20Airshow%20statics%20page%202.htm

http://www.air-and-space.com/2003%20Miramar%20Airshow%20statics%20page%202.htm


Radomeshttp://www.hiaper.ucar.edu/photo_gallery/031802.html

http://www.hiaper.ucar.edu/photo_gallery/031802.html


Radomes http://en.wikipedia.org/wiki/File:Navy-Radome.jpg


Radomes

http://www.collegeem.qc.ca/ena/avionique/photos/antennes/radar.htm

http://www.radome.net/nws.html


Example 3

A 10 GHz radar transmitter is used in the configuration shown below. Note that the radome-outside air boundary is identical to the boundary examined in example 2.

=9r

Region 3

=1r

d

Radar antenna

=1r

Region 2Region 1

radome outside air


Example 3

a. What is |E|/|H| at the z=0 boundary of example 2? (equivalent to the region 2-3 boundary in this problem). Compare it with the value in air. b. Now refer to the full radome problem. Where can you put the left boundary so that |E|/|H| in the radome matches that in the air on the left? For mechanical reasons, the radome must be more than 2 cm thick.c. What is for this value of d?d. What is if d is 0.2 mm thinner than designed?


Example 3


Summary of the radome problem

Zj dj dinpu t

2

3 2 2

2 3 2

tantan

Zjjinpu t

23 2

2 32

3

23

tantan

22

2d

Thus, since regions 1 and 3 are the same, the input impedance gives a perfect match and no reflection occurs.


Another common multiple boundary application

The radome is a half wavelength thick. We can also use a layer one quarter wavelength thick to eliminate reflections from a lens (at least at a particular frequency).

Zj dj dinpu t

2

3 2 2

2 3 2

tantan

2

24 2

d

2 1 3 Zj

jinpu t

23 2

2 3

22

31

2

2

tan

tan

Thus, since the input impedance is the same as the intrinsic impedance of region 1, we again have a perfect match and no reflection occurs.


http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/antiref.html#c3

Anti-Reflection Coating

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/antiref.html#c3



For an experiment at Oberlin, the coated lens was optimized for the middle of the visible spectrum so red and blue are reflected to form violet

AR coatings are similar to the coatings found on microscopes and camera lenses. They consist of several layers of metal oxides applied to the front and back lens surfaces. Because of the layering effect, AR coatings sometimes have a hint of green or purple color, depending on the individual manufacturer's formula. http://news.thomasnet.com/fullstory/23211/1782

http://www.oberlin.edu/physics/catalog/demonstrations/optics/coatedlens.html


Multiple vs Single Layer Coating

From left to right a lens without coating, single coated and multicoated. From the first to the third image the light transmission improved from 96 to 99.5%

http://www.astrosurf.org/lombry/reports-coating.htm

http://www.astrosurf.org/lombry/reports-coating.htm


Multiple vs Single Layer Coating

The principle of coatings is to reduce the light reflecting on the glass surface. If the coating has a quarter wavelength thickness and the glass yields an index of refraction higher than the one of the coating the two reflections are in phase opposition and cancel. For one multicoated surface the transmission can increase up to 3.5%. Single

Multiplehttp://www.kenrockwell.com/tech/lenstech.htm



One of the characteristics of an AR coating is the colour of the residual 1% reflection on the lens - sometimes called the bloom. On modern broadband Reflection Free lenses it can be tuned to be either a soft green or blue without compromising the quality of the anti-reflection properties. The AR bloom should not be confused with a permanent lens tint. An AR bloom is almost imperceptible and can only be seen when holding the lens up to sunlight or artificial light.

http://www.siltint.com/Ophthalmic_products/technical.htm

http://www.siltint.com/Ophthalmic_products/technical.htm


Question

From the point of view of an EE, CSE or EPE, what are we doing when we eliminate reflections?

Impedance Matching

A great deal of what we do as engineers is really impedance matching. When impedances match, we usually have the optimum operating conditions we seek.


Boundary between lossy media

• two lossy media

E

z=0~ ( ) ( )E z a E e ex

i z z1

0

1 1

c

c1

1

1

1 1 1 j

.

.

.

MEDIUM 1 ε1, μ1, σ1MEDIUM 2 ε2, μ2, σ2

E i

H i a i

E r

H ra r

E t

H t a t.

.

x

zy

Medium 1

H z aE

e ey

i

c

z z1

0 1 1

1

( ) ( )


Boundary between lossy media

~ ( ) E z a E exi z

2 0

2

c

c2

2

2

2 2 2 j

.

Medium 2

H z aE

ey

i

c

z2

0 2

2

( )

Γ and τ are real for lossless dielectric media

EE

r

ic c

c c

0

0

2 1

2 1

1

2 2

2 1

c

c c

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Fields and Waves I