1. Fields and Waves I Lecture 23 K. A. Connor
Reflection and Transmission at Normal Incidence
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

2. J. Darryl Michael – GE Global Research Center, Niskayuna, NY
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.
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4. Figure References for Previous Slide
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5. From H.G. Wells The Invisible Man Signet Classic, 2002
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6. Overview EM Waves in Lossless Media EM Waves in Lossy 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
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7. Normal incidence
MEDIUM η1
MEDIUM η2
incident plane wave
transmitted plane wave
reflected plane wave
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
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8. 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.
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9. Lossless media two lossless, homogenous, dielectric media . . . . .
MEDIUM 1 ε1, μ1
MEDIUM ε2, μ2 . . . . .
z=0
Incident wave
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10. Lossless media transmitted wave reflected wave 3 April 2019
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11. 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).
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12. Boundary conditions
Tangential component of the electric field should be continuous across the boundary
Tangential component of magnetic field should be continuous (no current source)
Note that inc + ref = trans
not inc = ref + trans
At the boundary (z=0): or or
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13. 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).
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14. Example 1
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15. Example 1
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16. Example 1
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17. 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.)
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18. reflection and transmission coefficients
Normally incident
Normally incident
Γ and τ are real for lossless dielectric media
Normally incident
For nonmagnetic media
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19. transmission line analogy
One to one correspondence between the transmission line parameters and plane wave parameters
incident and reflected waves create a standing wave pattern
Standing wave ratio
if η1= η Γ= S=1
η2= 0 (perfect conductor) Γ= S= ∞
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
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20. 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?
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21. Example 2
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22. Example 2
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23. Power flow in lossless media
The average power density flowing in medium 1
The average power density flowing in medium 2
For lossless media
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24. General Solution Ei & Er Et Hi & Hr Ht Boundary
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 …
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25. Lossless Transmission Lines Uniform Plane Waves in Lossless Media
Total voltage and current:
Total fields:
Phase velocity:
Propagation constant:
Characteristic/Intrinsic impedance:
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26. Lossless Transmission Lines Uniform Plane Waves in Lossless Media
Generalized Reflection Coefficient:
Total again:
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27. Lossless Transmission Lines Uniform Plane Waves in Lossless Media
Line or Wave Impedance:
Also:
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28. Lossless Transmission Lines Uniform Plane Waves in Lossless Media
At another location z’:
Continuity of Z(z) at a boundary between two regions:
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29. Lossless Transmission Lines Uniform Plane Waves in Lossless Media
Input impedance:
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30. Multiple Boundaries
z = 0
z = d z
Region 1
Region 2
Region 3 A C B G K F D E J H I
If A is incident, each reflection produces the additional waves shown.
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31. Multiple Boundaries
The waves take the usual general form in each region.
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32. Radomes http://igscb.jpl.nasa.gov/network/site/areq.html
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33. Radomes
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34. Radomes http://www.hiaper.ucar.edu/photo_gallery/031802.html
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35. Radomes http://en.wikipedia.org/wiki/File:Navy-Radome.jpg 3 April 2019
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36. Radomes
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37. 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.
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38. 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?
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39. Example 3
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40. Summary of the radome problem
Thus, since regions 1 and 3 are the same, the input impedance gives a perfect match and no reflection occurs.
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41. 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).
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.
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42. Anti-Reflection Coating
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43. Anti-Reflection Coating
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.
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
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44. 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%
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45. 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
Multiple
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46. Anti-Reflection Coating
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.
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47. 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.
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48. Boundary between lossy media
two lossy media
MEDIUM 1 ε1, μ1, σ1
MEDIUM ε2, μ2, σ2 . . . . .
z=0
Medium 1
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49. Boundary between lossy media
Medium 2 .
Γ and τ are real for lossless dielectric media
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