Fields and Waves I

Fields and Waves I Lecture 24 Plane Waves at Oblique Incidence K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, 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. Fields and Waves I

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 • Dielectric-Conductor Interface • Dielectric-Dielectric Interface • Multiple Boundaries • Plane Waves at Oblique Incidence Fields and Waves I

Example 1 – Arbitrary Propagation Angle The direction of E and of a electromagnetic wave with = 500nm are shown below. The wave is traveling through air. The electric field has a magnitude of 30 V/m. What are the E and H phasors? Fields and Waves I

Example 1 Fields and Waves I

Arbitrary Propagation Angle In phasor form we have had We can generalize this with Fields and Waves I

Arbitrary Propagation Angle For propagation in more than the z direction, let us consider just adding x propagation, since that is all we will need to do oblique incidence. where we have left unspecified the unit vectors for E & H Fields and Waves I

Oblique Incidence – Parallel Polarization For the first choice, we can assume that the electric field in directed in the plane of incidence. This is called parallel polarization since E is parallel to this plane. Note that H is only tangent to the boundary while E has both normal and tangential components. Ulaby Fields and Waves I

Oblique Incidence – Perpendicular Polarization For the second choice, we can assume that the electric field in directed out of the plane of incidence. This is called perpendicular polarization since E is perpendicular to this plane. Note that E is only tangential while H has both components. Ulaby Fields and Waves I

Oblique Incidence With the two possible polarizations, we have two sets of boundary conditions. Thus, they will behave differently. Note also that the combination of the two polarizations gives us all possible vector components for E and H. Now we must apply the boundary conditions to determine how the incident, reflected and transmitted waves relate to one another. VERY IMPORTANT POINT: Because of the x-directed propagation, the phase of the E and H fields vary along the boundary. Thus, our first task is to match the phase and then we will match the amplitudes. The matching of the phase will allow us to derive one of the most fundamental laws of optics. Fields and Waves I

Oblique Incidence – Matching the Phase of the Electric and Magnetic Fields at a Boundary The incident electric field: The reflected electric field: To match the phase of the terms at z = 0: Fields and Waves I

Oblique Incidence – Matching the Phase of the Electric and Magnetic Fields at a Boundary Thus, we have that the angle of incidence equals the angle of reflection, a result that all of us have seen before. Now, we need to see what happens to the transmitted angle. Ulaby Fields and Waves I

Oblique Incidence – Matching the Phase of the Electric and Magnetic Fields at a Boundary Consider now all three waves – incident, reflected and transmitted: Matching the phases at z = 0: Fields and Waves I

Oblique Incidence – Matching the Phase of the Electric and Magnetic Fields at a Boundary This is Snell’s Law: To put it in its more normal form: Fields and Waves I

Snell’s Law: There are many useful wave representations: Ulaby Fields and Waves I

Snell’s Law: Using the wave front representation, we can see that Snell’s Law is required to match the wave variations on the two sides of the boundary. Fields and Waves I

Applying the Boundary Conditions for Both Polarizations Gives the Reflection and Transmission Coefficients Ulaby Fields and Waves I

Applying the Boundary Conditions for Both Polarizations Gives the Reflection and Transmission Coefficients Perpendicular Polarization: Fields and Waves I

Applying the Boundary Conditions for Both Polarizations Gives the Reflection and Transmission Coefficients Parallel Polarization: Fields and Waves I

Example 2 – Oblique Incidence • A plane wave described by is incident upon a dielectric material with = 4. • Write in phasor form. • b. What are and ? • c. What are and ? • d. What are the reflection and transmission coefficients? • e. Write the total electric field phasors in both regions. Fields and Waves I

Critical Angle For total reflection: Ulaby Fields and Waves I

Example 3 – Snell’s Law and Critical Angle For visible light, the index of refraction for water is n = 1.33. If we put a light source 1 meter under water and observe it from above the surface of the water, what is the largest for which light will be transmitted? How large will the circle of illumination be? Fields and Waves I

Example 4 -- Polarization Consider the same material properties and incident angle as Example 2, but assume the opposite polarization. a. What are the reflection and transmission coefficients? Which polarization has a lower reflection coefficient (magnitude)? b. Now allow to vary. At what value of is the wave completely transmitted? (i.e. What's the Brewster angle?) Fields and Waves I

Example 4 Example 2 Fields and Waves I

Reflection as a Function of Angle Note that the reflection varies with angle. Perpendicular reflects more that Parallel. There is also an angle for which there is no reflection for parallel polarization. Ulaby Fields and Waves I

Brewster’s Angle for Parallel Polarization Fields and Waves I

Optical Fibers Light is guided down the fiber. Ulaby Fields and Waves I

Optical Fibers Cladding is added to eliminate surface problems since part of the wave actually propagates outside the core. Also note that the pulses spread and decay due to a variety of losses. Ulaby Fields and Waves I

Rensselaer & Other Info Sources • Prof. E. F. Schubert http://www.ecse.rpi.edu/~schubert/Light-Emitting-Diodes-dot-org/chap22/chap22.htm • Prof. D. J. Wagner http://www.rpi.edu/dept/phys/ScIT/ • Prof. F. Ulaby (From his CD) • http://www.amanogawa.com/index.html • Movies of Waves from Prof. H. C. Han at Iowa State http://www.ee.iastate.edu/%7Ehsiu/em_movies.html Fields and Waves I

From Prof. Schubert’s Notes This is why the sky is blue. Fields and Waves I

Power and Energy Note that power density (the Poynting Vector) is not necessarily conserved across the boundary. However, the total power is. Because of the boundary conditions, the Poynting Vector is conserved for perpendicular but not for parallel polarization. All formulas are summarized in Table 8-2 of Ulaby. Ulaby Fields and Waves I

Fields and Waves I