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**Electromagnetic (E-M) theory of waves at a dielectric interface**

While it is possible to understand reflection and refraction from Fermat’s principle, we need to use E-M theory in order to understand quantitatively the relationship between the incident, reflected, and transmitted radiant flux densities: We can accomplish this treatment by assuming incident monochromatic light waves which form plane waves with well defined k-vectors as shown in the diagram. The interface is shown with an origin and coordinates (x,y,z). Ir i r t ni nt ûn x y . z b Ii It We will consider E-field polarizations which are (i) in the plane of incidence and (ii) perpendicular to the plane of incidence, as shown below.

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**E-field is perpendicular to the plane-of incidence**

E-field is parallel to the plane of incidence

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**Maxwell’s Equations for time-dependent fields in matter**

D – Displacement field H – Magnetic Intensity P – Polarization M – Magnetization - Magnetic permeability - Permittivity e - Dielectric Susceptibility m - Magnetic Susc. g – Conductivity j – Current density

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**Summary of the boundary conditions for fields at an interface**

Side 1 Maxwell’s equations in integral form allow for the derivation of the boundary conditions for the total fields on both sides of a boundary. Boundary Side 2 Normal component of D is discontinuous by the free surface charge density Tangential components of E are continuous Normal components of B are continuous Tangential components of H are discontinuous by the free surface current density

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For dielectrics, j = 0. Therefore, the components of E and H that are tangent to the interface must be continuous across it. Since we have Ei, Er, and Et the continuity of E components yield: Note that i r ûn ni x y . z nt b t

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**Consider the expression on the interface (y = b) for all x, z and t**

Consider the expression on the interface (y = b) for all x, z and t. The above relationship must hold at all points and at any instant in time on the interface. Therefore Since then we have

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**Thus, at the interface plane**

r cos which is again Snell’s Law

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**Case 1: E Plane of incidence**

Continuity of the tangential components of E and H give Cosines cancel Using H = -1 B, the tangential components are

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**The last two equations give**

The symbol means E Plane of incidence. These are called the Fresnel equations; most often i t o. Let r = amplitude reflection coefficient and t = amplitude transmission coefficient. Then, the Fresnel equations appear as Note that t - r = 1

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**Case 2: E || Plane of Incidence**

Continuity of the tangential components of E: Continuity of the tangential components of -1 B:

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If both media forming the interface are non-magnetic i t o then the amplitude coefficients become Using Snell’s law the Fresnel Equations for dielectric media become Note that t - r = 1 holds for all i , whereas t|| + r|| = 1 is only true for normal incidence, i.e., i = 0.

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**Consider limiting cases for nearly normal incidence: i 0.**

In which case, we have: since Also, using the following identity with Snell’s law Therefore, the amplitude reflection coefficient can be written as:

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**In the limiting case for normal incidence i=t = 0, we have :**

Note that these equalities occur for near normal incidence as a consequence of the fact that the plane of incidence is no longer specified when i t 0. Consider a specific example of an air-glass interface: i ni = 1 We will consider a particular angle called the Brewster’s angle: p + t = 90 t nt = 1.5 External reflection nt > ni Internal reflection ni > nt At the polarization angle p, only the component of light polarized normal to the incident plane and therefore parallel to the surface will be reflected.

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**External Reflection (nt > ni) Internal Reflection (nt < ni)**

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**Concept of Phase Shifts () in E-M waves:**

Since when nt > ni and t < i as in the Air Glass interface, we expect a reversal of sign in the electric field for the Ecase when We need to define phase shift for two cases: When two fields E or B are to the plane of incidence, they are said to be (i) in-phase (=0) if the two E or B fields are parallel and (ii) out-of-phase ( = ) if the fields are anti-parallel. When two fields E or B are parallel to the plane of incidence, the fields are (i) in-phase if the y-components of the field are parallel and (ii) out-of-phase if the y-components of the field are anti-parallel.

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**Examples of Phase shifts for two particular cases:**

(b)

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**Analogy between a wave on a string and an E-M wave traversing the air-glass interface.**

Glass (n = 1.5) Air (n = 1) = = 0 Air (n = 1) Glass (n = 1.5) = = 0 Compare with the case of

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**Examples of phase-shifts using our air-glass interface:**

In order to understand these phase shifts, it’s important to understand the definition of .

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Reflected E-field orientations at various angles for the case of External Reflection (ni < nt). It is worth checking and comparing with the various plots for the phase shift on the previous slides.

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Reflected E-field orientations at various angles for the case of Internal Reflection (ni > nt). It is worth checking and comparing with the various plots for the phase shift on the previous slides.

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**Reflectance and Transmittance**

Remember that the power/area crossing a surface in vacuum (whose normal is parallel to the Poynting vector) is given by The radiant flux density or irradiance (W/m2) is Phase velocity From the geometry and total area A of the beam at the interface, the power (P) for the (i) incident, (ii) reflected and (ii) transmitted beams are:

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**Define Reflectance and Transmittance:**

Note that Conservation of Energy at the interface yields:

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Therefore, We can write this expression in the form of componets and ||: We must use the previously calculated values for

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**It’s possible to verify for the special case of normal incidence:**

Consider the case of Total Internal Reflection (TIR): t nt = 1 ni = 1.5 i

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