Review of Electromagnetic Field Theory

 For a source free region with permeability and permittivity , electromagnetic (EM) fields satisfy Maxwell's equations:

 where

 Note that for time varying fields, the electric and magnetic fields always appear in pairs.

 If we take the curl of equation-1 and substitute into equation-2, we get a form known as the homogeneous wave equation:

 Consider the simple special case of spatial variation in one dimension only (z direction), then the one-dimensional wave equation becomes:

 Assuming sinusoidal time dependence, this equation has the general solution:

 Using the complex phasor notation, and are understood and will not be written.

 Therefore the solution is two waves:

 The first term is a wave traveling in the positive z direction with a phase velocity of

 where for free space:

 The second term is a wave traveling in the negative z direction with the same velocity.

 This can be rewritten as:

 where is the phase coefficient

 The magnetic field is related to the electric field

 where

is the intrinsic impedance of the media

ohms

for free space

 Since the magnetic field is completely specified by the electric field and the media permittivity and permeability, it is redundant; and therefore, it is customary to perform our analysis in terms of only the electric field.

 When the wave propagates a distance z = , then the phase or

where f = frequency of sinusoid in Hz

 For free space: , m

 At any instant of time, the surface on which the phase of the electromagnetic field is a constant is defined by:

 which is the equation of a plane defined by Z = constant. i.e., a plane parallel to the xy plane. Waves of this type are known as "Uniform Plane Waves" or just "Plane Waves".

 Using terminology from the field of optics, the Index of Refraction of a medium is the phase velocity of the wave in free space (speed of light) divided by the phase velocity of the wave in the medium:

where is the relative permittivity of the medium and

is the relative permeability of the medium

 Properties of Plane Waves Summary

 1. Velocity of propagation is

 2. No electric or magnetic field in the direction of propagation.

i.e., Transverse Electric and magnetic fields

 3. Direction of propagation given by

 For plane waves propagating through a "slightly lossy medium", the complex permittivity (dielectric constant) of the medium is:

 where is the media conductivity

is the real part of the dielectric constant

is the imaginary part of the dielectric constant

 Remembering that is the phase coefficient for the propagating EM wave

where is the attenuation coefficient

 The solution to the wave equation becomes

 or for the positive z-direction traveling wave

 where accounts for the attenuation in the lossy medium

  Polarization of a plane wave is defined by the orientation of the electric field vector. If the Electric field is confined to a plane, it known as "plane polarized" or "linear polarization".

  For the general case of a plane wave propagating in the z-direction, which has both x and y components, the electric field components are:

 where is the wave number, 1/m

a = constant (phase difference)

 If E₁ and E₂ are equal amplitudes and a = ± 90°, the wave is circular polarized and the electric field vector rotates.

 Normal Incidence on a Lossless Dielectric

 The normal incidence voltage reflection coefficient is the ratio (complex i.e., magnitude and phase) of the reflected electric field to the incident field.

 Since the reflection coefficients are ratios of expressions involving the intrinsic impedances of the media, one may use relative values of the permittivity and permeability for the media since the

The voltage transmission coefficient is the ratio (complex i.e., magnitude and phase) of the transmitted electric field to the incident field.

 Kirchoff's Law of the conservation of energy

  Incident Power = Transmitted Power + Reflected Power

  Since the power reflection coefficient is:

 Kirchoff's Law becomes:

  The intrinsic impedance (just inside the media-2 side) of the media-1/media-2 interface viewed in the direction of media-3 is:

 Thus the reflection coefficient "as seen" by the incident wave (propagating in media-1) is a function of the intrinsic impedance's of the three media and the thickness of the media-2 (dielectric slab). For more layers than three, the input reflection coefficient is found by starting with the last layer (furthermost away in the propagation direction) and working with three layers at a time back to the input.

 Specular Reflection

 Consider the trace of a plane wave surface AA' which is just making contact with the reflecting plane surface at point-P. After a time interval delta-t, the plane wave propagates to position BB' where the reflection occurs at point-Q. Finally at a time interval 2 delta-t, the plane wave propagates to position BB' where the reflection occurs at point-R.

  The Law of Reflection: A plane wave is reflected from a plane surface with the angle of reflection equal to the angle of incidence.

 Arbitrary Incidence on a Lossless Dielectric

 Consider first the arbitrary polarization plane wave incident upon the interface of two semi-infinite media.

 The electric field of a plane wave lies in a plane perpendicular to its propagation direction. For an arbitrary polarization, the electric field can be decomposed into two linear orthogonal components (e.g., parallel to plane of incidence (vertical polarization) and perpendicular, to the plane of incidence (horizontal polarization).

 There will be a reflected wave at an angle (with respect to the normal to the interface) and a transmitted (or refracted) wave into the second medium at an angle . For continuity of the tangential E and H fields over the entire interface, the y coordinate variation of all three partial field must be the same which becomes:

 Therefore, the angle of incidence equals the angle of reflection

 and Snellís law of refraction

 wheren is the index of refraction

 Consider now the electric field polarization parallel (vertical polarization) to the incidence plane.

 For this case, the voltage reflection and transmission coefficients become

 Where is the intrinsic wave impedance for the refracted wave referred to the normal (Z) direction

 and the intrinsic impedance for medium-1 referred to the normal (Z) direction is

 For the special case where medium-1 is air and medium-2 has a permeability

 Where is the relative dielectric constant of medium-2 and

 Consider now the electric plane polarization perpendicular (horizontal polarization) to the incidence plane.

 For this case, the voltage reflection and transmission coefficient are

 Where

 And for the same special case above (air/dielectric interface)

 Arbitrary Incidence on a Lossless Dielectric

 Summary

 Vertical-polHorizontal-pol

 Total Transmission

 Polarizing Angle or Brewster Angle

 For the condition whereby , the reflection coefficient equals zero and no reflection occurs.

 For parallel polarization and dielectrics with , this condition is satisfied at a given incidence such that

 Where

  Note: Solution exist for both , therefore there is always some angle at which no reflection occurs.

  For perpendicular polarization there is no condition whereby for media with and ; therefore, the magnitude of the reflection coefficient is always greater than zero.

 Hence, a wave incident at with both polarization components present has some of the perpendicular polarization reflected (but none of the parallel polarization).

Therefore, the reflection wave is linearly polarized with the polarization perpendicular to the plane of incidence (i.e., horizontal polarization). Thus the origin of the name Polarization Angle.

 Total Reflection

 Examining the equations-22 & -27 for the voltage reflection coefficient, total reflection occurs when for = 0, , or pure imaginary. For example, = jX :

 For parallel (vertical) polarization, the

 becomes zero for some critical angle such that

 For perpendicular (horizontal) polarization, the

 becomes for this same condition

 For both polarizations,

 would be imaginary for and thus . For common dielectrics where

 when , the sine will be greater than unity. What this means is that the transmitted wave is no longer a plane wave propagating into region-2, and the field becomes:

 which is an exponentially damped field in the z-direction. Further the intrinsic impedance is imaginary, and there is no power flow in the z-direction. Thus the transition from wave refraction to total reflection is which exists only when the wave is propagating from a dense medium to optically rarer.

 From Snellís law, the angle of refraction would be for and would be imaginary for . From this point of view, we have no transfer of energy into the second medium. Although, there is no energy transfer there are finite values of fields which decay exponentially with distance into medium-2.

 Also, even though the reflection wave has the same amplitude as the incident wave, it does not in general have the same phase. The phase shift upon reflection is different for the perpendicular and parallel polarizations.

Note: the negative sign for the r12 means 180 deg phase shift upon reflection.

E1 = 1 @ 0⁰ E'1 = r12 x E1 E2 = t12 x E1

= -1/3 x 1 = -0.333 = 2/3 x 1 = 2/3 @ 0⁰ at Z=Z1

= 0.333 @ 180⁰ = 2/3 @ 180⁰ at Z=Z2

E3 = t21 x E2 E4 = r21 x E2 E5= t21 x E4

= 2/3 x 4/3 = 0.889 @ 180⁰ = 1/3 x 2/3 = 2/9 @ 180⁰ at Z=Z2 = 4/3 x 2/9 = 0.296 @ 360⁰ at Z=Z1

= 2/9 @ 360⁰ at Z=Z1

E6 = r21 x E4 E7 = t21 x E6 E8 = r21 x E6

= 1/3 x 2/9 = 2/27 @ 360⁰ at Z=Z1 = 4/3 x 2/27 = 0.099 @ 540⁰ at Z=Z2 = 1/3 x 2/27 = 2/81 @ 540⁰ at Z=Z2

= 2/27 @ 540⁰ at Z=Z2 = 2/81 @ 720⁰ at Z=Z1

E9= t21 x E8 E10 = r21 x E8 E11 = t21 x E10

= 4/3 x 2/81 = 0.033 = 1/3 x 2/81 = 2/243 @ 720⁰ at Z=Z1 = 4/3 x 2/243 = 0.011 @ 900⁰ at Z=Z2

@ 720⁰ at Z=Z1 = 2/243 @ 900⁰ at Z=Z2

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