1. 1 EM theory and its application to microwave remote sensing Chris Allen (callen@eecs.ku.edu) Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm

2. 2 Outline Plane wave propagation Lossless media Lossy media Polarization and coherence Fresnel reflection and transmission Layered media EM spectra, bands, and energy sources

3. 3 Plane wave propagation Plane wave propagation through lossless and lossy media is fundamental to microwave remote sensing. Consider the wave equation and plane waves in homogeneous unbounded, lossless media Plane waves – constant phase and amplitude in the plane Homogeneous – electrical and magnetic parameters do not vary with throughout the medium Beginning with Maxwell’s equations and assuming a homogeneous, source-free medium leads to the homogeneous wave equation where E is the electric field vector (V/m) [note that bolded symbols denote vectors]  is the medium’s magnetic permeability (H/m) [H: Henrys]  is the medium’s permittivity (F/m) [F: Farads]

4. 4 Plane wave propagation Assuming sinusoidal time dependence where  is the radian frequency (rad/s) r is the displacement vector and Re{  } is the real operator E(r,t) satisfies the wave equation if Using phasor representation (i.e., e.j  t is understood) and assuming a rectangular coordinate system, the solution has the general form of where E 0 is a constant vector and

5. 5 Plane wave propagation A more compact form results from letting where k is the propagation vector, and k = |k| is called the wave number (rad/m) resulting in Finally reintroducing the time dependence and expressing only the real-time field component yields This equation represents two waves propagating in opposite directions defined by the propagation vector k

6. 6 Plane wave propagation Rotating the Cartesian coordinate system such that the z axis aligns with k yields representing two waves propagating in the + z and – z directions. The argument of the cosine function contains two phase terms: the time phase,  t, and the space phase, kz. If we use the – part of the ± solution we have representing a wave whose constant phase moves in the positive z direction. As time increases, z must increase to maintain a constant phase argument.

7. 7 Plane wave propagation The time phase component is characterized by  where f is the frequency (Hz) and T is the time period (s). Similarly the space phase component depends on k where is the space period (m), or wavelength, in the medium which can also be expressed as

8. 8 Plane wave propagation Consider now the electric field’s phase for a positive- traveling wave, i.e.,  t – kz. A surface on which this phase is constant requires For any given time t, this surface represents a plane defined by z = constant, on which both the phase and amplitude are constant. As time progresses, this plane of constant phase and amplitude advances along the z axis, hence the name uniform plane wave. The rate at which this plane advances along the z axis is called the phase velocity, v (m/s)

9. 9 Plane wave propagation Given an uniform plane E-field solution to the wave equation, the H-field is found using Maxwell’s equations From the E-field component in the x-axis direction, E x, is found the H-field component in the y-axis direction, H y, as where  is the intrinsic impedance (  ) of the medium Note that E x and H y are related through the intrinsic impedance similar to how voltage and current in a circuit are related through Ohm’s law. Note also the orthogonality of the E, H, and k vectors.

10. 10 Plane waves in a lossy medium A lossy medium is characterized by its permeability, , permittivity, , and conductivity,  (S/m) [S: Siemens]. Maxwell’s equations for a source-free medium become And the corresponding wave equation remains where the wave number is Note that for a lossless medium, k is purely real when  = 0 and both  and  are real

11. 11 Plane waves in a lossy medium For a lossy medium k is complex due to   0 or either  or  are complex For lossless media the imaginary parts of the permeability and permittivity are zero. Non-zero imaginary terms (  > 0 and  > 0) represent mechanisms for converting a portion of the electromagnetic wave’s energy into heat, resulting in a loss of wave energy.

12. 12 Plane waves in a lossy medium Consider the complex electric field plane wave propagation along the positive z axis whereas for the lossless case k was real, in a lossy medium k is complex and is related to the propagation factor or propagation constant,  (1/m), by such that where  and  are real quantities and  is the attenuation constant (Np/m) and  is the phase constant (rad/m) [Np = Neper]

13. 13 Plane waves in a lossy medium Clearly for a wave travelling along the +z axis as z increases, the magnitude of the electric field decreases. The real time expression for the x-axis field component is The attenuation constant is the real part of jk The phase constant is the imaginary part of of jk Note: (Neper/m)  8.686 (dB/Neper) = (dB/m)

14. 14 Plane waves in a lossy medium In a lossy medium, the intrinsic impedance is also complex giving rise to a non-zero phase relationship between the E and H field components. While a medium’s loss may be due to its conductivity, or the imaginary components of permittivity or permeability, in most microwave remote sensing applications the magnetization loss (  ) is negligible. Exceptions include the ferrous-rich sands found in Hawaii and soils on the Martian surface. Therefore the  term will be neglected from now on. A material’s permeability is usually specified relative to that of free space,  o, (  o = 4   10 -7 H/m ), as  =  r  o and typically  r = 1

15. 15 Plane waves in a lossy medium A medium’s loss may be due to its conduction loss (  > 0 but finite) or its polarization loss (  > 0). Conduction loss is gives rise to a conduction current where electrical energy is converted to heat energy due to ohmic losses.

16. 16 Plane waves in a lossy medium Polarization loss is due to a displacement current, similar to current through a capacitor. For an ideal dielectric, equal amounts of energy are stored and released during each cycle. For lossy dielectrics, some of the stored energy is converted into heat. The imaginary part of the displacement current is in-phase from the E field, and hence contributes to real energy loss.

17. 17 Plane waves in a lossy medium For a sinusoidal time variation, there is a transition frequency,  t = 2  f t, where these two current components are equal

18. 18 Plane waves in a lossy medium For dielectric materials, these two loss mechanisms are often combined into a single imaginary component as The permittivity of dielectric materials is usually specified relative to the permittivity of free space,  o, where (  o = 8.854  10 -12 F/m ) as where

19. 19 Plane waves in a lossy medium Often instead of specifying the  r , a material’s loss tangent tan  where

20. 20 Plane waves in a lossy medium Clearly since the loss mechanism due to conductivity is frequency dependent, a medium’s loss characteristics may also be frequency dependent. At low frequencies where the conductivity introduces signficant loss, the intrinsic impedance and phase velocity will be frequency dependent. At frequencies where the conductive loss is dimished, the intrinsic impedance and phase velocity will become frequency independent.

21. 21 Plane waves in a lossy medium Low-loss media (i.e., tan   1) For the case of low-loss media, the expressions for v, , , and  can be simplified to be where c = 3  10 8 m/s where  o = 120    377 

22. 22 Plane waves in a lossy medium High-loss media (i.e., tan  >> 1) For the case of high-loss media, the expressions for ,  and  can be simplified to Also, when an electromagnetic wave impinges on a conducting medium, the field amplitude decreases exponentially with depth. At a depth termed the skin depth,  (m), the field amplitude is e -1 of its value at the surface, where

23. 23 Plane waves in a lossy medium General lossy media For media that is neither high loss nor low loss, the simplifying approximations  and  do not apply. For these cases we have where k o is the free-space wave number and o is the free-space wavelength It is sometimes useful to refer to a medium’s refractive index, n, where

24. 24 Plane waves in a lossy medium

25. 25 Plane waves in a lossy medium

26. 26 Polarization of plane waves and coherence For a +z-axis propagating uniform plane wave, the E-field components must lie in the xy plane For any point on the xy-plane, the E-field varies with time. The wave’s polarization is associated with the curve the tip of the E-field vector traces out. A straight line indicates a linear polarization (i.e.,  y = 0 or  180  ) with tilt angle 

27. 27 Polarization of plane waves and coherence A circle indicates circular polarization where E xo = E yo and  y =  90  Left-hand circular polarization (LCP), which results when  y = +90 , has the E-field rotating once each cycle in the direction the fingers of the left hand point when gripping the z-axis with the thumb in the +z direction. Right-hand circular polarization (RCP), which results when  y = -90 , has the E-field rotating in the same direction as the fingers when gripping the z-axis with the right hand so that the thumb is in the +z direction.

28. 28 Polarization of plane waves and coherence An elliptical pattern indicates elliptical polarization where E xo and E yo > 0 and  y  0,  90 , or 180  yet these variables remain constant over time. A wave is unpolarized when the amplitude and phase relationships of the orthogonal components are time varying.

29. 29 Polarization of plane waves and coherence Signals produced by single-frequency or multifrequency transmitters are typically completely polarized Signals emitted by physical objects, irregular terrain, or inhomogeneous media are usually broadband and are composed of many statistically independent waves with different polarizations If there is no correlation between the component waves of such a signal it is incoherent or unpolarized Between these two extremes (completely polarized and unpolarized) are the partially-polarized signals that result when polarized signals are scattered by random targets While characterization of completely polarized plane waves is fairly straightforward, the characterization of these partially-polarized signals is more challenging

30. 30 Polarization of plane waves and coherence An analysis technique to evaluate the state of polarization or degree of coherence of a plane wave involves the magnitude of the normalized cross-correlation of the x and y phasor components as where * denotes the complex conjugate and  denotes the average operator over some finite time interval T For completely polarized signals  xy = 1 while for incoherent or unpolarized signals  xy = 0. For partially-polarized or partially-coherent signals 0 <  xy < 1

31. 31 Polarization of plane waves and coherence To evaluate the degree of polarization (DOP), another measure of a signal’s polarization, involves the Stokes parameters The DOP is found by where DOP = 1 for a completely polarized signal, a DOP = 0 for an unpolarized signal, and for partially-polarized signals 0 < DOP < 1

32. 32 Polarization of plane waves and coherence The Poincaré sphere is a tool for visualizing the continuum of polarization states. Derived from the Stokes parameters, the sphere maps linear polarizations on the equator (LVP: linear vertical pol.; LHP: linear horizontal pol.) and the right (RCP) and left (LCP) circular polarizations at the north and south poles. Points opposite one another on the sphere represent orthogonal polarizations. Points representing completely- polarized signals lie on the sphere’s surface while points representing partially-polarized waves appear within the sphere.

33. 33 Electromagnetic phenomena Speed of light c : speed of light in vacuum ( 2.99792458  10 8 m/s  3  10 8 m/s ) n : refractive index of material ( n  1 ) v : speed of light in material ( m/s ), v = c/n Wavelength and frequency f : frequency ( Hz ) : wavelength ( m ) o : wavelength in free space ( m )  f : bandwidth in frequency ( Hz )  : bandwidth in wavelength ( m ) In vacuum, ( n = 1, v = c ) In a medium, ( n  1, v  c )

34. 34 Fresnel reflection and transmission Now consider the electromagnetic interactions as a plane wave impinges on a planar boundary between two different homogeneous media with semi-infinite extent Properties of interest include reflection, refraction, transmission Solutions are found by satisfying the requirement for the continuity of tangential E and H fields across the boundary Simplifying assumptions: Plane wave propagation Smooth, planar interface infinite in extent Linear, isotropic refractive indices Semi-infinite media

35. 35 Fresnel reflection and transmission Snell’s law n 1 : refractive index of medium 1 (incidence side) n 2 : refractive index of medium 2  i : incidence angle (measured from surface normal)  r : reflected angle (measured from surface normal)  t : transmitted angle (measured from surface normal) Requirement for continuity of tangential E and H fields across the boundary yields Reflected Transmitted

36. 36 Fresnel reflection and transmission Fresnel equations Predicts reflected and transmitted vector field quantities at a plane interface Satisfies requirement for continuity of tangential E and H fields across the boundary Consequently the interaction is polarization dependent Arbitrarily polarized incident plane wave can be decomposed into two orthogonal linear polarizations: perpendicular (  ) and parallel (//) Separate solutions for perpendicular and parallel cases Perpendicular and parallel refer to E -field orientation with respect to the plane of incidence (plane containing incident, reflected, and transmitted rays) These same polarizations have other names as well

37. 37 Fresnel reflection and transmission Perpendicular (horizontal) case Reflection coefficient (relates to field strength) (sometimes represented by , , or r ). Transmission coefficient (relates to field strength) Note that 1 + R  = T 

38. 38 Fresnel reflection and transmission Perpendicular (horizontal) case Reflectivity (relates to power or intensity) Transmissivity (relates to power or intensity) (sometimes represented by T ) or Note that   +   = 1 which satisfies the conservation of energy

39. 39 Fresnel reflection and transmission Parallel (vertical) case Reflection coefficient (relates to field strength) Transmission coefficient (relates to field strength) Note that 1 + R // = T //

40. 40 Fresnel reflection and transmission Parallel (vertical) case Reflectivity (relates to power or intensity) Transmissivity (relates to power or intensity) or Note that  // +  // = 1 which satisfies the conservation of energy

41. 41 Fresnel reflection and transmission Special cases Normal incidence (  i = 0 )  i =  r =  t = 0, cos  = 1 Brewster angle  B : angle where reflection coefficient for parallel (vertical) polarized field goes to zero i.e., at  =  B,  // = 0,  // = 1 (note polarization dependence)

42. 42 Fresnel reflection and transmission Special cases Critical angle  C : incidence angle at which total internal reflection occurs (for n 1 > n 2 ) i.e., at  C,  // =   = 1,  // =   = 0 (note polarization independence) Evanescent waves exist in medium 2, with imaginary propagation coefficients meaning they decay rapidly with distance z from the boundary.

43. 43 Fresnel reflection and transmission Normal incidence reflection coefficient for some typical geological contacts

44. 44 Fresnel reflection and transmission Example #1 Consider the case where a plane wave impinges on a planar boundary between homogenous ice (  = 3.14, n = 1.78) and air (  = 1, n = 1). From the formulas presented earlier Normal reflectivity = -11.0 dB Normal transmissivity = -0.4 dB Critical angle,  C = 34.4° Brewster angle,  B = 29.4°

45. 45 Fresnel reflection and transmission Fresnel reflection and transmission coefficients vs. incidence angle at an ice-air boundary (  ice = 3.14,  air = 1)

46. 46 Fresnel reflection and transmission Reflectivity and transmissivity expressed in linear units vs. incidence angle at an ice-air boundary (  ice = 3.14,  air = 1)

47. 47 Fresnel reflection and transmission Reflectivity and transmissivity expressed in decibels vs. incidence angle at an ice-air boundary (  ice = 3.14,  air = 1)

48. 48 Fresnel reflection and transmission Example #2 Consider the case where a plane wave impinges on a planar boundary between homogenous ice (  = 3.14, n = 1.78) and rock (  = 5, n = 2.24). From the formulas presented earlier Normal reflectivity = -19.1 dB Normal transmissivity = -0.1 dB Critical angle,  C = NA Brewster angle,  B = 51.6°

49. 49 Fresnel reflection and transmission Fresnel reflection and transmission coefficients vs. incidence angle at an ice-rock boundary (  ice = 3.14,  rock = 5)

50. 50 Fresnel reflection and transmission Reflectivity and transmissivity expressed in linear units vs. incidence angle at an ice-rock boundary (  ice = 3.14,  rock = 5)

51. 51 Fresnel reflection and transmission Reflectivity and transmissivity expressed in decibels vs. incidence angle at an ice-rock boundary (  ice = 3.14,  rock = 5)

52. 52 Layered media Now consider the case of multiple, planar layers and the associated composite reflection and transmission characteristics. Consider first the simplest case, a single layer of thickness d 1 sandwiched between two semi-infinite layers.

53. 53 Layered media (1 of 5)

54. 54 Layered media (2 of 5)

55. 55 Layered media (3 of 5)

56. 56 Layered media (4 of 5)

57. 57 Layered media (5 of 5) A similar approach can be developed for the V-polarized case.

58. 58 Layered media – example (1 of 4) Multilayer example: 1-layer case

59. 59 Layered media – example (2 of 4) Wavenumbers in media 1 and 2 are Matching tangential E -field components at each interface requires Matching tangential H -field components at each interface requires

60. 60 Layered media – example (3 of 4) These boundary matching conditions can be written in matix form as which leads to

61. 61 Layered media – example (4 of 4) And the R variables in the matrices are defined as To find the reflection and transmission coefficient for this 1-layer structure, we solve for R and T

62. 62 Electromagnetic spectrum

63. 63 Radio spectrum

64. 64 Atmospheric transmission at radio frequencies

65. 65 Reserved frequencies and band designations

66. 66 Generic energy band diagram

67. 67 Quantum energy levels Photon contains energy related to the electromagnetic frequency, (Hz) E = h where h is Planck’s constant h = 6.625 × 10 -34 J s h = 4.136 × 10 -15 eV s [Note: 1 eV = 1.6 × 10 -19 J ] At 1 GHz ( = 10 9 Hz), E = 4.136 × 10 -6 eV or 6.625 × 10 -25 J At 10 GHz ( = 10 10 Hz), E = 4.136 × 10 -5 eV or 6.625 × 10 -24 J Therefore at 1 GHz, a 10-mW signal contains more than 1.5 × 10 22 photons For comparison, one photon of visible light (  500 nm, = 600 THz) E = 2.5 eV or 4 × 10 -19 J Therefore, a 10-mW signal would contain more than 2.5 × 10 16 photons

68. 68 Relating EM band frequencies and wavelengths to mechanisms

69. 69 EM source mechanisms