Chapter 2 Electromagnetic Radiation Goal: The theory behind the electromagnetic radiation in remote sensing. 2.1 Maxwell Equations and Electromagnetic Waves Electromagnetic waves do not need a medium to propagate (no ether ) but are given by the changing electric and magnetic fields. They follow from the Maxwell equations: with: E: Electric field D: Displacement (D = ǫ 0 ǫ r E) H: Magnetic field B: Induction (B = µ 0 µ r H) J: Current density (J = σe) ρ: Charge density σ: Conductivity E = B t (2.1) H = D t + J (2.2) d = ρ (2.3) B = 0 (2.4) 11
12 CHAPTER 2. ELECTROMAGNETIC RADIATION µ: Permeability (µ = µ 0 µ r ) ǫ: Permittivity (dielectric constant, ǫ = ǫ 0 ǫ r ) So we can thus write: E = µ t H (2.5) H = σe + ǫ t E (2.6) and with the assumption of E = 0 (no free charges) it follows: 2 E = µǫ 2 E t 2 (2.7) This is a three-dimensional wave equation. Solutions are propagating planewaves. E.g., for the y-component of the electric field we can write: (and similarly for the x- and z-component) with: E 0 : Amplitude k: (complex) wave number ω: angular frequency E y = E 0 e i(kx ωt) (2.8) Inserting the wave ansatz of eq. 1.8 in eq. 1.7, gives: For the (phase) velocity of electromagnetic waves in vacuum it thus follows: c = ω k = 1 µ0 ǫ 0 (2.9a) This is the speed of light! In media the speed of light (= velocity of electromagnetic waves) will be reduced: c r = c (2.9b) ǫ r k 2 = ω 2 µǫ iωµσ ( = ω 2 µ ǫ i σ ) ω (2.9c) (2.9d)
2.1. MAXWELL EQ. & EM WAVES 13 Most of interest here haver µ r 1, so we can write: k 2 = ω 2 µ 0 ǫ 0 ǫ r (2.10) with an effective dielectric constant: ǫ r = ǫ r i σ ǫ 0 ω (2.11) In vacuum ǫ r = 1 and σ = 0, so that k 2 = ω 2 µ 0 ǫ 0 (2.12) The ratio between the speed of light in vacuum and the speed of light in a medium is the refractive index: n = c c r = ǫ r (2.13) The refractive index is thus a complex quantity that we can split into a real and an imaginary part: Writing for k: and inserting into the wave ansatz. gives: n = η + iχ (2.14) k = ωn c = ωη + iωχc (2.15) c E y = E 0 e i(kx ωt) (2.16) E y = E 0 e ωχx ωηx c e i( ωt) c (2.17) I.e., we see that the imaginary part of the refractive index (χ) describes an attenuation (or damping) of the wave in the medium.
14 CHAPTER 2. ELECTROMAGNETIC RADIATION by The distance d over which the electric field is fallen off to 1 e and called the skin depth. d = c ω χ is thus given (2.18) Example: Sea water at 20 C has a dielectric constant at 10 GHz of ǫ r = 52 37i: n = ǫ r = 7.6 2.4i χ = 2.4 d = c ω χ = 3 10 8 m/s2π 10 10 9 s 1 2.4 = 1.99 mm This means that sea water is virtually opaque at 10 GHz. Summary electromagnetic waves: Frequency ν = ω 2π Wavelength λ = c ν Wavenumber k = 2πn (if using the refractive index n here, then λ is the λ vacuum wave length). Note that wave number is proportional to frequency. Often another definition of the wave number is introduced as: ν = ν/c. (2.19) (The wavenumber ν is typically expressed in units of cm 1.) This is related to k by ν = k/2π. (Similarly as ν = ω/2π.) Energy of a photon E = hν, with h Planck s constant. (Please don t confuse with electric field vector!) Because ν = E hc wavenumber is also proportional to photon energy. (2.20)
2.2. POLARIZATION 15 2.2 Polarization For electromagnetic waves the vectors for the electric and magnetic fields are perpendicular to each other and also perpendicular to the direction of propagation: E B Direction of Propagation Figure 2.1: Polarization with electric and magnetic field. The electromagnetic wave is thus linearly polarized (with the E-vector in the x-y-plane here). Altough a single electromagnetic wave is linearly polarized, natural light is in general unpolarized (polarization is possible, but only a special case). Two waves with the same frequency and the same propa-
16 CHAPTER 2. ELECTROMAGNETIC RADIATION gation direction but different polarization planes for the E-vector superpose to a resulting E-vector. Depending on the phase difference of the two waves the resulting wave will be either linearly, elliptically or circularly polarized. (Individual electromagnetic waves also have a clear phase relationship, i.e., they are coherent. In contrast, natural light is incoherent.) 2.2.1 Stokes Parameter We can describe the state of polarization by the amplitudes of two orthogonal linear polarizations and the phase difference between them. A more convenient description uses the the so-called 4 Stokes parameters I, Q, U, and V. Assume that we have an instrument that measures light intensity (we will shortly define more precisely what intensity means) using a polarizer where the angle by which we turn the polarizer is given by φ: Stokes Parameter Intensity I I max I/2 I min ö 0 Angle ö Figure 2.2: The special stokes parameter I. I(φ) = 1 2 (Ī + I cos 2 (φ φ0 ) ) (2.21) with: Ī = I max + I min I = I max I min (2.22a) (2.22b)
2.2. POLARIZATION 17 then: so that: Q I cos 2φ 0 U I sin 2φ 0 I(φ) = 1 (Ī ) + Q cos 2φ + U sin 2φ 2 (2.22c) (2.22d) (2.23) Now we introduce a wave plate with retardations η and measure: I(φ, ǫ) = 1 2 (Ī + Q cos 2φ + (U cosǫ V sin ǫ) sin 2φ ) (2.24) We can determine I, Q, U and V from four measurement of the intensity: I(φ = 0, ǫ = 0) = 1/2(I + Q) I(φ = π/2, ǫ = 0) = 1/2(I Q) I(φ = π/4, ǫ = 0) = 1/2(I + U) I(φ = π/4, ǫ = π/2) = 1/2(I V ) (2.25a) (2.25b) (2.25c) (2.25d) Q and U describe the degree of linear polarization and V describes the degree of circular polarization. If light is unpolarized then Q = U = V = 0. We can define the degree of polarization as: Q2 + U 2 + V 2 (2.26) I 2.2.2 Radiometric Definitions The energy flux for electromagnetic wave is given the time average of the Poynting Vector S: S = E H (2.27) The intensity I (more precise definition will follow) is then given by the time average of the Poynting Vector. Side remark: The solid angle: The treatment of the radiation field requires us to consider the amount of radiant energy with a certain solid angle Ω.
18 CHAPTER 2. ELECTROMAGNETIC RADIATION Ù r surface area r Figure 2.3: The solid angle over the surface area. The solid angle is defined as, Ω = area r 2 and given in sterradians (sr.). The solid angle spanned by a full sphere is 4πsr. This is in analogy with the ordinary plane angle in radians (rad), where the angle φ can be defined as, φ = length r and where the angle over a full circle is 2π rad. ö r (arc-) lenght r Figure 2.4: Location of the φ in plane. 2.2.3 Monochromatic Radiance (also called intensity) I ν = = de ν dt da dωdν energy time area solid angle frequency (2.28a) (2.28b)
2.2. POLARIZATION 19 typically given in units of J s 1 m 2 sr 1 Hz 1 = W m 2 sr 1 Hz 1 with: E ν : (monochromatic) radiant energy (J) t: time (s) A: irradiated area (m 2 ) Ω: solid angle (sr) ν: frequency (Hz = s 1 ) 2.2.4 Monochromatic Irradiance (Also known as flux density.) F ν = I ν cosθ dω (2.29) with units of: Wm 2 Hz 1. 2π dù È I í Figure 2.5: The orientaion of dω in the flux density. I.e., the irradiance (flux density) is the normal component of the radiance (intensity) integrated over one hemisphere. 2.2.5 Total Flux Density The total flux density (also called total irradiance) is obtained by integrating the flux density over all frequencies: with units of: W m 2. F = 0 F ν dν (2.30)
20 CHAPTER 2. ELECTROMAGNETIC RADIATION 2.2.6 Total Flux The total flux (also known as the radiant power) is given by integrating the total flux density over the irradiated surface area: Φ = F da (2.31) with units of W. Caution: The definition of these quantities (radiance, irradiance, flux, etc.) may vary in different text books! 2.3 Interaction of Electromagnetic Radiation with Matter Matter can react with electromagnetic radiation by: reflection absorption transmission emission Incoming Reflected Absorbed Transmitted Figure 2.6: The interaction of electromagnetic radiation with matter. The fraction of the incoming radiation being reflected is described by the reflectivity coefficient ρ that is dimensionless and between 0 and 1. E.g., a
2.4. BLACK BODY & PLANCK 21 reflectivity of ρ = 1 means all incoming radiation is reflected. Similarly there are the absorptivity (α), emissivity (e) and transmissivity (τ) coefficients. The sum of reflection, absorption and transmission is 1: α + ρ + τ = 1 (2.32) Furthermore, from Kirchhoff s law follows that always the emissivity coefficient equals the absorptivity coefficient. e = α (2.33) I.e., a body can only emit radiation where it also absorbs radiation. 2.4 Black Body Radiation, Planck s Law All bodies with e 0 emit radiation, called thermal radiation. For a perfectly black body (i.e., α = e = 1) the emitted radiance I ν is given by Planck s law: I ν = L ν = 2hν3 c 2 1 e hν kt 1 (2.34) typically with units of W m 2 sr 1 Hz 1, or expressed in terms of wavelength rather than frequency: L λ = 2hc2 1 (2.35) λ 5 e hc λkt 1 in units of W m 3 sr 1. Here k is the Boltzmann constant and h is Planck s constant k = 1.38 10 23 J K 1 (2.36) h = 6.63 10 34 Js (2.37) (From ν = c λ follows dν = c λ 2 dλ and thus L ν dν = L λ dλ (2.38).
22 CHAPTER 2. ELECTROMAGNETIC RADIATION 10 10 Brightness [Wm -2 Hz -1 sr -1 ] 10 5 10 0 10-5 10-10 10 Bio. K 1Bio. K 10 Mio. K 1 Mio. K 100000 K 10000 K 6000 K 1000 K 100 K 10-15 10-20 10 5 10 10 10 15 10 20 Frequency [Hz] Figure 2.7: Planck curves for several temperatures in dependence to the frequency.
2.4. BLACK BODY & PLANCK 23 10 40 Brightness [Wm -2 m -1 sr -1 ] 10 30 10 20 10 10 10 0 10 Bio. K 1Bio. K 10 Mio. K 1 Mio. K 100000 K 10000 K 6000 K 1000 K 100 K 10-10 10 5 10 0 10-5 Wavelength [m] 10-10 10-15 Figure 2.8: Planck curves for several temperatures in dependence to the wavelength. The maximum of the Planck curve is at or ν max = 0.941 3kT h [Hz] (2.39) 2.989 10 3 λ max = [m] (2.40) T This is Wien s displacement law. Examples: For T 300 K (typical temperature of the Earth) we get λ max 10 µm and for T 6000 K (approximately the temperature of the sun) we get λ max 480, nm. Integrating the Planck function over all wavelength (or frequencies) we get the Stefan-Boltzmann law: L = 0 L λ dλ = 2k4 π 4 15c 2 h 3T4 (2.41)
24 CHAPTER 2. ELECTROMAGNETIC RADIATION and by integrating over all directions we get: 2π 0 dφ π/2 0 sin θ cosθl dθ = σt 4 (2.42) (units: W m 2 ) with: σ = 5.670 10 8 W m 2 K 4 (2.43). 2.5 Brightness Temperature At microwave frequencies (more specifically for hν kt) the exponential in the Planck function can be approximated by: e hν kt 1 + hν kt (2.44) so that: This is the Rayleigh-Jeans approximation. L ν 2kν2 c 2 T (2.45) Side remark: Note that the Rayleigh-Jeans law does not contain the Planck constant! This is historically of interest, as the Rayleigh-Jeans approximation was formulated before the Planck law.
2.5. BRIGHTNESS TEMPERATURE 25 Different Approximations for Radiation Rayleigh approxim. Brightness [Wm -2 Hz -1 sr -1 ] 10-10 10-15 Wien s approxim. Planck approxim. 10-20 10 6 10 8 10 10 10 12 10 14 10 16 Frequency [Hz] Figure 2.9: The different approximations for blackbody radiation by Planck, Rayleigh and Wien. In this regime (hν kt) radiance is proportional to temperature. I.e., we can use temperature to measure radiance. By using the Rayleigh-Jeans approximation, we can define the so called black-body brightness temperature. T B = c2 2kν 2L ν (2.46) This is the temperature a black body would have emitting the same radiance.
26 CHAPTER 2. ELECTROMAGNETIC RADIATION