54A Quick Look at THz Attenuation Mechanisms and Propagation(1) Absorption- Conversion of electromagnetic radiation into heat(a) Conduction current (Joule) heating: Pdiss= J·E (e.g., losses in undepleted semiconductors)(overbar denotes time average)(b) Polarization current heating Pdiss= (dP/dt)· E(e.g., polar liquids, such as water in a microwave oven)(2) Scattering(a) Non-resonant internal scattering from inhomogeneous dielectrics(e.g., composite materials such as RT-Duroid)(b) Resonant scattering from particles of dimension ~λ (e.g., raindrops)(c) Mixture of specular scattering from smooth surfaces anddiffuse scattering from rough surfaces (in infrared, more surfacestend to be diffuse; in RF bands more surfaces tend to be specular)Two types of attenuation, both more common than in lower RF bands:Notes #3, ECE594I, Fall 2009, E.R. Brown55Radiative TransferΩrΒSΒ04dB JBdrραπ+=B(r)= BSexp(-τ) + (ρJ/4πα)[1 – exp(-τ)]Homogeneous Solution Particular Solutionα→ attenuation constant ; ρ→density ; J → emission coefficient 0()Lrdrτα=∫“Optical Depth”L1D Radiative Transport EquationSpecial Cases: (1) j = 0 ⇒ B = BSexp(-τ) Beer-Lambert Law(2) B = uniform ⇒ dB = 0 ⇒ B = (ρJ/4πα) Kirchoff’s Law Good Introductory Reference: J.D. Kraus, “Radio Astronomy” (McGraw-Hill, New York, 1966).Notes #3, ECE594I, Fall 2009, E.R. Brown56exp( )tAiITzIα≡= −⋅Describes attenuation effects for many THz point sensors (active and passive), and for remote active systems. Especially useful when intervening materialis a linear, isotropic, and homogeneous (LIH) dielectric, or an array ofindependent scatterers:Τ → transmission, αA→ attenuation coefficientFor absorption, αΑ= αABS, the absorption coefficient (a characteristic of the material, and generally a function of frequency)For scattering αΑ= ρσS, ρ being the density and σSthe scattering cross sectionIi→ incident intensity, It→ transmitted intensityBeer-Lambert LawNotes #3, ECE594I, Fall 2009, E.R. Brown57Atmospheric Attenuation SimulationSimulation Tool:• Windows®-based atmospheric propagation tool called PcLnWin (Ontar, Corp.)• FASCODE radiative transfer engine developed by U.S. Air Force Geophysics Lab in 1970s.• HITRAN96 database (> 1 million molecular lines) developed by molecular chemists and maintained by U.S. Air Force (Hanscom AFB)Atmospheric Conditions• Seven “standard” molecules: H20, CO2, O3, N2O, CO, CH4, O2• 1976 U.S. standard atmospheric model @ sea level- Temperature = 288 K, Pressure = 1 atm, Humidity = 46%Between 300 GHz and 2.0 THz, water vapor dominates the attenuationby way of molecular dipole transitionsNotes #3, ECE594I, Fall 2009, E.R. Brown581E-301E-281E-261E-241E-221E-20100 200 300 400 500 600 700 800 900 1000Line Strength [cm-1/(molecule-cm-2)]Frequency [GHz]183.4325.4380.5 448.3548.0475.0 621.1752.6916.8971.0988.6557.3Stick Diagram of Atmospheric Molecular LinesNotes #3, ECE594I, Fall 2009, E.R. Brown59Why is Water Vapor Absorption So Strong and So Complicated ?Notes #3, ECE594I, Fall 2009, E.R. Brown• Photon absorption or emission can occur of ∆l = 1, for which -++p• For all polar molecules in the vapor state, THz transitions can occur between angularmomentum eigenstates of the entire molecule if the energy difference between eigenstates = hν, and if the change in angular momentum along the axis of photonpropagation = h/2πIgnoring spin, UX=2IXL222(1)Lll=+h22[( 1)( 2) ( 1)] ( 1)2XUllll lIIω∆= + +− + = +=hhh• For symmetric rotors (e.g., carbom monoxide CO, hydrogen chloride HCl, etc.), two of themoments (along two orthogonal axes) are identical and the third is trivially small, so thatIX→ moment of inertia for rotation about a given axis, xL → orbital angular momentum eigenvalue• Although highly polar, water is asymmetric rotor meaning that no twomoments of inertia are equal. This follows from its non-colinearalignment between the three atoms+-RI = µR2This creates a “ladder” of photon frequencies as shown to the right2/ Ih22/Ihl =0l =1l =2...• Any of these purely rotational transitions will be easily observed ifωτ >> 1 where τ is the momentum relaxation time (or dephasing time)• Generally, τ > 100 ps or more under STP conditions, so THz transitions are obvious601.E-011.E+001.E+011.E+021.E+031.E+041.E+051.E+060 0.5 1 1.5 2Frequency [THz]Absorption [dB/km]Atmospheric Simulation, Propagation “Windows”650 GHz “Window”Notes #3, ECE594I, Fall 2009, E.R. Brown860 GHz “Window”260 GHz “Window”61Equivalent Attenuation Length1.0E-011.0E+001.0E+011.0E+021.0E+031.0E+041.0E+051.0E+06100 400 700 1000 1300 1600 1900Frequency [GHz]1/e Attenuation Length [m]850-GHz“window”660-GHz“window”1.49 THz“window”* Since Lambert-Beer law of transmission, T = exp(-αz) appliesthe 1/e attenuation length is a “natural” metric*Notes #3, ECE594I, Fall 2009, E.R. Brown6246.31.92059.91.869150.01.718163.61.67152.91.41271.61.16461.61.0989.40.98912.70.75818.40.558Peak Attenuation [dB/m]Freq [THz]Ten Strongest Water Vapor Lines in THz Region(the big “whoppers”)Notes #3, ECE594I, Fall 2009, E.R. Brown63011(/)jεεεενν∞∞−=++e.g., for pure water at 300 K, single Debye relaxation frequency ν1= 17.5 GHz, and the dielectric constants ε0and ε∞are 79.7 and 5.26, respectively(2) Debye model of polar liquids (single component):Absorption Models for Liquids and Solids(1) Drude model of conductors :01jσσωτ=−backjσεεω=−011121(/)1(/)jjεεεεεενννν∞∞−−=+ +++Single Debye Double Debye(3) Independent of absorption model, Maxwell’s eqns predict:1/22'2 1 ( ''/ ') 12µεαω εε⎧⎫⎡⎤=+−⎨⎬⎣⎦⎩⎭'''jεεε≡+τ→momentum relaxation time~ 10 fs in common metals @300 K> 100 fs in semiconductors @300 KwhereNotes #3, ECE594I, Fall 2009, E.R. Brown6401020304050607080901 10 100 1000Frequency [GHz]Real {ε}(ε0= 78.2, ε∞= 4.9, τ = 8.1 ps, all at 25oC) T=10oC=20oC=30oC=40oC=50oC=60oCSingle-Debye Model for Liquid WaterNotes #3, ECE594I, Fall 2009, E.R. Brown650501001502002503000 200 400 600 800 1000Frequency [GHz]Absorption Coefficient [1/cm]T=60oC50oC40oC30oC20oC10oC(ε0= 78.2, ε∞= 4.9, τ = 8.1 ps, all at 25oC) Single-Debye Model for Liquid Water (cont)Notes #3, ECE594I, Fall 2009, E.R. Brown66Scattering from a Conducting SphereRayleigh limit, σ→(a/λ)4First Mie resonance[L.V. Blake, “Radar Range Performance Analysis,” (ArtechHouse, Norwoood, MA,