Reflection and Refraction of Electromagnetic Waves - Lecture Notes | PHYS 436, Study notes of Physics

Download Reflection and Refraction of Electromagnetic Waves - Lecture Notes | PHYS 436 and more Study notes Physics in PDF only on Docsity! UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 1 LECTURE NOTES 8.5 Reflection and Refraction of EM Waves at the Boundary of a Dispersive/Absorbing/Conducting Medium Consider a situation where monochromatic plane EM waves are incident on a boundary between two media {located at z = 0 and lying in the x-y plane} as shown in the figure below. For the sake of simplicity, the 1st medium (z < 0) is linear/homogeneous/isotropic, non-absorbing / non-dispersive and non-magnetic. The 2nd medium is also linear/homogeneous/isotropic and non-magnetic, but is absorbing/dispersive and conductive. Because of the above-stated EM properties of the two media, in medium (1) the incident and reflected wavevectors inck and reflk are purely real, whereas in medium (2), the transmitted wavevector is complex: ( ) ( ) ( )trans trans transk k iω ω κ ω= + . Note that the monochromatic plane EM wave(s) have the same frequencyω , independent of the medium they are propagating in. THE ELECTRIC FIELDS: ( ) ( ) ( ) ( ) ( ) ( ) ,Medium (1) (non-absorbing) , inc inc refl i k r t inc o k r t refl orefl E r t E r e E r t E r e ω ω − − ⎧ =⎪ ⎨ ⎪ =⎩ i i ,inc reflk k ⇐ real, constant wavevectors ( ) ( ) ( )( )Medium 2) ,(absorbing / conducting) trans trans i k r t trans oE r t E r e ω ω−⎧ =⎨ ⎩ i ( ) ( ) ( )trans trans transk k iω ω κ ω= + ⇐ On the boundary/interface (lying in the x-y plane at z = 0) we must have (for arbitrary times, t): ( ) ( )inc refli k r t i k r te eω ω− −=i i and: ( ) ( )( ) ( )( ) ( )transinc trans transi k r ti k r t i k r t re e e eω ωω ω ω κ ω−− − −= =ii i i ⇒ inc reflk r k r=i i and: ( ) ( ) ( )( ) ( ) ( )inc trans trans trans trans transk r k r k i r k r i rω ω κ ω ω κ ω= = + = +i i i i i complex wavevector UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 2 On the interface/boundary lying in the x-y plane at z = 0: The 1st equation: inc reflk r k r=i i gives usual Law of Reflection: sin sininc inc refl reflk r k rθ θ= but: 1 1inc reflk v k vω ω= = = because both the incident and reflected waves are in the same non-dispersive/non-absorbent medium {medium (1)}. ⇒ sin sininc reflθ θ= ⇒ inc reflθ θ= The 2nd equation: ( ) ( ) ( )( ) ( ) ( )inc trans trans trans trans transk r k r k i r k r i rω ω κ ω ω κ ω= = + = +i i i i i , after equating real and imaginary parts, gives: ( ) ( )Re : inc transk r k rω =i i and ( ) ( )Im : 0 trans rκ ω = i ⇒ In general, ( ) ( ) and trans transk ω κ ω are not parallel to each other!! i.e. In general, ( ) ( ) and trans transk ω κ ω will point in different directions!! Why/How??? Physically, the requirement that ( ) 0trans rκ ω =i on the interface/boundary {lying in the x-y plane at z = 0} means that ( ) ( )( )Imtrans transkκ ω ω= must be ⊥ to the boundary (i.e. ˆtrans zκ + ), since the position vector r {pointing from the origin ϑ ( )0,0,0 to an arbitrary point ( ), , 0x y z = on the boundary} lies in the x-y plane. Inside Absorbing/Conducting Medium (2) (i.e. z > 0): Because trans trans transk k iκ= + , then ( ) ( ) ( ) ( ) ( ), trans transtrans trans trans i k r t i k r tz trans o oE z t E r e E r e e ω ωκ− −−= = i i , Thus, we see that: ( )Imtrans transkκ = defines planes ( to the boundary/interface) of constant electric field amplitude in medium (2). ( )ˆˆ Imtrans transkκ = is the unit normal to the planes of constant electric field amplitude in medium (2). Furthermore: ( )Retrans transk k= defines planes of constant phase in medium (2) ( )ˆ ˆRetrans transk k= is the unit normal to the planes of constant phase in medium (2) {n.b. in general, planes of constant phase could be in any direction, depending on the material!} See the following figure for a explicit diagram of exactly what is occurring in this physics problem: n.b. , and inc refl transθ θ θ are defined with respect to the ẑ+ unit normal of the interface/boundary. UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 5 Thus, for ( ) ( ) ( )trans trans transk k iω ω κ ω= + and ( ) ( ) ( )2 2 2n n iω ω η ω= + we have the complex relations: 1.) ( ) ( )( ) ( ) ( )( )2 2 2 2 22 2trans trans ok n kω κ ω ω η ω− = − with vacuum wavenumber 2o o k c π ω λ ≡ = 2.) ( ) ( ) ( ) ( ) 22 2costrans trans trans ok n kω κ ω θ ω η ω= and vacuum wavelength o c fλ ≡ , 2 fω π= We also have the relation: 3.) ( )sin sininc inc trans transk kθ ω θ= where: 1 1inc ok n k nc ω⎛ ⎞= =⎜ ⎟ ⎝ ⎠ Inserting relation 3.) into relations 1.) and 2.) above, after some algebra these relations yield the following relation: ( ) ( ) ( ) ( ) ( ) ( )( ) ( )2 2 22 2 2 2 22 2 1 12 2 1 1 2 cos sin sintrans trans trans o inc o inc n in n k i n k n k n n ω η ω ω η ω ω ω θ κ ω θ θ − + + = − = − {n.b. if medium (2) is L/H/I non-conductive/non-magnetic/non-dispersive medium (i.e. like medium (1)), then 2 0transκ η= = and it is easy to show that this relation then reduces to ( )sin sininc inc trans transk k fcnθ θ ω= ≠ } Let us define: ( ) ( ) ( ) ( ) ( ) ( ) ( )( )2 22 2 2 2 22 2 2 2 1 1 1 2n inn n n n n ω η ω ω η ωω ω ω − + ≡ = =a ⇐ complex! Then: ( ) ( ) ( )2 21cos sintrans trans trans o inck i n kω θ κ ω ω θ+ = −a We define the Law of Complex Refraction {for this particular boundary/interface situation} as: ( ) ( )1 2sin sininc transn nθ ω θ ω= where: ( )transθ ω ≡ complex angle: ( ) ( ) ( )trans trans transiθ ω θ ω ω≡ + Θ with: ( ) ( )( )Retrans transθ ω θ ω≡ and: ( ) ( )( )Imtrans transω θ ωΘ ≡ Physically, ( ) ( )( )Retrans transθ ω θ ω≡ has the usual physical meaning (except that it is now frequency-dependent), whereas ( ) ( )( )Imtrans transω θ ωΘ ≡ has no simple/easy physical meaning. The Law of Complex Refraction can be rewritten as: ( ) ( ) ( )( ) ( ) 2 2 2 2 1 1 sin sin inc trans n n n n ω ω θω ω θ ω ≡ = =a UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 6 Then: ( ) ( )2 2 2sin sintrans incω θ ω θ=a ⇒ ( ) ( )( )2 2 21 cos sintrans incω θ ω θ− =a ⇒ ( )( ) ( )2 2 21 cos sintrans incθ ω θ ω− = a ⇒ ( ) ( )( )2 2cos 1 sintrans incθ ω θ ω= − a But: ( ) ( ) ( ) ( ) ( )( )2 2 2 21 1cos sin 1 sintrans trans trans o inc o inck i n k n kω θ κ ω ω θ ω θ ω+ = − = −a a a But: ( ) ( )( )2 2cos 1 sintrans incθ ω θ ω= − a {from above} ∴ ( ) ( ) ( ) ( )( ) ( ) ( )2 21 1cos 1 sin costrans trans trans o inc o transk i n k n kω θ κ ω ω θ ω ω θ ω+ = − =a a a i.e. ( ) ( ) ( ) ( )1cos costrans trans trans o transk i n kω θ κ ω ω θ ω+ = a Solve for ( )ωa : ( ) ( ) ( )( ) ( )2 11 cos cos trans trans trans o trans k i n nn k ω θ κ ω ω ω θ ω + = =a ⇒ ( ) ( ) ( )( )2 cos cos trans trans trans o trans k i n k ω θ κ ω ω θ ω + = The {complex} and E B fields involved at the interface are: Incident wave: ( ) ( ) ( ), inc inc i k r t inc oE r t E r e ω− = i ( ) ( )1, ,inc inc incB r t k E r tω= × Reflected wave: ( ) ( ) ( ), refl refl i k r t refl oE r t E r e ω− = i ( ) ( )1, ,refl refl incB r t k E r tω= × Transmitted wave: ( ) ( ) ( ), trans trans i k r t trans oE r t E r e ω− = i ( ) ( )1, ,trans trans transB r t k E r tω= × = ( ) ( )( )1 , ,trans trans trans transk E r t i E r tκω × + × The boundary conditions at the interface {lying in the x-y plane at z = 0} are: BC 1) (normal D continuous): 1 1 2 2E Eε ε ⊥ ⊥= ( 0freeσ = on the interface/boundary) BC 2) (tangential E continuous): 1 2E E= BC 3) (normal B continuous): 1 2B B ⊥ ⊥= BC 4) (tangential H continuous): 1 2 1 2 1 1B B μ μ = ( 0freeK = on the interface/boundary) ⇒ 1 2B B= if 1 2 oμ μ μ (medium (1) and medium (2) both non-magnetic) n.b. this form of B – takes care of everything!!! UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 7 On the interface/boundary at z = 0 (for any arbitrary (x,y,z = 0) and time t): TE Polarization Case: BC 2) inc refl transo o o E E E+ = 2o ok cω π λ= = , o c fλ = BC 4) cos cos cos inc refl transo inc o refl o trans B B Bθ θ θ+ = 1inc ok n k= , 1refl ok n k= and inc reflθ θ= = ( )cos cos cosinc refl trans transinc o inc refl o refl trans o trans trans ok E k E k E i Eθ θ θ κ− + = − + = ( ) ( )1 cos cosinc refl transo o o inc trans trans trans on k E E k i Eθ θ κ− − = − + = ( ) ( )1 cos cosinc refl transo inc o o trans trans trans on k E E k i Eθ θ κ+ − = + + or: ( ) 1 cos cosinc refl trans trans trans trans o o o o inc k iE E E n k θ κ θ ⎛ ⎞+ − = ⎜ ⎟ ⎝ ⎠ but from BC 2) trans inc reflo o o E E E= + ∴ ( ) ( ) 1 cos cosinc refl inc refl trans trans trans o o o o o inc k iE E E E n k θ κ θ ⎛ ⎞+ − = +⎜ ⎟ ⎝ ⎠ Skipping the details of the algebra, but using: ( ) ( ) ( )( ) ( )2 11 cos cos trans trans trans o trans k i n nn k ω θ κ ω ω ω θ ω + = =a It can be shown that: TE Polarization: ( ) ( ) cos cos cos cos refl inc o inc trans o inc transTE E E θ ω θ θ ω θ ⎛ ⎞ − ⎜ ⎟ = ⎜ ⎟ +⎝ ⎠ a a Similarly, it can be shown that: TM Polarization: ( ) ( ) cos cos cos cos refl inc o inc trans o inc transTM E E ω θ θ ω θ θ ⎛ ⎞ − + ⎜ ⎟ = ⎜ ⎟ +⎝ ⎠ a a Reflectance / Reflection Coefficient: 22 refl refl inc inc o o o o E E R E E ⎛ ⎞ = =⎜ ⎟⎜ ⎟ ⎝ ⎠ n.b. these have the identical functional forms of those the lossless dielectric case! UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 10 For even higher frequencies, but Pω ω , but where 1 jω ω of {all of} the bound/valence band resonances in the metal, the complex electric permittivity is given approximately by: ( ) 2 1 PTot o ωε ω ε ω ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ for oω γ , 1 jω ω of valence band resonances, but Pω ω . Visible light penetrates only a very short distance, ( ) ( )1sc vis vis Pcδ ω κ ω ω= into the metal and is almost entirely reflected. When the frequency of the incident EM wave is increased still further, into the UV and x-ray region then Pω ω≥ and the metal suddenly becomes transparent – the transmittance T increases from zero and the reflectance 1R T= − therefore decreases. A Simplified Model of EM Wave Propagation in the Earth’s Ionosphere and Magnetosphere Propagation of EM waves in the earth’s ionosphere is very similar to that in a tenuous plasma, however, the earth’s weak DC magnetic dipole field: 40.3 Gauss 0.3 10 Tesla 30 Tesla earthB μ −≈ = × = at the earth’s surface significantly changes the nature of EM wave propagation in the earth’s ionosphere, and thus cannot be neglected in the theory formalism. Consider a tenuous electronic (i.e. e− -only) plasma of uniform number density with a strong, static and uniform magnetic field oB B= with monochromatic plane EM waves propagating in the direction parallel to ˆoB B z= + . If the displacement amplitude r of the electronic motion is small and damping/collisions are neglected, then the approximate equation of motion is given by the following inhomogeneous 2nd order differential equation: ( ) ( ) ( ), , i te om r r t eB r r t eE r e ω−− × = − Note that we can safely neglect the influence of the magnetic Lorentz force term ev B− × acting on the electrons associated with the B -field of the EM wave, as long as EM oB B . We specifically/deliberately consider here circularly polarized monochromatic plane EM waves propagating in the ẑ+ direction ( )oB B= , which in complex notation can be written succinctly as: LCP ( ) ( ) ( )1 2ˆ ˆ, ,E r t i E r t= ∈ ± ∈ where the polarization vectors are e.g 1ˆ x̂∈ = and 2ˆ ŷ∈ = RCP UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 11 If the monochromatic plane EM wave’s polarization vectors are 1ˆ x̂∈ = and 2ˆ ŷ∈ = and ˆoB B z= then we see that ( )1̂ ˆB xε⊥ = and ( )2ˆ ˆB yε⊥ = . The magnetic Lorentz force term ( ) ( )( )ˆ, ,o oeB r r t eB z r r t− × = − × can then only have components in the x-y plane - i.e. it can only have components along the ˆ ˆx y− or 1 2ˆ ˆ∈ −∈ axes. A steady-state solution to the above 2nd order inhomogeneous differential equation for the electron’s displacement amplitude ( )er r at the space point r is: ( ) ( ) ( )e e B er r E r m ω ω ω = ∓ i.e. ( ) ( ) ( ) ( ) , i t i te e e B er r t r r e E r e m ω ω ω ω ω − −= = ∓ where B o eeB mω ≡ = electron precession frequency spiraling around the magnetic field lines and the ∓ sign depends on the handedness of the circular polarization {TBD, momentarily}. We can understand this relation better in the rest frame of electrons precessing with frequency Bω about the direction of ˆoB B z= (= direction of propagation of the EM wave) – the static B - field is eliminated – it is replaced by a rotating electric field of effective frequency ( )Bω ω∓ , where again the ∓ sign depends on the handedness of the circular polarization. The harmonic oscillation of each electron’s displacement ( ) ( ), i te er r t r r e ω−= also constitutes an oscillating electric dipole moment ( ) ( ) ( ), , i te ep r t er r t er r e ω−= = , and thus results in a corresponding macroscopic electric polarization ( ),r tΡ (= electric dipole moment/unit volume), ( ) ( ), ,er t n p r tΡ = where en = electron # density and corresponding ( ) ( ) ( ), ,o er t E r tε χ ωΡ = and thus has a corresponding macroscopic electric permittivity ( ) ( )( )1o eε ω ε χ ω= + . For circularly-polarized monochromatic plane EM waves propagating ˆoB B z= the macroscopic electric permittivity is: ( ) ( ) 2 1 Po B ωε ω ε ω ω ω ± ⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎝ ⎠∓ where: 2 2 e P o e n e m ω ε ⎛ ⎞ ≡ ⎜ ⎟ ⎝ ⎠ and: oB e eB m ω = where the upper sign (−) in the denominator is for a LCP EM wave, the lower sign (+) in the denominator is for a RCP EM wave. For circularly-polarized monochromatic plane EM waves propagating anti- ˆoB B z= the macroscopic electric permittivity is: ( ) ( ) 2 1 Po B ωε ω ε ω ω ω ± ⎛ ⎞= −⎜ ⎟⎜ ⎟±⎝ ⎠ where the upper sign (+) in the denominator is for a LCP EM wave, the lower sign (−) in the denominator is for a RCP EM wave. UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 12 ⇒ LCP and RCP monochromatic plane EM waves propagate differently in a tenuous electronic plasma, depending on whether the EM wave propagation direction is || to (or anti-||) to B . ⇒ The earth’s ionosphere is bi-refringent !!! If the direction of EM wave propagation not perfectly || to (or anti-||) to B , then simply replace cosB Bω ω→ Θ in the above formulae, where Θ ≡ opening angle between propagation wavevector and k B , i.e. ( )ˆˆ ˆ coso o ok B k B z kB k z kB= = = Θi i i ⇒ A tenuous electronic plasma is also anisotropic !!! A typical maximum number density of free electrons in the tenuous electronic plasma of the earth’s ionosphere is 10 12~ 10 10en − electrons/m 3. ⇒ Corresponds to a plasma frequency of 2 6 76 10 6 10p e o en e mω ε= × − × (radians/sec). ⇒ The precession frequency of electrons in this plasma, in the earth’s magnetic field is: ( ) 65.3 10B o eeB mω = × (radians/sec) for 30 Teslao earthB B μ= . ( ) ( ) 2 : 1 Po B k B ωε ω ε ω ω ω ± ⎛ ⎞ = −⎜ ⎟⎜ ⎟ ⎝ ⎠∓ ( ) ( ) 2 anti- : 1 Po B k B ωε ω ε ω ω ω ± ⎛ ⎞ = −⎜ ⎟⎜ ⎟±⎝ ⎠ UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 15 Finally, we consider the complex index of refraction ( ) ( ) ( )n n iω ω η ω= + or equivalently, the complex wave number, ( ) ( ) ( )k k iω ω κ ω= + of pure water (H2O): ( ) ( )k nc ωω ω⎛ ⎞= ⎜ ⎟ ⎝ ⎠ The top graph in the figure below shows ( ) .n f vs f , the bottom graph shows the absorption coefficient, ( )2 2 vs. , and f E eV c γ ωα κ η λ⎛ ⎞≡ = ⎜ ⎟ ⎝ ⎠ . Note that both plots are log-log plots!!! 191 1.6 10eV J− = × UIUC Physics 436 EM Fields & Sources II Spring Semester, 2008 Lect. Notes 8.5 Prof. Steven Errede © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2008. All Rights Reserved. 16 Note the following aspects of the above plots for pure H2O: • At low frequencies ( ) ( ) ( ) ( )( )29 81!!!o en f n f f K fε ε = = arises from partial orientation of the permanent electric dipole moment p of the H2O molecule (Langevin equation) – partial orientation is due to finite-temperature thermal energy density fluctuations…. • The ( ) vs. n f f curve falls smoothly through the infrared region – ∃ some “glitches” in ( ) ( ) and n f fη due to molecular vibrational excitations/resonances in infrared region!! • ∃ more resonances in the UV region – due to excitations in the oxygen atom • The absorption coefficient α is very small at low frequencies, but starts to rise steeply at 810f Hz . At ( )12 4 110 ~ far infrared , ~ 10 m 100scf Hz mα δ μ− ⇒ in H2O!!! ⇒ In the microwave region, ∃ strong absorption by H2O → can use for microwave ovens!!! ⇒ Strong absorption by H2O limited the trend of RADAR { During WWII} of going to shorter and shorter wavelengths, to achieve better spatial resolution . . . • In the infrared region, the absorption coefficient for H2O is very large, due to vibrational resonances of the H2O molecule, 4 110 mα − . • In the visible light region, there are no resonances of the H2O molecule, so the absorption coefficient α drops by ~ 7-8 orders of magnitude {!!!} Thus in the visible light region H2O/water is transparent/invisible. • However, getting into the UV region, ∃ oxygen atom resonances (due to inner L, K-shell electrons), thus α rises again dramatically, even higher, 6 110 mα − in the UV region. ⇒ ∃an absorption window in the visible light region: 144 8 10 Hz− × - not very wide!!! red light blue/violet light 750 nmRλ = 375 nmBVλ = ⇒ The H2O absorption window is of fundamental importance to the evolution of life on earth Life started off in the water/ocean, aquatic critter vision/sight developed in that environment and specifically in thre H2O absorption window, where significant amounts of EM energy are present {thanks to the sun!} to be of use/benefit for survival… ⇒ The co-incidence of the H2O absorption window and our (and other creature’s) ability today to see in the visible light region of the EM spectrum is not a mere coincidence! ⇒ Green grass/plants at the center of visible light absorption window! Because green = reflected light, plants have absorption in both the red and blue/violet regions. ⇒ On either side of the H2O absorption window there is not much/very little infrared or UV radiation in water after ~ few ~ 100IRsc mδ μ ~ 1 UV sc mδ μ - because strongly attenuated !!!