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Reflection refraction

The document discusses the principles of reflection and refraction in optics, detailing historical contributions from notable figures such as Claudius Ptolemy, Pierre de Fermat, and Christiaan Huygens. It outlines key laws, including Huygens' principle and Fermat's principle, alongside the development of these concepts using both wave and electromagnetic theories. The document emphasizes the mathematical formulations related to optical paths, phase-matching conditions, and the relationships between angles and refractive indices.

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1 REFLECTION & REFRACTION SOLO HERMELIN Updated: 4.11.06http://www.solohermelin.com
2 SOLO TABLE OF CONTENT REFLECTION & REFRACTION History of Reflection and Refraction Huygens Principle Reflections Laws Development Using Huygens Principle Refractions Laws Development Using Huygens Principle Fermat’s Principle Reflections Laws Development Using Fermat’s Principle Refractions Laws Development Using Fermat’s Principle Reflections and Refractions Laws Development Using the Electromagnetic Approach Monochromatic Wave Equations Phase-Matching Conditions Fresnel Conditions Energy Reflected and Refracted for Normal Polarization Malus-Dupin Theorem Stokes Treatment of Reflection and Refraction References
3 REFLECTION & REFRACTIONSOLO Introduction
4 REFLECTION & REFRACTIONSOLO History of Reflection & Refraction 100-170A.D. Claudius Ptolemey Alexandria “Optics” 130 A.D. Tabulated angle of incidence and refraction for several media. c. 300 B.C. Euclid Greece “Optica” 280 B.C. Rectilinear propagation of Light. Law of Reflection. Light originate in the eye, illuminates the object seen, and then returns to the eye.
5 REFLECTION & REFRACTIONSOLO History of Reflection & Refraction Willebrord van Roijen Snell 1580-1626 Professor at Leyden, experimentally discovered the law of refraction in 1621 René Descartes 1596-1650 Was the first to publish the law of refraction in terms of sinuses in “La Dioptrique” in 1637. Descartes assumed that the component of velocity of light parallel to the interface was unaffected, obtaining ti vv θθ sinsin 21 = from which 1 2 sin sin v v t i = θ θ correct 1 2 sin sin n n t i = θ θ Descartes deduced wrong
6 REFLECTION & REFRACTIONSOLO History of Reflection & Refraction Pierre de Fermat 1601-1665 Principle of Fermat (1657) of the extremality of (usually a minimum) optical path enables the derivation of reflection and refraction laws. ∫ 2 1 P P dsn Christiaan Huygens 1629-1695 In a communication to the Académie des Science in 1678 reported his wave theory (published in his “Traité de Lumière” in 1690). He considered that light is transmitted through an all-pervading aether that is made up of small elastic particles, each of each can act as a secondary source of wavelets. On this basis Huygens explained many of the known properties of light, including the double refraction in calcite. Augustin Jean Fresnel 1788-1827 Presented the laws which enable the calculation of the intensity and polarization of reflected and refracted light in 1823.
7 REFLECTION & REFRACTIONSOLO Huygens Principle Christiaan Huygens 1629-1695 Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space. “We have still to consider, in studying the spreading of these waves, that each particle of matter in which a wave proceeds not only communicates its motion to the next particle to it, which is on the straight line drawn from the luminous point, but it also necessarily gives a motion to all the other which touch it and which oppose its motion. The result is that around each particle there arises a wave of which this particle is a center.” Huygens visualized the propagtion of light in terms of mechanical vibration of an elastic medium (ether).
8 REFLECTION & REFRACTIONSOLO Reflection Laws Development Using Huygens Principle Suppose a planar incident wave AB is moving toward the boundary AC between two media. The velocity of light in the first media is v1. The incident rays are reflected at the boundary AB. At the time the incident ray passing through B is reaching the boundary at C, the reflected ray at A will reach D and the ray passing through F will be reflected at G and reaches H. According to Huygens Principle a reflected wavefront CHD, normal to the reflected rays AD, GH is formed and CBGHFGAD =+= ADCABC ∆=∆ DCABAC ri ∠=∠= θθ &From the geometry DCABAC ∠=∠ ri θθ =
9 REFLECTION & REFRACTIONSOLO Refraction Laws Development Using Huygens Principle Suppose a planar incident wave AB is moving toward the boundary AC between two media. The velocity of light in the first media is v1. The incident rays are refracted at the boundary AB. At the time the incident ray passing through B is reaching the boundary at C, the refracted ray at A will reach E and the ray passing through F will be refracted at G and reaches H. According to Huygens Principle a reflected wavefront CH’E, normal to the refracted rays AD, GH’ is formed and tvCBtvAE 12 === ECABAC ti ∠=∠= θθ &From the geometry ( ) ( ) ACAEECAACBCBAC /sin&/sin =∠=∠ 2 1 sin sin v v EA BC t i == θ θ
10 SOLO Fermat’s Principle (1657) The Principle of Fermat (principle of the shortest optical path) asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length). ∫ 2 1 P P dsn An other form of the Fermat’s Principle is: Principle of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum). REFLECTION & REFRACTION
11 SOLO 1. The optical path is reflected at the boundary between two regions ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   In this case we have and21 nn = ( ) ( ) ( ) 0ˆˆ 21 21 =⋅−=⋅        − rdssrd sd rd sd rd rayray   We can write the previous equation as: i.e. is normal to , i.e. to the boundary where the reflection occurs. 21 ˆˆ ss − rd  ( ) 0ˆˆˆ 2121 =−×− ssn REFLECTION & REFRACTION Reflection Laws Development Using Fermat Principle ri θθ = Incident ray and Reflected ray are in the same plane normal to the boundary. This is equivalent with: &
12 SOLO 2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1) ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd  ( ) ( ) sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21  == Therefore is normal to . 2211 ˆˆ snsn − rd  Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn − 21 ˆ −n ( ) 0ˆˆˆ 221121 =−×− snsnn We recovered the Snell’s Law from Geometrical Optics REFLECTION & REFRACTION Refraction Laws Development Using Fermat Principle ti nn θθ sinsin 21 = Incident ray and Refracted ray are in the same plane normal to the boundary. &
13 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations we haveUsing: 1ˆˆ&ˆˆ 0 =⋅== kkknkkk εµω         =⋅∇ =⋅∇ −=×∇ =×∇ 0 0 H E HjE EjH ωµ ωε            =⋅ =⋅ =× −=× 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hk Ek HEk EHk ε µ µ ε        =⋅− =⋅− −=×− =×− ⇒ ⋅− ⋅− ⋅−⋅− ⋅−⋅− −=∇ ⋅−⋅− 0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj           ωµ ωε ( ) ∗∗ ⋅==⋅==+= HHwEEwwcn k wwcn k S meme 22 & 2 ˆ 2 ˆ µε Time Average Poynting Vector of the Planar Wave Reflections and Refractions Laws Development Using the Electromagnetic Approach
14 SOLO REFLECTION & REFRACTION Consider an incident monochromatic planar wave ( ) ( ) c n k eEkH eEE iiii rktj iii rktj ii ii ii 1 00 11 0011 0 0 ω εµ εµ εµωεµω µ ε ω ω === ×= = ⋅− ⋅−     The monochromatic planar reflected wave from the boundary is ( ) ( ) 1 1 1 1 0 0 & n c v vc n k eEkH eEE r rr rktj rrr rktj rr rr rr === ×= = ⋅− ⋅− ω ω µ ε ω ω     The monochromatic planar refracted wave from the boundary is ( ) ( ) 2 2 2 2 0 0 & n c v vc n k eEkH eEE t tt rktj ttt rktj tt tt tt === ×= = ⋅− ⋅− ω ω µ ε ω ω     Reflections and Refractions Laws Development Using the Electromagnetic Approach
15 SOLO REFLECTION & REFRACTION The Boundary Conditions at z=0 must be satisfied at all points on the plane at all times, implies that the spatial and time variations of This implies that Phase-Matching Conditions ( ) ( ) ( ) yxteEeEeE z rktj t z rktj r z rktj i ttrrii ,,,, 0 0 0 0 0 0 ∀ = ⋅− = ⋅− = ⋅−   ωωω ( ) ( ) ( ) yxtrktrktrkt z tt z rr z ii ,, 000 ∀⋅−=⋅−=⋅− ===  ωωω ttri ∀=== ωωωω ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  must be the same Reflections and Refractions Laws Development Using the Electromagnetic Approach
16 SOLO REFLECTION & REFRACTION tri nnn θθθ sinsinsin 211 == Phase-Matching Conditions ( ) ( )      −+= ++= zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1 θαθααω θαθααω   ( ) ( )         +=⋅ +=⋅ =⋅ = = = yyx c n rk yx c n rk y c n rk ttt z t irr z r i z i ˆsinsincos sinsincos sin 2 0 1 0 1 0 θααω θααω θω    ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  2 π αα == tr ttri ∀=== ωωωω x∀ y∀ Coplanar Snell’s Law ( )     ++= −= zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1   θθω Given: Let find: Reflections and Refractions Laws Development Using the Electromagnetic Approach
17 SOLO REFLECTION & REFRACTION Second way of writing phase-matching equations ri θθ = 11 22 2 1 1 2 sin sin εµ εµ θ θ === v v n n t iRefraction Law Reflection Law Phase-Matching Conditions ( )     ++= −= zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1   θθω ( ) ( )      −+= ++= zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1 θαθααω θαθααω   ( ) ( )[ ] ( ) ( )[ ]      −+−=−× −+−=−× ynnyn c kkz ynnyn c kkz ittrti irrrri ˆsinsinsinˆcosˆ ˆsinsinsinˆcosˆ 122 111 θθαα ω θθαα ω   ttri ∀=== ωωωω We can see that ( ) ( )         === =−×=−× ωωωω tri tiri kkzkkz 0ˆˆ            === == == ωωωω θθθ παα tri tri tr nnn sinsinsin 2/ 211 Reflections and Refractions Laws Development Using the Electromagnetic Approach
18 SOLO REFLECTION & REFRACTION ri θθ = 11 22 2 1 1 2 sin sin εµ εµ θ θ === v v n n t iRefraction Law Reflection Law Phase-Matching Conditions (Summary) ttri ∀=== ωωωω ( ) ( )         === =−×=−× ωωωω tri tiri kkzkkz 0ˆˆ            === == == ωωωω θθθ παα tri tri tr nnn sinsinsin 2/ 211 ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  ( ) ( ) ( ) yxtrktrktrkt z tt z rr z ii ,, 000 ∀⋅−=⋅−=⋅− ===  ωωω Vector Notation Scalar Notation Reflections and Refractions Laws Development Using the Electromagnetic Approach
19 SOLO REFLECTION & REFRACTION ( ) 0ˆ 2121  =−×− EEn ( ) 0ˆ 2121  =−×− HHn ( ) 0ˆ 2121 =−⋅− DDn  ( ) 0ˆ 2121 =−⋅− BBn  Boundary conditions for a source-less boundary ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ In our case ( ) ttrrii tri EkHEkEkH EEEEE   ×=×+×= =+= ˆ&ˆˆ & 2 2 2 1 1 1 21 µ ε µ ε Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations Boundary conditions
20 SOLO REFLECTION & REFRACTION ( ) 0ˆ 00021  =−+×− tri EEEn 0 111 ˆ 0 2 0 1 0 1 21  =      ×−×+××− ttrrii EkEkEkn µµµ ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆ 00021 =×−×+×⋅− ttrrii EkEkEkn  Using ,ˆ,ˆ,ˆˆ 221111 1 ttrriii kkkkkk c n k εµωεµωεµωω ====  ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations
21 SOLO REFLECTION & REFRACTION ( ) ( ) ( ) ( ) ( ) ( ) ( ) xEknEEknxEknEEkn xEknEEnkEkn tttttirrrrri iiiiiiii ir i ˆcosˆˆˆˆ&ˆcosˆˆˆˆ ˆcosˆˆˆˆˆˆ 0 cos 2100210 cos 210021 0 cos 210 0 021021 θθ θ θθ θ =⋅−=××−=⋅−=×× =⋅−⋅=×× −− − −− −−−         ( ) ( ) ( ) ( ) ( ) ( ) tttttt rrrrrriiiiii EEzzEkn EEzzEknEEzzEkn θθ θθθθ sinsinˆˆˆˆ sinsinˆˆˆˆ&sinsinˆˆˆˆ 00021 0002100021 =−⋅−=×⋅ =−⋅−=×⋅=−⋅−=×⋅ − −−   zn ˆˆ 21 −=− ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ 1 2 3 4 zykzykzyk tttiiriii ˆcosˆsin&ˆcosˆsin&ˆcosˆsin θθθθθθ −=+=−=  Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach
22 SOLO REFLECTION & REFRACTION ( ) 0coscoscos 0 2 2 00 1 1 =−− ttirii EEE θ µ ε θθ µ ε 1 0000  =−+ tri EEE2 ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ 1 2 3 4 ( ) 0 sinsin sinsin 0sinsinsin 000 2211 0220011 =−+⇒ = = =−+ tri ti ri ttrrii EEE EEE θεµθεµ θθ θεµθθεµ4 Identical to 2 3 00 = Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach
23 SOLO REFLECTION & REFRACTION ( ) 0cos 1 cos 1 0 00 22 2 00 00 11 1 21 =−− tt n iri n EEE θ εµ εµ µ θ εµ εµ µ  1 0000 =−+ tri EEE2 From and ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =      = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 2 2 1 1 1 1 0 0 + =      = ⊥ ⊥ For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i = θ θ ( ) ( )ti ti i r E E r θθ θθ + − −=      = ⊥ ⊥ sin sin 0 0 ( )ti it i t E E t θθ θθ + =      = ⊥ ⊥ sin cossin2 0 0 1 2 Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Reflections and Refractions Laws Development Using the Electromagnetic Approach
24 SOLO REFLECTION & REFRACTION ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =      = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 2 2 1 1 1 1 0 0 + =      = ⊥ ⊥ For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i = θ θ ( ) ( )ti ti i r E E r θθ θθµµ + − −=      = = ⊥ ⊥ sin sin21 0 0 ( )ti it i t E E t θθ θθµµ + =      = = ⊥ ⊥ sin cossin221 0 0 Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations
25 SOLO REFLECTION & REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ( ) ( ) ( ) ( ) ( ) yEEknyEEkn yEknEEnkEkn ttirri iiiiiii ii ˆˆˆ&ˆˆˆ ˆˆˆˆˆˆˆ 00210021 0 cos 210 sin 021021 −=××−=×× −=⋅−⋅=×× −− − − −−      θθ ( ) ( ) ( ) 0ˆˆˆˆˆˆ 021021021 =×⋅=×⋅=×⋅ −−− ttrrii EknEknEkn  zn ˆˆ 21 −=− zykzykzyk tttiiriii ˆcosˆsin&ˆcosˆsin&ˆcosˆsin θθθθθθ −=+=−=  xEEnxEEnxEEn tttirriii ˆcosˆ&ˆcosˆ&ˆcosˆ 002100210021 θθθ =×−=×=× −−−  tttirriii EEnEEnEEn θθθ sinˆ&sinˆ&sinˆ 002100210021 −=⋅−=⋅−=⋅ −−−  Reflections and Refractions Laws Development Using the Electromagnetic Approach
26 SOLO REFLECTION & REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ 1 2 3 4 ( ) 0sinsin 02001 =−+ ttiri EEE θεθε3 ( )[ ] 0ˆcoscos 000 =−− xEEE ttiri θθ2 ( ) ( ) 0 11 0ˆ 0 00 22 2 00 00 11 1 0 2 2 00 1 1 21 =−+=      −+ t n ri n tri EEEoryEEE   µε µε µµε µε µµ ε µ ε 1 4 00 = Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach
27 SOLO REFLECTION & REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ( ) ( ) 0sinsin sin sin / / sinsin 0ˆ 02001 2 1 22 11 2211 0 2 2 00 1 1 =−+ =⇒= =      −+ ttiri t i ti tri EEE yEEE θεθε θε θε µε µε θεµθεµ µ ε µ ε  1 Identical to 3 We have two independent equations ( ) 0coscos 000 =−− ttiri EEE θθ2 ( ) 00 2 2 00 1 1 =−+ tri E n EE n µµ 1 ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =      = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =      = Reflections and Refractions Laws Development Using the Electromagnetic Approach
28 SOLO REFLECTION & REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =      = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =      = For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i = θ θ ( ) ( )ti ti i r E E r θθ θθµµ + − =      = = tan tan21 ||0 0 || ( ) ( )titi it i t E E t θθθθ θθµµ −+ =      = = cossin cossin221 ||0 0 || Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations
29 SOLO REFLECTION & REFRACTION ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =        = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =        = ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =        = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 2 2 1 1 1 1 0 0 + =        = ⊥ ⊥ The equations of reflection and refraction ratio are called Fresnel Equations, that first developed them in a slightly less general form in 1823, using the elastic theory of light. Augustin Jean Fresnel 1788-1827 The use of electromagnetic approach to prove those relations, as described above, is due to H.A. Lorentz (1875) Reflections and Refractions Laws Development Using the Electromagnetic Approach Hendrik Antoon Lorentz 1853-1928
30 SOLO REFLECTION & REFRACTION Discussion of Fresnel Equations it n ni θθ θ 2 1 0→ = 0→iθ [ ] [ ] ti ti nn nn rr ii θθ θθ θθ + − = + − =−= =⊥= 21 12 00|| [ ] [ ] ti t nn n tt ii θθ θ θθ + = + == =⊥= 22 21 1 00|| 1 2 sin sin n n t i = θ θ Snell’s law Reflections and Refractions Laws Development Using the Electromagnetic Approach n2 / n1 =1.5 n2 / n1 =1/1.5 0cos90 =→→ ii θθ  [ ] [ ] 19090|| −=−= =⊥=  ii rr θθ [ ] [ ] 09090|| == =⊥=  ii tt θθ ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
31 SOLO REFLECTION & REFRACTION Discussion of Fresnel Equations 1 2 sin sin n n t i = θ θ Snell’s law Reflections and Refractions Laws Development Using the Electromagnetic Approach David Brewster 1781-1868 David Brewster , “On the laws which regulate the polarization of light by reflection from transparent bodies”, Philos. Trans. Roy. Soc., London 105, 125-130, 158-159 1815). In contrast r|| changes sign (for both n2> n1 and n2<n1) when tan(θi+θt)=∞ → θi + θt=π/2 The incident angle, θi, when this occurs is denoted θp and is referred as polarization or Brewster angle (after David Brewster who found it in 1815). n2 / n1 =1.5 n2 / n1 =1/1.5 For n2>n1 we have from Snell’s law θi > θt ,therefore r┴ is negative for all values of θi. it LawsSnell i nnn it θθθ θθ cossinsin 2 90 2 ' 1 −= ==  pi θθ = → 1 2 tan n n p =θ ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
32 SOLO REFLECTION & REFRACTION Discussion of Fresnel Equations (continue) n2 / n1 =1.5 Reflections and Refractions Laws Development Using the Electromagnetic Approach Brewster Angle n1 < n2
33 SOLO REFLECTION & REFRACTION Discussion of Fresnel Equations (continue) n2 / n1 =1/1.5 Reflections and Refractions Laws Development Using the Electromagnetic Approach 1 2 sin sin n n t i = θ θ Snell’s law For n2<n1 we have from Snell’s law θi < θt therefore when θi increases, θt increases until it reaches 90° (no refraction and total reflection ). The incident angle when this occurs is denoted θic and is referred as the critical angle.       = − 1 21 sin n n icθCritical Angle Brewster Angle ( ) ( ) 1 2/sin 2/sin21 2/ 0 0 = + − −=        = = = ⊥ ⊥ πθ πθµµ πθ i i i r t E E r ( ) ( ) 1 2/tan 2/tan21 2/ ||0 0 || = + − =        = = = πθ πθµµ πθ i i i r t E E r n1 > n2
34 SOLO REFLECTION & REFRACTION n1 > n2 Total Reflection Reflections and Refractions Laws Development Using the Electromagnetic Approach For n1>n2 and θi > θic we have total reflection. 1sinsin 21 2 1 nn it ici n n > > >= θθ θθFrom Snell’s law therefore is no solution for θt , but we can use for 1sinsin1cos 2 2 2 12 −      −=−= itt n n i θθθ ( ) ( )2 21 2 2 21 2 21 21 0 0 /sincos /sincos coscos coscos21 nni nni nn nn E E r ii ii ti ti i r −− −+ = + − =        = = ⊥ ⊥ θθ θθ θθ θθµµ ( ) ( ) ( ) ( )2 21 22 21 2 21 22 21 12 12 ||0 0 || /sincos/ /sincos/ coscos coscos21 nninn nninn nn nn E E r ii ii ti ti i r −− −+ = + − =        = = θθ θθ θθ θθµµ We can see that ( )⊥⊥ = ϕjr exp ( )|||| exp ϕjr = ( ) i i nn θ θϕ cos /sin 2 tan 2 21 2 − =      ⊥ ( ) ( ) i i nn nn θ θϕ cos/ /sin 2 tan 2 21 2 21 2 || − =      n2 / n1 =1/1.5
35 SOLO REFLECTION & REFRACTION Phase Shifts ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21 n2 / n1 =1/1.5 n2 / n1 =1.5 On can see that and are always positive, therefore is no phase difference between incidence and refracted waves. ⊥t ||t On the other hand and can be either positive or negative depending of the sign of (θi-θt). ⊥r || r The phase change of the reflected wave, in the cases where refraction is possible, can be either π or 0, depending on whether the index n1 of the medium in which the wave originates is less or greater than n2 of the medium in which it travels.
36 SOLO REFLECTION & REFRACTION Phase Shifts ||i E  ⊥i E  ⊥rE  ik  rk  tk  Boundary 21ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21ˆ −n Boundary Plan of incidence iE  ||i E  ⊥i E  ||rE  ⊥r E  pi nn < pi θθ = p n i n p n in ||iE  ⊥i E  ⊥rE  ik  rk  tk  Boundary 21ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence iE  ||iE  ⊥i E  ||rE  ⊥r E  pi nn < pi θθ > pn in p n i n ||r E  rE  n2 / n1 =1.5 n1 < n2 The phase change of the reflected wave is: - π for , - 0 for 0 ≤ θi ≤ θp, and π for θi > θp for ⊥ r || r ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
37 SOLO REFLECTION & REFRACTION Phase Shifts n2 / n1 =1/1.5 n1 > n2 The phase change of the reflected wave is: -0 for 0 ≤ θi ≤ θc and changes from 0 to π for θi > θc for - π for 0 ≤ θi ≤ θp, π for θp>θi > θc for ⊥r || r ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
38 SOLO REFLECTION & REFRACTION Phase Shifts ||i E  ⊥iE  ⊥rE  ik  rk  tk  Boundary 21 ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence i E  ||i E  ⊥iE  ||r E  ⊥rE  pi nn < pi θθ = pn in pn i n ||i E  ⊥iE  ⊥rE  ik  rk  tk  Boundary 21 ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence iE  ||iE  ⊥iE  ||r E  ⊥rE  pi nn < pi θθ > p n in pn i n ||r E  rE  n1 > n2 n1 < n2
39 SOLO REFLECTION & REFRACTION Energy Reflected and Refracted for Normal Polarization 2 01 12 ˆ ⊥ ⊥ = i i i E n ck S ε  Time Average Poynting Vectors (Irradiances) of the Planar waves are 2 01 12 ˆ ⊥ ⊥ = r r r E n ck S ε  2 02 22 ˆ ⊥ ⊥ = t t t E n ck S ε  2 2 2 1 1 2 2 2 1 1 2 0 2 0 coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 2 2 1 1 21 12 2 1 1 2 021 2 012 coscos cos cos cos2 cos cos ˆ ˆ       +       == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti i t i ii tt i t nn n nn En En zS zS T θ µ θ µ θε θε θ µ θε θε     titi cn cn i t i nn n nn n nn θθ µµ θθ µεµ εµ µθε θε θ µ coscos4coscos 1 4 cos cos cos2 21 21 2211 122 1 2 1 21 12 2 1 1 22 1 22 2 ==      2 2 2 1 1 2 2 1 1 2 021 2 012 coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance Reflections and Refractions Laws Development Using the Electromagnetic Approach
40 SOLO REFLECTION & REFRACTION 2 2 2 1 1 2 2 2 1 1 2 0 2 0 coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 2 2 1 1 2 2 1 1 2 021 2 012 coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance We can see that 1=+ ⊥⊥ TR Energy Reflected and Refracted for Normal Polarization Reflections and Refractions Laws Development Using the Electromagnetic Approach
41 SOLO REFLECTION & REFRACTION Energy Reflected and Refracted for Parallel Polarization 2 ||01 1 || 2 ˆ i i i E n ck S ε=  Time Average Poynting vector of the Planar waves are 2 ||01 1 || 2 ˆ r r r E n ck S ε=  2 ||02 2 || 2 ˆ t t t E n ck S ε=  2 1 1 2 2 2 1 1 2 2 2 ||0 2 ||0 || || || coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 1 1 2 2 21 12 2 1 1 2 ||021 2 ||012 || || || coscos cos cos cos2 cos cos ˆ ˆ       +       == ⋅ ⋅ = ti i t i ii tt i t nn n nn En En zS zS T θ µ θ µ θε θε θ µ θε θε     titi cn cn i t i nn n nn n nn θθ µµ θθ µεµ εµ µθε θε θ µ coscos4coscos 1 4 cos cos cos2 21 21 2211 122 1 2 1 21 12 2 1 1 22 1 22 2 ==      2 1 1 2 2 2 2 1 1 2 ||021 2 ||012 || || || coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance Reflections and Refractions Laws Development Using the Electromagnetic Approach
42 SOLO REFLECTION & REFRACTION 2 1 1 2 2 2 1 1 2 2 2 ||0 2 ||0 || || || coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   Reflectance Transmittance 2 1 1 2 2 2 2 1 1 2 ||021 2 ||012 || || || coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   We can see that 1|||| =+TR Average Poynting vector of the Planar waves are Reflections and Refractions Laws Development Using the Electromagnetic Approach
43 SOLO REFLECTION & REFRACTION ( ) ( )ti ti i r zS zS R θθ θθµµ + − = ⋅ ⋅ = = 2 2 || || || tan tan ˆ ˆ 21   ( ) ( )titi ti i t zS zS T θθθθ θθµµ −+ = ⋅ ⋅ = = 22 || || || cossin 2sin2sin ˆ ˆ 21   1|||| =+TR Reflections and Refractions Laws Development Using the Electromagnetic Approach ( ) ( )ti ti i r zS zS R θθ θθµµ + − = ⋅ ⋅ = = ⊥ ⊥ ⊥ 2 2 sin sin ˆ ˆ 21   ( )ti ti i t zS zS T θθ θθµµ + = ⋅ ⋅ = = ⊥ ⊥ ⊥ 2 sin 2sin2sin ˆ ˆ 21   1=+ ⊥⊥ TR Summary
44 SOLO REFLECTION & REFRACTION Reflections and Refractions Laws Development Using the Electromagnetic Approach || R ⊥ R || R ⊥ R
45 Malus-Dupin TheoremSOLO Étienne Louis Malus 1775-1812 A surface passing through the end points of rays which have traveled equal optical path lengths from a point object is called an optical wavefront. 1808 1812 If a group of ray is such that we can find a surface that is orthogonal to each and every one of them (this surface is the wavefront), they are said to form a normal congruence. The Malus-Dupin Theorem (introduced in 1808 by Malus and modified in 1812 by Dupin) states that: “The set of rays that are orthogonal to a wavefront remain normal to a wavefront after any number of refraction or reflections.” Charles Dupin 1784-1873 Using Fermat principle [ ] [ ]'' BQBAVApathoptical == [ ] [ ] ( )2 '' εOAVAAQA += VQ=ε is a small quantity [ ] [ ] ( )2 '' εOBQBAQA += Since ray BQ is normal to wave W at B [ ] [ ] ( )2 εOBQAQ += [ ] [ ] ( )2 '' εOQBQA += ray BQB’ is normal to wave W’ at B’ Proof for Refraction: n 'n P Q VA P’ A' B B' Wavefront from P Wavefront to P'
46 Stokes Treatment of Reflection and RefractionSOLO An other treatment of reflection and refraction was given by Sir George Stokes. Suppose we have an incident wave of amplitude E0i reaching the boundary of two media (where n1 = ni and n2 = nt) at an angle θ1. The amplitudes of the reflected and transmitted (refracted) waves are, E0i·r and E0i·t, respectively (see Fig. a). Here r (θ1) and t (θ2) are the reflection and transmission coefficients. According to Fermat’s Principle the situation where the rays direction is reversed (see Fig. b) is also permissible. Therefore we have two incident rays E0i·r in media with refraction index n1 and E0i·t in media with refraction index n2. E0i·r is reflected, in media with refraction index n1, to obtain a wave with amplitude (E0i·r )·t and refracted, in media with refraction index n2, to obtain a wave with amplitude (E0i·r )·r (see Fig. c). E0i·t is reflected, in media with refraction index n2, to obtain a wave with amplitude (E0i·t )·r’ and refracted, in media with refraction index n1, to obtain a wave with amplitude (E0i·t )·t’ (see Fig. c). θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
47 Stokes Treatment of Reflection and RefractionSOLO To have Fig. c identical to Fig. b the following conditions must be satisfied: ( ) ( ) ( ) ( ) iii ErrEttE 0110120 ' =+ θθθθ ( ) ( ) ( ) ( ) 0' 220210 =+ θθθθ rtEtrE ii Hence: ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ rr rrtt −= =+ Stokes relations θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
48 Stokes Treatment of Reflection and RefractionSOLO ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ rr rrtt −= =+ Stokes relations Let check that Fresnel Equation do satisfy Stokes relations ( ) 2211 11 2 coscos cos221 θθ θ θ µµ nn n t + = = ⊥ 2112 11 || coscos cos221 θθ θµµ nn n t + = = ( ) 2211 2211 1 coscos coscos21 θθ θθ θ µµ nn nn r + − = = ⊥ ( ) 2112 2112 1|| coscos coscos21 θθ θθ θ µµ nn nn r + − = = ( ) 2211 22 1 coscos cos2 ' 21 θθ θ θ µµ nn n t + = = ⊥ ( ) 2211 1122 2 coscos coscos ' 21 θθ θθ θ µµ nn nn r + − = = ⊥ 1 ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ ⊥⊥ ⊥⊥⊥⊥ −= =+ rr rrtt We can see that: 2 2112 22 || coscos cos2 ' 21 θθ θµµ nn n t + = = ( ) 2112 1221 1|| coscos coscos ' 21 θθ θθ θ µµ nn nn r + − = = ( ) ( ) ( ) ( ) ( ) ( )1||2|| 1||1||1||2|| ' 1' θθ θθθθ rr rrtt −= =+ We can see that: θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
49 SOLO REFLECTION & REFRACTION Thinks to complete Photons and Laws of Reflection & Refraction (Hecht & Zajac pp. 93) http://physics.nad.ru/Physics/English/index.htm
50 ELECTROMAGNETICSSOLO References J.D. Jackson, “Classical Electrodynamics”, 3rd Ed., John Wiley & Sons, 1999 R. S. Elliott, “Electromagnetics”, McGraw-Hill, 1966 J.A. Stratton, “Electromagnetic Theory”, McGraw-Hill, 1941 W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison- Wesley, 1962 F.T. Ulaby, R.K. More, A.K. Fung, “Microwave Remote Sensors Active and Passive”, Addson-Wesley, 1981 A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”, John Wiley & Sons, 1988
51 SOLO References Foundation of Geometrical Optics [1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, 6th Ed., Pergamon Press, 1980, Ch. 3 and App. 1 [2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, pp. 692-694
52 SOLO References Foundation of Geometrical Optics [3] E.Hecht, A. Zajac, “Optics ”, 3th Ed., Addison Wesley Publishing Company, 1997, [4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd Ed., John Wiley & Sons, 1986
53 ELECTROMAGNETICSSOLO References 1.W.K.H. Panofsky & M. Phillips, “Classical Electricity and Magnetism”, 2.J.D. Jackson, “Classical Electrodynamics”, 3.R.S. Elliott, “Electromagnetics”, 4.A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,
54 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions ( ) ( ) ldtHtHhldtHldtHldH h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbHHnldnbHHldtHH e ˆˆˆˆˆˆ 21212121121 ⋅=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) ekHHn  =−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +=⋅ → S e C Sd t D JdlH    ( ) dlbkbdlh t D JSd t D J e h e S e ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e  
55 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 1) ( ) ( ) ldtEtEhldtEldtEldE h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbEEnldnbEEldtEE m ˆˆˆˆˆˆ 21212121121 ⋅−=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) mkEEn  −=−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +−=⋅ → S m C Sd t B JdlE    ( ) dlbkbdlh t B JSd t B J m h m S m ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      FARADAY’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m  
56 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 2) ( ) ( ) SdnDnDhSdnDSdnDSdD h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnDDSdnDD eσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) eDDn σ=−⋅− 2121 ˆ  ( ) dSdShdv e h e V e σρρ 0→ ==∫∫∫ GAUSS’ LAW - ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ∫∫∫∫∫ =• V e S dvSdD ρ 
57 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 3) ( ) ( ) SdnBnBhSdnBSdnBSdB h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnBBSdnBB mσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) mBBn σ=−⋅− 2121 ˆ  ( ) dSdShdv m h m V m σρρ 0→ ==∫∫∫ GAUSS’ LAW – MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ ∫∫∫∫∫ =• V m S dvSdB ρ 
58 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (summary) ( ) mkEEn  −=−×− 2121 ˆ FARADAY’S LAW ( ) ekHHn  =−×− 2121 ˆ AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e   [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m   ( ) eDDn σ=−⋅− 2121 ˆ  GAUSS’ LAW ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ( ) mBBn σ=−⋅− 2121 ˆ  GAUSS’ LAW MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ Fresnel Equations
January 4, 2015 59 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

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Reflection refraction

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    2 SOLO TABLE OF CONTENT REFLECTION& REFRACTION History of Reflection and Refraction Huygens Principle Reflections Laws Development Using Huygens Principle Refractions Laws Development Using Huygens Principle Fermat’s Principle Reflections Laws Development Using Fermat’s Principle Refractions Laws Development Using Fermat’s Principle Reflections and Refractions Laws Development Using the Electromagnetic Approach Monochromatic Wave Equations Phase-Matching Conditions Fresnel Conditions Energy Reflected and Refracted for Normal Polarization Malus-Dupin Theorem Stokes Treatment of Reflection and Refraction References
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    4 REFLECTION & REFRACTIONSOLO Historyof Reflection & Refraction 100-170A.D. Claudius Ptolemey Alexandria “Optics” 130 A.D. Tabulated angle of incidence and refraction for several media. c. 300 B.C. Euclid Greece “Optica” 280 B.C. Rectilinear propagation of Light. Law of Reflection. Light originate in the eye, illuminates the object seen, and then returns to the eye.
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    5 REFLECTION & REFRACTIONSOLO Historyof Reflection & Refraction Willebrord van Roijen Snell 1580-1626 Professor at Leyden, experimentally discovered the law of refraction in 1621 René Descartes 1596-1650 Was the first to publish the law of refraction in terms of sinuses in “La Dioptrique” in 1637. Descartes assumed that the component of velocity of light parallel to the interface was unaffected, obtaining ti vv θθ sinsin 21 = from which 1 2 sin sin v v t i = θ θ correct 1 2 sin sin n n t i = θ θ Descartes deduced wrong
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    6 REFLECTION & REFRACTIONSOLO Historyof Reflection & Refraction Pierre de Fermat 1601-1665 Principle of Fermat (1657) of the extremality of (usually a minimum) optical path enables the derivation of reflection and refraction laws. ∫ 2 1 P P dsn Christiaan Huygens 1629-1695 In a communication to the Académie des Science in 1678 reported his wave theory (published in his “Traité de Lumière” in 1690). He considered that light is transmitted through an all-pervading aether that is made up of small elastic particles, each of each can act as a secondary source of wavelets. On this basis Huygens explained many of the known properties of light, including the double refraction in calcite. Augustin Jean Fresnel 1788-1827 Presented the laws which enable the calculation of the intensity and polarization of reflected and refracted light in 1823.
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    7 REFLECTION & REFRACTIONSOLO HuygensPrinciple Christiaan Huygens 1629-1695 Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space. “We have still to consider, in studying the spreading of these waves, that each particle of matter in which a wave proceeds not only communicates its motion to the next particle to it, which is on the straight line drawn from the luminous point, but it also necessarily gives a motion to all the other which touch it and which oppose its motion. The result is that around each particle there arises a wave of which this particle is a center.” Huygens visualized the propagtion of light in terms of mechanical vibration of an elastic medium (ether).
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    8 REFLECTION & REFRACTIONSOLO ReflectionLaws Development Using Huygens Principle Suppose a planar incident wave AB is moving toward the boundary AC between two media. The velocity of light in the first media is v1. The incident rays are reflected at the boundary AB. At the time the incident ray passing through B is reaching the boundary at C, the reflected ray at A will reach D and the ray passing through F will be reflected at G and reaches H. According to Huygens Principle a reflected wavefront CHD, normal to the reflected rays AD, GH is formed and CBGHFGAD =+= ADCABC ∆=∆ DCABAC ri ∠=∠= θθ &From the geometry DCABAC ∠=∠ ri θθ =
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    9 REFLECTION & REFRACTIONSOLO RefractionLaws Development Using Huygens Principle Suppose a planar incident wave AB is moving toward the boundary AC between two media. The velocity of light in the first media is v1. The incident rays are refracted at the boundary AB. At the time the incident ray passing through B is reaching the boundary at C, the refracted ray at A will reach E and the ray passing through F will be refracted at G and reaches H. According to Huygens Principle a reflected wavefront CH’E, normal to the refracted rays AD, GH’ is formed and tvCBtvAE 12 === ECABAC ti ∠=∠= θθ &From the geometry ( ) ( ) ACAEECAACBCBAC /sin&/sin =∠=∠ 2 1 sin sin v v EA BC t i == θ θ
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    10 SOLO Fermat’s Principle (1657) ThePrinciple of Fermat (principle of the shortest optical path) asserts that the optical length of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length). ∫ 2 1 P P dsn An other form of the Fermat’s Principle is: Principle of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum). REFLECTION & REFRACTION
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    11 SOLO 1. The opticalpath is reflected at the boundary between two regions ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   In this case we have and21 nn = ( ) ( ) ( ) 0ˆˆ 21 21 =⋅−=⋅        − rdssrd sd rd sd rd rayray   We can write the previous equation as: i.e. is normal to , i.e. to the boundary where the reflection occurs. 21 ˆˆ ss − rd  ( ) 0ˆˆˆ 2121 =−×− ssn REFLECTION & REFRACTION Reflection Laws Development Using Fermat Principle ri θθ = Incident ray and Reflected ray are in the same plane normal to the boundary. This is equivalent with: &
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    12 SOLO 2. The opticalpath passes between two regions with different refractive indexes n1 to n2. (continue – 1) ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd  ( ) ( ) sd rd s sd rd s rayray 2 :ˆ, 1 :ˆ 21  == Therefore is normal to . 2211 ˆˆ snsn − rd  Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have rd  2211 ˆˆ snsn − 21 ˆ −n ( ) 0ˆˆˆ 221121 =−×− snsnn We recovered the Snell’s Law from Geometrical Optics REFLECTION & REFRACTION Refraction Laws Development Using Fermat Principle ti nn θθ sinsin 21 = Incident ray and Refracted ray are in the same plane normal to the boundary. &
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    13 ELECTROMAGNETICSSOLO To satisfy theMaxwell equations for a source free media we must have: Monochromatic Planar Wave Equations we haveUsing: 1ˆˆ&ˆˆ 0 =⋅== kkknkkk εµω         =⋅∇ =⋅∇ −=×∇ =×∇ 0 0 H E HjE EjH ωµ ωε            =⋅ =⋅ =× −=× 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hk Ek HEk EHk ε µ µ ε        =⋅− =⋅− −=×− =×− ⇒ ⋅− ⋅− ⋅−⋅− ⋅−⋅− −=∇ ⋅−⋅− 0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj           ωµ ωε ( ) ∗∗ ⋅==⋅==+= HHwEEwwcn k wwcn k S meme 22 & 2 ˆ 2 ˆ µε Time Average Poynting Vector of the Planar Wave Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    14 SOLO REFLECTION &REFRACTION Consider an incident monochromatic planar wave ( ) ( ) c n k eEkH eEE iiii rktj iii rktj ii ii ii 1 00 11 0011 0 0 ω εµ εµ εµωεµω µ ε ω ω === ×= = ⋅− ⋅−     The monochromatic planar reflected wave from the boundary is ( ) ( ) 1 1 1 1 0 0 & n c v vc n k eEkH eEE r rr rktj rrr rktj rr rr rr === ×= = ⋅− ⋅− ω ω µ ε ω ω     The monochromatic planar refracted wave from the boundary is ( ) ( ) 2 2 2 2 0 0 & n c v vc n k eEkH eEE t tt rktj ttt rktj tt tt tt === ×= = ⋅− ⋅− ω ω µ ε ω ω     Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    15 SOLO REFLECTION &REFRACTION The Boundary Conditions at z=0 must be satisfied at all points on the plane at all times, implies that the spatial and time variations of This implies that Phase-Matching Conditions ( ) ( ) ( ) yxteEeEeE z rktj t z rktj r z rktj i ttrrii ,,,, 0 0 0 0 0 0 ∀ = ⋅− = ⋅− = ⋅−   ωωω ( ) ( ) ( ) yxtrktrktrkt z tt z rr z ii ,, 000 ∀⋅−=⋅−=⋅− ===  ωωω ttri ∀=== ωωωω ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  must be the same Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    16 SOLO REFLECTION &REFRACTION tri nnn θθθ sinsinsin 211 == Phase-Matching Conditions ( ) ( )      −+= ++= zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1 θαθααω θαθααω   ( ) ( )         +=⋅ +=⋅ =⋅ = = = yyx c n rk yx c n rk y c n rk ttt z t irr z r i z i ˆsinsincos sinsincos sin 2 0 1 0 1 0 θααω θααω θω    ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  2 π αα == tr ttri ∀=== ωωωω x∀ y∀ Coplanar Snell’s Law ( )     ++= −= zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1   θθω Given: Let find: Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    17 SOLO REFLECTION &REFRACTION Second way of writing phase-matching equations ri θθ = 11 22 2 1 1 2 sin sin εµ εµ θ θ === v v n n t iRefraction Law Reflection Law Phase-Matching Conditions ( )     ++= −= zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1   θθω ( ) ( )      −+= ++= zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1 θαθααω θαθααω   ( ) ( )[ ] ( ) ( )[ ]      −+−=−× −+−=−× ynnyn c kkz ynnyn c kkz ittrti irrrri ˆsinsinsinˆcosˆ ˆsinsinsinˆcosˆ 122 111 θθαα ω θθαα ω   ttri ∀=== ωωωω We can see that ( ) ( )         === =−×=−× ωωωω tri tiri kkzkkz 0ˆˆ            === == == ωωωω θθθ παα tri tri tr nnn sinsinsin 2/ 211 Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    18 SOLO REFLECTION &REFRACTION ri θθ = 11 22 2 1 1 2 sin sin εµ εµ θ θ === v v n n t iRefraction Law Reflection Law Phase-Matching Conditions (Summary) ttri ∀=== ωωωω ( ) ( )         === =−×=−× ωωωω tri tiri kkzkkz 0ˆˆ            === == == ωωωω θθθ παα tri tri tr nnn sinsinsin 2/ 211 ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  ( ) ( ) ( ) yxtrktrktrkt z tt z rr z ii ,, 000 ∀⋅−=⋅−=⋅− ===  ωωω Vector Notation Scalar Notation Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    19 SOLO REFLECTION &REFRACTION ( ) 0ˆ 2121  =−×− EEn ( ) 0ˆ 2121  =−×− HHn ( ) 0ˆ 2121 =−⋅− DDn  ( ) 0ˆ 2121 =−⋅− BBn  Boundary conditions for a source-less boundary ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ In our case ( ) ttrrii tri EkHEkEkH EEEEE   ×=×+×= =+= ˆ&ˆˆ & 2 2 2 1 1 1 21 µ ε µ ε Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations Boundary conditions
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    20 SOLO REFLECTION &REFRACTION ( ) 0ˆ 00021  =−+×− tri EEEn 0 111 ˆ 0 2 0 1 0 1 21  =      ×−×+××− ttrrii EkEkEkn µµµ ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆ 00021 =×−×+×⋅− ttrrii EkEkEkn  Using ,ˆ,ˆ,ˆˆ 221111 1 ttrriii kkkkkk c n k εµωεµωεµωω ====  ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations
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    21 SOLO REFLECTION &REFRACTION ( ) ( ) ( ) ( ) ( ) ( ) ( ) xEknEEknxEknEEkn xEknEEnkEkn tttttirrrrri iiiiiiii ir i ˆcosˆˆˆˆ&ˆcosˆˆˆˆ ˆcosˆˆˆˆˆˆ 0 cos 2100210 cos 210021 0 cos 210 0 021021 θθ θ θθ θ =⋅−=××−=⋅−=×× =⋅−⋅=×× −− − −− −−−         ( ) ( ) ( ) ( ) ( ) ( ) tttttt rrrrrriiiiii EEzzEkn EEzzEknEEzzEkn θθ θθθθ sinsinˆˆˆˆ sinsinˆˆˆˆ&sinsinˆˆˆˆ 00021 0002100021 =−⋅−=×⋅ =−⋅−=×⋅=−⋅−=×⋅ − −−   zn ˆˆ 21 −=− ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ 1 2 3 4 zykzykzyk tttiiriii ˆcosˆsin&ˆcosˆsin&ˆcosˆsin θθθθθθ −=+=−=  Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    22 SOLO REFLECTION &REFRACTION ( ) 0coscoscos 0 2 2 00 1 1 =−− ttirii EEE θ µ ε θθ µ ε 1 0000  =−+ tri EEE2 ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ 1 2 3 4 ( ) 0 sinsin sinsin 0sinsinsin 000 2211 0220011 =−+⇒ = = =−+ tri ti ri ttrrii EEE EEE θεµθεµ θθ θεµθθεµ4 Identical to 2 3 00 = Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    23 SOLO REFLECTION &REFRACTION ( ) 0cos 1 cos 1 0 00 22 2 00 00 11 1 21 =−− tt n iri n EEE θ εµ εµ µ θ εµ εµ µ  1 0000 =−+ tri EEE2 From and ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =      = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 2 2 1 1 1 1 0 0 + =      = ⊥ ⊥ For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i = θ θ ( ) ( )ti ti i r E E r θθ θθ + − −=      = ⊥ ⊥ sin sin 0 0 ( )ti it i t E E t θθ θθ + =      = ⊥ ⊥ sin cossin2 0 0 1 2 Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    24 SOLO REFLECTION &REFRACTION ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =      = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 2 2 1 1 1 1 0 0 + =      = ⊥ ⊥ For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i = θ θ ( ) ( )ti ti i r E E r θθ θθµµ + − −=      = = ⊥ ⊥ sin sin21 0 0 ( )ti it i t E E t θθ θθµµ + =      = = ⊥ ⊥ sin cossin221 0 0 Assume is normal o plan of incidence (normal polarization) E  xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥  Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations
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    25 SOLO REFLECTION &REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ( ) ( ) ( ) ( ) ( ) yEEknyEEkn yEknEEnkEkn ttirri iiiiiii ii ˆˆˆ&ˆˆˆ ˆˆˆˆˆˆˆ 00210021 0 cos 210 sin 021021 −=××−=×× −=⋅−⋅=×× −− − − −−      θθ ( ) ( ) ( ) 0ˆˆˆˆˆˆ 021021021 =×⋅=×⋅=×⋅ −−− ttrrii EknEknEkn  zn ˆˆ 21 −=− zykzykzyk tttiiriii ˆcosˆsin&ˆcosˆsin&ˆcosˆsin θθθθθθ −=+=−=  xEEnxEEnxEEn tttirriii ˆcosˆ&ˆcosˆ&ˆcosˆ 002100210021 θθθ =×−=×=× −−−  tttirriii EEnEEnEEn θθθ sinˆ&sinˆ&sinˆ 002100210021 −=⋅−=⋅−=⋅ −−−  Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    26 SOLO REFLECTION &REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ( ) 0ˆ 00021  =−+×− tri EEEn [ ] 0ˆ/ˆ/ˆ/ˆ 02201101121  =×−×+××− ttrrii EkEkEkn µεµεµε ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn  εµεµεµ 1 2 3 4 ( ) 0sinsin 02001 =−+ ttiri EEE θεθε3 ( )[ ] 0ˆcoscos 000 =−− xEEE ttiri θθ2 ( ) ( ) 0 11 0ˆ 0 00 22 2 00 00 11 1 0 2 2 00 1 1 21 =−+=      −+ t n ri n tri EEEoryEEE   µε µε µµε µε µµ ε µ ε 1 4 00 = Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach
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    27 SOLO REFLECTION &REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ( ) ( ) 0sinsin sin sin / / sinsin 0ˆ 02001 2 1 22 11 2211 0 2 2 00 1 1 =−+ =⇒= =      −+ ttiri t i ti tri EEE yEEE θεθε θε θε µε µε θεµθεµ µ ε µ ε  1 Identical to 3 We have two independent equations ( ) 0coscos 000 =−− ttiri EEE θθ2 ( ) 00 2 2 00 1 1 =−+ tri E n EE n µµ 1 ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =      = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =      = Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 28.
    28 SOLO REFLECTION &REFRACTION Assume is parallel to plan of incidence (parallel polarization) E  ( ) ( ) ( )zyEE zyEE zyEE tttt rrrr iiii ˆsinˆcos ˆsinˆcos ˆsinˆcos 0||0 0||0 0||0 θθ θθ θθ += +−= +=    ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =      = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =      = For most of media μ1= μ2 , and using refraction law: 1 2 sin sin n n t i = θ θ ( ) ( )ti ti i r E E r θθ θθµµ + − =      = = tan tan21 ||0 0 || ( ) ( )titi it i t E E t θθθθ θθµµ −+ =      = = cossin cossin221 ||0 0 || Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations
  • 29.
    29 SOLO REFLECTION &REFRACTION ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =        = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =        = ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =        = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 2 2 1 1 1 1 0 0 + =        = ⊥ ⊥ The equations of reflection and refraction ratio are called Fresnel Equations, that first developed them in a slightly less general form in 1823, using the elastic theory of light. Augustin Jean Fresnel 1788-1827 The use of electromagnetic approach to prove those relations, as described above, is due to H.A. Lorentz (1875) Reflections and Refractions Laws Development Using the Electromagnetic Approach Hendrik Antoon Lorentz 1853-1928
  • 30.
    30 SOLO REFLECTION &REFRACTION Discussion of Fresnel Equations it n ni θθ θ 2 1 0→ = 0→iθ [ ] [ ] ti ti nn nn rr ii θθ θθ θθ + − = + − =−= =⊥= 21 12 00|| [ ] [ ] ti t nn n tt ii θθ θ θθ + = + == =⊥= 22 21 1 00|| 1 2 sin sin n n t i = θ θ Snell’s law Reflections and Refractions Laws Development Using the Electromagnetic Approach n2 / n1 =1.5 n2 / n1 =1/1.5 0cos90 =→→ ii θθ  [ ] [ ] 19090|| −=−= =⊥=  ii rr θθ [ ] [ ] 09090|| == =⊥=  ii tt θθ ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 31.
    31 SOLO REFLECTION &REFRACTION Discussion of Fresnel Equations 1 2 sin sin n n t i = θ θ Snell’s law Reflections and Refractions Laws Development Using the Electromagnetic Approach David Brewster 1781-1868 David Brewster , “On the laws which regulate the polarization of light by reflection from transparent bodies”, Philos. Trans. Roy. Soc., London 105, 125-130, 158-159 1815). In contrast r|| changes sign (for both n2> n1 and n2<n1) when tan(θi+θt)=∞ → θi + θt=π/2 The incident angle, θi, when this occurs is denoted θp and is referred as polarization or Brewster angle (after David Brewster who found it in 1815). n2 / n1 =1.5 n2 / n1 =1/1.5 For n2>n1 we have from Snell’s law θi > θt ,therefore r┴ is negative for all values of θi. it LawsSnell i nnn it θθθ θθ cossinsin 2 90 2 ' 1 −= ==  pi θθ = → 1 2 tan n n p =θ ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 32.
    32 SOLO REFLECTION &REFRACTION Discussion of Fresnel Equations (continue) n2 / n1 =1.5 Reflections and Refractions Laws Development Using the Electromagnetic Approach Brewster Angle n1 < n2
  • 33.
    33 SOLO REFLECTION &REFRACTION Discussion of Fresnel Equations (continue) n2 / n1 =1/1.5 Reflections and Refractions Laws Development Using the Electromagnetic Approach 1 2 sin sin n n t i = θ θ Snell’s law For n2<n1 we have from Snell’s law θi < θt therefore when θi increases, θt increases until it reaches 90° (no refraction and total reflection ). The incident angle when this occurs is denoted θic and is referred as the critical angle.       = − 1 21 sin n n icθCritical Angle Brewster Angle ( ) ( ) 1 2/sin 2/sin21 2/ 0 0 = + − −=        = = = ⊥ ⊥ πθ πθµµ πθ i i i r t E E r ( ) ( ) 1 2/tan 2/tan21 2/ ||0 0 || = + − =        = = = πθ πθµµ πθ i i i r t E E r n1 > n2
  • 34.
    34 SOLO REFLECTION &REFRACTION n1 > n2 Total Reflection Reflections and Refractions Laws Development Using the Electromagnetic Approach For n1>n2 and θi > θic we have total reflection. 1sinsin 21 2 1 nn it ici n n > > >= θθ θθFrom Snell’s law therefore is no solution for θt , but we can use for 1sinsin1cos 2 2 2 12 −      −=−= itt n n i θθθ ( ) ( )2 21 2 2 21 2 21 21 0 0 /sincos /sincos coscos coscos21 nni nni nn nn E E r ii ii ti ti i r −− −+ = + − =        = = ⊥ ⊥ θθ θθ θθ θθµµ ( ) ( ) ( ) ( )2 21 22 21 2 21 22 21 12 12 ||0 0 || /sincos/ /sincos/ coscos coscos21 nninn nninn nn nn E E r ii ii ti ti i r −− −+ = + − =        = = θθ θθ θθ θθµµ We can see that ( )⊥⊥ = ϕjr exp ( )|||| exp ϕjr = ( ) i i nn θ θϕ cos /sin 2 tan 2 21 2 − =      ⊥ ( ) ( ) i i nn nn θ θϕ cos/ /sin 2 tan 2 21 2 21 2 || − =      n2 / n1 =1/1.5
  • 35.
    35 SOLO REFLECTION &REFRACTION Phase Shifts ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21 n2 / n1 =1/1.5 n2 / n1 =1.5 On can see that and are always positive, therefore is no phase difference between incidence and refracted waves. ⊥t ||t On the other hand and can be either positive or negative depending of the sign of (θi-θt). ⊥r || r The phase change of the reflected wave, in the cases where refraction is possible, can be either π or 0, depending on whether the index n1 of the medium in which the wave originates is less or greater than n2 of the medium in which it travels.
  • 36.
    36 SOLO REFLECTION &REFRACTION Phase Shifts ||i E  ⊥i E  ⊥rE  ik  rk  tk  Boundary 21ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21ˆ −n Boundary Plan of incidence iE  ||i E  ⊥i E  ||rE  ⊥r E  pi nn < pi θθ = p n i n p n in ||iE  ⊥i E  ⊥rE  ik  rk  tk  Boundary 21ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence iE  ||iE  ⊥i E  ||rE  ⊥r E  pi nn < pi θθ > pn in p n i n ||r E  rE  n2 / n1 =1.5 n1 < n2 The phase change of the reflected wave is: - π for , - 0 for 0 ≤ θi ≤ θp, and π for θi > θp for ⊥ r || r ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 37.
    37 SOLO REFLECTION &REFRACTION Phase Shifts n2 / n1 =1/1.5 n1 > n2 The phase change of the reflected wave is: -0 for 0 ≤ θi ≤ θc and changes from 0 to π for θi > θc for - π for 0 ≤ θi ≤ θp, π for θp>θi > θc for ⊥r || r ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 38.
    38 SOLO REFLECTION &REFRACTION Phase Shifts ||i E  ⊥iE  ⊥rE  ik  rk  tk  Boundary 21 ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence i E  ||i E  ⊥iE  ||r E  ⊥rE  pi nn < pi θθ = pn in pn i n ||i E  ⊥iE  ⊥rE  ik  rk  tk  Boundary 21 ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence iE  ||iE  ⊥iE  ||r E  ⊥rE  pi nn < pi θθ > p n in pn i n ||r E  rE  n1 > n2 n1 < n2
  • 39.
    39 SOLO REFLECTION &REFRACTION Energy Reflected and Refracted for Normal Polarization 2 01 12 ˆ ⊥ ⊥ = i i i E n ck S ε  Time Average Poynting Vectors (Irradiances) of the Planar waves are 2 01 12 ˆ ⊥ ⊥ = r r r E n ck S ε  2 02 22 ˆ ⊥ ⊥ = t t t E n ck S ε  2 2 2 1 1 2 2 2 1 1 2 0 2 0 coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 2 2 1 1 21 12 2 1 1 2 021 2 012 coscos cos cos cos2 cos cos ˆ ˆ       +       == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti i t i ii tt i t nn n nn En En zS zS T θ µ θ µ θε θε θ µ θε θε     titi cn cn i t i nn n nn n nn θθ µµ θθ µεµ εµ µθε θε θ µ coscos4coscos 1 4 cos cos cos2 21 21 2211 122 1 2 1 21 12 2 1 1 22 1 22 2 ==      2 2 2 1 1 2 2 1 1 2 021 2 012 coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 40.
    40 SOLO REFLECTION &REFRACTION 2 2 2 1 1 2 2 2 1 1 2 0 2 0 coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 2 2 1 1 2 2 1 1 2 021 2 012 coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance We can see that 1=+ ⊥⊥ TR Energy Reflected and Refracted for Normal Polarization Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 41.
    41 SOLO REFLECTION &REFRACTION Energy Reflected and Refracted for Parallel Polarization 2 ||01 1 || 2 ˆ i i i E n ck S ε=  Time Average Poynting vector of the Planar waves are 2 ||01 1 || 2 ˆ r r r E n ck S ε=  2 ||02 2 || 2 ˆ t t t E n ck S ε=  2 1 1 2 2 2 1 1 2 2 2 ||0 2 ||0 || || || coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 1 1 2 2 21 12 2 1 1 2 ||021 2 ||012 || || || coscos cos cos cos2 cos cos ˆ ˆ       +       == ⋅ ⋅ = ti i t i ii tt i t nn n nn En En zS zS T θ µ θ µ θε θε θ µ θε θε     titi cn cn i t i nn n nn n nn θθ µµ θθ µεµ εµ µθε θε θ µ coscos4coscos 1 4 cos cos cos2 21 21 2211 122 1 2 1 21 12 2 1 1 22 1 22 2 ==      2 1 1 2 2 2 2 1 1 2 ||021 2 ||012 || || || coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 42.
    42 SOLO REFLECTION &REFRACTION 2 1 1 2 2 2 1 1 2 2 2 ||0 2 ||0 || || || coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   Reflectance Transmittance 2 1 1 2 2 2 2 1 1 2 ||021 2 ||012 || || || coscos coscos4 cos cos ˆ ˆ       + == ⋅ ⋅ = ti ti ii tt i t nn nn En En zS zS T θ µ θ µ θθ µµ θε θε   We can see that 1|||| =+TR Average Poynting vector of the Planar waves are Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 43.
    43 SOLO REFLECTION &REFRACTION ( ) ( )ti ti i r zS zS R θθ θθµµ + − = ⋅ ⋅ = = 2 2 || || || tan tan ˆ ˆ 21   ( ) ( )titi ti i t zS zS T θθθθ θθµµ −+ = ⋅ ⋅ = = 22 || || || cossin 2sin2sin ˆ ˆ 21   1|||| =+TR Reflections and Refractions Laws Development Using the Electromagnetic Approach ( ) ( )ti ti i r zS zS R θθ θθµµ + − = ⋅ ⋅ = = ⊥ ⊥ ⊥ 2 2 sin sin ˆ ˆ 21   ( )ti ti i t zS zS T θθ θθµµ + = ⋅ ⋅ = = ⊥ ⊥ ⊥ 2 sin 2sin2sin ˆ ˆ 21   1=+ ⊥⊥ TR Summary
  • 44.
    44 SOLO REFLECTION &REFRACTION Reflections and Refractions Laws Development Using the Electromagnetic Approach || R ⊥ R || R ⊥ R
  • 45.
    45 Malus-Dupin TheoremSOLO Étienne LouisMalus 1775-1812 A surface passing through the end points of rays which have traveled equal optical path lengths from a point object is called an optical wavefront. 1808 1812 If a group of ray is such that we can find a surface that is orthogonal to each and every one of them (this surface is the wavefront), they are said to form a normal congruence. The Malus-Dupin Theorem (introduced in 1808 by Malus and modified in 1812 by Dupin) states that: “The set of rays that are orthogonal to a wavefront remain normal to a wavefront after any number of refraction or reflections.” Charles Dupin 1784-1873 Using Fermat principle [ ] [ ]'' BQBAVApathoptical == [ ] [ ] ( )2 '' εOAVAAQA += VQ=ε is a small quantity [ ] [ ] ( )2 '' εOBQBAQA += Since ray BQ is normal to wave W at B [ ] [ ] ( )2 εOBQAQ += [ ] [ ] ( )2 '' εOQBQA += ray BQB’ is normal to wave W’ at B’ Proof for Refraction: n 'n P Q VA P’ A' B B' Wavefront from P Wavefront to P'
  • 46.
    46 Stokes Treatment ofReflection and RefractionSOLO An other treatment of reflection and refraction was given by Sir George Stokes. Suppose we have an incident wave of amplitude E0i reaching the boundary of two media (where n1 = ni and n2 = nt) at an angle θ1. The amplitudes of the reflected and transmitted (refracted) waves are, E0i·r and E0i·t, respectively (see Fig. a). Here r (θ1) and t (θ2) are the reflection and transmission coefficients. According to Fermat’s Principle the situation where the rays direction is reversed (see Fig. b) is also permissible. Therefore we have two incident rays E0i·r in media with refraction index n1 and E0i·t in media with refraction index n2. E0i·r is reflected, in media with refraction index n1, to obtain a wave with amplitude (E0i·r )·t and refracted, in media with refraction index n2, to obtain a wave with amplitude (E0i·r )·r (see Fig. c). E0i·t is reflected, in media with refraction index n2, to obtain a wave with amplitude (E0i·t )·r’ and refracted, in media with refraction index n1, to obtain a wave with amplitude (E0i·t )·t’ (see Fig. c). θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
  • 47.
    47 Stokes Treatment ofReflection and RefractionSOLO To have Fig. c identical to Fig. b the following conditions must be satisfied: ( ) ( ) ( ) ( ) iii ErrEttE 0110120 ' =+ θθθθ ( ) ( ) ( ) ( ) 0' 220210 =+ θθθθ rtEtrE ii Hence: ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ rr rrtt −= =+ Stokes relations θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
  • 48.
    48 Stokes Treatment ofReflection and RefractionSOLO ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ rr rrtt −= =+ Stokes relations Let check that Fresnel Equation do satisfy Stokes relations ( ) 2211 11 2 coscos cos221 θθ θ θ µµ nn n t + = = ⊥ 2112 11 || coscos cos221 θθ θµµ nn n t + = = ( ) 2211 2211 1 coscos coscos21 θθ θθ θ µµ nn nn r + − = = ⊥ ( ) 2112 2112 1|| coscos coscos21 θθ θθ θ µµ nn nn r + − = = ( ) 2211 22 1 coscos cos2 ' 21 θθ θ θ µµ nn n t + = = ⊥ ( ) 2211 1122 2 coscos coscos ' 21 θθ θθ θ µµ nn nn r + − = = ⊥ 1 ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ ⊥⊥ ⊥⊥⊥⊥ −= =+ rr rrtt We can see that: 2 2112 22 || coscos cos2 ' 21 θθ θµµ nn n t + = = ( ) 2112 1221 1|| coscos coscos ' 21 θθ θθ θ µµ nn nn r + − = = ( ) ( ) ( ) ( ) ( ) ( )1||2|| 1||1||1||2|| ' 1' θθ θθθθ rr rrtt −= =+ We can see that: θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
  • 49.
    49 SOLO REFLECTION &REFRACTION Thinks to complete Photons and Laws of Reflection & Refraction (Hecht & Zajac pp. 93) http://physics.nad.ru/Physics/English/index.htm
  • 50.
    50 ELECTROMAGNETICSSOLO References J.D. Jackson, “ClassicalElectrodynamics”, 3rd Ed., John Wiley & Sons, 1999 R. S. Elliott, “Electromagnetics”, McGraw-Hill, 1966 J.A. Stratton, “Electromagnetic Theory”, McGraw-Hill, 1941 W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison- Wesley, 1962 F.T. Ulaby, R.K. More, A.K. Fung, “Microwave Remote Sensors Active and Passive”, Addson-Wesley, 1981 A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”, John Wiley & Sons, 1988
  • 51.
    51 SOLO References Foundation of GeometricalOptics [1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, 6th Ed., Pergamon Press, 1980, Ch. 3 and App. 1 [2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, pp. 692-694
  • 52.
    52 SOLO References Foundation of GeometricalOptics [3] E.Hecht, A. Zajac, “Optics ”, 3th Ed., Addison Wesley Publishing Company, 1997, [4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd Ed., John Wiley & Sons, 1986
  • 53.
    53 ELECTROMAGNETICSSOLO References 1.W.K.H. Panofsky &M. Phillips, “Classical Electricity and Magnetism”, 2.J.D. Jackson, “Classical Electrodynamics”, 3.R.S. Elliott, “Electromagnetics”, 4.A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,
  • 54.
    54 SOLO ELECTROMAGNETICS BOUNDARYCONDITIONS Boundary Conditions ( ) ( ) ldtHtHhldtHldtHldH h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbHHnldnbHHldtHH e ˆˆˆˆˆˆ 21212121121 ⋅=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) ekHHn  =−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +=⋅ → S e C Sd t D JdlH    ( ) dlbkbdlh t D JSd t D J e h e S e ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e  
  • 55.
    55 SOLO ELECTROMAGNETICS BOUNDARYCONDITIONS Boundary Conditions (continue – 1) ( ) ( ) ldtEtEhldtEldtEldE h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbEEnldnbEEldtEE m ˆˆˆˆˆˆ 21212121121 ⋅−=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) mkEEn  −=−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +−=⋅ → S m C Sd t B JdlE    ( ) dlbkbdlh t B JSd t B J m h m S m ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      FARADAY’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m  
  • 56.
    56 SOLO ELECTROMAGNETICS BOUNDARYCONDITIONS Boundary Conditions (continue – 2) ( ) ( ) SdnDnDhSdnDSdnDSdD h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnDDSdnDD eσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) eDDn σ=−⋅− 2121 ˆ  ( ) dSdShdv e h e V e σρρ 0→ ==∫∫∫ GAUSS’ LAW - ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ∫∫∫∫∫ =• V e S dvSdD ρ 
  • 57.
    57 SOLO ELECTROMAGNETICS BOUNDARYCONDITIONS Boundary Conditions (continue – 3) ( ) ( ) SdnBnBhSdnBSdnBSdB h S 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ → ∫∫  where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and 21 ˆ,ˆ nn 2121 ˆˆˆ −=−= nnn - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n ( ) ( ) SdSdnBBSdnBB mσ=⋅−=⋅− −2121121 ˆˆ  Since this must be true for any dS on the boundary between regions (1) and (2) we must have: ( ) mBBn σ=−⋅− 2121 ˆ  ( ) dSdShdv m h m V m σρρ 0→ ==∫∫∫ GAUSS’ LAW – MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ ∫∫∫∫∫ =• V m S dvSdB ρ 
  • 58.
    58 SOLO ELECTROMAGNETICS BOUNDARYCONDITIONS Boundary Conditions (summary) ( ) mkEEn  −=−×− 2121 ˆ FARADAY’S LAW ( ) ekHHn  =−×− 2121 ˆ AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e   [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m   ( ) eDDn σ=−⋅− 2121 ˆ  GAUSS’ LAW ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ( ) mBBn σ=−⋅− 2121 ˆ  GAUSS’ LAW MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ Fresnel Equations
  • 59.
    January 4, 201559 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

Editor's Notes

  • #35 J.P.Mathieu, “Optics”, Pergamon Press, 1975, pp.52-57 E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  • #36 E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  • #37 E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  • #38 E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  • #39 E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  • #46 Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.105-106 V.N. Mahajan, “Optical Imaging and Aberrations” Part I, Ray Geometrical Optics, SPIE, 1998, pp.11,12 M. Born, E. Wolf, “Principles of Optics”, Pergamon Press, 5th Ed., 1975, pp.130-132
  • #47 Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93
  • #48 Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93
  • #49 Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93