13. Fresnel's Equations For Reflection And Transmission

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13. Fresnel's Equations for Reflectionand TransmissionIncident, transmitted, and reflected beamsBoundary conditions: tangential fields are continuousReflection andtransmissioncoefficientsThe "Fresnel Equations"Brewster's AngleTotal internal reflectionPower reflectanceand transmittanceAugustin Fresnel1788-1827

Posing the problemWhat happens when light, propagating in auniform medium, encounters a smooth interfacewhich is the boundary of another medium (with adifferent refractive index)?k-vector of theincident lightnincidentntransmittedboundaryFirst we need todefine someterminology.

Definitions: Plane of Incidence andplane of the interfacePlane of incidence (in thisillustration, the yz plane) is theplane that contains the incidentand reflected k-vectors.yzPlane of the interface (y 0, the xz plane) is the plane thatdefines the interface between the two materialsx

Definitions: “S” and “P” polarizationsA key question: which way is the E-field pointing?There are two distinct possibilities.1. “S” polarization is the perpendicular polarization, andit sticks up out of the plane of incidenceIHere, the plane ofincidence (z 0) is theplane of the diagram.zT2. “P” polarization is the parallelpolarization, and it lies parallelto the plane of incidence.yRxThe plane of the interface (y 0)is perpendicular to this page.

Definitions: “S” and “P” polarizationsNote that this is a different use of the word “polarization”from the way we’ve used it earlier in this class.reflecting mediumreflected lightThe amount of reflected (and transmitted) light isdifferent for the two different incident polarizations.

Fresnel Equations—Perpendicular E fieldAugustin Fresnel was the first to do this calculation (1820’s).We treat the case of s-polarization first: kiBiEi krEr i rBrInterfaceBeam geometry forlight with its electricfield sticking up out ofthe plane of incidence(i.e., out of the page)yniz tEtBt ktntxthe xz plane (y 0)

Boundary Condition for the ElectricField at an Interface: s polarizationThe Tangential Electric Field is ContinuousIn other words,The component ofthe E-field that lies inthe xz plane iscontinuous as youmove across theplane of the interface. kiBiEiEr i rxz krBrniInterfaceHere, all E-fields arein the z-direction,which is in the planeof the interface.So:yEi(y 0) Er(y 0) Et(y 0) tEtBt ktnt(We’re not explicitly writingthe x, z, and t dependence,but it is still there.)

Boundary Condition for the MagneticField at an Interface: s polarizationyxzThe Tangential Magnetic Field* is ContinuousIn other words,The total B-field in theplane of the interface iscontinuous. ki iBi i i rInterfaceHere, all B-fields are inthe xy-plane, so we takethe x-components:ErEi krBr tEtBt kt–Bi(y 0) cos i Br(y 0) cos r –Bt(y 0) cos t*It's really the tangential B/ , but we're using i t 0nint

Reflection and Transmission forPerpendicularly Polarized LightIgnoring the rapidly varying parts of the light wave and keepingonly the complex amplitudes:E0i E0 r E0t B0i cos( i ) B0 r cos( r ) B0t cos( t )But B E /(c0 / n) nE / c0 and i r .Substituting into the second equation:ni ( E0 r E0i ) cos( i ) nt E0t cos( t )Substituting for E0t using E0i E0 r E0t :ni ( E0 r E0i ) cos( i ) nt ( E0 r E0i ) cos( t )

Reflection & Transmission Coefficientsfor Perpendicularly Polarized LightRearranging ni ( E0 r E0i ) cos( i ) nt ( E0 r E0i ) cos( t ) yields:E0 r ni cos( i ) nt cos( t ) E0i ni cos( i ) nt cos( t ) Solving for E0 r / E0i yields the reflection coefficient :r E0 r / E0i ni cos( i ) nt cos( t ) / ni cos( i ) nt cos( t ) Analogously, the transmission coefficient, E0t / E0i , ist E0t / E0i 2ni cos( i ) / ni cos( i ) nt cos( t ) These equations are called the Fresnel Equations forperpendicularly polarized (s-polarized) light.

Fresnel Equations—Parallel electric fieldNow, the case of P polarization: kiBiEiBr i r kryni Er tBtEt ktxNote that Hechtuses a differentnotation for thereflected field,which is confusing!InterfaceBeam geometryfor light with itselectric fieldparallel to theplane of incidence(i.e., in the page)zntOurs is better!This leads to adifference in thesigns of someequations.Note that the reflectedmagnetic field must point into the screen to achieve E B k for the reflected wave. The x with a circlearound it means “into the screen.”

Reflection & Transmission Coefficientsfor Parallel Polarized LightFor parallel polarized light,andB0i B0r B0tE0icos( i) E0rcos( r) E0tcos( t)Solving for E0r / E0i yields the reflection coefficient, r :r E0 r / E0i ni cos( t ) nt cos( i ) / ni cos( t ) nt cos( i ) Analogously, the transmission coefficient, t E0t / E0i, ist E0t / E0i 2ni cos( i ) / ni cos( t ) nt cos( i ) These equations are called the Fresnel Equations forparallel polarized (p-polarized) light.

To summarize incidentwaveplane ofincidenceinterfaceincidentwaveplane ofincidenceinterfaceE-field vectors are red.k vectors are black.transmitted wavetransmitted waves-polarized light:p-polarized light:ni cos( i ) nt cos( t )r ni cos( i ) nt cos( t )ni cos( t ) nt cos( i )r ni cos( t ) nt cos( i )2ni cos( i )t ni cos( i ) nt cos( t )2ni cos( i )t ni cos( t ) nt cos( i )And, for both polarizations:ni sin( i ) nt sin( t )

Reflection Coefficients for anAir-to-Glass InterfaceThe two polarizations areindistinguishable at 0 Total reflection at 90 for both polarizations.Zero reflection for parallelpolarization at:“Brewster's angle”The value of this angledepends on the value ofthe ratio ni/nt: Brewster tan-1(nt/ni)For air to glass(nglass 1.5),this is 56.3 .Sir David Brewster1781 - 1868Reflection coefficient, r1.0nair 1 nglass 1.5Brewster’s angle.5r 0!r 0r -.5-1.00 30 60 Incidence angle, i90

Reflection Coefficients for aGlass-to-Air InterfaceTotal internal reflectionabove the "critical angle" crit sin-1(nt /ni) 41.8 for glass-to-air(The sine in Snell's Lawcan't be greater than one!)1.0Reflection coefficient, rnglass nairCriticalangler .5Total ler -1.00 30 60 Incidence angle, i90

The obligatory java applet.http://www.ub.edu/javaoptics/docs applets/Doc PolarEn.html

2 c I n 0 0 E0 2 Reflectance (R)R Reflected Power / Incident Power winint i rI r ArI i AiA AreawiBecause the angle of incidence the angle of reflection,the beam’s area doesn’t change on reflection.Also, n is the same for both incident and reflected beams.So:R r2sinceE0 rE0i22 r2

2 c I n 0 0 E0 2 Transmittance (T)I t AtT Transmitted Power / Incident Power I i AiIf the beamhas width wi:wi inint twtA AreaAt wt cos( t ) Ai wi cos( i )The beam expands (or contracts) in one dimension on refraction.2 0 c0 ntE0t2 nEwt nt wt 2 wt I t At 2 0ttT t 22I i Ai 0 c0 wi ni E0i wi ni winE i 0i2 nt cos t 2T t ni cos i sinceE0tE0i22 t2

Reflectance and Transmittance for anAir-to-Glass InterfacePerpendicular polarization1.0Parallel polarization1.0TT.5Brewster’sangle.5R0R00 30 60 Incidence angle, i90 0 30 60 Incidence angle, iNote that it is NOT true that: r t 1.But, it is ALWAYS true that: R T 190

Reflectance and Transmittance for aGlass-to-Air InterfacePerpendicular polarization1.0Parallel polarization1.0TT.5.5R0R00 30 60 90 0 Incidence angle, i30 60 90 Incidence angle, iNote that the critical angle is the same for both polarizations.And still,R T 1

Reflection at normal incidence, i 0When i 0, the Fresnelequations reduce to: nt ni R nt ni 2For an air-glass interface (ni 1 and nt 1.5),R 4% and T 96%The values are the same, whicheverdirection the light travels, from air toglass or from glass to air.This 4% value has big implicationsfor photography.“lens flare”T 4 nt ni nt ni 2

Where you’ve seen Fresnel’s Equations in actionWindows look like mirrors at night(when you’re in a brightly lit room).One-way mirrors (used by police tointerrogate bad guys) are just partialreflectors (actually, with a very thinaluminum coating).Disneyland puts ghouls next to you inthe haunted house using partialreflectors (also aluminum-coated oneway mirrors).Smooth surfaces can produce prettygood mirror-like reflections, eventhough they are not made of metal.

Fresnel’s Equations in opticsOptical fibers onlywork because of totalinternal reflection.Many lasers use Brewster’sangle components to avoidreflective losses:R 100%0% reflection!Laser medium R 90%0% reflection!

Reflection coefficient, r 1.0.5 0-.5-1.0 r r 0 30 60 90 Brewster’s angle Total internal reflection Critical angle Critical angle Total internal reflection above the "critical angle" crit sin-1(n t /n i) 41.8 for glass-to-air n glass n air (The sine in Snell's Law can't be greater than one!) Reflection Coefficients for a .

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