classical electromagnetism and optics

7/30/2019 Classical Electromagnetism and Optics

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Chapter 2Classical ElectromagnetismandOpticsTheclassicalelectromagneticphenomenaarecompletelydescribedbyMaxwellsEquations. Thesimplestcasewemayconsideristhatofelectrodynamicsofisotropicmedia2.1 Maxwells Equations of Isotropic MediaMaxwellsEquationsare

DH =t + J, (2.1a)

BE = t , (2.1b)

D = , (2.1c)B = 0. (2.1d)

ThematerialequationsaccompanyingMaxwellsequationsare: D = 0E+ P , (2.2a) B = 0H+ M. (2.2b)

Here, E andH aretheelectricandmagneticfield, D thedielectricflux, Bthemagneticflux,Jthecurrentdensityoffreechareges,is the free charge density,P isthepolarization,andM themagnetization.

13


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14 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSNote, it isEqs.(2.2a)and(2.2b)whichmakeelectromagnetisman inter

estingand

always

ahot

topic

with

never

ending

possibilities.

All

advances

in

engineeringofartificalmaterialsorfindingofnewmaterialproperties,suchassuperconductivity,bringnewlife,meaningandpossibilitiesintothisfield.

BytakingthecurlofEq. (2.1b)andconsidering E = E E,

whereistheNablaoperatorandtheLaplaceoperator,weobtain

!

E P

E 0t j+ 0 t + t =t M+ E (2.3)andhence ! 1 2 j 2

c20t2

E = 0 t +t2P +t M+ E . (2.4)withthevacuumvelocityoflight s

1c0 = . (2.5)

00Fordielectricnonmagneticmedia,whichweoftenencounterinoptics,withno free charges and currents due to free charges, there is M = 0, J = 0,= 0,whichgreatlysimplifiesthewaveequationto 1 2 2

c2t2 E

= 0t2P + E . (2.6)

0

2.1.1 Helmholtz EquationIn general, the polarization in dielectric media may have a nonlinear andnonlocaldependenceonthefield. Forlinearmediathepolarizabilityofthemediumisdescribedbyadielectricsusceptibility (r, t)Z Z

P(r,t) = 0 dr0dt0 (rr0, t t0) E (r0, t0) . (2.7)


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152.1. MAXWELLSEQUATIONSOFISOTROPICMEDIAThe polarization in media with a local dielectric suszeptibility can be described

by

ZP(r,t) = 0 dt0 (r,tt0) E (r,t0) . (2.8)

This relationship further simplifies for homogeneous media, where the susceptibilitydoesnotdependonlocationZ

P(r,t) = 0 dt0 (tt0) E (r,t0) . (2.9)whichleadstoadielectricresponsefunctionorpermittivity

(t) = 0((t) + (t)) (2.10)and with it to Z

D(r,t) = dt0 (tt0) E(r,t0) . (2.11)Insucha linearhomogeneousmediumfollows fromeq.(2.1c) forthecaseofnofreecharges Z

dt0 (tt0) (E(r,t0)) = 0. (2.12)This iscertainly fulfilled for E = 0, which simplifiesthe wave equation

(2.4)further 1 2 2

E= 0 P . (2.13)2c0t2 t2 This isthewaveequationdrivenbythepolarizationofthemedium. Ifthemediumislinearandhasonlyaninducedpolarization,completelydescribedinthetimedomain(t) orinthefrequencydomainbyitsFouriertransform,the complex susceptibility () = r() 1 with the relative permittivityr() = ()/0,weobtain inthefrequencydomainwiththeFouriertransformrelationship

Z+eE(z, ) = E(z, t)ejtdt, (2.14)

eP() = 0()eE(), (2.15)


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16 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS

where,the

tildes

denote

the

Fourier

transforms

in

the

following.

Substituted

into(2.13)

+2 Ee () = 200()Ee(), (2.16)2c0weobtain

+2

2(1 + () E

e() = 0, (2.17)

c0withtherefractiveindexn() and1 + () = n()2 resultsintheHelmholtzequation

2 e+ E() = 0, (2.18)c2wherec() = c0/n() isthevelocityof light inthemedium. ThisequationisthestartingpointforfindingmonochromaticwavesolutionstoMaxwellsequationsinlinearmedia,aswewillstudyfordifferentcasesinthefollowing.Also,sofarwehavetreatedthesusceptibility() asarealquantity,whichmaynotalwaysbethecaseaswewillseelaterindetail.2.1.2 Plane-Wave Solutions (TEM-Waves) and Com-

plex NotationTherealwaveequation(2.13) fora linearmediumhasrealmonochromaticplane wave solutions Ek(r,t), which can be be written most efficiently intermsofthecomplexplane-wavesolutionsEk(r,t) accordingtoh i n o1

Ek(r,t) = Ek(r,t) + Ek(r,t) =


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172.1. MAXWELLSEQUATIONSOFISOTROPICMEDIArelationshipbetweenwavevectorandfrequencynecessarytosatisfythewaveequation

2

|k|2 =c()

= k()2. (2.21)2

Thus,thedispersionrelationisgivenby

k() = n(). (2.22)c0

withthewavenumberk= 2/, (2.23)

where isthewavelengthofthewave inthemediumwithrefractive indexn, theangular frequency,k thewavevector. Note,thenatural frequencyf = /2. From E = 0, foralltime,weseethatke. Substitutionoftheelectricfield2.19intoMaxwellsEqs. (2.1b)resultsinthemagneticfieldh i1

Hk(r,t) = Hk(r,t) + Hk(r,t) (2.24)2with

Hk(r,t) = Hk ej(tkr) h(k). (2.25)Thiscomplexcomponentofthe magneticfieldcanbedetermined fromthecorrespondingcomplexelectricfieldcomponentusingFaradayslaw

jk Ek ej(tkr) e(k) = j0Hk(r,t), (2.26)or

Hk(r,t) = Ek ej(tkr)ke= Hkej(tkr)h (2.27)0with

kh(k) = e(k) (2.28)|k|

andHk =

|k

0

|Ek = Z

1FEk. (2.29)

ThecharacteristicimpedanceoftheTEM-waveistheratiobetweenelectricandmagneticfieldstrength r

ZF = 0c= 0 = 1ZF0 (2.30)0r n


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c

y

x

z

E

H

Figure2.1: Transverseelectromagneticwave(TEM)[6]

withtherefractiveindexn=r andthefreespaceimpedancerZF0 = 0 377. (2.31)0

Notethatthevectorse,handk formanorthogonaltrihedral,eh, ke, kh. (2.32)

That is why we call these waves transverse electromagnetic (TEM) waves.Weconsidertheelectricfieldofamonochromaticelectromagneticwavewithfrequency and electric field amplitude E0, which propagates in vacuumalongthez-axis, and ispolarizedalong thex-axis, (Fig. 2.1), i.e. |kk| =ez,ande(k) = ex. Then we obtain from Eqs.(2.19) and (2.20)

E(r,t) = E0cos(tkz)ex, (2.33)andsimiliarforthemagneticfield

H(r,t) = E0 cos(tkz)ey, (2.34)ZF0

see Figure 2.1.Note, that for a backward propagating wave with E(r,t) = E ejt+j

kr ex, and H(r,t) = H ej(t+kr) ey, there is a sign change for the magneticfield

H= |k| E, (2.35)0

sothatthe(k, H)alwaysformarighthandedorthogonalsystem.E,


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192.1. MAXWELLSEQUATIONSOFISOTROPICMEDIA2.1.3 Poynting Vectors, Energy Density and IntensityThe table below summarizes the instantaneous and time averaged energycontentandenergytransportrelatedtoanelectromagneticfield

Quantity Realfields Complexfields Electricandmagneticenergydensity

we = 12E D= 120rE2wm = 12 H B= 120rH2w= we+ wm

we = 140rE 2wm = 140r

H 2w= we+ wm

Poyntingvector S= EH T = 1EH2

Poyntingtheorem divS+ Ej+ w t

= 0 divT+ 12Ej+

+2j( wmwe) = 0 Intensity I=S = cw I= Re{ T}= cw

Table2.1: PoyntingvectorandenergydensityinEM-fieldsFor a plane wave with an electric field E(r,t) = Eej(tkz) ex we obtain

fortheenergydensityinunitsof[J/m3]w= 1

2r0|E|2, (2.36)the

complex

Poynting

vector

T = 1

2ZF|E|2 ez, (2.37)

andtheintensityinunitsof[W/m2]I= 1

2ZF|E|2 =1

2ZF|H|2. (2.38)2.1.4 Classical PermittivityInthissectionwewanttogetinsightintopropagationofanelectromagneticwavepacketinanisotropicandhomogeneousmedium,suchasaglassopticalfiberduetothe interactionofradiationwiththemedium. Theelectromagneticpropertiesofadielectricmedium is largelydeterminedbytheelectricpolarization inducedbyanelectricfield inthemedium. Thepolarization is


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Figure2.2: Classicalharmonicoscillatormodelforradiationmatterinteraction

definedasthetotal induceddipolemomentperunitvolume. Weformulatethisdirectlyinthefrequencydomaine dipolemoment e eP() =

volume = N hp()i = 0e()E(), (2.39)where N is density of elementary units and hpi is the average dipole moment of the unit (atom, molecule, ...). In an isotropic and homogeneousmediumtheinducedpolarizationisproportionaltotheelectricfieldandtheproportionalityconstant,

e(),iscalledthesusceptibilityofthemedium.

As it turns out (justification later), an electron elastically bound to apositivelychargedrestatomisnotabadmodelforunderstandingtheinteraction of light with matter at very low electricfields, i.e. thefields do notchangetheelectrondistributionintheatomconsiderablyorevenionizetheatom, seeFigure 2.2. This model is called Lorentz model after the famousphysicist A. H. Lorentz (Dutchman) studying electromagnetic phenomenaattheturnofthe19thcentury. Healso foundtheLorentzTransformationandInvarianceofMaxwellsEquationswithrespecttothesetransformation,whichshowedthepathtoSpecialRelativity.

Theequationofmotionforsuchaunitisthedampedharmonicoscillatordriven by an electricfield in one dimension, x. At optical frequencies, thedistanceofelongation,x,ismuchsmallerthananopticalwavelength(atomshave dimensions on the order of a tenth of a nanometer, whereas opticalfieldshavewavelengthontheorderofmicrons)andtherefore,wecanneglectthe spatial variation of the electricfield during the motion of the chargeswithin an atom (dipole approximation, i.e. E(r,t) = E(rA, t) = E(t)ex).


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2.1. MAXWELLSEQUATIONSOFISOTROPICMEDIA 21Theequationofmotionis

md2xdt2 + 2

0Qm

dxdt + m20x= e0E(t), (2.40)

whereE(t) = Eejt.Here, m isthemassof theelectronassumingthethatthe rest atom has infinite mass, e0 the charge of the electron, 0 is theresonancefrequencyoftheundampedoscillatorandQthequalityfactoroftheresonance,whichdeterminesthedampingoftheoscillator. Byusingthetrialsolutionx(t) = xejt,weobtainforthecomplexamplitudeofthedipolemomentp withthetimedependentresponsep(t) = e0x(t) = pejt

0e

p = m2002) + 2j E. (2.41)(2 Q

Note, that we included ad hoc a damping term in the harmonic oscillatorequation. Atthispointitisnotclearwhatthephysicaloriginofthisdamping term is and we will discuss this at length later in chapter 4. For themoment, we can view this term simply as a consequence of irreversible interactionsoftheatomwithitsenvironment. Wethenobtainfrom(2.39)forthesusceptibility

20Ne

m 01

() = (2.42)0(202) + 2jQor

2pe() = (2.43),

(202) + 2jQ0withp calledtheplasmafrequency,whichisdefinedas2p = Ne20/m0. Figure2.3showsthenormalizedrealandimaginarypart,e() = er() + jei()of the classical susceptibility (2.43). Note, that there is a small resonanceshift(almostinvisible)duetotheloss. Offresonance,theimaginarypartapproacheszeroveryquickly. Notsotherealpart,whichapproachesaconstantvalue 2p/20 belowresonancefor 0,andapproacheszerofaraboveresonance,butmuchslowerthantheimaginarypart. Aswewillsee later,thisisthereasonwhythereare low loss, i.e. transparent,mediawithrefractiveindex very much differentfrom1.


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Figure 2.3: Real part (dashed line) and imaginary part (solid line) of thesusceptibilityoftheclassicaloscillatormodelforthedielectricpolarizability.2.1.5 Optical PulsesOptical pulses are wave packets constructed by a continuous superpositionof monochromatic plane waves. Consider a TEM-wavepacket, i.e. a superpositionofwaveswithdifferent frequencies, polarized along the x-axis andpropagatingalongthez-axis

Z dE(r,t) = Ee()ej(tK()z) ex. (2.44)20Correspondingly,themagneticfieldisgivenbyZ

H(r,t) = d Ee()ej(tK()z) ey (2.45)2ZF()0

Again, the physical electric and magneticfields are real and related to thecomplexfieldsby

1

E(r,t) = E(r,t) + E(r,t) (2.46)2 1H(r,t) = H(r,t) + H(r,t) . (2.47)

2Here,|E()|ej() isthecomplexwaveamplitudeoftheelectromagneticwaveat frequency and K() = /c() = n()/c0 the wavenumber, where,


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232.1. MAXWELLSEQUATIONSOFISOTROPICMEDIA

0

|E( )|

|A( )|

0

Figure 2.4: Spectrum of an optical wave packet described in absolute andrelativefrequenciesn()isagaintherefractive indexofthemedium

n2() = 1 + (), (2.48)candc0 arethevelocityoflightinthemediumandinvacuum,respectively.Theplanesofconstantphasepropagatewiththephasevelocityc ofthewave.

ThewavepacketconsistsofasuperpositionofmanyfrequencieswiththespectrumshowninFig. 2.4.At a given point in space, for simplicity z = 0, the complexfield of apulseisgivenby(Fig. 2.4) Z

E(z= 0, t) = 1 E()ejtd. (2.49)2 0

Optical pulses often have relatively small spectral width compared tothe center frequency of the pulse 0, as it is illustrated in the upper partof Figure 2.4. For example typical pulses used in optical communicationsystems for 10Gb/s transmission speed are on the order of 20ps long andhaveacenterwavelengthof= 1550nm. Thusthespectralwithisonlyonthe order of 50GHz, whereas the center frequency of the pulse is 200THz,i.e. thebandwidth is4000smallerthanthecenter frequency. Insuchcasesit isusefultoseparatethecomplexelectricfield in Eq. (2.49) into a carrier frequency0 and an envelope A(t)andrepresenttheabsolute frequencyas


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24 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS= 0+ .We can then rewrite Eq.(2.49) as

Z1 E(0+ )ej(0+)td (2.50)E(z = 0, t) = 2 0

= A(t)ej0t.Theenvelope,seeFigure2.8,isgivenbyZ

Z1

A()ejtd (2.51)A(t) = 2 0

1A()ejtd, (2.52)=

2 where A() is the spectrum of the envelope with, A() = 0 for 0.Tobephysicallymeaningful, thespectralamplitudeA() must be zero fornegative frequencies less than or equal to the carrier frequency, see Figure2.8. Note, that waves with zero frequency can not propagate, since thecorrespondingwavevector iszero. Thepulseand itsenvelopeareshown inFigure2.5.

Figure2.5: Electricfieldandenvelopeofanopticalpulse.Table2.2showspulseshapeandspectraofsomeoftenusedpulsesaswell

asthepulsewidthandtimebandwidthproducts. ThepulsewidthandbandwidthareusuallyspecifiedastheFullWidthatHalfMaximum(FWHM)of2the intensity in the time domain, , and the spectral densityA(t) A()| |inthe frequencydomain, respectively. ThepulseshapesandcorrespondingspectratothepulseslistedinTable2.2areshowninFigs2.6and2.7.

2


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2.1. MAXWELLSEQUATIONSOFISOTROPICMEDIA 25PulseShape FourierTransformR

PulseWidth Time-BandwidthProduct

A(t) A() = a(t)ejtdt t tfGaussian: e t222 2e1222 2ln2 0.441

HyperbolicSecant:sech(t

) 2 sech

2 1.7627 0.315

Rect-function:=

1, |t| /20, |t|> /2

sin(/2)/2 0.886

Lorentzian: 11+(t/)2 2e

| | 1.287 0.142Double-Exp.: e|t|

1+()2 ln2 0.142Table 2.2: Pulse shapes, corresponding spectra and time bandwidth products.

Image removed for copyright purposes.

Figure2.6: Fouriertransformstopulseshapeslistedintable2.2[6].


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Figure2.7: Fouriertransformstopulseshapes listed intable2.2continued[6].

2.1.6 Pulse PropagationHaving a basic model for the interaction of light and matter at hand, viasection 2.1.4, we can investigate what happens if an electromagnetic wavepacket, i.e. an optical pulse propagates through such a medium. We startfrom Eqs.(2.44) to evaluate the wave packet propagation for an arbitrary


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272.1. MAXWELLSEQUATIONSOFISOTROPICMEDIApropagationdistancez

ZE(z,t) = 1 E()ej(tK()z)d. (2.53)

2 0AnalogoustoEq.(2.50) forapulseatagivenposition,wecanseparateanoptical pulse into a carrier wave at frequency 0 and a complex envelopeA(z, t),

E(z, t) = A(z,t)ej(0tK(0)z). (2.54)Byintroducingtheoffsetfrequency, the offsetwavenumberk() andspectrumoftheenvelopeA()

= 0, (2.55)k() = K(0+ ) K(0), (2.56)A() = E(= 0+ ). (2.57)

theenvelopeatpropagationdistancez,seeFig.2.8,isexpressedasZA(z, t) = 1 A()ej(tk()z)d, (2.58)

2 with the same constraints on the spectrum of the envelope as before, i.e.the

spectrum

of

the

envelope

must

be

zero

for

negative

frequencies

beyond

the carrier frequency. Depending on the dispersion relation k(), (see Fig.2.9),.the pulse will be reshaped during propagation as discussed in the followingsection.2.1.7 DispersionThedispersionrelationindicateshowmuchphaseshifteachfrequencycomponentexperiencesduringpropagation. Thesephaseshifts,ifnotlinearwithrespecttofrequency,willleadtodistortionsofthepulse. Ifthepropagationconstantk() isonlyslowlyvaryingoverthepulsespectrum, it isusefultorepresentthepropagationconstant,k(),ordispersionrelationK() by itsTaylorexpansion,seeFig. 2.9,

k00 k(3)k() = k0+ 2+ 3+ O(4). (2.59)

2 6


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Figure2.8: Electricfield and pulse envelope in time domain.

Figure 2.9: Taylor expansion of dispersion relation at the center frequencyofthewavepacket.

Iftherefractiveindexdependsonfrequency,thedispersionrelationisnolongerlinearwithrespecttofrequency,seeFig. 2.9andthepulsepropagationaccordingto(2.58)canbeunderstoodmosteasilyinthefrequencydomain

A(z,)=

jk()A(z, ). (2.60)

z TransformationofEq.()intothetimedomaingives

nXk(n)A(z, t) =j j A(z,t). (2.61)

z n! tn=1


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292.1. MAXWELLSEQUATIONSOFISOTROPICMEDIAIfwekeeponlythefirstterm,thelinearterm,inEq.(2.59),thenweobtainforthe

pulse

envelope

from

(2.58)

by

defi

nitionof

the

group

velocity

at

frequency

0

g0 = 1/k0 =

dk() d =01

(2.62)A(z,t) = A(0, t z/g0). (2.63)

Thusthederivativeofthedispersionrelationatthecarrierfrequencydeterminesthepropagationvelocityoftheenvelopeofthewavepacketorgroupvelocity,whereastheratiobetweenpropagationconstantandfrequencydeterminesthephasevelocityofthecarrier

1Togetridofthetrivialmotionofthepulseenvelopewiththegroupvelocity,weintroducetheretardedtimet0 = tz/vg0. Withrespecttothisretardedtimethepulseshapeis invariantduringpropagation, ifweapproximatethedispersionrelationbytheslopeatthecarrierfrequency

A(z, t) = A(0, t0). (2.65)Note, if we approximate the dispersion relation by its slope at the carrierfrequency,

i.e.

we

retain

only

the

first

term

in

Eq.(2.61),

we

obtain

K(0)p0 = 0/K(0) = (2.64).0

A(z, t) 1 A(z,t)+ = 0, (2.66)

z g0 t and(2.63)isitssolution. If,wetransformthisequationtothenewcoordinatesystem

z0 = z, (2.67)t0 = tz/g0, (2.68)

with 1

z = z 0 g0t0, (2.69)

= (2.70)t t0


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30 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSthetransformedequationis

A(z0, t0) = 0. (2.71)z 0

Sincez isequaltoz0 wekeepz inthefollowing.If the spectrum of the pulse is broad enough, so that the second order

term in (2.59) becomes important, the pulse will no longer keep its shape.Whenkeepinginthedispersionrelationtermsuptosecondorder itfollowsfrom(2.58)and(2.69,2.70)

A(z,t0) k002A(z, t0)= j . (2.72)

z 2 t02This is the first non trivial term in the wave equation for the envelope.Because of the superposition principle, the pulse can be thought of to bedecomposed intowavepackets(sub-pulses)withdifferentcenterfrequencies.Now,thegroupvelocitydependsonthespectralcomponentofthepulse,seeFigure2.10,whichwillleadtobroadeningordispersionofthepulse.

A( )

0 11

2

S

pectrum

12

3

k21

vg1 vg3

vg2

DispersionRelation

Figure2.10: Decompositionofapulseintowavepacketswithdifferentcenterfrequency. Ina medium with dispersion the wavepackets move at differentrelativegroupvelocity.

Fortunately, for a Gaussian pulse, the pulse propagation equation 2.72canbesolvedanalytically. Theinitialpulseisthenoftheform

E(z = 0, t) = A(z= 0, t)ej0t (2.73) 1 t02

A(z = 0, t = t0) = A0exp (2.74)2 2


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2.1. MAXWELLSEQUATIONSOFISOTROPICMEDIA 31Eq.(2.72) is most easily solved in the frequencydomain since it transformsto

A(z,)

= jk002A(z,), (2.75)z 2

withthesolutionk002

A(z, ) = A(z= 0, )exp j

2 z

. (2.76)The pulse spectrum aquires a parabolic phase. Note, that here is theFourierTransformvariableconjugatetot0 ratherthant. TheGaussianpulsehastheadvantagethatitsFouriertransformisalsoaGaussian

A(z= 0, ) = A02exp

2

122 (2.77)and,therefore,inthespectraldomainthesolutionatanarbitraypropagationdistancez is

2A(z, ) = A02exp1

2

2+ jk00z . (2.78)

TheinverseFouriertransformisanalogously2 1 t02A(z,t0) = A0

(2+ jk00z)1/2

exp

2 (2 + jk00z)

(2.79)Theexponentcanbewrittenasrealandimaginarypartandwefinallyobtain

2 1 2t02 1 t02A(z, t0) = A0

1/2exp

" + j k00z #2 2(2+ jk00z) 2 4+ (k00z) 2 4+ (k00z)(2.80)

As we see from Eq.(2.80) during propagation the FWHM of the Gaussiandeterminedby

" #(0 /2)2exp FWHM = 0.5 (2.81)2 4+ (k00z)changesfrom

F W H M = 2

ln 2 (2.82)


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Amptu

32 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSatthestartto

s 22

1 + = 2ln 2 k00L0FWHM (2.83)2s k00L2= FWHM 1 +

atz= L.Forlargestretchingthisresultsimplifiesto0FWHM = 2

ln 2

for

k

00L k00L.1 (2.84)

2The strongly dispersed pulse has a width equal to the difference in groupdelayoverthespectralwidthofthepulse.

Figure 2.11showsthe evolutionof themagnitudeof the Gaussianwavepacketduringpropagationinamediumwhichhasnohigherorderdispersioninnormalizedunits. Thepulsespreadscontinuously.

1

0.8

0.6

0.4

0.2

0

0.5

Distance z 1

1.5 -6-4

-20

Time

24

6

Figure 2.11: Magnitude of the complex envelope of a Gaussian pulse,|A(z,t0)|,inadispersivemedium.


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332.1. MAXWELLSEQUATIONSOFISOTROPICMEDIAAs discussed before, the origin of this spreading is the group velocity

dispersion(GVD),

k

00= 0.

The

group

velocity

varies

over

the

pulse

spectrum

6significantly leadingtoagroupdelaydispersion(GDD)afterapropagation

distancez= Lofk00L= 0,forthedifferentfrequencycomponents. Thisleads6tothebuild-upofchirpinthepulseduringpropagation. Wecanunderstandthischirpby lookingattheparabolicphasethatdevelopsoverthepulse intimeatafixedpropagationdistance. Thephaseis,seeEq.(2.80)

1 k00L 1 t02(z= L,t0) = arctan

+ k00L . (2.85)

2 2 2 4+ (k00L)2

(a) Phase

Time t

k'' < 0

k'' > 0

Front Back

Instantaneous

Frequency

Time t

k'' < 0

k'' > 0

(b)

Figure2.12:

(a)

Phase

and

(b)

instantaneous

frequency

of

aGaussian

pulse

duringpropagationthroughamediumwithpositiveornegativedispersion.

This parabolic phase, see Fig. 2.12 (a), can be understood as a localyvarying frequency in the pulse, i.e. the derivative of the phase gives the


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34 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSinstantaneousfrequencyshiftinthepulsewithrespecttothecenterfrequency

k00L(z= L, t0) = (L,t0) = t0 (2.86)t0 4+ (k00L)2

see Fig.2.12 (b). The instantaneous frequency indicates that for a mediumwith positive GVD, ie. k00 > 0, the low frequencies are in the front of thepulse, whereas the high frequencies are in the back of the pulse, since thesub-pulses with lower frequencies travel faster than sub-pulses with higherfrequencies. Theoppositeisthecasefornegativedispersivematerials.

Itisinstructiveforlaterpurposes,thatthisbehaviourcanbecompletelyunderstoodfromthecenterofmassmotionofthesub-pulses,seeFigure2.10.Note,wecanchooseasetofsub-pulses, withsuchnarrowbandwidth, thatdispersiondoes not matter. Inthe time domain, these pulses are of coursevery long, because of the time bandwidth relationship. Nevertheless, sincethey all have different carrier frequencies, they interfere with each other insuchawaythatthesuperpositionisaverynarrowpulse. Thisinterference,becomes destroyed during propagation, since the sub-pulses propagate atdifferentspeed, i.e. theircenterofmasspropagatesatdifferentspeed.

5

k">0k" 1 inamediumwithnegativedispersion.

Time,t


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352.1. MAXWELLSEQUATIONSOFISOTROPICMEDIAThedifferentialgroupdelayTg() = k00Lofasub-pulsewithitscen

ter

frequency

diff

erent

from

0,

is

due

to

its

diff

erential

group

velocity

vg() = vg0Tg()/Tg0 = vg20k00.Note,thatTg0 = L/vg0.Thisisillustrated inFigure2.13byplotingthetrajectoryoftherelativemotionofthecenterofmassofeachsub-pulseasafunctionofpropagationdistance,whichasymptoticallyapproachestheformulaforthepulsewidthofthehighlydispersed pulse Eq.(2.84). If we assume that the pulse propagates through a negative dispersive medium following the positive dispersive medium, thegroup velocity of each sub-pulse is reversed. The sub-pulses propagate towards each otheruntil they all meet at one point (focus) to produce againashortandunchirpedinitialpulse,seeFigure2.13. Thisisaverypowerfultechnique to understand dispersive wave motion and as we will see in thenextsectionistheconnectionbetweenrayopticsandphysicaloptics.2.1.8 Loss and GainIf the medium considered has loss, described by the imaginary part of thedielectricsusceptibility,see(2.43)andFig. 2.3,wecanincorporatethislossintoacomplexrefractiveindex

n() = nr() + jni() (2.87)via

qn() = 1 + e(). (2.88)

Foranopticallythinmedium,i.e. e 1 thefollowingapproximationisveryuseful

e()n() 1 + . (2.89)

2As one can show (in Recitations) the complex susceptibility (2.43) can beapproximated close to resonance, i.e. 0, by the complex Lorentzianlineshape

e() = j0 , (2.90)1 + jQ00

pwhere 0 =Q222 will turn out to be related to the peak absorption of the 0

line,whichisproportionaltothedensityofatoms,0 isthecenterfrequency


24/159

36 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSand = Q0 is the half width half maximum (HWHM) linewidth of thetransition.

The

real

and

imaginary

part

of

the

complex

Lorentzian

are

(0)0 e () = 2, (2.91)r

1 + 0

ei() = 0 2, . (2.92)01 +

In the derivation of the wave equation for the pulse envelope (2.61) insection2.1.7, therewas norestrictiontoareal refractive index. Therefore,thewaveequation(2.61)alsotreatsthecaseofacomplexrefractive index.If we assume a medium with the complex refractive index (2.89), then thewavenumber isgivenby

1K() = 1 + (er() + jei()) . (2.93)c0 2

Sincewe introducedacomplexwavenumber, wehavetoredefine thegroupvelocity as the inverse derivative of the real part of the wavenumber withrespecttofrequency. Atlinecenter,weobtain

g1 = Kr()

=

11 0 0. (2.94)

0 c0 2Thus, foranarrowabsorption line,0 >0and 0 1,theabsolutevalueof the group velocity can become much larger than the velocity of light invacuum. The opposite is true for anamplifying medium, 0


25/159

372.1. MAXWELLSEQUATIONSOFISOTROPICMEDIAIn the envelope equation (2.60) for a wavepacket with carrier frequency 0 =

0and

K0

=

0the loss leads

to

a term

of the

form

c0A(z, ) 0K02

Inthetimedomain,weobtainuptosecondorderintheinverselinewidth= (0+ )A(z,) = A(z, ). (2.96)

z 1 + (loss)

A(z,t0) z (loss) = 0K0

1 + 1 22t2

A(z,t0), (2.97)

which corresponds to a parabolic approximation of the line shape at linecenter, (Fig. 2.3). Aswewillsee later, foranamplifyingopticaltransitionweobtainasimilarequation. Weonlyhavetoreplacethelossbygain

1 + whereg =0K0 isthepeakgainat linecenterperunit lengthandg istheHWHMlinewidthofatransitionprovidinggain.2.1.9 Sellmeier Equation and Kramers-Kroenig Rela-

tionsThe linearsusceptibility isthe frequencyresponseor impulseresponse ofalinearsystemtoanapplied electricfield, see Eq.(2.41). For areal physicalsystemthisresponse iscausal,andthereforerealand imaginarypartsobeyKramers-KroenigRelations Z

r() = 2 i(

)2d= nr2() 1, (2.99) 2

0

Z

i() =

2

2

r(

)2d. (2.100)

0For optical media these relations have the consequence that the refractiveindex and absorption of a medium are not independent, which can oftenbe exploited to compute the index from absorption data or the other way

2A(z, t0) 1 0), (2.98)A(z,t= g2t2gz (gain)


26/159

38 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSaround. TheKramers-KroenigRelationsalsogiveusagoodunderstandingof

the

index

variations

in

transparent

media,

which

means

the

media

are

used

inafrequencyrange farawayfromresonances. Thenthe imaginarypartofthesusceptibilityrelatedtoabsorptioncanbeapproximatedbyX

i() = Ai(i) (2.101)i

and theKramers-Kroenig relation results intheSellmeier Equation fortherefractiveindex

X i

X

n2() = 1 + Ai2i

2 = 1 + ai2

i2. (2.102)i i

This formula is very useful infitting the refractive index of various mediaover a large frequency range with relatively few coefficients. For exampleTable2.3showsthesellmeiercoefficientsforfusedquartzandsapphire.

FusedQuartz Sapphirea1 0.6961663 1.023798a2 0.4079426 1.058364a3 0.8974794 5.28079221 4.679148103 3.775881032

2 1.3512063102 1.22544102

23 0.9793400102 3.213616102Table2.3: TablewithSellmeiercoefficientsforfusedquartzandsapphire.

Ingeneral,eachabsorptionlinecontributesacorrespondingindexchangeto the overall optical characteristics of amaterial, see Fig. 2.14. Atypicalsituation foramaterialhavingresonances intheUVandIR,suchasglass,isshowninFig. 2.15. AsFig. 2.15shows,duetotheLorentzianlineshape,that outside of an absorption line the refractive index is always decreasingas a function of wavelength. This behavior is called normal dispersion andtheoppositebehaviorabnormaldispersion.

dn 0 : abnormaldispersiond


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392.1. MAXWELLSEQUATIONSOFISOTROPICMEDIAThisbehaviorisalsoresponsibleforthemostlypositivegroupdelaydispersion

over

the

transparency

range

of

amaterial,

as

the

group

velocity

or

group

dndelay dispersion is closely related to d. Fig.2.16 shows the transparency

rangeofsomeoftenusedmedia.

Figure 2.14: Each absorption line must contribute to an index change viatheKramers-Kroenigrelations.

Figure2.15: Typcialdistributionofabsorptionlinesinamediumtransparentinthevisible.


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DispersionCharacteristic Definition Comp. fromn()mediumwavelength: n n n()wavenumber: k 2

n2 n()

phasevelocity: p k c0n()groupvelocity: g ddk;d= 22c0d c0n 1ndnd1groupvelocitydispersion: GVD d2k

d23

2c20

d2nd2

groupdelay: Tg = Lg = dd dd = d(kL)d nc0 1ndndLgroupdelaydispersion: GDD dTg

d = d2(kL)d2 32c20

d2nd2L


Figure2.16: Transparencyrangeofsomematerialsaccordingto[6],p. 175.Often the dispersion GVD and GDD needs to be calculated from the

Sellmeierequation,i.e. n().ThecorrespondingquantitiesarelistedinTable2.4. The computations are done by substituting the frequency with thewavelength.

Table2.4: Tablewithimportantdispersioncharacteristicsandhowtocomputethemfromthewavelengthdependentrefractiveindexn().


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2.2. ELECTROMAGNETICWAVESANDINTERFACES 412.2 Electromagnetic Waves and InterfacesManymicrowaveandopticaldevicesarebasedonthecharacteristicsofelectromagnetic waves undergoing reflection or transmission at interfaces betweenmediawith different electric or magnetic propertiescharacterizedbyand,seeFig. 2.17. Withoutrestrictionwecanassumethattheinterfaceisthe(x-y-plane)andtheplaneofincidenceisthe(x-z-plane). Anarbitraryincident plane wave can always be decomposed into two components. Onecomponenthas itselectricfieldparalleltothe interfacebetweenthemedia,i.e. itispolarizedparalleltotheinterfaceanditiscalledthetransverseelectric(TE)-wave oralsos-polarized wave. Theothercomponent is polarizedintheplaneofincidenceanditsmagneticfield is in the plane ofthe interface between the media. This wave is called the TM-wave or also p-polarizedwave. Themostgeneralcaseofan incidentmonochromaticTEM-wave isalinearsuperpositionofaTEandaTM-wave.

a) Reflection of TE-Wave b) Reflection of TM-Wav

, 1 1 1, 1

x

ErEiH

krHi ki i r

t

zEt

kt 2 2,

x

Er

Ei

kr

H

i ki i r

t

z Etkt

H

r

2 2,

r

Ht Ht

Figure 2.17: a) Reflection of a TE-wave at an interface, b) Reflection ofaTM-waveataninterface

Thefieldsforbothcasesaresummarizedintable2.5


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42 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSTE-wave TM-waveEi =Ei e

j(t

kir)

ey Ei =Ei ej(tk

ir)

eiHi =Hi ej(tkir)hi Hi =Hi ej(tkir)eyEr =Er ej(tkrr)ey Er =Er ej(t

krr)r er

Hr =Hr ej(tkrr)hr Hr =Er ej(tkrr)eyEt =Et ej(tktr)ey Et =Et ej(tktr)etHt =Ht ej(tktr)ht Ht =Ht ej(tktr)ey

Table2.5: ElectricandmagneticfieldsforTE- andTM-waves.withwavevectorsofthewavesgivenby

ki = kr =k011,kt = k022,

ki,t = ki,t(sini,t ex+ cos i,t ez),kr = ki(sinr excosr ez),

andunitvectorsgivenbyhi,t = cosi,t ex+ sin i,t ez,hr = cos r ex+ sin r ez,

ei,t = hi,t = cos i,t exsin i,t ez,er = hr =cosr exsinr ez.

2.2.1 Boundary Conditions and Snells lawFrom6.013,weknowthatStokesandGaussLawfortheelectricandmagnetic fields require constraints on some of the field components at mediaboundaries. In the absence of surface currents and charges, the tangentialelectric and magnetic fields as well as the normal dielectric and magneticfluxeshavetobecontinuouswhengoingfrommedium1 intomedium2forall

times

at

each

point

along

the

surface,

i.e.

z= 0

E/Hi,x/ye

j(tki,xx)+E/Hr,x/yej(tkr,xx) =E/Hi,x/yej(tkt,xx). (2.103)Thisequationcanonlybefulfilledatalltimesifandonlyifthex-componentofthek-vectorsforthereflectedandtransmittedwaveareequalto(match)


31/159

432.2. ELECTROMAGNETICWAVESANDINTERFACESthecorrespondingcomponentoftheincidentwave

ki,x =kr,x =kt,x (2.104)This phase matching condition is shown in Fig. 2.18 for the case22 >

11 orkt > ki.1,1

xkr

ki i r

t

z

kt

2 2,

Figure2.18: PhasematchingconditionforreflectedandtransmittedwaveThe phase matching condition Eq(2.104) results in r = i = 1 and

Snellslawfortheanglet =2 ofthetransmittedwave

11sint = sini (2.105)22

orforthecaseofnonmagneticmediawith1 =2 =0sint = n1 sini (2.106)

n2


32/159

44 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS2.2.2 Measuring Refractive Index with Minimum De-

viationSnellslawcanbeusedformeasuringtherefractiveindexofmaterials. Consideraprismpreparedfromamaterialwithunknownrefractiveindexn(),seeFig. 2.19(a).


Figure 2.19: (a) Beam propagating through a prism. (b) For the case ofminimumdeviation[3]p. 65.

The prism is mounted on a rotation stage as shown in Fig. 2.20. Theangle of incidence is then varied with a fixed incident beam path andthe transmitted light is observed on a screen. If one starts of with normalincidenceonthefirstprismsurfaceonenoticesthatafterturningtheprismone goes through a minium for the deflectionangleofthebeam. ThisbecomesobviousfromFig. 2.19(b). Thereisanangleofincidencewherethebeampaththroughtheprismissymmetric. Iftheinputangleisvariedaroundthispoint, itwouldbe identicaltoexchangethe inputandoutputbeams. Fromthatweconcludethatthedeviationmust go through an extremum at the symmetry point, see Figure 2.21. It can be shown (Recitations), that therefractiveindexisthendeterminedby

sin(min)+minn= 2 . (2.107)

sin(min)2If themeasurement isrepeated forvariouswavelengthof the incidentradiation the complete wavelength dependent refractive index is characterized,seeforexample,Fig. 2.22.


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452.2. ELECTROMAGNETICWAVESANDINTERFACES

Figure 2.20: Refraction of a Prism with n=1.731 for different angles of incidence alpha. Theangle of incidence is stepwise increasedbyrotatingtheprismclockwise. Theangle of transmissionfirst increases. After the angleforminimumdeviation is reachedthetransmissionangle startsto decrease[3]p67.



34/159



Figure2.21: Deviationversusincidentangel[1]

Figure2.22: Refractive indexasafunctionofwavelengthforvariousmediatransmissiveinthevisible[1],p42.



35/159

472.2. ELECTROMAGNETICWAVESANDINTERFACES2.2.3 Fresnel ReflectionAfterunderstandingthedirectionofthereflectedandtransmittedlight,formulas for how much light is reflected and transmitted are derived by evaluating the boundary conditions for the TE and TM-wave. According toEqs.(2.103)and(2.104)weobtainforthecontinuityofthetangentialEandHfields:

TE-wave(s-pol.) TM-wave(p-pol.)Ei+Er =Et EicosiErcosr =EtcostHi cosiHr cosr =Ht cost Hi+Hr =Ht

(2.108)

qIntroducingthecharacteristicimpedancesinbothhalfspacesZ1/2 = 01/2,01/2and the impedances that relate the tangential electric and magnetic fieldcomponents ZTE/TM in both half spaces the boundary conditions can be1/2rewrittenintermsoftheelectricormagneticfieldcomponents.

TE-wave(s-pol.) TM-wave(p-pol.)ZT E1/2 = Ei/tHi/t cosi/t = Z1/2cos1/2 ZT M1/2 = Ei/tcosi/tHi/t =Z1/2cos1/2Ei+Er =Et HiHr = ZT M2ZT M

1 HtEi

Er = ZT E 1ZT E

2 Et Hi+Hr =Ht(2.109)

AmplitudeReflection and Transmission coefficientsFrom these equations we can easily solve for the reflected and transmittedwave amplitudes in terms of the incident wave amplitudes. By dividingbothequationsbytheincidentwaveamplitudesweobtainfortheamplitudereflectionandtransmissioncoeffcients. Note,thatreflectionandtransmissioncoefficientsaredefinedintermsoftheelectricfieldsfortheTE-waveandintermsofthemagneticfieldsfortheTM-wave.

TE-wave(s-pol.)

TM-wave

(p-pol.)

rT E= Er

Ei;tT E= EtEi rT M = HrHi;tT M = HtHi1 + rT E =tT E 1rT M = ZT M2

ZT M1 t

T M1 rT E = ZT E 1

ZT E 2 t

T E 1 + rT M =tT M(2.110)


36/159

48 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSorinbothcasestheamplitudetransmissionandreflectioncoefficientsare

2 2ZTE/TMtTE/TM =

ZTE/TM = TE/TM2/1 TE/TM (2.111)1/2 Z1 +Z21 + TE/TM

Z2/1

TE/TM TE/TMrTE/TM = Z2/1 Z1/2 (2.112)

TE/TM TE/TMZ1 +Z2

Despitethesimplicityoftheseformulas,theydescribealreadyanenormouswealth of phenomena. To get some insight, consider the case of purely dielectricandlosslessmediacharacterizedbyitsrealrefractiveindicesn1 andn2.

Then

Eqs.(2.111)

and

(2.112)

simplify

for

the

TE

and

TM

case

to

TE-wave(s-pol.) TM-wave(p-pol.)ZT E1/2 = Z1/2cos1/2 = Z0n1/2cos1/2rT E= n1cos1n2cos2n1cos1+n2cos2

ZT M1/2 =Z1/2cos1/2rT M = n2cos2 n1cos1n2

cos2+ n1cos1

= Z0n1/2 cos1/2

tT E= 2n1cos1n1cos1+n2cos2 tT M = 2

n2cos2

n2cos2+ n1cos1

(2.113)

Figure2.23showstheevaluationofEqs.(2.113)forthecaseofareflectionattheinterfaceofairandglasswithn2 > n1 and(n1 = 1, n2 = 1.5).

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

56.3

B

tTM

tTE

rTM

rTE

Amplituderefl.andtransm.coefficients

0 15 30 45 60 75 90Incident angle (deg.)

Figure 2.23: The amplitude coefficients of reflection and transmission as afunctionof incident angle. Thesecorrespondtoexternal reflectionn2 > n1atanair-glasinterface(n1 = 1, n2 = 1.5).


37/159

492.2. ELECTROMAGNETICWAVESANDINTERFACESForTE-polarizedlightthereflectedlightchangessignwithrespecttothe

incidentlight

(re

fl

ectionat

the

optically

more

dense

medium).

This

is

not

so for TM-polarized light under close to normal incidence. It occurs onlyfor angles larger than B, which is called the Brewster angle. So for TM-polarized light the amplitude reflection coefficient is zero at the Brewsterangle. Thisphenomenawillbediscussedinmoredetail later.

This behavior changes drastically if we consider the opposite arrangement of media, i.e. we consider the glass-air interface with n1 > n2, seeFigure2.24. ThentheTM-polarized lightexperiencesa-phaseshiftuponreflection close to normal incidence. For increasing angle of incidence thisreflection coefficient goes through zero at the Brewster angle B0 differentfrombefore. However,forlargeenoughangleofincidencethereflectioncoefficientreachesmagnitude1andstaysthere. Thisphenomenoniscalledtotalinternal reflectionandthe angle where this occursfirst is the critical anglefortotalinternalreflection,tot. Totalinternalreflectionwillbediscussedinmoredetaillater.

-0.2

0.0

0.2

0.40.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

B

tot

rTE

rTM

tTE

tTM

Amplituderefl.

andtransm.coefficients

0 15 3033.7 41.8 45

Incident angle (deg.)

Figure 2.24: The amplitude coefficients of reflection and transmission as afunction of incident angle. These correspond to internal reflection n1 > n2ataglas-airinterface(n1 = 1.5, n2 = 1).

Power reflection and transmissioncoefficientsOftenwearenotinterestedintheamplitudebutratherintheopticalpowerreflectedortransmittedinabeamoffinitesize,seeFigure2.25.


38/159


Figure 2.25: Reflection and transmission of an incident beam offinite size[1].

Note,thattogetthepowerinabeamoffinitesize,weneedtointegratedthecorrespondingPoyntingvectoroverthebeamarea,whichmeansmultiplicationbythebeamcrosssectionalareaforahomogenousbeam. Sincetheangleof incidenceandreflectionareequal,i =r =1 thisbeamcrosssectionalareadropsoutinreflection

2TE/TM TE/TM TE/TM

)

2However, due to the different angles for the incident and the transmitted beamt =2 =1,wearriveat6

TE/TMTTE/TM = ItTE/TMAcost (2.115)Ii Acosr

1

Ir Acosi ZZ2 1RTE/TM TE/TM (2.114)= = r =TE/TM TE/TM TE/TM

I Acosr Z1 +Z2i

) (( cos2 1 1 2tTE/TMRe Re= .cos1 Z2/1 Z1/2



39/159

512.2. ELECTROMAGNETICWAVESANDINTERFACESUsinginthecaseofTE-polarization Z1/2 =ZT E andanalogouslyforTM-

cos1/2 1/2=ZT MpolarizationZ1/2cos1/2 1/2,weobtain

1 ) ( 4ZTE/TM1 2/1

TTE/TM = Re (2.116)Re 2TE/TM Z TE/TM TE/TMZ1 +Z1/2 2Note,forthecasewherethecharacteristicimpedancesarecomplexthiscannot be further simplified. Ifthecharacteristic impedancesarereal, i.e. themediaarelossless,thetransmissioncoefficientsimplifiesto

4ZTE/TM

ZTE/TM

1/2 2/1T

TE/TM 2 (2.117)= Z1TE/TM+Z2TE/TM .Tosummarizeforlosslessmediathepowerreflectionandtransmissioncoefficientsare

TE-wave(s-pol.) TM-wave(p-pol.)ZT E1/2 = Z1/2cos1/2 = Z0n1/2cos1/2 ZT M1/2 =Z1/2cos1/2 = Z0n1/2 cos1/2 RT E=

n2cos2n1cos1n1cos1+n2cos2

2 RT M =

n2cos2 n1cos1

n2cos2+ n1cos1

2

TT E=4n1cos1n2cos2

|n1cos1+n2cos2|2 TT M =4 n2

cos2n1

cos1 n2cos2+ n1cos12

TT E+RT E = 1 TT M+RT M = 1

(2.118)

A few phenomena that occur upon reflection at surfaces between differentmediaareespeciallynoteworthyandneedamoreindepthdiscussionbecausethey enhance or enable the construction of many optical components anddevices.2.2.4 Brewsters AngleAs Figures 2.23 and 2.24 already show, for light polarized parallel to theplane of incidence, p-polarized light, the reflection coefficient vanishes at a givenangleB,calledtheBrewsterangle. UsingSnellsLawEq.(2.106),

n2 sin1= , (2.119)

n1 sin2


40/159

52 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSwecanrewritethereflectionandtransmissioncoefficientsinEq.(2.118)onlyin

terms

of

the

angles.

For

example,

we

find

for

the

refl

ectioncoe

ffi

cient2 2

sin cos2 1 =

=

cos2 sin1 cos2n2n1 cos1

+cos2cos1

RT M (2.120)=sin1 +cos2sin2 cos1

n2n1

2sin21sin22(2.121)

sin21+ sin 22where we used in the last step in addition the relation sin2= 2 sin cos.ThusbyforcingRT M = 0,theBrewsterangleisreachedfor

sin21,Bsin22,B = 0 (2.122)

or

21,B =22,B or1,B +2,B = (2.123)2

This relation is illustrated in Figure 2.26. The reflected and transmittedbeams are orthogonal to each other, so that the dipoles induced in themedium by the transmitted beam, shown as arrows in Fig. 2.26, can notradiate into the direction of the reflectedbeam. This isthephysical originof the zero in the reflection coefficient, only possible for a p-polarized orTM-wave.

Therelation(2.123)canbeusedtoexpresstheBrewsterangleasafunctionoftherefractiveindices,becauseifwesubstitute(2.123)intoSnellslawweobtain

sin1 n2=

sin2 n1sin1,B2 1,B cos1,B

sin1,B= = tan 1,B,

sin

ortan1,B = n2. (2.124)

n1

UsingtheBrewsterangleconditiononecaninsertanopticalcomponentwitha refractive index n = 1 into a TM-polarized beam in air without having6reflections, see Figure 2.27. Note, this is not possible for a TE-polarizedbeam.


41/159

532.2. ELECTROMAGNETICWAVESANDINTERFACESReflection of TM-Wave at Brewsters Angle

x

Ei E

r

krki

z Etkt

1,

2,

1,

Figure 2.26: Conditions for reflection of a TM-Wave at Brewsters angle. The reflected and transmitted beams are orthogonal to eachother, so thatthedipolesexcited inthemediumbythetransmittedbeamcannotradiateintothedirectionofthereflectedbeam.


Figure2.27: AplateunderBrewstersangledoesnotreflectTM-light. Theplatecanbeusedasawindowtointroducegasfilledtubesintoalaserbeamwithoutinsertionloss(ideally),[6]p. 209.


42/159

54 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS2.2.5 Total Internal ReflectionAnother striking phenomenon, see Figure 2.24, occurs for the case wherethebeamhitsthesurfacefromthesideoftheopticallydensermedium, i.e.n1 > n2. There is obviously a critical angle of incidence, beyond which alllightisreflected. Howcanthatoccur? This iseasytounderstandfromthephasematchingdiagramatthesurface,seeFigure2.18,whichisredrawnforthiscaseinFigure2.28.

n n2> 1

xkrki i r

tot

z

k2

k1

Figure2.28: Phasematchingdiagramfortotal internalreflection.Thereisnorealwavenumberinmedium2possibleassoonastheangleof

incidencebecomeslargerthanthecriticalanglefortotalinternalreflectioni > tot (2.125)

withsintot =

n2. (2.126)n1

Figure2.29showstheangleofrefractionand incidence forthetwocasesofexternalandinternalreflection,whentheangleof incidenceapproachesthecriticalangle.


43/159

552.2. ELECTROMAGNETICWAVESANDINTERFACES


Figure2.29: Relationbetweenangleofrefractionandincidenceforexternalrefractionandinternalrefraction([6],p. 11).


Figure2.30: Relationbetweenangleofrefractionandincidenceforexternalrefractionandinternalrefraction([1],p. 81).

Total internal reflectionenablesbroadbandreflectors. Figure2.30shows


44/159

56 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSagain what happens when the critical angle of reflection is surpassed. Figure

2.31

shows

how

total

internal

refl

ectioncan

be

used

to

guide

light

via

reflectionataprismorbymultiplereflectionsinawaveguide.


Figure2.31: (a) Total internal reflection, (b) internal reflection in aprism,(c) Raysare guided bytotal internal reflection fromthe internal surface ofanopticalfiber([6]p. 11).

Figure2.32showstherealizationofaretroreflector,whichalwaysreturnsaparallelbeamindependentoftheorientationoftheprism(infacttheprismcan be a real 3D-corner so that the beam is reflected parallel independentfromthepreciseorientationofthecornercube). Asurfacepatternedbylittlecornercubesconstitutea"catseye"usedontraffic signs.


Figure2.32: Totalinternalreflectioninaretroreflector.Moreonreflectingprismsanditsusecanbefoundin[1],pages131-136.


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572.2. ELECTROMAGNETICWAVESANDINTERFACESEvanescent WavesWhat is thefield in medium 2 when total internal reflection occurs? Is itidenticaltozero? Itturnsoutphasematchingcanstilloccurifthepropagationconstantinz-directionbecomesimaginary,k2z =j2z,becausethenwecanfulfillthewaveequationinmedium2. Thisisequivalenttothedispersionrelation

k2 +k2 =k22x 2z 2,orwithk2x =k1x =k1sin1,weobtainfortheimaginarywavenumber

q2z = k12sin21k22, (2.127)p

= k1 sin21sin2tot. (2.128)

Theelectricfieldinmedium2isthen,fortheexampleforaTE-wave,givenby

Et = Et ey ej(tktr)

, (2.129)Et ey ej(tk2,xx)e2zz. (2.130)

Thusthewavepenetrates intomedium2exponentiallywitha1/e-depth,givenby

1 1= = p (2.131)

2z k1 sin21sin2tot

Figure2.33showsthepenetrationdepthasa functionofangleof incidenceforasilica/airinterfaceandasilicon/airinterface. Thefiguredemonstratesthatlightfrominsideasemiconductormaterialwitharelativelyhighindexaroundn=3.5ismostlycapturedinthesemiconductormaterial(Problemoflightextractionfromlightemittingdiodes(LEDs)),seeproblemset2.


46/159


0.4

0.2

0.0Penetrationdepth,m

806040200

n1=3.5n2=1n1=1.45n2=1

Angle of incidence,o

Figure2.33: Penetrationdepthfortotalinternalreflectionatasilica/airandasilicon/airinterfacefor= 0.633nm.

As the magnitude of the reflection coefficient is 1 for total internal reflection,thepowerflowingintomedium2mustvanish,i.e. thetransmissioniszero. Note,thatthetransmissionandreflectioncoefficients inEq.(2.113)canbe used beyond the critical angle for total internal reflection. We onlyhavetobeawarethattheelectricfield in medium 2 has an imaginary dependence intheexponentforthez-direction, i.e. k2z =k2cos2 =j2z.Thuscos2 inallformulasforthereflectionandtransmissioncoefficientshastobereplacedbytheimaginarynumber

k2z k1

pcos2 = =j sin21sin2tot (2.132)

k2 k2n1p= jn2 sin

21sin2tots 2sin1

= jsintot 1.


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592.2. ELECTROMAGNETICWAVESANDINTERFACESThenthereflectioncoefficientsinEq.(2.113)changetoall-passfunctions

TE-wave(s-pol.) TM-wave(p-pol.)rT E = n1cos1n2cos2

n1cos1+n2cos2r rT M = n2cos2 n1cos1n2

cos2+ n1cos1rrT E = cos1+jn2n1 sin1sintot

21

cos1jn2n1r

sin1sintot

21r

rT M = cos1+jn1n2 sin1sintot21

cos1jn1n2r

sin1sintot

21r

tanT E 2 = 1cos1 n2n1 sin1sintot

21 tanT M

2 = 1cos1 n1n2 sin1sintot2

1(2.133)

Thus the magnitude of the reflection coefficient is 1. However, there is anon-vanishingphaseshift forthe lightfieldupontotal internal reflection,denotedasT E andT M inthetableabove. Figure2.34showsthesephaseshiftsfortheglass/airinterfaceandforbothpolarizations.

200

100

0PhaseinReflection,

o

806040200

n1=1.45

n2=1

TE-Wave(s-polarized)

TM-Wave(p-polarized)

Brewsterangle

Angle of incidence,o

Figure 2.34: Phase shifts for TE- and TM- wave upon reflection from asilica/airinterface,withn1 = 1.45andn2 = 1.Goos-Haenchen-ShiftSofar,welookedonlyatplanewavesundergoingreflectionatsurfaceduetototal internalreflection. Ifabeamoffinitetransversesize isreflected from


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60 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSsuchasurfaceitturnsoutthatitgetsdisplacedbyadistancez,seeFigure2.35

(a),

called

Goos-Haenchen-Shift.


Figure2.35: (a)Goos-HaenchenShiftandrelatedbeamdisplacementuponreflectionofabeamwithfinitesize; (b)Accumulation ofphaseshifts inawaveguide.

Detailedcalculationsshow(problemset2),thatthedisplacementisgivenby

z= 2TE/TMtan1, (2.134)asifthebeamwasreflectedatavirtuallayerwithdepthTE/TMintomedium2. Itturnsout,thatforTE-waves

T E=, (2.135)where

is

the

penetration

depth

according

to

Eq.(2.131)

for

evanescent

waves. ButforTM-wavesT M = 2 (2.136)

n11 + sin211n2


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612.2. ELECTROMAGNETICWAVESANDINTERFACESThese shiftsaccumulate whenthebeam is propagating in awaveguide, seeFigure

2.35

(b)

and

is

important

to

understand

the

dispersion

relations

of

waveguidemodes. TheGoos-Haenchenshiftcanbeobservedbyreflectionataprismpartiallycoatedwithasilverfilm,seeFigure2.36. Thepartreflectedfromthesilverfilmisshiftedwithrespecttothebeamreflectedduetototalinternalreflection,asshowninthefigure.


Figure 2.36: Experimental proof of the Goos-Haenchen shift by total internal

refl

ection

at

aprism,

that

is

partially

coated

with

silver,

where

the

penetrationoflightcanbeneglected. [3]p. 486.

Frustratedtotal internal reflectionAnother proof for the penetration of light into medium 2 in the case oftotal internalreflectioncanbeachievedbyputtingtwoprisms,wheretotalinternal reflection occurs back to back, see Figure 2.37. Then part of thelight, depending on the distance between the two interfaces, is convertedbackintoapropagatingwavethatcanleavethesecondprism. Thiseffectiscalledfrustratedinternalreflectionand itcanbeusedasabeamsplitterasshowninFigure2.37.


50/159



Figure2.37:

Frustrated

total

internal

refl

ection.

Partof

the

light

is

picked

upbythesecondsurfaceandconvertedintoapropagatingwave.

2.3 Mirrors, Interferometers and Thin-FilmStructures

Oneofthemoststrikingwavephenomenais interference. Manyopticaldevicesarebasedontheconceptofinterferingwaves,suchaslowlossdielectricmirrorsandinterferometersandotherthin-filmopticalcoatings. Afterhavingaquicklookintothephenomenonofinterference,wewilldevelopeapowerfulmatrix formalism that enables us to evaluate efficiently many optical (alsomicrowave)systemsbasedoninterference.2.3.1 Interference and CoherenceInterferenceInterferenceofwaves isaconsequenceofthe linearityofthewaveequation(2.13). Ifwehavetwoindividualsolutionsofthewaveequation

E1(r,t) = E1cos(1tk1 r+1)e1, (2.137)E2(r,t) = E2cos(2tk1 r+2)e2, (2.138)

with arbitrary amplitudes, wave vectors and polarizations, the sum of thetwofields(superposition)isagainasolutionofthewaveequation

E(r,t) = E1(r,t) + E2(r,t). (2.139)


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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES63If we lookat the intensity, wich is proportional to theamplitude square ofthe

total

field

2E(r,t)2 = E1(r,t) + E2(r,t) , (2.140)wefind

E(r,t)2 = E1(r,t)2+ E2(r,t)2+ 2E1(r,t) E2(r,t) (2.141)with E2

E1(r,t)2 = 1 1 + cos 2(1tk1 r+ 1) , (2.142)2

E2

E2(r,t)2 = 2 1 + cos 2(2t r+ 2) , (2.143)k2 2E1(r,t) E2(r,t) = (e1 e2) E1E2cos(1tk1 r+ 1) (2.144)

cos(2tk2 r+ 2)1

E1(r,t) E2(r,t) =2

(e1 e2) E1E2 (2.145) cos

(12) t

k1k2 r+ (12) (2.146)

+cos (1+ 2) t k1+ k2 r+ (1+ 2)Since at optical frequencies neither our eyes nor photo detectors, can everfollowtheopticalfrequencyitselfandcertainlynottwiceaslargefrequencies,wedroptherapidlyoscillatingterms. Orinotherwordswelookonlyonthecycle-averagedintensity,whichwedenotebyabar

E2 E2E(r,t)2 = 1 + 2 + (e1 e2) E1E2

2 2 cos (12) t k1k2 r+ (12) (2.147)

Dependingonthefrequencies1 and2 andthedeterministicandstochasticproperties ofthe phases 1 and2, wecandetect thisperiodicallyvaryingintensity pattern called interference pattern. Interference of waves can bebestvisuallizedwithwaterwaves,seeFigure2.38. Note,however,thatwaterwavesareascalarfield,whereastheEM-wavesarevectorwaves. Therefore,the interference phenomena of EM-waves are much richer in nature than


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64 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSfor water waves. Notice, from Eq.(2.147), it follows immedicatly that theinterference

vanishes

in

the

case

of

orthogonally

polarized

EM-waves,

because

ofthescalarproduct involved. Also, ifthe frequenciesofthewavesarenotidentical,theinterferencepatternwillnotbestationaryintime.


Figure2.38: Interferenceof water waves fromtwopointsources inarippletank[1]p. 276.

If the frequencies are identical, the interference pattern depends on thewave vectors, see Figure 2.39. The interference pattern which has itself awavevectorgivenby

k1k2 (2.148)showsaperiodof

2 . (2.149)= k1k2


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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES65

k1

k2

Wavefronts

Lines of constantdifferentialphase

Figure 2.39: Interference pattern generated by two monochromatic planewaves.

CoherenceTheabilityofwavestogenerateaninterferencepattern iscalledcoherence.Coherencecanbequantifiedbothtemporallyorspatially. Forexample,ifweareatacertainpositionrintheinterferencepatterndescribedbyEq.(2.147),wewillonlyhavestationaryconditionsoveratimeinterval

2Tcoh


54/159

66 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSIt is by no means trivial to arrive at a light source and an experimen

talsetup

that

enables

good

coherence

and

strong

interference

of

the

beams

involved.

Interferenceofbeamscanbeusedtomeasurerelativephaseshiftsbetweenthem which may be proportional to a physical quantity that needs to bemeasured. Suchphaseshiftsbetweentwobeamscanalsobeusedtomodulatethe light output at a given position in space via interference. In 6.013, wehavealreadyencounteredinterferenceeffectsbetweenforwardandbackwardtravelingwavesontransmissionlines. Thisisverycloselyrelatedtowhatweuseinoptics,therefore,wequicklyrelatetheTEM-waveprogagationtothetransmissionlineformalismdevelopedinChapter5of6.013.2.3.2 TEM-Waves and TEM-Transmission LinesThemotionofvoltageV andcurrentI alongaTEMtransmissionlinewithaninductanceL0 andacapacitanceC perunitlengthissatisfies0

V(t,z) I(t, z)= L0 (2.150)

z t I(t,z) V(t,z)

= C0 (2.151)z t

Substitutionoftheseequationsintoeachotherresultsinwaveequationsforeitherthevoltageorthecurrent

2V(t,z) 1 2V(t, z)z 2 c2 t2 = 0, (2.152)

2I(t, z) 1 2I(t, z)z 2 c2 t2 = 0, (2.153)

where c = 1/L0C0 is the speed of wave propagation on the transmissionline. Theratiobetweenvoltageandcurrent formonochomaticwaves isthepcharacteristicimpedanceZ= L0/C0.

Theequationsofmotionfortheelectricandmagneticfieldofax-polarizedTEM wave according to Figure 2.1, with Efield along the x-axis and H-fi

eldsalong

the

y- axis

follow

directly

from

Faradays

and

Amperes

law

E(t, z) H(t, z)

= , (2.154)z t

H(t, z) E(t, z)= , (2.155)

z t


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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES67which are identical to the transmission line equations (2.150) and (2.151).Substitution

of

these

equations

into

each

other

results

again

in

wave

equations

forelectricandmagneticfieldspropagatingatthespeedoflightc= 1/pandwithcharacteristicimpedanceZF = /.

The solutions of the wave equationare forward andbackward travelingwaves,whichcanbedecoupledbytransformingthefieldstotheforwardandbackwardtravelingwaves r

Aeffa(t,z) = (E(t,z) + ZF oH(t,z)) , (2.156)

2ZF

rAeff

b(t,z) = 2ZF (E(t,z) ZF oH(t,z)) , (2.157)

whichfulfilltheequations 1

+ a(t,z) = 0, (2.158)z c t 1

z c t b(t,z) = 0. (2.159)Note,weintroducedthatcrosssectionAeff suchthat|a|2 isproportionaltothetotalpowercarriedbythewave. Clearly,thesolutionsare

a(t,z) = f(t

z/c0), (2.160)b(t,z) = g(t+ z/c0), (2.161)

whichresemblestheDAlembertsolutionsofthewaveequationsfortheelectricandmagneticfield s

ZF oE(t,z) = (a(t,z) + b(t,z)) , (2.162)

2Aeffs1

H(t,z) = (a(t,z) b(t,z)) . (2.163)2ZF oAeff

Here, the forward and backward propagatingfields are already normalizedsuchthatthePoyntingvectormultipliedwiththeeffectiveareagivesalreadythetotalpowertransportedbythefieldsintheeffectivecrosssectionAeff

P = S(Aeff ez) = AeffE(t,z)H(t,z) = |a(t,z)|2|b(t,z)|2. (2.164)


56/159

68 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSIn 6.013, itwas shown that the relationbetweensinusoidal current and

voltagewaves

V(t,z) = Re V(z)ejt andI(t, z) = Re I(z)ejt (2.165)

along the transmission line orcorrespondingelectric andmagneticfields inone dimensional wave propagation is described by a generalized compleximpedance Z(z) that obeys certain transformation rules, see Figure 2.40(a).

Figure2.40: (a)Transformationofgeneralizedimpedancealongtransmissionlines,(b)Transformationofgeneralizedimpedanceaccrossfreespacesectionswithdifferentcharacterisitcwaveimpedancesineachsection.

Alongthefirsttransmissionline,whichisterminatedbyaloadimpedance,thegeneralizedimpedancetransformsaccordingto

Z1(z) = Z1 Z0jZ1tan(k1z) (2.166)Z1

jZ0

tan(k1z)withk1 =k0n1 andalongthesecondtransmissionlinethesameruleappliesasanexample

Z2(z) = Z2 Z1(L1) jZ2tan(k2z) (2.167)Z2jZ1(L1)tan(k2z)


57/159

2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES69withk2 =k0n2.Note,thatthemediacanalsobe lossy,thenthecharacteristic

impedances

of

the

transmission

lines

and

the

propagation

constants

are

already themselves complex numbers. The same formalism can be used tosolvecorrespondingonedimensionalEM-wavepropagationproblems.Antireflection CoatingThe task of an antireflection (AR-)coating, analogous to load matching intransmission linetheory, istoavoidreflectionsbetweenthe interfaceoftwomedia with different optical properties. One method of course could be toplace the interface at Brewsters angle. However, this is not always possible. Lets assume we want to put a medium with index n into a beam under normal

incidence,

without

having

refl

ections

on

the

air/medium

interface.

Themediumcanbeforexamplealens. This isexactlythesituationshownin Figure 2.40 (b). Z2 describes the refractive index of the lense material,e.g. n2 = 3.5 for a silicon lense, we can deposit on the lens a thin layerof material with indexn1 corresponding to Z1 andthis layershouldmatchto the free space index n0 = 1 or impedance Z0 = 377. Using (2.166) weobtain

Z2 =Z1(L1) = Z1Z0jZ1tan(k1L1) (2.168)Z1jZ0tan(k1L1)Ifwechooseaquarterwavethickmatchinglayerk1L1 =/2,thissimplifiestothefamousresult

Z2Z2 = 1, (2.169)Z0

orn1 = n2n0 andL1 = . (2.170)4n1

Thus a quarter wave AR-coating needs a material which has an indexcorrespondingtothegeometricmeanof thetwomediatobematched. Inthecurrentexamplethiswouldben2 =3.51.872.3.3 Scattering and Transfer MatrixAnother formalism to analyze optical systems (or microwave circuits) canbe formulated using the forward and backward propagating waves, whichtransformmuchsimpleralongahomogenoustransmissionlinethanthetotalfields, i.e. the sumof forward adnbackward waves. However, at interfaces


58/159

70 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSscatteringofthesewavesoccurswhereasthetotalfieldsarecontinuous. Formonochromatic

forward

and

backward

propagating

waves

a(t,z) = a(z)ejt andb(t,z) = b(z)ejt (2.171)

propagating inz-directionoveradistancez withapropagationconstantk,wefindfromEqs.(2.158)and(2.159)

a(z) ejkz 0 a(0)

= jkz . (2.172)b(z) 0 e b(0)A piece of transmission line is a two port. The matrix transforming theamplitudesofthewavesattheinputport(1)tothoseoftheoutputport(2)iscalledthetransfermatrix,seeFigure2.41

T

Figure2.41: DefinitionofthewaveamplitudesforthetransfermatrixT.

For example, from Eq.(2.172) follows that the transfer matrix for freespacepropagationis

ejkz 0T=

0 ejkz . (2.173)This formalism can be expanded to arbitrary multiports. Because of itsmathematical properties the scattering matrix that describes the transformationbetweenthe incomingandoutgoingwaveamplitudesofamultiportisoftenused,seeFigure2.42.


59/159


S

Figure2.42: Scatteringmatrixanditsportdefinition.Thescatteringmatrixdefinesalineartransformationfromtheincoming

totheoutgoingwavesb= Sa, witha= (a1,a2,...)T ,b= (b1,b2,...)T . (2.174)

Note,thatthemeaningbetweenforwardandbackwardwavesno longercoincides with a and b, a connection, which is difficult to maintain if severalports come in from many differentdirections.

ThetransfermatrixT hasadvantages, ifmanytwoportsareconnectedin series with each other. Then the total transfer matrix is the product oftheindividualtransfermatrices.2.3.4 Properties of the Scattering MatrixPhysialpropertiesofthesystemreflectitselfinthemathematicalpropertiesofthescatteringmatrix.ReciprocityAsystemwithconstantscalardielectricandmagneticpropertiesmusthaveasymmetricscatteringmatrix(withoutproof)

S= ST. (2.175)


60/159

72 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSLosslessnessInalosslesssystemthetotalpowerflowingintothesystemmustbeequaltothepowerflowingoutofthesysteminsteadystate

2b 2 (2.176)|a| = ,

i.e.S

+S= 1 orS1=S+. (2.177)

Thescatteringmatrixofalosslesssystemmustbeunitary.Time ReversalTo find the scattering matrix of the time reversed system, we realize thatincoming waves become outgoing waves under time reversal and the otherway around, i.e. the meaning of aandb is exchanged and on top of it thewavesbecomenegativefrequencywaves.

aej(tkz) timereversal aej(tkz). (2.178)To obtain the complex amplitude of the corresponding positive frequencywave,weneedtotakethecomplexconjugatevalue. Sotoobtaintheequations forthetimereversedsystemwehavetoperformthe followingsubstitutions

Originalsystem Timereversedsystem= Sb S1 . (2.179)ab= Sa a b=

2.3.5 BeamsplitterAsanexample,welookatthescatteringmatrixforapartiallytransmittingmirror, which could be simply formed by the interface between two mediawith different refractive index, which we analyzed in the previous section,seeFigure 2.43. (Note, for brevityweneglect thereflections at the normalsurfaceinputtothemedia,orweputanAR-coatingonthem.) Inprinciple,thisdevicehasfourportsandshouldbedescribedbya4x4matrix. However,most often only one of the waves is used at each port, as shown in Figure2.43.


61/159


Figure2.43: Portdefinitionsforthebeamsplitter

Thescatteringmatrixisdeterminedbyb= Sa, witha = (a1,a2)T , b = (b3, b4)T (2.180)

and S=

jtr jt

r , withr2+ t2 = 1. (2.181)

Thematrix

Swas

obtained

using

using

the

S-matrix

properties

described

above. From Eqs.(2.113) we could immediately identify r as a function ofthe refractive indices, angle of incidence and the polarization used. Note,that theoff-diagonal elements ofSare identical, which is a consequence ofreciprocity. Thatthemaindiagonalelementsare identical isaconsequenceof unitarity for a lossless beamsplitter and furthermore t= 1 r2. For agivenfrequencyrandt canalwaysbemaderealbychoosingproperreferenceplanesatthe inputandtheoutputofthebeamsplitter. Beamsplitterscanbemadeinmanyways,seeforexampleFigure2.37.

2.3.6 InterferometersHavingavaliddescriptionofabeamsplitterathand,wecanbuildandanalyzevarioustypesofinterferometers,seeFigure2.44.


62/159



Figure 2.44: Different types of interferometers: (a) Mach-Zehnder Interferometer;(b)MichelsonInterferometer;(c)SagnacInterferometer[6]p. 66.

Each of these structures has advantages and disadvantages dependingon the technology they are realized. The interferometer in Figure 2.44 (a)is called Mach-Zehnder interferometer, the one in Figure 2.44 (b) is calledMichelsonInterferometer. IntheSagnacinterferometer,Figure2.44(c)bothbeamsseeidenticalbeampathandthereforeerrorsinthebeampathcanbebalance outandonlydifferentialchanges due toexternal influences leadtoan

output

signal,

for

example

rotation,

see

problem

set

3.

Tounderstandthe lighttransmissionthroughaninterferometerweana

lyzeas anexampletheMach-Zehnder interferometer shown inFigure2.45.If we excite input port 1 with a wave with complex amplitude a0 and no

inputatport2andassume50/50beamsplitters,thefirstbeamsplitterwill


63/159


3

4

Figure2.45: Mach-ZehnderInterferometerproducetwowaveswithcomplexamplitudes

b3 =12a0 (2.182)b4 =j12a0

Duringpropagationthroughthe interferometerarms, bothwavespickupaphasedelay3 =kL3 and4 =kL4,respectively

a5 =12a0ej3,a6 =j12a0ej4. (2.183)

Afterthesecondbeamsplitterwiththesamescatteringmatrixasthefirstej3 ,

a0 ej3

Thetransmittedpowertotheoutputportsis2

one,weobtain1 ej4

+ej4b7 = a0 (2.184)2=j12

b8 .2 22

4| = |a0| |a0|

21ej(34)1 + ej(34)

|b7 [1cos(34)],= (2.185)

2| |

Thetotaloutputpowerisequaltotheinputpower,asitmustbeforalosslesssystem. However,dependingonthephasedifference=34 betweenboth arms, the power is split differently betweenthe twooutput ports, seeFigure2.46.Withproperbiasing, i.e. 34 =/2 + ,thedifference in

2 22 = |a0|4 |a0|2b8 [1+cos(3

4)].=


64/159

1.0

0.8

0.6

0.4

0.2

0.0

Outputpower

|b7|2

|b8|

2


-3 -2 -1 0 1 2 3

Figure 2.46: Output power from the two arms of an interfereometer as afunctionofphasedifference.output power between the two arms can be made directly proportional tothephasedifference.

Opening up the beamsize in the interferometer and placing optics intothebeamenablestovisualizebeamdistortionsduetoimperfectopticalcomponents,seeFigures2.47and2.48.


Figure2.47: Twyman-GreenInterferometertotestopticsquality[1]p. 324.


65/159



Figure 2.48: Interference pattern with a hot iron placed in one arm of theinterferometer([1],p. 395).

2.3.7 Fabry-Perot ResonatorInterferometerscanactasfilters. Thephasedifferencebetweentheinterferometerarmsdependsonfrequency,therefore,thetransmissionfrominputtooutputdependsonfrequency,seeFigure2.46.However,thefilterfunctionisnotverysharp. Thereason forthis is that onlyatwobeam interference isused. Muchmorenarrowbandfilterscanbeconstructedbymultipass interferencessuchas inaFabry-PerotResonator, seeFigure2.49. ThesimplestFabryPerotisdescribedbyasequenceofthreelayerswhereatleastthemiddlelayerhasanindexdifferentfromtheothertwolayers,suchthatreflectionsoccurontheseinterfaces.


66/159


n2n1 n3

1 2S S21

Figure2.49: Multipleintereferences inaFabryPerotresonator. InthesimplestimplementationaFabryPerotonlyconsistsofasequenceofthreelayerswithdifferentrefractiveindexsothattworeflectionsoccurwithmultipleinterferences. Each of this discontinuites can be described by a scatteringmatrix.

Any kind of device that has reflections at two parallel interfaces mayact as a Fabry Perot such as two semitransparent mirrors. A thin layerof material against air can act as a Fabry-Perot and is often called etalon.Given the reflectionand transmissioncoefficientsat the interfaces 1 and2,we can write down the scattering matrices for both interfaces according toEqs.(2.180)and(2.181).

b1 r1 jt1 a1 b3 r2 jt2 a3b2

=jt1 r1 a2 and b4 = jt2 r2 a4 .

(2.186)If we excite the Fabry-Perot with a wave from the right with amplitude.a1 = 0,thenafractionofthatwavewillbetransmittedtotheinterfaceinto6theFabry-Perotaswave b2 and part will be already reflectedinto b1,

(0)

b1 = r1a1. (2.187)Thetransmittedwavewillthenpropagateandpickupaphasefactorej/2,with= 2k2Landk2 = 2n2,

a3=jta1ej/2. (2.188)


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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES79Afterpropagationitwillbereflectedofffromthesecondinterfacewhichhasa re

fl

ectioncoe

ffi

cient b3

2 = = r2. (2.189)a3

a4

=0Thenthereflectedwaveb3 propagatesbacktointerface1,pickingupanother

(1)phasefactorej/2 resultinginanincomingwaveafteroneroundtripofa2 =jt1r2e

ja1. Upon reflectionon interface1,partofthiswave istransmittedleadingtoanoutput

(1)b1 =jt1jt1r2eja1. (2.190)

Thepartialwavea(1)2 isreflectedagainandafteranotherroundtripitarrives(2)

at interface 1 as a2 = (r1r2) ej jt1r2eja1. Part of this wave is transmittedandpartof itisreflectedbacktogothroughanothercycle. Thusintotal ifwesumupallpartialwavesthatcontributetotheoutputatport1oftheFabry-Perotfilter,weobtain

X (n)b1 = b1

n=0 !X= r1t21r2ej r1r2ej a1

n=0

ej

= r1t21r21 r1r2ej a1

= r1r2ej a1 (2.191)1 r1r2ej Note, that the coefficient in front of Eq.(2.191) is the coefficient S11 of thescatteringmatrixoftheFabry-Perot. Inasimilarmanner,weobtain

b

b3

4

= S

aa

1

2

(2.192)1 r1r2ej t1t2ej/2and

S=1 r1r2ej

t1t2ej/2 r2r1ej

(2.193)In the following, we want to analyze the properties of the Fabry-Perot for thecaseofsymmetricreflectors,i.e. r1 = r2 andt1 = t2.Thenweobtainfor


68/159

80 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSthepowertransmissioncoefficientoftheFabry-Perot, S21 2 intermsofthe

2 | |power

refl

ectivityof

the

interfaces

R

=

r

|S21 2| = 1R1Rej 2 2(1R)

2+ 4Rsin2(/2) (2.194)= (1R)2Figure2.50showsthetransmission |S21|

=R. oftheFabry-Perot interferometerforequalreflectivities r1 2 = r2 2| | | |1.0

0.8

0.6

0.4

0.2

0.0Fabry-PerotTransmissio

n,

|S21|2

R=0.1

R=0.5

R=0.7

R=0.9

FSR

fFWHM

0.0 0.5 1.0 1.5 2.0 2.5 3.0(f -fm)/ FSR

Figure 2.50: Transmission of a lossless Fabry-Perot interferometer with2 2|r1| =|r2| =RAtvery lowreflectivityRofthemirrorthetransmissionisalmostevery

where 1, there is only a slight sinusoidal modulation due to thefirst orderinterferences which are periodically in phase and out of phase, leading to100% transmission or small reflection. For large reflectivity R, due to thethenmultipleinterferenceoperationoftheFabry-PerotInterferometer,verynarrowtransmissionresonances emerge at frequencies, where the roundtrip phaseintheresonatorisequaltoamultipleof2

2f= n22L= 2m, (2.195)

c0


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21 3 4

2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES81whichoccursatacomboffrequencies,seeFigure2.51

fm =m c0 . (2.196)2n2L

Figure 2.51: Developement of a set of discrete resonances in a onedimensionsalresonator.

On a large frequency scale, a set of discrete frequencies, resonances ormodesarise. Thefrequencyrangebetweenresonances iscalledfreespectralrange(FSR)oftheFabry-PerotInterferometer

c0 1F SR = = , (2.197)2n2L TR

which is the inverse roundtrip time TR of the light in the one-dimensonalcavityor resonator formed by the mirrors. Thefiltercharacteristic of eachresonancecanbeapproximatelydescribedbyaLorentzianlinederivedfromEq.(2.194)bysubstitutingf =fm+f withf FSR,

2 (1R)22|S21| = (1R) + 4R sin2

m2+ 2f

/2

F SR

1 2R f 2, (2.198)1 +

1R F SR 1 f 2, (2.199)1 +

fF W H M /2

1.0

0.8

0.6

0.4

0.2

0.0Fabry

-PerotTransmission,

|S21

|2

5mf

am

fmfm-1 fm+1 fm+2

am-1 am+1 am+2

frequency


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82 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSwhereweintroducedtheFWHMofthetransmissionfilter

F SR fF W H M = , (2.200)F

withthefinesseoftheinterferometerdefinedas

R F =

1R . (2.201)TThe last simplificationisvalidforahighlyreflectingmirrorR1andT isthemirrortransmission. Fromthisrelation it is immediatelyclearthatthefinessehastheadditionalphysicalmeaningoftheopticalpowerenhancementinside

the

Fabry-Perot

at

resonance

besides

the

factor

of

,

since

the

power

inside the cavity must be larger by 1/T, if the transmission through theFabry-Perotisunity.2.3.8 Quality Factor of Fabry-Perot ResonancesAnother quantity often used to characterize a resonator or a resonance isits quality factor Q, which is defined as the ratio between the resonancefrequencyandthedecayratefortheenergystoredintheresonator,whichisalsooftencalledinversephotonlifetime,1ph

Q=phfm. (2.202)Letsassume,energyisstoredinoneoftheresonatormodeswhichoccupiesarangeoffrequencies[fmFSR/2, fm+FSR/2]asindicatedinFigure2.52.Thenthefourierintegral Z +FSR/2

am(t) = b2(f)ej2(ffm)t df, (2.203)FSR/2

2

b2(f)

oftheforwardtravelingwaveintheresonatorgivesthemodeamplitudeofthem-thmodeanditsmagnitudesquareistheenergystoredinthemode. Note,thatwecouldhavetakenanyoftheinternalwavesa2,b2, a3, andb3. The time dependentfieldwecreatecorrespondstothefieldoftheforwardorbackwardtravelingwaveatthecorrespondingreferenceplaneintheresonator.

where isnormalizedsuchthatitdescribesthepowerspectraldensity


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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES831.0

0.8

0.6

0.4

0.2

0.0Transmission,|S21|2

am(t) am+1(t)am-1(t)

0.0 0.5 1.0 1.5 2.0 2.5 3.0(f -fm)/ FSR

Figure2.52: Integrationoverallfrequencycomponentswithinthefrequencyrange[fmFSR/2, fm+FSR/2]definesamodeamplitudea(t)withaslowtimedependence

We now make a "Gedanken-Experiment". We switch on the incomingwaves a1() and a4() to load the cavity with energy and evaluate the internal wave b2(). Instead of summing up all the multiple reflections likewe did in constructing the scattering matrix (2.192), we exploit our skillsin analyzing feedback systems, which the Fabry-Perot filter is. The scatteringequationsset forcebythetwo scattering matrices characterizingtheresonator mirrors in the Fabry-Perot can be visuallized by the signalflowdiagraminFigure2.53

~

j t+

r

+

a2

_~

a1_

~ b2_

b1

_~

j t

r

j t+

r

+

a4_

~

a3_

~b4

_~

b3

_~

j t

r

e-j/2

e-j/2

S1 S2

Figure 2.53: Representation of Fabry-Perot resonator by a signalflow diagram


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84 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSForthetasktofindtherelationshipbetweenthe internalwaves feedby

theincoming

wave

only

the

dashed

part

of

the

signal

flow

is

important.

The

internal feedback loopcanbeclearlyrecognizedwithaclosed looptransferfunction

r2ej,whichleadstotheresonancedenominator

1 r2ej ineveryelementoftheFabry-Perotscatteringmatrix(2.192). UsingBlacksformulafrom6.003andthesuperpositionprinciplewe immediatelyfind fortheinternalwave

b2 = 1 rjt

2ej

a1+ rej/2a4

. (2.204)Closetooneoftheresonance frequencies,= 2fm + ,usingt= 1 r2,(2.204)canbeapproximatedby

mejTR/2b2() 1 +j1

jR TR

a1() + r(1) a4(), (2.205)R

j

1 +jTR/T a1() + r(1)mejTR/2a4() (2.206)

forhighreflectivityR.Multiplicationofthisequationwiththeresonantdenominator

(1 +jTR/T) b2() ja1() + r(1)mejTR/2a4() (2.207)and inverse Fourier-Transform in the time domain, while recognizing thattheinternalfieldsvanishfaroffresonance,i.e.Z + FSR Z

am(t) = b2()ejt d= +b2()ejt d, (2.208)F SR

weobtainthe followingdifferentialequation forthemodeamplitudeslowlyvarying in time

dTR a (t) = T(am(t) +ja1(t) +j(1)ma4(tTR/2)) (2.209)dt m


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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES85withtheinputfields

Z + FSR a1/4(t) = a1/4()ejt d. (2.210) FSR

Despitethepaintoderivethisequationthephysicalinterpretationisremarkablysimpleandfarreachingaswewillseewhenweapplythisequationlateron to many different situations. Lets assume, we switch off the loading ofthecavityatsomepoint,i.e. a1/4(t) = 0,thenEq.(2.209)resultsin

am(t) = am(0)et/(TR/T) (2.211)And

the

power

decays

accordingly

|a (t)|2 =|a (0)|2et/(TR/2T) (2.212)m m

twice as fast as the amplitude. The energy decay time of the cavity is often calledthecavityenergydecaytime,orphotonlifetime,ph,whichishere

TRph = .

2TNote, the factor of twocomes fromthe fact that eachmirror of the Fabry-PerothasatransmissionT perroundtriptime.For exampl a L= 1.5mlongcavitywithmirrorsof0.5%transmission,i.e. TR = 10nsand2T = 0.01hasaphotonlifetimeof1s. Itneedshundredbouncesonthemirrorforaphotontobeessentiallylostfromthecavity.

Highestqualitydielectricmirrorsmayhaveareflectionlossofonly105...6,thisisnotreallytransmissionbutratherscatteringlossinthemirror. Suchhighreflectivitymirrorsmayleadtotheconstructionofcavitieswithphotonlifetimesontheorderofmilliseconds.

Now,thatwehaveanexpressionfortheenergydecaytimeinthecavity,wecanevaluatethequalityfactoroftheresonator

mQ

=

fmph = . (2.213)2T

Again for a resonator with the same parameters as before and at opticalfrequencies of 300THz corresponding to 1m wavelength, we obtain Q =2 108.


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86 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICSThin-FilmFilters

Transfermatrixformlismisanefficientmethodtoanalyzethereflectionandtransmission properties of layered dielectric media, such as the one shown

Using the transfer matrix method, it is an easy task tocomputethetransmissionandreflectioncoefficientsofastructurecomposedof layerswitharbitrary indices andthicknesses. Aprominent example ofathin-filmfilterareBraggmirrors. Thesearemadeofaperiodicarrangement

re

efl

ct

t

i

iv

y

oftwo layers with low and high index n1 andn2,respectively. Formaximumreflectionbandwidth,thelayerthicknessesarechosentobequarterwaveforthewavelengthmaximumreflectionoccures,n1d1

Figure 2.54: Thin-Film dielectric mirror composed of alternating high and

AsanexampleFigure2.55showsthereflectionfromaBraggmirrorwith4 for a center wavelength of 0

Figure2.55: Reflectivityofan8pairquarterwaveBraggmirrorwithn14designedforacenterwavelengthof800nm. Themirroris

embeddedinthesamelowindexmaterial.

2.3.9

in Figure 2.54.

=0/4andn2d2 =0/4~~

n1 n1 n1n2n2 n2

d1 d1 d1d2 d2 d2

_b2_a1

....

~ ~_b1 _a2

lowindexlayers.

The layer= 1.45, n2 = 2. = 800nm.n1thicknessesarethend1 =134nmandd2 =83nm.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0500 600 700 800 900 1000 1100 1200 1300

lambda

=45andn2 = 2.1.


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872.4. PARAXIALWAVEEQUATIONANDGAUSSIANBEAMS2.4 ParaxialWaveEquationandGaussianBeamsSo far, we have only treated optical systems operating with plane waves,which is an idealization. In reality plane waves are impossible to generatebecause of there infiniteamount of energy requiredto do so. The simplest(approximate)solutionofMaxwellsequationsdescribingabeamoffinitesizeistheGaussianbeam. InfactmanyopticalsystemsarebasedonGaussianbeams. Most lasers are designed to generate a Gaussian beam as output.GaussianbeamsstayGaussianbeamswhenpropagatinginfreespace. However,duetoitsfinitesize,diffractionchangesthesizeofthebeamandlensesareimployedtoreimageandchangethecrosssectionofthebeam. Inthissection,wewanttostudythepropertiesofGaussianbeamsanditspropagationandmodificationinopticalsystems.2.4.1 Paraxial Wave EquationWestartfromtheHelm

classical electromagnetism and optics

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