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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18 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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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20 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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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22 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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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26 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
Image removed for copyright purposes.
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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28 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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
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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
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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)
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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
40 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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)
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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
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44 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS2.2.2 Measuring Refractive Index with Minimum De-
viationSnellslawcanbeusedformeasuringtherefractiveindexofmaterials. Consideraprismpreparedfromamaterialwithunknownrefractiveindexn(),seeFig. 2.19(a).
Image removed for copyright purposes.
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.
Image removed for copyright purposes.
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46 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
Image removed for copyright purposes.
Figure2.21: Deviationversusincidentangel[1]
Figure2.22: Refractive indexasafunctionofwavelengthforvariousmediatransmissiveinthevisible[1],p42.
Image removed for copyright purposes.
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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)
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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).
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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.
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50 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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
Image removed for copyright purposes.
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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
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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.
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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.
Image removed for copyright purposes.
Figure2.27: AplateunderBrewstersangledoesnotreflectTM-light. Theplatecanbeusedasawindowtointroducegasfilledtubesintoalaserbeamwithoutinsertionloss(ideally),[6]p. 209.
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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.
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552.2. ELECTROMAGNETICWAVESANDINTERFACES
Image removed for copyright purposes.
Figure2.29: Relationbetweenangleofrefractionandincidenceforexternalrefractionandinternalrefraction([6],p. 11).
Image removed for copyright purposes.
Figure2.30: Relationbetweenangleofrefractionandincidenceforexternalrefractionandinternalrefraction([1],p. 81).
Total internal reflectionenablesbroadbandreflectors. Figure2.30shows
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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.
Image removed for copyright purposes.
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.
Image removed for copyright purposes.
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.
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58 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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.
Image removed for copyright purposes.
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.
Image removed for copyright purposes.
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.
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62 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
Image removed for copyright purposes.
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.
Image removed for copyright purposes.
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
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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)
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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)
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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
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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.
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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES71
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)
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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.
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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES73
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.
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74 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
Image removed for copyright purposes.
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
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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES75
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)].=
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1.0
0.8
0.6
0.4
0.2
0.0
Outputpower
|b7|2
|b8|
2
76 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
-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.
Image removed for copyright purposes.
Figure2.47: Twyman-GreenInterferometertotestopticsquality[1]p. 324.
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2.3. MIRRORS,INTERFEROMETERSANDTHIN-FILMSTRUCTURES77
Image removed for copyright purposes.
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.
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78 CHAPTER2. CLASSICALELECTROMAGNETISMANDOPTICS
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
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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
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