Fundamentals of Modern Optics Winter Term 2012/2013 Prof. Thomas Pertsch Abbe School of Photonics Friedrich-Schiller-Universität Jena

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1 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox Fundamentals of Moden Optis Winte Tem /3 Pof. Thomas Petsh Abbe Shool of Photonis Fiedih-Shille-Univesität Jena This sipt is based on the letue seies Theoetishe Optik by Pof. Falk Ledee at the FSU Jena and adapted to English by Pof. Stefan Skupin fo the intenational eduation pogam of the Abbe Shool of Photonis. Table of ontent. Intodution (Ray optis - geometial optis) Intodution Postulates Simple ules fo popagation of light Simple optial omponents Ray taing in inhomogeneous media (gaded-index - GRIN optis) Ray equation The eikonal equation Matix optis The ay-tansfe-matix Maties of optial elements Casaded elements Optial fields in dispesive and isotopi media Maxwell s equations Adaption to optis Tempoal dependene of the fields Maxwell s equations in Fouie domain Fom Maxwell s equations to the wave equation Deoupling of the vetoial wave equation Optial popeties of matte Basis Dieleti polaization and suseptibility Condutive uent and ondutivity The genealized omplex dieleti funtion Mateial models in time domain The Poynting veto and enegy balane Time aveaged Poynting veto Time aveaged enegy balane Nomal modes in homogeneous isotopi media Tansvesal waves Longitudinal waves Plane wave solutions in diffeent fequeny egimes Time aveaged Poynting veto of plane waves Beams and pulses - analogy of diffation and dispesion Popagation of stationay beams in homogeneous isotopi media Popagation of Gaussian beams Gaussian optis... 8 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox.5.4 Gaussian modes in a esonato Pulse popagation (The Kames-Konig elation) Diffation theoy Inteation with plane masks Popagation using diffeent appoximations The geneal ase - small apetue Fesnel appoximation (paaxial appoximation) paaxial Faunhofe appoximation (fa field appoximation) Faunhofe diffation at plane masks (paaxial) Faunhofe diffation patten Examples: Remaks on Fesnel diffation Fouie optis - optial filteing Imaging of abitay optial field with thin lens Tansfe funtion of a thin lens optial imaging Optial filteing and image poessing The 4f-setup Examples: The polaization of eletomagneti waves Intodution Polaization of nomal modes in isotopi media Polaization states Piniples of Optis in Cystals Suseptibility and Dieleti Tenso The optial lassifiation of ystals the index ellipsoid nomal modes in anisotopi media Nomal modes popagating in pinipal dietions Nomal modes fo abitay popagation dietion dispesion elation speial ase: uniaxial ystals Optial Fields in Isotopi, Dispesive and Pieewise Homogeneous Media Basis definition of the poblem Deoupling of the vetoial wave equation intefaes and symmeties tansition onditions fields in a laye system matix method fields in one homogeneous laye the fields in a system of layes Refletion Tansmission Poblem fo Laye Systems geneal laye systems single intefae Peiodi multi-laye systems - Bagg-mios - D photoni ystals Faby-Peot-esonatos Guided Waved in Laye Systems Field stutue of guided waves dispesion elation fo guided waves guided waves at intefae - sufae polaiton guided waves in a laye film waveguide... 3

2 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 3 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox how to exite guided waves Statistial optis - oheene theoy Basis Statistial popeties of light Intefeene of patially oheent light Intodution 'optique' (Geek) loe of light 'what is light'? Is light a wave o a patile (photon)? D.J. Lovell, Optial Anedotes Light is the oigin and equiement fo life photosynthesis 9% of infomation we get is visual A) Oigin of light atomi system detemines popeties of light (e.g. statistis, fequeny, line width) optial system othe popeties of light (e.g. intensity, duation, ) invention of lase in 958 vey impotant development Shawlow and Townes, Phys. Rev. (958). lase atifiial light soue with new and unmathed popeties (e.g. oheent, dieted, foused, monohomati) appliations of lase: fibe-ommuniation, DVD, sugey, miosopy, mateial poessing,...

3 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 5 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 6 C) Light an modify matte light indues physial, hemial and biologial poesses used fo lithogaphy, mateial poessing, o modifiation of biologial objets (bio-photonis) Fibe lase: Limpet, Tünnemann, IAP Jena, ~kw CW (wold eod) B) Popagation of light though matte light-matte inteation dispesion diffation absoption satteing fequeny spatial ente of wavelength spetum fequeny fequeny spetum Hole dilled with a fs lase at Institute of Applied Physis, FSU Jena. matte is the medium of popagation the popeties of the medium (natual o atifiial) detemine the popagation of light light is the means to study the matte (spetosopy) measuement methods (intefeomete) design media with desied popeties: glasses, polymes, semiondutos, ompounded media (effetive media, photoni ystals, meta-mateials) Two-dimensional photoni ystal membane.

4 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 7 D) Optis in ou daily life Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 8 E) Optis in teleommuniations tansmitting data (Teabit/s in one fibe) ove tansatlanti distanes A small stoy desibing the impotane of light fo eveyday life, whee all things whih ely on optis ae maked in ed. m teleommuniation fibe is installed evey seond.

5 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 9 F) Optis in mediine, life sienes Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox G) Optial sensos and light soues new light soues to edue enegy onsumption new pojetion tehniques Deutshe Zukunftspeis 8 - IOF Jena + OSRAM

6 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox H) Mio- and nano-optis ulta small amea Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox I) Relativisti optis Inset inspied amea system develop at Faunhofe Institute IOF Jena

7 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 3 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 4 K) What is light? eletomagneti wave (= 3* 8 m/s) amplitude and phase omplex desiption polaization, oheene Region Spetum of Eletomagneti Radiation Wavelength (nanometes) Wavelength (entimetes) Fequeny (Hz) Enegy (ev) Radio > 88 > < 3 x 9 < -5 Miowave x 9-3 x Infaed x -5 3 x x 4. - Visible x -5-4 x x x 4-3 Ultaviolet 4-4 x x 4-3 x X-Rays x 7-3 x Gamma Rays <. < -9 > 3 x 9 > 5 L) Shemati of optis quantum optis eletomagneti optis wave optis geometial optis geometial optis << size of objets daily expeienes optial instuments, optial imaging intensity, dietion, oheene, phase, polaization, photons wave optis size of objets intefeene, diffation, dispesion, oheene lase, hologaphy, esolution, pulse popagation intensity, dietion, oheene, phase, polaization, photons eletomagneti optis efletion, tansmission, guided waves, esonatos lase, integated optis, photoni ystals, Bagg mios... intensity, dietion, oheene, phase, polaization, photons quantum optis small numbe of photons, flutuations, light-matte inteation intensity, dietion, oheene, phase, polaization, photons in this letue eletomagneti optis and wave optis no quantum optis advaned letue

8 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 5 M) Liteatue Fundamental. Saleh, Teih, 'Fundamenals of Photonis', Wiley, 99. Mansuipu, 'Classial Optis and its Appliations', Cambidge, 3. Heht, 'Optik', Oldenboug, 4. Menzel, 'Photonis', Spinge, 5. Lipson, Lipson, Tannhäuse, 'Optik'; Spinge, Bon, Wolf, 'Piniples of Optis', Pegamon 7. Sommefeld, 'Optik' Advaned. W. Silvast, 'Lase Fundamentals',. Agawal, 'Fibe-Opti Communiation Systems', Wiley 3. Band, 'Light and Matte', Wiley, 6 4. Kathe, Mülle, 'Integiete Optik', Teubne 5. Diels, Rudolph, 'Ultashot Lase Pulse Phenomena', Aademi 6. Yaiv, 'Optial Eletonis in moden Communiations', Oxfod 7. Snyde, Love, 'Optial Waveguide Theoy', Chapman&Hall 8. Röme, 'Theoetial Optis', Wiley,5. Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 6. (Ray optis - geometial optis) The topi of Ray optis geometial optis is not oveed in the ouse Fundamentals of moden optis. This topi will be oveed athe by the ouse Intodution to optial modeling. The following pat of the sipt whih is devoted to this topi is just inluded in the sipt fo onsisteny.. Intodution Ray optis o geometial optis is the simplest theoy fo doing optis. In this theoy, popagation of light in vaious optial media an be desibed by simple geometial ules. Ray optis is based on a vey ough appoximation (, no wave phenomena), but we an explain almost all daily life expeienes involving light (shadows, mios, et.). In patiula, we an desibe optial imaging with ay optis appoah. In isotopi media, the dietion of ays oesponds to the dietion of enegy flow. What is oveed in this hapte? It gives fundamental postulates of the theoy. It deives simple ules fo popagation of light (ays). It intodues simple optial omponents. It intodues light popagation in inhomogeneous media (gaded-index (GRIN) optis). It intodues paaxial matix optis.. Postulates A) Light popagates as ays. Those ays ae emitted by light-soues and ae obsevable by optial detetos. B) The optial medium is haateized by a funtion n(), the so-alled efative index (n() - meta-mateials n() <) n n speed of light in the medium n C) optial path length delay i) homogeneous media nl ii) inhomogeneous media B n() ds D) Femat s piniple A

9 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 7 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 8 B n() ds A Rays of light hoose the optial path with the shotest delay..3 Simple ules fo popagation of light A) Homogeneous media n = onst. minimum delay = minimum distane Rays of light popagate on staight lines. B) Refletion by a mio (metal, dieleti oating) The efleted ay lies in the plane of inidene. The angle of efletion equals the angle of inidene. C) Refletion and efation by an intefae Inident ay efleted ay plus efated ay The efleted ay obeys b). The efated ay lies in the plane of inidene. ii) Paaboli mio Paallel ays onvege in the foal point (foal length f). Appliations: Telesope, ollimato iii) Ellipti mio Rays oiginating fom foal point P onvege in the seond foal point P The angle of efation depends on the angle of inidene and is given by Snell s law: n sin n sin no infomation about amplitude atio..4 Simple optial omponents A) Mio i) Plana mio Rays oiginating fom P ae efleted and seem to oiginate fom P. iv) Spheial mio Neithe imaging like elliptial mio no fousing like paaboli mio paallel ays oss the optial axis at diffeent points onneting line of intesetions of ays austi

10 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 9 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox paallel, paaxial ays onvege to the foal point f = (-R)/ onvention: R < - onave mio; R > - onvex mio. fo paaxial ays the spheial mio ats as a fousing as well as an imaging optial element. paaxial ays emitted in point P ae efleted and onvege in point P C) Spheial intefae (paaxial) paaxial imaging n n n y (*) n n R (imaging fomula) z z ( R) paaxial imaging: imaging fomula and magnifiation m = -z /z (poof given in exeises) B) Plana intefae Snell s law: nsin nsin fo paaxial ays: nn extenal efletion ( n n ): ay efated away fom the intefae intenal efletion ( n n ): ay efated towads the intefae total intenal efletion (TIR) fo: n sin sin TIR n n n n n (imaging fomula) z z R n z m (magnifiation) n z (Poof: exeise) if paaxiality is violated abeation ays oming fom one point of the objet do not inteset in one point of the image (austi) D) Spheial thin lense (paaxial)

11 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox two spheial intefaes (R, R, ) apply (*) two times and assume y=onst ( small) vaiation of the path: () s () s B L nds nds A B A ngad n y with foal length: n f f R R z (imaging fomula) m (magnifiation) z z f z (ompae to spheial mio).5 Ray taing in inhomogeneous media (gaded-index - GRIN optis) n() - ontinuous funtion, fabiated by, e.g., doping uved tajetoies gaded-index laye an at as, e.g., a lens.5. Ray equation Stating point: we minimize the optial path o the delay (Femat) omputation: B n() ds A B A L n s ds ds d d d d d d d d d d ds ds ds ds d d ds ds ds ds d d ds ds ds B d d Lgad n n ds ds ds A integation by pats and A,B fix B d d gad n n ds ds ds A L fo abitay vaiation d d gad n n ay equation ds ds Possible solutions: A) tajetoy x(z), y(z) and ds dz dx dz dy dz solve fo x(z), y(z) paaxial ays (ds dz )

12 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 3 d dx dn nx, y, z dz dz dx d dy dn nx, y, z dz dz dy B) homogeneous media staight lines C) gaded-index laye n(y) - paaxial, SELFOC dy paaxial and dz dz ds n ( y) n y n( y) n y fo a d dy d dy d y d y dn( y) ny ny ny ds ds dz dz dz dz n y dy fo n(y)-n <<: d y dz y yz ( ) y osz sinz dy ( z) ysinzosz dz.5. The eikonal equation bidge between geometial optis and wave eikonal S() = onstant planes pependiula to ays fom S() we an detemine dietion of ays gad S() (like potential) gads n Remak: it is possible to deive Femat s piniple fom eikonal equation geometial optis: Femat s o eikonal equation S B B S gad S ds n ds B A eikonal optial path length phase of the wave A A Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 4.6 Matix optis tehnique fo paaxial ay taing though optial systems popagation in a single plane only ays ae haateized by the distane to the optial axis (y) and thei inlination () two algebai equation x matix Advantage: we an tae a ay though an optial system of many elements by multipliation of maties..6. The ay-tansfe-matix in paaxial appoximation: y Ay B Cy D A=: same same y fousing D=: same y same ollimation y A B y A B C D M C D.6. Maties of optial elements A) fee spae d M B) efation on plana intefae

13 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 5 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 6 M n n C) efation on spheial intefae M n n nr n n D) thin lens M f E) efletion on plana mio M F) efletion on spheial mio (ompae to lens) M R.6.3 Casaded elements yn A By A B N C D M C D M=M N.M M. Optial fields in dispesive and isotopi media. Maxwell s equations Ou geneal stating point is the set of Maxwell s equations. They ae the basis of the eletomagneti appoah to optis developed in this letue... Adaption to optis Maxwell s equations (maosopi) B ot E div D ext(,) t t D ot H jmak(,) t div B t eleti field E [V/m] magneti flux density B [Vs/m ] o [tesla] o magneti indution dieleti flux density D [As/m ] magneti field H [A/m] extenal hage density ext (,) t [As/m 3 ] maosopi uent density jmak (,) t [A/m ] Auxiliay fields D E P H B M dieleti polaization P [As/m ], magneti polaization M [Vs/m ] (magnetization) eleti onstant (vauum pemittivity) As/Vm magneti onstant (vauum pemeability) 4 7 Vs/Am

14 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 7 Light matte inteation The effet of the medium gives ise to polaization P f E and magnetization M f B. In ode to solve Maxwell s equations we need mateial models desibing those quantities. In optis, we geneally deal with non-magnetizable media M (exeptions ae metamateials with M ). In Maxwell s equations we also have so-alled soue tems: fee hage density ext (,) t [As/m 3 ] maosopi uent density jmak(,) t jond (,) t jonv (,) t [A/m ] ondutive uent density j (,) t f E ond onvetive uent density j (,) t (,) t v(,) t onv ext In optis, we geneally have no fee hages whih hange at speeds oesponding to the fequeny of light: (,) t j (,) t ext onv With the above simplifiations, we an fomulate Maxwell s equations in the ontext of optis: H ot E div E divp(, t) t P E ot H j (, t) div H(,) t t t In optis, the medium (o moe peisely the mathematial mateial model) detemines the dependene of the polaization on the eleti field, PE, ( ) and the dependene of the (ondutive) uent density on the eleti field, j( E ). One we have speified those elations, we an solve Maxwell s equations onsistently. Example: In vauum, both polaization and uent density ae zeo, and we an solve Maxwell s equations dietly (most simple mateial model). Remak: We an define a bound hage density Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 8 (,) t divp(,) t b and a bound uent density P jb (,) t t This essentially means that we an desibe the same physis in two diffeent ways (see genealized omplex dieleti funtion below). Complex field fomalism: Maxwell s equations ae also valid fo omplex fields and ae easie to solve This fat an be exploited to simplify alulations, beause it is easie to deal with omplex exponential funtions [exp(ix)] than with tigonometi funtions [os(x) and sin(x)]. onvention in this letue eal physial field: E (,) t omplex mathematial epesentation: E E (,) t E E (,) t Re E This means in geneal: Fo alulation we use the omplex fields [ E ] and fo physial esults we go bak to eal fields by simply omitting the imaginay pat. This woks beause Maxwell s equations ae linea and no multipliations of fields ou. Theefoe, be aeful when multipliations of fields ae equied go bak to eal quantities befoe! (elevant fo, e.g., alulation of Poynting veto, see Chapte below)... Tempoal dependene of the fields When it omes to time dependene of the eletomagneti field, we an distinguish two diffeent types of light: A) monohomati light stationay fields hamoni dependene on tempoal oodinate exp( i t) phase is fixed oheent, infinite wave tain e.g.: E E ()exp( i t) Monohomati light appoximates vey well the typial output of a ontinuous wave (CW) lase. One we know the fequeny we have to ompute the spatial dependene of the (stationay) fields only. B) polyhomati light non-stationay fields finite wave tain

15 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 9 With the help of Fouie tansfomation we an deompose the fields into infinite wave tains and use A) (see next setion) E E (, )exp( it) d E (, ) (,)exp( t t) dt E i..3 Maxwell s equations in Fouie domain We want to plug the Fouie deompositions of ou fields into Maxwell s equations in ode to get a moe simple desiption. Fo this pupose, we need to know how a time deivative tansfoms into Fouie spae. Hee we used integation by pats:, exp, exp (, ) dt t i t i dt t i t i E t E E FT ule: i t Now we an wite Maxwell s equations in Fouie domain: ot E (, ) ih (, ) div E (, ) div P (, ) ot H (, ) j ip (, ) i E (, ) div H (, )..4 Fom Maxwell s equations to the wave equation Fom Maxwell s equations it is staight fowad to deive the wave equation by using the two ul equations. A) Time domain deivation We stat fom applying the ul a seond time on ot E(,) t H (,) (,) (,) t P t E t otot E ot t t j t t And find the wave equation fo the eleti field otot E (,) t j P(,) t E t The blue tems equie knowledge of the mateial model. Additionally, we have to make sue that all othe Maxwell s equations ae fulfilled, in patiula: div E(,) t P(,) t One we know the eleti field, we an easily ompute the magneti field: H ot E(,) t t t t Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 3 Remaks: analog poedue possible fo H, i.e., we an deive a wave equation fo the magneti field geneally, the wave equation fo E is moe onvenient, beause we have P (E) given fom the mateial model howeve, fo inhomogeneous media H an be the bette hoie fo the numeial solution of the patial diffeential equation analog poedue possible fo H E geneally, wave equation fo E is moe onvenient, beause P (E) given fo inhomogeneous media H an be bette hoie B) Fequeny domain deivation We an do the same poedue in the Fouie domain and find otot E E j P and div (, ) (, ) E P magneti field: i H (, ) ot E (, ) (, ) (, ) i (, ) (, ) tansfeing the esults fom the Fouie domain to the time domain -i t fo stationay fields: take solution and multiply by e. fo non-stationay fields and linea media invese Fouie tansfomation E E (, )exp( i t) d..5 Deoupling of the vetoial wave equation Fo abitay isotopi media geneally all 3 field omponents ae oupled. Fo poblems with tanslational invaiane in at least one dietion, as e.g. fo homogeneous infinite media, layes o intefaes, thee an be deoupling of the omponents. Let s assume invaiane in the y-dietion and popagation only in the x-z-plane. Then all spatial deivatives along the y-dietion disappea and the opeatos in the wave equation simplify. geneally all field omponents ae oupled fo tanslational invaiane in e.g. the y-dietion and popagation only in the x-z-plane / y

16 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 3 Ex Ez () x x z E x () E y E x E z () E z x z z ot ot E gad div E E deomposition of eleti field E x EE E E E y, E E z x () with Nabla opeato (), and Laplae z x z gives two unoupled wave equations () E E (, ) ij (, ) P(, ) () () () E E (, ) gad div E ij (, ) P(, ) popeties popagation of pependiulaly polaized fields teated sepaately altenative notations: s TE (tansvesal eleti) p TM (tansvesal magneti) E and E an be. Optial popeties of matte In this hapte we will deive a simple mateial model fo the polaization and the uent density. The basi idea is to wite down an equation of motion fo a single exemplay haged patile and assume that all othe patiles of the same type behave similaly. Moe peisely, we will use a diven hamoni osillato to desibe the motion of bound hages giving ise to a polaization of the medium. Fo fee hages leading eventually to a uent density we will use the same model but without estoing foe. In the liteatue, this simple appoah is often alled the Dude-Loentz model (named afte Paul Dude and Hendik Antoon Loentz)... Basis We ae looking fo PE ( ) and j( E ). In geneal, this leads to a many body poblem in solid state theoy whih is athe omplex. Howeve, in many ases phenomenologial models ae suffiient to desibe the neessay Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 3 phenomena. We use the simplest appoah, the so-alled Dude-Loentz model fo fee o bound hage aies (eletons): ensemble of non-oupling, diven, and damped hamoni osillatos fee hage aies: metals and exited semiondutos (intaband) bound hage aies: dieleti media and semiondutos (inteband) The Dude-Loentz model eates a link between ause (eleti field) and effet (indued polaization o uent). Beause the esulting elations PE ( ) and j( E ) ae linea (no E et.), we an use linea esponse theoy. Fo the polaization PE ( ) (fo j( E ) vey simila): desiption in both time and fequeny domain possible in time domain we intodue the esponse funtion E medium (esponse funtion) P t P(,) t R (, tt) E (, t) dt i ij j j with ˆR being a nd ank tenso i x, y, z and summing ove j x, y, z in fequeny domain we intodue the suseptibility E (, ) medium (suseptibility) P (, ) P(, ) (, ) E (, ) i ij j j esponse funtion and suseptibility ae linked via Fouie tansfom (onvolution theoem) Rij () t ij ( )exp( t) d i Obviously, things look fiendlie in fequeny domain. Using the wave equation fom befoe we find (assumption of no uent) otot E E P (, ) (, ) (, ) E E gaddive P (, ) (, ) (, ) (, ) and fo auxiliay fields D (, ) E (, ) P (, )

17 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 33 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 34 The geneal esponse funtion and the espetive suseptibility given above simplifies fo etain popeties of the medium: Diffeent types of media A) linea, homogenous, isotopi, non-dispesive media (most simple but vey unphysial ase) homogenous (, ) ( ) isotopi (, ) (, ) ij ij ij ij non-dispesive (, ) () instantaneous: R (,) t ()() t (, ) is a sala onstant ij ij ij P (, ) E (, ) P E (unphysial!) D (, ) E (, ) D E Maxwell: div D div E(, ) E (, ) E (, ) E E t appoximation only fo a etain fequeny ange, beause all media ae dispesive based on an unphysial mateial model B) linea, homogeneous, isotopi, dispesive media ( ) P (, ) ( ) E (, ), D (, ) ( ) E (, ) div D(, ) div E(, ) fo ( ). (, ) E (, ) Helmholtz equation E This desiption is suffiient fo many mateials. C) linea, inhomogeneous, isotopi, dispesive media (, ) P (, ) (, ) E (, ), D (, ) (, ) E (, ). div D (, ) div D (, ) (, ) div E (, ) E (, ) gad (, ), gad (, ) div E (, ) E (, ). (, ) ij ij gad (, ) E (, ), E (, ) gad E (, ) (, ) All field omponents ouple. D) linea, homogeneous, anisotopi, dispesive media ( ) Pi(, ) ij( ) Ej(, ) j see hapte on ystal optis D(, ) ( ) E (, ). i ij j j This is the wost ase fo a linea esponse of the media. Befoe we stat witing down the atual mateial model equations, let us summaize what we want to do: What physis do we want to onside? I) Bound eletons and lattie vibations The ontibutions of bound eletons and lattie vibations in dieletis and semiondutos give ise to the polaization P. The lattie vibations (phonons) ae the ioni pat of the mateial model. Beause of the lage mass 3 of the ions ( mass of eleton) the esulting osillations will be small. Geneally speaking, phonons ae esponsible fo themal popeties of the medium. Howeve, some phonon modes may ontibute to optial popeties, but they have small dispesion (weak dependene on fequeny ). Undestanding the eletoni tansitions of bound eletons equies quantum theoetial teatment, whih allows an auate omputation of the tansition fequenies. Howeve, a (phenomenologial) lassial teatment of the osillation of bound eletons is possible and useful. II) Fee eletons The ontibution of fee eletons in metals and exited semiondutos gives ise to a uent density j. We assume a so-alled (inteation-)fee eleton gas, whee the eleton hages ae neutalized by the bakgound ions. Only ollisions with ions and elated damping of the eleton motion will be onsideed. We will look at the ontibutions fom I) and II) sepaately, and join the esults late... Dieleti polaization and suseptibility Let us fist fous on bound hages (ions, eletons). In the so-alled Dude model, the eleti field E gives ise to a displaement s of haged patiles fom thei equilibium positions. In the easiest appoah this an be modeled by a diven hamoni osillato: ij

18 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 35 q t t m s g s s E esonane fequeny (eletoni tansition) damping g hage q mass m The indued eleti dipole moment due to the displaement is given by p(,) t qs(,), t We futhe assume that all bound hages of the same type behave idential, i.e., we teat an ensemble of non-oupled, diven, and damped hamoni osillatos. Then, the dipole density (polaization) is given by P Np(,) t Nqs(,) t Hene, the govening equation fo the polaization P eads t g qn P P Pt (,) E (,) (,) t m t f E t en with osillato stength f, fo q=-e (eletons) m This equation is easy to solve in Fouie domain: P (, ) i gp (, ) P (, ) f E (, ) f P (, ) E (, ). ig In geneal we have seveal types of osillatos in a medium, i.e., seveal diffeent esonane fequenies. Nevetheless, sine in a good appoximation they do not influene eah othe, all these diffeent osillatos ontibute individually to the polaization. Hene the model an be onstuted by simply summing up all ontibutions. seveal esonane fequenies f j P (, ) (, ) (, ) E E j j ig j f j. j j ig j is the omplex, fequeny dependent suseptibility D (, ) E (, ) E (, ) E (, ) Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 36 is the omplex fequeny dependent dieleti funtion Example: (hee is plotted by eta and kappa i )..3 Condutive uent and ondutivity Let us now desibe the esponse of a fee eleton gas with positively haged bakgound (no inteation). Again we use the model of a diven hamoni osillato, but this time with esonane fequeny. This oesponds to the ase of zeo estoing foe. e s g s E (,), t t t m The esulting indued uent density is given by j(,) t Ne s(,) t t and the govening dynami equation eads as t j j m E E en with plasma fequeny p f m Again we solve this equation in Fouie domain: en g p i j j E (, ) g (, ) p (, ) p j (, ) E (, ) E (, ). g i Hee we intodued the omplex fequeny dependent ondutivity

19 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 37 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 38 p p i. g i ig Remaks on plasma fequeny We onside a loud of eletons and positive ions desibed by the total hage density in thei self-onsistent field E. Then we find aoding to Maxwell: dive(,) t (,) t Fo old eletons, and beause the total hage is zeo, we an use ou damped osillato model fom befoe to desibe the uent density (only eletons move): g t j j E p Now we apply divegene opeato and plug in fom above (ed tems): div j gdivj pdive(,) t p (,) t t With the ontinuity equation fo the hage density (fom Maxwell's equations) divj, t We an substitute the divegene of the uent density and find: t t g p g, p t t hamoni osillato equation Hene, the plasma fequeny p is the eigen-fequeny of suh a hage density...4 The genealized omplex dieleti funtion In the setions above we have deived expessions fo both polaization (bound hages) and ondutive uent density (fee hages). Let us now plug ou j(, ) and P (, ) into the wave equation (in Fouie domain) otot E (, ) E (, ) P (, ) ij (, ) ( ) i E (, ) Hene we an ollet all tems popotional to E (, ) and wite i otot E (, ) ( ) E (, ) otot E (, ) ( ) (, ) E Hee, we intodued the genealized omplex dieleti funtion i ( ) ( ) ( ) i( ) So, in geneal we have f j p ( ), j j g g i j i beause (fom befoe) f j j j ig j i ig, p. ( ) ontains ontibutions fom vauum, phonons (lattie vibations), bound and fee eletons. Some speial ases fo mateials in the IR und VIS: A) Dieletis (insulatos) nea phonon esonane (IR) If we ae inteested in dieletis (insulatos) nea phonon esonane in the infaed spetal ange we an simplify the dieleti funtion as follows: f j f ( ) j j ig j ig f, ig The ontibution of vauum and eletoni tansitions show almost no fequeny dependene (dispesion) in this egime and an be expessed as a onstant. Let us study the eal and the imaginay pat of the esulting ( ) sepaately: vauum and eletoni tansitions ( ) ( ) i ( ) ( ) i ( ) f ( ), g j

20 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 39 ( ) gf. g esonane fequeny width of esonane peak g Loentz uve stati dieleti onstant in the limit : so alled longitudinal fequeny L : ( ) f ( ) : absoption and dispesion appea always togethe Example: single esonane 8 L ε ε Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 4 L. B) Dieleti media in visible spetal ange Dieleti media in visible spetal ange an be deibed by a so-alled double esonane model (phonon in IR and eletoni tansition in UV). fp fe ( ), ig ig p p e e p e ontibution of vauum and othe (fa away) esonanes. ε' 4.8 ε ε ω ω L ω.6 VIS -4 nea esonane we find ( ) (damping without absoption if '' ) nomal dispesion ( )/, anomalous dispesion ( )/ Example: shap esonane fo undamped osillato g ε ε elation between esonane fequeny and longitudinal fequeny L (Lyddane-Sahs-Telle elation) f ( L ) L ε ω ω L ω, f (fom above) The genealization of this appoah in the tanspaent spetal ange leads to so-alled Sellmeie fomula: f j j ( ), j j desibes many media vey well (dispesion of absoption is negleted) osillato stengths and esonane fequenies ae fit paametes C) Metals in visible spetal ange If we want to desibe metals in visible spetal ange we find ( ). g p i ω in 5 s - p p gp ( ), ( ). g g Metals show a lage negative eal pat of the dieleti funtion ( )

21 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox Mateial models in time domain Let us now tansfom ou esults of the mateial models bak to time domain. In Fouie domain we found fo homogeneous and isotopi media: D (, ) ( ) E (, ) P (, ) ( ) E (, ). The esponse funtion (o Geen's funtion) Rt () is then given by Rt () ( )exp td. i ( ) Rt ( )exp tdt i To pove this, we an use the onvolution theoem VIS ω in 5 s - -i ( ) -i E ( t i tdt -i td P (, t) P (, )exp t d E (, )exp t d ( ), )exp exp Now we swith the ode of integation, and identify the esponse funtion R (ed tems): ( )exp ( tt) d (, t ) dt -i E Rt ( t) E (, t) dt Fo a delta exitation in the eleti field we find the esponse o Geens funtion as the polaization: E (, t) e ( tt ) P R( tt) e Geen's funtion. Fo instantaneous (o non-dispesive) media we find: Rt () () t P, t E, t ε ε ω P -g Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 4 Examples A) dieleti media f RP () t exp td exp td, i g i i Using the esidual theoem we an find: B) metals f g exp t sin t t Rt () t t with f g P exp ( tt)sin ( tt) (, t) dt E g 4 p Rj () t exp td exp td, i i g i Using again the esidual theoem we an find: p expgt t Rt () g t t j exp g( tt) E (, t) dt p.3 The Poynting veto and enegy balane.3. Time aveaged Poynting veto The enegy flux of the eletomagneti field is given by the Poynting veto S. In patie, we always measue the enegy flux though a sufae (deteto), S n, whee n is the nomal veto of sufae. To be moe peise, the Poynting veto S E(,) t H (,) t gives the momentay enegy flux. Note that we have to use the eal eleti and magneti fields, beause a podut of fields ous. In optis we have to onside the following time sales: 4 optial yle T / s pulse duation T p in geneal Tp T

22 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 43 duation of measuement T m in geneal Tm T Hene, in geneal the deteto does not eognize the fast osillations of the i t optial field e (optial yles) and delives a time aveaged value. Fo the situation desibed above, the eleto-magneti fields fatoize in slowly vaying envelopes and fast aie osillations: (,)exp t t.. E i E Fo suh pulses, the momentay Poynting veto eads: S E(,) t H (,) t 4 E H E H (,)exp t t (, t 4 E H E H i )expi t (,)os t t E H E H si E H n t. We find that the momentay Poynting veto has some slow ontibutions whih hange ove time sales of the pulse envelope T p, and some fast ontibutions os t, sint hanging ove time sales of the optial yle T. Now, a measuement of the Poynting veto ove a time inteval T m leads to a time aveage of S ttm / S (,') t dt' T S ttm / m The fast osillating tems ~os t and ~sin t anel by the integation sine the pulse envelope does not hange muh ove one optial yle. Hene we get only a ontibution fom the slow tem: Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 44 ttm / S (, t) (, t ') (, t ') dt ' T tt / E H m m Let us now have a look at the speial (but impotant) ase of stationay (monohomati) fields. Then, the pulse envelope does not depend on time at all (infinitely long pulses): E (,') t E (), H (,') t H () S () (). E H This is the definition fo the optial intensity I S. We see that an intensity measuement destoys infomation on the phase. I S measuement destoys phase infomation.3. Time aveaged enegy balane Let us motivate a little bit futhe the onept of the Poynting veto. It appeas natually in the Poynting theoem, the equation fo the enegy balane of the eletomagneti field. The Poynting theoem an be deived dietly fom Maxwell s equations. We multiply the two ul equations by H esp. E : (note that we use eal fields): H ote H H t E E E oth E ( j P) t t Next, we subtat the two equations and get HotE E oth E E H H E( j P). t t t This equation an be simplified by using the following veto identity: div E H H ote E oth Finally, with E E E we find Poynting's theoem t t E H dive H E j P (*) t t t This equation has the geneal fom of a balane equation, hee it epesents the enegy balane. Apat fom the appeaane of the Poynting veto (enegy flux), we an identify the vauum enegy density u E H. The ight-hand-side of Poynting's theoem ontains the so-alled soue tems.

23 Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 45 whee u E H vauum enegy density In the ase of stationay fields and isotopi media (simple but impotant) E(,) t ()exp.. E i t H(,) t ()exp t.. H i Time aveaging of the left hand side of Poynting s theoem (*) yields: E () () H div E H div t t E H div S(,). t Note that the time deivative emoves stationay tems in E (,) t and H (,) t. Time aveaging of the ight hand side of Poynting s theoem yields (soue tems): j (,) t P (,) t t E Sipt Fundamentals of Moden Optis, FSU Jena, Pof. T. Petsh, FoMO_Sipt_--9.dox 46 This shows that a nonzeo imaginay pat of epsilon ( ) auses a dain of enegy flux. In patiula, we always have, othewise thee would be gain of enegy. In patiula nea esonanes we have and theefoe absoption. Futhe insight into the meaning of div S gives the so-alled divegene theoem. If the enegy of the eleto-magneti field is flowing though some volume, and we wish to know how muh enegy flows out of a etain egion within that volume, then we need to add up the soues inside the egion and subtat the sinks. The enegy flux is epesented by the (time aveaged) Poynting veto, and the Poynting veto's divegene at a given point desibes the stength of the soue o sink thee. So, integating the Poynting veto's divegene ove the inteio of the egion equals the integal of the Poynting veto ove the egion's bounday. div S dv S n da V A 4 it it i t ( ) E( ) e i ( ) E ( ) e.. E() e.. Now we use ou genealized dieleti funtion: i i E() exp it.. exp t.. 4 E() i i.. 4 E()E() Again, all fast osillating tems exp t i anel due to the time aveage. Finally, splitting into eal and imaginay pat yields.. ( ) ( ). 4 i i E()E() E E Hene, the divegene of the time aveaged Poynting veto is elated to the imaginay pat of the genealized dieleti funtion: div S E() E ().