Grundkonzepte der Optik Sommersemester 2014

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1 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox Grundkonzepte der Optik Sommersemester 4 Prof. Thomas Pertsh Abbe Shool of Photonis, Friedrih-Shiller-Universität Jena Table of ontent. Introdution Ray optis - geometrial optis Introdution Postulates Simple rules for propagation of light Simple optial omponents Ray traing in inhomogeneous media (graded-index - GRIN optis) Ray equation The eikonal equation....6 Matrix optis The ray-transfer-matrix Matries of optial elements Casaded elements Optial fields in dispersive and isotropi media Maxwell s equations Adaption to optis Temporal dependene of the fields Maxwell s equations in Fourier domain From Maxwell s equations to the wave equation Deoupling of the vetorial wave equation Optial properties of matter Basis Dieletri polarization and suseptibility Condutive urrent and ondutivity The generalized omplex dieletri funtion Material models in time domain The Poynting vetor and energy balane Time averaged Poynting vetor Time averaged energy balane Normal modes in homogeneous isotropi media Transversal waves Longitudinal waves Plane wave solutions in different frequeny regimes Time averaged Poynting vetor of plane waves The Kramers-Kronig relation Beams and pulses - analogy of diffration and dispersion Diffration of monohromati beams in homogeneous isotropi media Arbitrarily narrow beams (the general ase) Fresnel- (paraxial) approximation The paraxial wave equation Propagation of Gaussian beams Propagation in paraxial approximation Propagation of Gauss beams with q-parameter formalism Gaussian optis Gaussian modes in a resonator Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox.9 Dispersion of pulses in homogeneous isotropi media Pulses with finite transverse width (pulsed beams) Infinite transverse extension - pulse propagation Example : Gaussian pulse without hirp Example : Chirped Gaussian pulse Diffration theory Interation with plane masks Propagation using different approximations The general ase - small aperture Fresnel approximation (paraxial approximation) Paraxial Fraunhofer approximation (far field approximation) Non-paraxial Fraunhofer approximation Fraunhofer diffration at plane masks (paraxial) Fraunhofer diffration pattern Remarks on Fresnel diffration Fourier optis - optial filtering Imaging of arbitrary optial field with thin lens Transfer funtion of a thin lens Optial imaging Optial filtering and image proessing The 4f-setup Examples of aperture funtions Optial resolution The polarization of eletromagneti waves Introdution Polarization of normal modes in isotropi media Polarization states Priniples of optis in rystals Suseptibility and dieletri tensor The optial lassifiation of rystals The index ellipsoid Normal modes in anisotropi media Normal modes propagating in prinipal diretions Normal modes for arbitrary propagation diretion Normal surfaes of normal modes Speial ase: uniaxial rystals Optial fields in isotropi, dispersive and pieewise homogeneous media Basis Definition of the problem Deoupling of the vetorial wave equation Interfaes and symmetries Transition onditions Fields in a layer system matrix method Fields in one homogeneous layer The fields in a system of layers Refletion transmission problem for layer systems General layer systems Single interfae Periodi multi-layer systems - Bragg-mirrors - D photoni rystals Fabry-Perot-resonators Guided waves in layer systems Field struture of guided waves... 75

2 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox Dispersion relation for guided waves Guided waves at interfae - surfae polariton Guided waves in a layer film waveguide how to exite guided waves This sript originates from the leture series Theoretishe Optik given by Falk Lederer at the FSU Jena for many years between 99 and. Later the sript was adapted by Stefan Skupin and Thomas Pertsh for the international eduation program of the Abbe Shool of Photonis. Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 4. Introdution 'optique' (Greek) lore of light 'what is light'? Is light a wave or a partile (photon)? D.J. Lovell, Optial Anedotes Light is the origin and requirement for life photosynthesis 9% of information we get is visual A) What is light? 8 eletromagneti wave ( 3 m/ s) amplitude and phase omplex desription polarization, oherene Region Spetrum of Eletromagneti Radiation Wavelength [nm] Wavelength [m] (nm= -9 m) Frequeny [Hz] (THz= Hz) Energy [ev] Radio > 8 > - < 3 x 9 < -5 Mirowave x 9-3 x Infrared x -7 3 x x 4. - Visible x -7-4 x x x 4-3 Ultraviolet 4-4 x x 4-3 x X-Rays x 7-3 x Gamma Rays <. < - > 3 x 9 > 5

3 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 5 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 6 laser artifiial light soure with new and unmathed properties (e.g. oherent, direted, foused, monohromati) appliations of laser: fiber-ommuniation, DVD, surgery, mirosopy, material proessing,... Fiber laser: Limpert, Tünnermann, IAP Jena, ~kw CW (world reord) C) Propagation of light through matter light-matter interation (G: Liht-Materie-Wehselwirkung) B) Origin of light atomi system determines properties of light (e.g. statistis, frequeny, line width) optial system other properties of light (e.g. intensity, duration, ) invention of laser in 958 very important development dispersion diffration absorption sattering frequeny spatial enter of wavelength spetrum frequeny frequeny spetrum matter is the medium of propagation the properties of the medium (natural or artifiial) determine the propagation of light light is the means to study the matter (spetrosopy) measurement methods (interferometer) design media with desired properties: glasses, polymers, semiondutors, ompounded media (effetive media, photoni rystals, meta-materials) Shawlow and Townes, Phys. Rev. (958).

4 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 7 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 8 E) Optial teleommuniation transmitting data (Terabit/s in one fiber) over transatlanti distanes Two-dimensional photoni rystal membrane. D) Light an modify matter light indues physial, hemial and biologial proesses used for lithography, material proessing, or modifiation of biologial objets (bio-photonis) Hole drilled with a fs laser at Institute of Applied Physis, FSU Jena. m teleommuniation fiber is installed every seond.

5 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 9 F) Optis in mediine and life sienes Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox G) Light sensors and light soures new light soures to redue energy onsumption new projetion tehniques Deutsher Zukunftspreis 8 - IOF Jena + OSRAM

6 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox H) Miro- and nano-optis ultra small amera Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox I) Relativisti optis Inset inspired amera system develop at Fraunhofer Institute IOF Jena

7 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 3 J) Shemati of optis quantum optis eletromagneti optis wave optis geometrial optis geometrial optis << size of objets daily experiene optial instruments, optial imaging intensity, diretion, oherene, phase, polarization, photons G: Intensität, Rihtung, Kohärenz, Phase, Polarisation, Photon wave optis size of objets interferene, diffration, dispersion, oherene laser, holography, resolution, pulse propagation intensity, diretion, oherene, phase, polarization, photons Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 4 no quantum optis advaned leture K) Literature Fundamental. Saleh, Teih, 'Fundamenals of Photonis', Wiley (99) in German: "Grundlagen der Photonik" Wiley (8). Heht, 'Opti', Addison-Wesley () in German: "Optik", Oldenbourg (5) 3. Mansuripur, 'Classial Optis and its Appliations', Cambridge () 4. Menzel, 'Photonis', Springer () 5. Lipson, Lipson, Tannhäuser, 'Optik'; Springer (997) 6. Born, Wolf, 'Priniples of Optis', Pergamon 7. Sommerfeld, 'Optik' Advaned. W. Silvast, 'Laser Fundamentals',. Agrawal, 'Fiber-Opti Communiation Systems', Wiley 3. Band, 'Light and Matter', Wiley, 6 4. Karthe, Müller, 'Integrierte Optik', Teubner 5. Diels, Rudolph, 'Ultrashort Laser Pulse Phenomena', Aademi 6. Yariv, 'Optial Eletronis in modern Communiations', Oxford 7. Snyder, Love, 'Optial Waveguide Theory', Chapman&Hall 8. Römer, 'Theoretial Optis', Wiley,5. eletromagneti optis refletion, transmission, guided waves, resonators laser, integrated optis, photoni rystals, Bragg mirrors... intensity, diretion, oherene, phase, polarization, photons quantum optis small number of photons, flutuations, light-matter interation intensity, diretion, oherene, phase, polarization, photons in this leture eletromagneti optis and wave optis

8 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 5. Ray optis - geometrial optis. Introdution Ray optis or geometrial optis is the simplest theory for doing optis. In this theory, propagation of light in various optial media an be desribed by simple geometrial rules. Ray optis is based on a very rough approximation (, no wave phenomena), but we an explain almost all daily life experienes involving light (shadows, mirrors, et.). In partiular, we an desribe optial imaging with ray optis approah. In isotropi media, the diretion of rays orresponds to the diretion of energy flow. What is overed in this hapter? It gives fundamental postulates of the theory. It derives simple rules for propagation of light (rays). It introdues simple optial omponents. It introdues light propagation in inhomogeneous media (graded-index (GRIN) optis). It introdues paraxial matrix optis.. Postulates A) Light propagates as rays. Those rays are emitted by light-soures and are observable by optial detetors. B) The optial medium is haraterized by a funtion n(r), the so-alled refrative index (n(r) - meta-materials n(r) <) n n speed of light in the medium n C) optial path length delay i) homogeneous media nl ii) inhomogeneous media B n() r ds D) Fermat s priniple A B n() r ds A Rays of light hoose the optial path with the shortest delay. Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 6.3 Simple rules for propagation of light A) Homogeneous media n = onst. minimum delay = minimum distane Rays of light propagate on straight lines. B) Refletion by a mirror (metal, dieletri oating) The refleted ray lies in the plane of inidene. The angle of refletion equals the angle of inidene. C) Refletion and refration by an interfae Inident ray refleted ray plus refrated ray The refleted ray obeys b). The refrated ray lies in the plane of inidene. The angle of refration depends on the angle of inidene and is given by Snell s law: n sin n sin no information about amplitude ratio..4 Simple optial omponents A) Mirror i) Planar mirror Rays originating from P are refleted and seem to originate from P. ii) Paraboli mirror Parallel rays onverge in the foal point (foal length f). Appliations: Telesope, ollimator

9 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 7 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 8 iii) Ellipti mirror Rays originating from foal point P onverge in the seond foal point P iv) Spherial mirror Neither imaging like elliptial mirror nor fousing like paraboli mirror parallel rays ross the optial axis at different points onneting line of intersetions of rays austi parallel, paraxial rays onverge to the foal point f = (-R)/ onvention: R < - onave mirror; R > - onvex mirror. for paraxial rays the spherial mirror ats as a fousing as well as an imaging optial element. paraxial rays emitted in point P are refleted and onverge in point P (imaging formula) z z ( R) paraxial imaging: imaging formula and magnifiation m = -z /z (proof given in exerises) B) Planar interfae Snell s law: nsin nsin for paraxial rays: nn external refletion ( n n ): ray refrated away from the interfae internal refletion ( n n ): ray refrated towards the interfae total internal refletion (TIR) for: n sin sin TIR n C) Spherial interfae (paraxial) paraxial imaging

10 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 9 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox n n n y (*) n n R y with foal length: n f f R R z (imaging formula) m (magnifiation) z z f z (ompare to spherial mirror).5 Ray traing in inhomogeneous media (graded-index - GRIN optis) n() r - ontinuous funtion, fabriated by, e.g., doping urved trajetories graded-index layer an at as, e.g., a lens n n n n (imaging formula) z z R n z m (magnifiation) n z (Proof: exerise) if paraxiality is violated aberration rays oming from one point of the objet do not interset in one point of the image (austi) D) Spherial thin lense (paraxial).5. Ray equation Starting point: we minimize the optial path or the delay (Fermat) omputation: B n() r ds A B A L nr s ds variation of the path: r() s r () s two spherial interfaes (R, R, ) apply (*) two times and assume y=onst ( small)

11 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox B L nds nds A B A ngrad nr r r r ds d d d dr dr d r d r dr dr dr ds ds ds ds dr dr ds ds ds ds dr dr ds ds ds B dr dr Lgrad nr n ds ds ds A integration by parts and A,B fix B d dr grad n n ds ds ds r A L for arbitrary variation d dr grad n n ray equation ds ds Possible solutions: A) trajetory x(z), y(z) and ds dz dx dz dy dz solve for x(z), y(z) paraxial rays (ds dz ) d dx dn nx, y, z dz dz dx d dy dn nx, y, z dz dz dy B) homogeneous media straight lines C) graded-index layer n(y) - paraxial, SELFOC Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox for n(y)-n <<: dy paraxial and dz dz ds n ( y) n y n( y) n y for a d dy d dy d y d y dn( y) ny ny ny ds ds dz dz dz dz n y dy d y dz y yz ( ) y osz sinz dy ( z) ysinzosz dz.5. The eikonal equation bridge between geometrial optis and wave eikonal S(r) = onstant planes perpendiular to rays from S(r) we an determine diretion of rays grad S(r) (like potential) grads r nr Remark: it is possible to derive Fermat s priniple from eikonal equation geometrial optis: Fermat s or eikonal equation S r B B S r grad S r ds n r ds B A eikonal optial path length phase of the wave A.6 Matrix optis tehnique for paraxial ray traing through optial systems propagation in a single plane only rays are haraterized by the distane to the optial axis (y) and their inlination () two algebrai equation x matrix Advantage: we an trae a ray through an optial system of many elements by multipliation of matries. A

12 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 3 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 4.6. The ray-transfer-matrix D) thin lens M n n nr n n M f E) refletion on planar mirror in paraxial approximation: 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. Matries of optial elements A) free spae M F) refletion on spherial mirror (ompare to lens) M R.6.3 Casaded elements yn A By A B N C D M C D M=M N.M M d M B) refration on planar interfae M n n C) refration on spherial interfae

13 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 5. Optial fields in dispersive and isotropi media. Maxwell s equations Our general starting point is the set of Maxwell s equations. They are the basis of the eletromagneti approah to optis whih is developed in this leture... Adaption to optis The notation of Maxwell s equations is different for different disiplines of siene and engineering whih rely on these equations to desribe eletromagnet phenomena at different frequeny ranges. Even though Maxwell's equations are valid for all frequenies, the physis of light matter interation is different for different frequenies. Sine light matter interation must be inluded in the Maxwell's equations to solve them onsistently, different ways have been established how to write down Maxwell's equations for different frequeny ranges. Here we follow a notation whih was established for a onvenient notation at frequenies lose to visible light. Maxwell s equations (marosopi) In a rigorous way the eletromagneti theory is developed starting from the properties of eletromagneti fields in vauum. In vauum one ould write down Maxwell's equations in there so-alled pure mirosopi form, whih inludes the interation with any kind of matter based on the onsideration of point harges. Obviously this is inadequate for the desription of light in ondensed matter, sine the number of point harges whih would need to be taken into aount to desribe a marosopi objet, would exeed all imaginable omputational resoures. To solve this problem one uses an averaging proedure, whih summarizes to influene of many point harges on the eletromagneti field in a homogeneously distributed response of the solid state on the exitation by the light. In turn, also the eletromagneti fields are averaged over some adequate volume. For optis this proedure is justified, sine any kind of available experimental detetor ould not resolve the very fine spatial details of the fields in between the point harges, e.g. ions or eletrons, whih are lost by this averaging. These averaged eletromagneti equations have been rigorously derived in a number of fundamental text books on eletro-dynami theory. Here we will not redo this derivation. We will rather start diretly from the averaged Maxwell's equations equation. Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 6 Br (,) t rot Er (,) t div Dr (,) t ext(,) rt t Dr (,) t rot Hr (,) t jmakr(,) rt div Br (,) t t eletri field (G: elektrishes Feld) Er (,) t [V/m] magneti flux density (magneti indution) (G: magnetishe Flussdihte oder magnetishe Induktion) Br (,) t [Vs/m ] or [tesla] eletri flux density (eletri displaement field) (G: elektrishe Flussdihte oder dielektrishe Vershiebung) Dr (,) t [As/m ] magneti field (G: magnetishes Feld) Hr (,) t [A/m] external harge density ext (,) r t [As/m 3 ] marosopi urrent density jmakr (,) r t [A/m ] Auxiliary fields The "ost" of the introdution of marosopi Maxwell's equations is the ourrene of two additional fields, the dieletri flux density Dr (,) t and the magneti field Hr (,) t. These two fields are related to the eletri field Er (,) t and magneti flux density Br (,) t by two other new fields. Dr (,) t Er (,) t Pr (,) t Hr (,) t Br (,) t Mr (,) t dieletri polarization (G: dielektrishe Polarisation) Pr (,) t [As/m ], magneti polarization (magnetization) (G: Magnetisierung) Mr (,) t [Vs/m ] eletri onstant (vauum permittivity) (G: Vakuumpermittivität) As/Vm magneti onstant (vauum permeability) (G: Vakuumpermeabilität) 4 7 Vs/Am Light matter interation In order to solve this set of equations, i.e. Maxwell's equations and auxiliary field equations one needs to onnet the dieletri flux density Dr (,) t and the

14 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 7 magneti field Hr (,) t to the eletri field Er (,) t and the magneti flux density Br (,) t. This is ahieved by modeling the material properties by introduing the material equations. The effet of the medium gives rise to polarization Pr (,) t f E and magnetization Mr (,) t f B. In order to solve Maxwell s equations we need material models desribing these quantities. In optis, we generally deal with non-magnetizable media Mr (,) t (exeptions are metamaterials with Mr (,) t ). Furthermore we need to introdue soures of the fields into our model. This is ahieved by the so-alled soure terms whih are inhomogeneities and hene they define unique solutions of the equations. free harge density (G: Dihte freier Ladungsträger) ext (,) r t [As/m 3 ] marosopi urrent density (G: makroskopishe Stromdihte) jmakr(,) r t jond (,) r t jonv (,) r t [A/m ] ondutive urrent density (G: Konduktionsstromdihte) j (,) r t f E ond onvetive urrent density (G: Konvektionsstromdihte) j (,) r t (,) r t v(,) r t onv ext In optis, we generally have no free harges whih hange at speeds omparable to the frequeny of light: ( r, t) j ( r, t) ext onv With the above simplifiations, we an formulate Maxwell s equations in the ontext of optis: Hr (,) t rot Er (,) t div Er (,) t divp(, r t) t Pr (,) t Er (,) t rot Hr (,) t jr (, t) div H(,) r t t t In optis, the medium (or more preisely the mathematial material model) determines the dependene of the polarization on the eletri field PE ( ) and the dependene of the (ondutive) urrent density on the eletri field j( E ). One we have speified these relations, we an solve Maxwell s equations onsistently. Example: Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 8 In vauum, both polarization and urrent density are zero, and we an solve Maxwell s equations diretly (most simple material model). Remark: We an define a bound harge density (G: Dihte gebundener Ladungsträger) (,) r t divp(,) r t b and a bound urrent density (G: Stromdihte gebundener Ladungsträger) Pr (,) t jb (,) r t t This essentially means that we an desribe the same physis in two different ways (see generalized omplex dieletri funtion below). Complex field formalism (G: komplexer Feld-Formalismus): Maxwell s equations are also valid for omplex fields and are easier to solve This fat an be exploited to simplify alulations, beause it is easier to deal with omplex exponential funtions (exp( ix )) than with trigonometri funtions [os(x) and sin(x)]. onvention in this leture real physial field: E (,) r t r omplex mathematial representation: Er (,) t.. They are related by E (,) r t Er (,) t E (,) r t Re Er r (,) t Remark: This relation an be defined differently in different textbooks. This means in general: For alulations we use the omplex fields [ Er (, t)] and for physial results we go bak to real fields by simply omitting the imaginary part. This works beause Maxwell s equations are linear and no multipliations of fields our. Therefore, be areful when multipliations of fields are required go bak to real quantities before! This is relevant for, e.g., alulation of Poynting vetor, see Chapter below. Temporal dependene of the fields When it omes to time dependene of the eletromagneti field, we an distinguish two different types of light: A) monohromati light stationary fields harmoni dependene on temporal oordinate

15 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 9 exp( i t) phase is fixed oherent, infinite wave train e.g.: Er (, t) Er ( )exp( i t) Monohromati light approximates very well the typial output of a ontinuous wave (CW) laser. One we know the frequeny we have to ompute the spatial dependene of the (stationary) fields only. B) polyhromati light non-stationary fields finite wave train With the help of Fourier transformation we an deompose the fields into infinite wave trains and use all the results from ase A) (see next setion)..3 Er (,) t Er (, )exp( it) d Er (, ) (, t)exp( t) dt Er i Remark: The position of the sign in the exponent and the fator / an be defined differently in different textbooks. Maxwell s equations in Fourier domain We want to plug the Fourier deompositions of our fields into Maxwell s equations in order to get a more simple desription. For this purpose, we need to know how a time derivative transforms into Fourier spae. Here we used integration by parts:, exp, exp (, ) dt t i t i dt t i t i Er t Er Er rule: t i FT Now we an write Maxwell s equations in Fourier domain: rot Er (, ) ihr (, ) div Er (, ) div Pr (, ) rot Hr (, ) jr (, ) ipr (, ) i Er (, ) div Hr (, )..4 From Maxwell s equations to the wave equation Maxwell's equations provide the basis to derive all possible mathematial solutions of eletromagneti problems. However very often we are interested just in the radiation fields whih an be desribed more easily by an adapted equation, whih is the so-alled wave equation. From Maxwell s equations it is straight forward to derive the wave equation by using the two url equations. Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 3 A) Time domain derivation We start from applying the url operator ( rot ) a seond time on rot Er (, t) and substitute rot H with the other Maxwell equation Hr (,) (,) (,) (,) t t Pr (,) t t E r t rotrot E r rot t t j r t t And find the wave equation for the eletri field rotrot Er (,) t (,) t jr (,) t P(,) r t E r t The blue terms require knowledge of the material model. Additionally, we have to make sure that all other Maxwell s equations are fulfilled, in partiular: div E(,) r t P(,) r t One we have solved the wave equation, we know the eletri field. From that we an easily ompute the magneti field: Hr (,) t rot E(,) r t t Remarks: An analog proedure is possible for H, i.e., we an derive a wave equation for the magneti field. Generally, the wave equation for E is more onvenient, beause the material model defines PE ( ). However, for inhomogeneous media H an be the better hoie for the numerial solution of the partial differential equation sine it forms a hermitian operator. analog proedure possible for H E generally, wave equation for E is more onvenient, beause PE ( ) given for inhomogeneous media H an be better hoie B) Frequeny domain derivation We an do the same proedure in the Fourier domain and find rotrot E r E r j r P r and div (, ) (, ) E r P r magneti field: t t (, ) (, ) i (, ) (, )

16 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 3 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 3..5 i Hr (, ) rot Er (, ) transferring the results from the Fourier domain to the time domain -i t for stationary fields: take solution and multiply by e. for non-stationary fields and linear media inverse Fourier transformation Er (,) t Er (, )exp( i t) d Deoupling of the vetorial wave equation So far we have seen that for the general problem of eletromagneti waves all 3 field omponents of the eletri or the magneti field are oupled. Hene we have to solve a vetorial wave equation for the general problem. However, it would be desirable to express problems also by salar equation sine they are muh easier to solve. For problems with translational invariane in at least one diretion, as e.g. for homogeneous infinite media, layers or interfaes, this an be ahieved sine the vetorial omponents of the fields an be deoupled. Let s assume invariane in the y-diretion and propagation only in the x-zplane. Then all spatial derivatives along the y-diretion disappear ( / y ) and the operators in the wave equation simplify. E x E z () x x z E x () E y E x E z () E z x z z rot rot E grad div E E The deoupling beomes visible when the three omponents of the general vetorial field are deomposed in the following way. deomposition of eletri field E x EE E E y, E z E E x () with Nabla operator (), and Laplae z x z Hene we obtain two wave equations for the gives two deoupled wave equations E and E fields. () E( r, ) E (, ) r i j ( r, ) P( r, ) E ( r, ) E ( r, ) grad div E j ( r, ) P ( r, ) () () () i These two wave equations are independent as long as the material response, whih is expressed by j and P, does not ouple the respetive field omponents by some anisotropi response. Properties propagation of perpendiularly polarized fields E and E an be treated separately propagation of E is desribed by salar equation similarly the other field omponents an be desribed by a salar equation for H alternative notations: s TE (transversal eletri) p TM (transversal magneti). Optial properties of matter In this hapter we will derive a simple material model for the polarization and the urrent density. The basi idea is to write down an equation of motion for a single exemplary harged partile and assume that all other partiles of the same type behave similarly. More preisely, we will use a driven harmoni osillator model to desribe the motion of bound harges giving rise to a polarization of the medium. For free harges we will use the same model but without restoring fore, leading eventually to a urrent density. In the literature, this simple approah is often alled the Drude-Lorentz model (named after Paul Drude and Hendrik Antoon Lorentz)... Basis We are looking for PE ( ) and je ( ). In general, this leads to a many body problem in solid state theory whih is rather omplex. However, in many ases phenomenologial models are suffiient to desribe the neessary phenomena. As already pointed out above, we use the simplest approah, the so-alled Drude-Lorentz model for free or bound harge arriers (eletrons). assume an ensemble of non-oupling, driven, and damped harmoni osillators free harge arriers: metals and exited semiondutors (intraband) bound harge arriers: dieletri media and semiondutors (interband)

17 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 33 The Drude-Lorentz model reates a link between ause (eletri field) and effet (indued polarization or urrent). Beause the resulting relations PE ( ) and j( E ) are linear (no E et.), we an use linear response theory. For the polarization PE ( ) (for j( E ) very similar): desription in both time and frequeny domain possible In time domain: we introdue the response funtion (G: Responsfunktion) Er (,) t medium (response funtion) Pr (,) t t P(,) r t R (, r tt) E (, r t) dt i ij j j with ˆR being a nd rank tensor i x, y, z and summing over j xyz,, In frequeny domain: we introdue the suseptibility (G: Suszeptibilität) Er (, ) medium (suseptibility) Pr (, ) P(, r ) (, r ) E (, r ) i ij j j response funtion and suseptibility are linked via Fourier transform (onvolution theorem) Rij () t ij ( )exp( t) d i Obviously, things look friendlier in frequeny domain. Using the wave equation from before and assuming that there are no urrents ( j ) we find rotrot E r E r P r or (, ) (, ) (, ) E r E r graddive r P r (, ) (, ) (, ) (, ) and for auxiliary fields Dr (, ) Er (, ) Pr (, ) The general response funtion and the respetive suseptibility given above simplifies for ertain properties of the medium: Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 34 Simplifiation of the wave equation for different types of media A) linear, homogenous, isotropi, non-dispersive media (most simple but very unphysial ase) homogenous ij(, r ) ij( ) isotropi ij (, r ) (, r ) ij non-dispersive ij (, r ) ij () r instantaneous: Rij (,) r t ij ()() r t (Attention: This is unphysial!) ij (, r ) is a salar onstant frequeny domain time domain desription Pr (, ) Er (, ) Pr (,) t Er (,) t (unphysial!) Dr (, ) Er (, ) Dr (,) t Er (,) t Maxwell: div D div E(, r ) for ( ) Er (, ) Er (, ) Er (,) t Er (,) t t approximation is valid only for a ertain frequeny range, beause all media are dispersive based on an unphysial material model B) linear, homogeneous, isotropi, dispersive media ( ) Pr (, ) ( ) Er (, ) Dr (, ) ( ) Er (, ) div D(, r ) div E(, r ) for ( ) (, ) Er (, ) Helmholtz equation Er This desription is suffiient for many materials. C) linear, inhomogeneous, isotropi, dispersive media (, r ) Pr (, ) (, r) Er (, ), Dr (, ) (, r) Er (, ). div Dr (, ) div Dr (, ) (, r) div Er (, ) Er (, ) grad (, r), grad (, r ) div Er (, ) Er (, ). (, r )

18 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 35 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 36 grad (, r ) Er (, ) r, Er (, ) grad Er (, ) (, r ) All field omponents ouple. D) linear, homogeneous, anisotropi, dispersive media ( ) Pi(, r ) ij( ) Ej(, r ) j see hapter on rystal optis D(, r ) ( ) E (, r ). i ij j j This is the worst ase for a medium with linear response. Before we start writing down the atual material model equations, let us summarize what we want to do: What kind of light-matter interation do we want to onsider? I) Interation of light with bound eletrons and the lattie The ontributions of bound eletrons and lattie vibrations in dieletris and semiondutors give rise to the polarization P. The lattie vibrations (phonons) are the ioni part of the material model. Beause of the large mass 3 of the ions ( mass of eletron) the resulting osillation frequenies will be small. Generally speaking, phonons are responsible for thermal properties of the medium. However, some phonon modes may ontribute to optial properties, but they have small dispersion (weak dependene on frequeny ). Fully understanding the eletroni transitions of bound eletrons requires quantum theoretial treatment, whih allows an aurate omputation of the transition frequenies. However, a (phenomenologial) lassial treatment of the osillation of bound eletrons is possible and useful. II) Interation of light with free eletrons The ontribution of free eletrons in metals and exited semiondutors gives rise to a urrent density j. We assume a so-alled (interation-)free eletron gas, where the eletron harges are neutralized by the bakground ions. Only ollisions with ions and related damping of the eletron motion will be onsidered. We will look at the ontributions from I) and II) separately, and join the results later... Dieletri polarization and suseptibility Let us first fous on bound harges (ions, eletrons). In the so-alled Drude model, the eletri field Er (,) t gives rise to a displaement sr (,) t of harged ij partiles from their equilibrium positions. In the easiest approah this an be modeled by a driven harmoni osillator: q t t m sr (,) t g sr (,) t sr (,) t Er (,) t resonane frequeny (eletroni transition) damping g harge q mass m The indued eletri dipole moment due to the displaement of harged partiles is given by pr (,) t qsr (,), t We further assume that all bound harges of the same type behave idential, i.e., we treat an ensemble of non-oupled, driven, and damped harmoni osillators. Then, the dipole density (polarization) is given by Pr (,) t Npr (,) t Nqsr (,) t Hene, the governing equation for the polarization Pr (,) t reads as (,) t g Pr Pr (,) t Prt (,) qn Er (,) (,) t t m t f Er t en with osillator strength f, for q=-e (eletrons) m This equation is easy to solve in Fourier domain: Pr (, ) i gpr (, ) Pr (, ) f Er (, ) f Pr (, ) Er (, ) ig f with Pr (, ) ( )Er (, ) ( ) i g In general we have several different types of osillators in a medium, i.e., several different resonane frequenies. Nevertheless, sine in a good approximation they do not influene eah other, all these different osillators ontribute individually to the polarization. Hene the model an be onstruted by simply summing up all ontributions. several resonane frequenies f j Pr (, ) (, ) (, ) Er Er j j ig j

19 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 37 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 38 f j j j ig j is the omplex, frequeny dependent suseptibility Dr (, ) Er (, ) Er (, ) Er (, ) is the omplex frequeny dependent dieletri funtion Example: (plotted for eta and kappa with i )..3 Condutive urrent and ondutivity Let us now desribe the response of a free eletron gas with positively harged bakground (no interation). Again we use the model of a driven harmoni osillator, but this time with resonane frequeny. This orresponds to the ase of zero restoring fore. e sr (,) t g sr (,) t Er (,), t t t m The resulting indued urrent density is given by jr (,) t Ne sr (,) t t and the governing dynami equation reads as en (,) t g (,) t (,) t p (,) t jr jr Er Er t m en with plasma frequeny p f m Again we solve this equation in Fourier domain: i jr jr Er (, ) g (, ) p (, ) p jr (, ) Er (, ) Er (, ). g i Here we introdued the omplex frequeny dependent ondutivity p p i. g i ig Remarks on plasma frequeny We onsider a loud of eletrons and positive ions desribed by the total harge density in their self-onsistent field E. Then we find aording to Maxwell: dive(,) r t (,) r t For old eletrons, and beause the total harge is zero, we an use our damped osillator model from before to desribe the urrent density (only eletrons move): g (,) t t j j E r p Now we apply divergene operator and plug in from above (red terms): div j gdivj pdive(,) r t p (,) r t t With the ontinuity equation for the harge density (from Maxwell's equations) divj, t We an substitute the divergene of the urrent density and find: t t g p t t g p harmoni osillator equation Hene, the plasma frequeny p is the eigen-frequeny of suh a harge density...4 The generalized omplex dieletri funtion In the setions above we have derived expressions for both polarization (bound harges) and ondutive urrent density (free harges). Let us now plug our jr (, ) and Pr (, ) into the wave equation (in Fourier domain)

20 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 39 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 4 rotrot E(, r ) E(, r ) P(, r ) ij(, r ) ( ) i Er (, ) Hene we an ollet all terms proportional to Er (, ) and write i rotrot Er (, ) ( ) Er (, ) rotrot Er (, ) ( ) (, ) Er Here, we introdued the generalized omplex dieletri funtion i ( ) ( ) ( ) i( ) So, in general we have beause (from before) f j p ( ), j j g g i j i f j j j ig j i ig, p. ( ) ontains ontributions from vauum, phonons (lattie vibrations), bound and free eletrons. Some speial ases for materials in the infrared and visible spetral range: A) Dieletris (insulators) in the infrared (IR) spetral range near phonon resonane If we are interested in dieletris (insulators) near phonon resonane in the infrared spetral range we an simplify the dieletri funtion as follows: f j ( ) j j j ig f ( ) ig f with j and ig The ontribution from eletroni transitions shows almost no frequeny dependene (dispersion) in this frequeny range far away from the eletroni resonanes. hene it an be expressed together with the vauum ontribution as a onstant. Let us study the real and the imaginary part of the resulting ( ) separately: vauum and eletroni transitions ( ) ( ) i ( ) ( ) i ( ) f ( ), g ( ) gf. Lorentz urve g properties: resonane frequeny: width of resonane peak: g 8 4 ε -4 stati dieletri onstant in the limit : f so alled longitudinal frequeny L : ( ) ( ) : absorption and dispersion appear always together near resonane we find ( ) (damping, i.e. deay of field, without absorption if '' ) frequeny range with normal dispersion: ( )/ frequeny range with anomalous dispersion: ( )/ Simplified example: sharp resonane for undamped osillator g ε ε ε ω ω L ω L

21 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 4 relation between resonane frequeny and longitudinal frequeny L (Lyddane-Sahs-Teller relation) f ( L ) L ε ε. L ig ε ω ω L ω, f (from above) B) Dieletris in the visible (VIS) spetral range Dieletri media in visible (VIS) spetral range an be desribed by a soalled double resonane model, where a phonon resonane exists in the infrared (IR) and an eletroni transition exists in the ultraviolet (UV). fp fe ( ), with p e ig p p e e ontribution of vauum and other (far away) resonanes. ε' Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 4 f j ( ), j j j with j being the number of resonanes taken into aount desribes many media very well (dispersion of absorption is negleted) osillator strengths and resonane frequenies are often fit parameters to math experimental data C) Metals in the visible spetral range If we want to desribe metals in the visible spetral range we find p ( ) ig with p p gp ( ), ( ). g g Metals show a large negative real part of the dieletri funtion ( ) whih gives rise to deay of the fields. Eventually this results in refletion of light at metalli surfaes. - ε ε ω P -g VIS ω in 5 s VIS ω in 5 s - The generalization of this approah in the transparent spetral range leads to the so-alled Sellmeier formula...5 Material models in time domain Let us now transform our results of the material models bak to time domain. In Fourier domain we found for homogeneous and isotropi media: Dr (, ) ( ) Er (, ) Pr (, ) ( ) Er (, ). The response funtion (or Green's funtion) Rt () in the time domain is then given by Rt () ( )exp td ( ) Rt ( )exp t dt i i

22 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 43 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 44 To prove this, we an use the onvolution theorem -i ( ) -i Er ( t i tdt -i td Pr (, t) Pr (, )exp t d Er (, )exp t d ( ), )exp exp Now we swith the order of integration, and identify the response funtion R (red terms): ( )exp ( tt) d (, t) dt -i Er Rt ( t) Rt ( t) Er (, t) dt For a delta exitation in the eletri field we find the response or Greens funtion as the polarization: Er (, t) e ( tt ) Pr (,) t R( t t) e Green's funtion Examples A) instantaneous media (unphysial simplifiation) For instantaneous (or non-dispersive) media, whih annot not really exist in nature, we would find: Rt () () t Pr, t Er, t (unphysial!) B) dieletris f RP() t exp td exp td, i g i i Using the residual theorem we find: C) metals f g exp t sin t t Rt () t t with f g Pr (,) t exp ( tt)sin ( tt) (, t) dt Er g 4 p Rj () t exp td exp td, i i g i p expgt t Rt () g t t jr (,) t exp g( tt) Er (, t) dt p.3 The Poynting vetor and energy balane.3. Time averaged Poynting vetor The energy flux of the eletromagneti field is given by the Poynting vetor S. In pratie, we always measure the energy flux through a surfae (detetor), S n, where n is the normal vetor of surfae. To be more preise, the Poynting vetor Sr (,) t Er(,) rt Hr (,) rt gives the momentary energy flux. Note that we have to use the real eletri and magneti fields, beause a produt of fields ours. In optis we have to onsider the following time sales: 4 optial yle: T / s pulse duration: T p in general Tp T duration of measurement: T m in general Tm T Hene, in general the detetor does not reognize the fast osillations of the i t optial field e (optial yles) and delivers a time averaged value. For the situation desribed above, the eletro-magneti fields fatorize in slowly varying envelopes and fast arrier osillations: (,)exp t t.. (,) t Er i Er r For suh pulses, the momentary Poynting vetor reads: Using again the residual theorem we find:

23 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 45 Sr (,) t Er(,) rt Hr(,) rt (,) (,) (,) (,) 4 Er t H rt E rt Hr t (,) t (,)exp t t (,) t (, t 4 Er Hr E r H (,) t (,) t (,) t (,)os t t E r H r Er Hr (,) t (,) t si E r H r n t. i r )expi t We find that the momentary Poynting vetor has some slow ontributions whih hange over time sales of the pulse envelope T p, and some fast ontributions os t, sint hanging over time sales of the optial yle T. Now, a measurement of the Poynting vetor over a time interval T m leads to a time average of Sr (,) t. ttm / Sr (,) t (,') t dt' T Sr ttm / m The fast osillating terms ~os t and ~sin t anel by the integration sine the pulse envelope does not hange muh over one optial yle. Hene we get only a ontribution from the slow term. ttm / Sr (, t) (, t ') (, t ') dt ' T tt / Er H r m m Let us now have a look at the speial (but important) ase of stationary (monohromati) fields. Then, the pulse envelope does not depend on time at all (infinitely long pulses). Er (,') t Er (), Hr (,') t Hr () Sr (,) t () (). Er H r This is the definition for the optial intensity I Sr (,) t. We see that an intensity measurement destroys information on the phase. I Sr (,) t measurement destroys phase information.3. Time averaged energy balane Let us motivate a little bit further the onept of the Poynting vetor. Some interesting insight on the energy flow of light and hene also on the transport of information an be obtained from the Poynting theorem, whih is the equation for the energy balane of the eletromagneti field. The Poynting Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 46 theorem an be derived diretly from Maxwell s equations. We multiply the two url equations by H r resp. E r (note that we use real fields): HrrotEr Hr Hr t Er Er ErrotHr Er ( jr Pr) t t Next, we subtrat the two equations and get Hr roter Er rothr Er Er Hr Hr Er( jr Pr). t t t This equation an be simplified by using the following vetor identity: diver Hr Hr roter Er rothr Finally, with E E E we find Poynting's theorem r t r t r E H dive H E j P (*) t r t r r r r r t r This equation has the general form of a balane equation. Here it represents the energy balane. Apart from the appearane of the Poynting vetor (energy flux), we an identify the vauum energy density u Er H r. The right-hand-side of the Poynting's theorem ontains the so-alled soure terms. where u Er H r vauum energy density In the ase of stationary fields and isotropi media (simple but important) Er(,) r t ()exp.. E r i t Hr(,) r t ()exp t.. H r i Time averaging of the left hand side of Poynting s theorem (*) yields: E (,) t (,) t (,) t (,) t () () r r Hr r div Er r Hr r div t t E r H r div S(,). r t Note that the time derivative removes stationary terms in E r (,) r t and H r (,) r t. Time averaging of the right hand side of Poynting s theorem yields (soure terms):

24 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 47 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 48 jr (,) r t Pr (,) r t (,) t t Er r 4 ( ) ) i t ( ) ( ) i t E( r e i Ere.. E(r) e i t.. Now we use our generalized dieletri funtion: i i E(r) exp it.. exp t.. 4 E(r) i i.. 4 E(r)E(r) Again, all fast osillating terms exp t i anel due to the time average. Finally, splitting into real and imaginary part yields.. ( ) ( ). 4 i i E(r)E(r) E r E r Hene, the divergene of the time averaged Poynting vetor is related to the imaginary part of the generalized dieletri funtion: div S E() r E (). r This shows that a nonzero imaginary part of epsilon ( ) auses a drain of energy flux. In partiular, we always have, otherwise there would be gain of energy. In partiular near resonanes we have and therefore absorption. Further insight into the meaning of div S gives the so-alled divergene theorem. If the energy of the eletro-magneti field is flowing through some volume, and we wish to know how muh energy flows out of a ertain region within that volume, then we need to add up the soures inside the region and subtrat the sinks. The energy flux is represented by the (time averaged) Poynting vetor, and the Poynting vetor's divergene at a given point desribes the strength of the soure or sink there. So, integrating the Poynting vetor's divergene over the interior of the region equals the integral of the Poynting vetor over the region's boundary. V div S dv S n da A

25 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 49.4 Normal modes in homogeneous isotropi media Using the linear material models whih we disussed in the previous hapters we an now look for self-onsistent solutions to the wave equation inlude the material response. It is onvenient to use the generalized omplex dieletri funtion for the derivative of the solution of the wave equation i ( ) ( ) ( ) i ( ) We will do our analysis in Fourier domain. In partiular, we will fous on the most simple solution to the wave equation in Fourier domain, the so-alled normal modes. We will see later that it is possible to onstrut general solutions from the normal modes. The wave equation in Fourier domain reads rotrot Er (, ) ( ) Er (, ) Aording to Maxwell the solutions have to fulfill additionally the divergene equation: ( ) div E( r, ) In general, this additional ondition implies that the eletri field is free of divergene: ( ) div E( r, ) (normal ase) Let us for a moment assume that we already know that we an find plane wave solutions of the following form in the frequeny domain: Er (, ) E( )exp i kr, k = unknown omplex wave-vetor The orresponding stationary field in time domain is given by: Er (,) t Eexp i kr t monohromati plane wave normal mode This is a monohromati plane wave, the simplest solution we an expet, a so-alled normal mode. Then, the divergene ondition implies that those waves are transversal k E( ) transverse wave If we split the omplex wave vetor into real and imaginary part we an define: o planes of onstant phase kr ' onst. k k' ik'', Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 5 o planes of onstant amplitude k''r onst. In the following we will all the solutions A) if those planes are idential homogeneous waves B) if those planes are perpendiular evanesent waves C) otherwise inhomogeneous waves We will see that in dieletris we an find a seond, exoti type of wave solutions: At L ( L ), so-alled longitudinal waves k E( ) appear..4. Transversal waves As pointed out above, for L the eletri field beomes free of divergene: ( )div (, ) Er div Er (, ) Then, the wave equation redues to the Helmholtz equation: Er (, ) ( ) Er (, ). Hene, we have three salar equations for Er (, ) (from Helmholtz), and together with the divergene ondition we are left with two independent field omponents. We will now onstrut solutions using the plane wave ansatz: Er (, ) E( )expi kr Immediately we see that the wave is transversal: dive( r, ) ik E( r, ) k E( ). Hene, we have to solve k ( ) ( ) E and ke ( ). whih leads to the following dispersion relation k k kx ky kz ( ) We see that the so-alled wave-number k( ) ( ) is a funtion of the frequeny. We an onlude that transversal plane waves are solutions to Maxwell's equations in homogeneous, isotropi media, only if the dispersion relation k( ) is fulfilled. In general, k=ki k is omplex. Alternatively it is sometimes useful to introdue the omplex refrative index (if k k): k( ) ( ) nˆ ( ) n( ) i ( )

26 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 5 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 5 However, instead of assuming that nˆ( ) and ( ) are just the same, one should learly distinguish between the two. While ( ) is a property of the medium, nˆ( ) is a property of a partiular type of the eletromagneti field in the medium, i.e. a property of the infinitely extended monohromati plane wave. Er (, ) E( )expi kr With the knowledge of the eletri field we an ompute the magneti field if desired:.4. i Hr (, ) rot Er (, ) k E( )exp ikr Hr (, ) H( )exp ikr, with H( ) k E( ) Longitudinal waves Let us now have a look at the rather exoti ase of longitudinal waves. Those waves an only exist for ( ) in dieletris at the longitudinal frequeny L. In this ase, we annot onlude that div E(, r ), and the wave equation reads (the l.h.s. vanishes beause ( ) ): rotrot E(, r L ) As for the transversal waves we try the plane wave ansatz and assume k to be real. Er (, ) E( )expi kr With rot E ( )exp kr k E ( )exp kr i i i we get from the wave equation: kk E(, r L) Now we deompose the eletri field into transversal and longitudinal omponents with respet to the wave vetor: Er (, ) E( )exp ikr E ( )exp ikr E( )exp ikr with, E ( ) k and E ( ) k This deomposed field is inserted into the wave equation: k k E E expikr k ke expikrkk Eexpikr Sine the ross produt of k with the longitudinal field E ( ) is trivially zero the remaining wave equation is: k E Hene the transversal field E must vanish and the only remaining field omponent is the longitudinal field E ( ) : Er (, ) E( )exp kr L L i.4.3 Plane wave solutions in different frequeny regimes The dispersion relation for plane wave solutions k k kx ky kz ( ) ditates the (omplex) wavenumber k only. Thus, different solutions for the omplex wave vetor k=ki k are possible. In addition, the generalized dieletri funtion ( ) is omplex. In this hapter we will disuss possible senarios and resulting plane wave solutions. A) Positive real valued epsilon ' This is the regime favorable for optis. We have transpareny, and the frequeny is far from resonanes. The dispersion relation gives k k' k'' k ' k'' ( ) ( ) k ' k'' i n There are two possibilities to fulfill this ondition, either k'' or k' k''. A.) Real valued wave-vetor k'' In this ase the wave vetor is real and we find the dispersion relation k( ) n( ) n( ) n Beause k'' these waves are homogeneous, i.e. planes of onstant phase are parallel to the planes of onstant amplitude. This is trivial, beause the amplitude is onstant. Example : single resonane in dieletri material for lattie vibrations (phonons)

27 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 53 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 54 ε ε ω ω = k ω ω p Now the imaginary part of ( ) is negleted, whih mathematially orresponds to an undamped resonane ( ) ( ) f We an invert the dispersion relation k( ) ( ) ( k) : ω f ω + ω Example : free eletrons for plasma and metal Again the imaginary part of ( ) is negleted ( ) ( ) p ε We again invert the dispersion relation k( ) ( ) ( k) : ω = ω = k ε ε k k A.) Complex valued wave-vetor k' k'' The seond possibility to fulfill the dispersion relation leads to a omplex wave-vetor and so-alled evanesent waves. We find k This means that k' k'' ( ) and therefore k'' k' k k'' and ' k k We will disuss the importane of evanesent waves in the next hapter, where we will study the propagation of arbitrary initial field distributions. What is interesting to note here is that evanesent waves an have ' arbitrary large k k, whereas the homogeneous waves of ase A.) ( ' k'' ) obey k k. If we plug our findings into the plane wave ansatz we get: for the evanesent waves: i k ' rex ( ) r Er (, ) E( ) exp p k'' The planes defined by the equation k''( )r = onst. are the so-alled planes of onstant amplitude, those defined by k'( )r = onst. are the planes of onstant phase. Beause of k' k'' these planes are perpendiular to eah other. The fator exp k''( )r leads to exponential growth of evanesent waves in homogeneous spae. Therefore, evanesent waves an't be physially justified normal modes of homogeneous spae and an only exist in inhomogeneous spae, where the exponential growth is suppressed, e.g. at interfaes. B) Negative real valued epsilon ( ) ( ) This situation (negative but real ( ) an our near resonanes in dieletris ( L ) or below the plasma frequeny ( p ) in metals. Then the dispersion relation gives k

28 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 55 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 56 k k' k'' i k' k'' ( ) ε ε ε ε ω ω As in the previous ase A), the imaginary term has to vanish and k' k''. Again this an be ahieved by two possibilities. B.) k' k'' ( ) Er (, ) expk''r strong damping B.) k' k'' k' k'' evanesent waves k k' k'' ( ) k'' ( ). k' As above, these evanesent waves exist only at interfaes (like for ( ) ( ) ). The interesting point is that here we find evanesent waves for all values of k'. In partiular, ase B.) ( k' ) is inluded. Hene, we an onlude that for ( ) ( ) we find only evanesent waves! C) Complex valued epsilon ( ) This is the general ase, whih is relevant partiularly near resonanes. From our (optial) point of view only weak absorption is interesting. Therefore, in the following we will always assume ( ) ( ). As we an see in the following sketh, we an have ( ), ( ), or ( ), ( ). Let us further onsider only the important speial ase of quasi-homogeneous plane waves, i.e., k' and k'' are almost parallel. Then, it is onvenient to use the omplex refrative index k ik k ( ) ( ) nˆ ( ) n( ) i ( ) Sine k' and k'' are almost parallel: k' n( ), k'' ( ) The dispersion relation in terms of the omplex refrative index gives k k ( ) n( ) i( ) Here we have ( ) ( ) i( ) n ( ) ( ) in( ) ( ), ( ) n ( ) ( ) and therefore ( ) n( ) ( ) n ( ) sgn /, ( ) sgn /. Two important limiting ases of quasi-homogeneous plane waves: C.),,, (dieletri media) ( ) n( ) ( ), ( ) ( )

29 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 57 In this regime propagation dominates ( n( ) ( ) ), and we have weak absorption: k' k'' k' k'', ( ), ( ). k' k'' k' k'' ( ) k' n( ) ( ), k'' ( ) ( ) k' and k'' almost parallel homogeneous waves in homogeneous, isotropi media, next to resonanes, we find damped, homogeneous plane waves, k' k ek with e k being the unit vetor along k Er (, ) E( )exp ikre( )expi ner k exp. er k C.),,, (metals and dieletri media in so-alled Reststrahl domain) ( ) n( ), ( ) ( ), ( ) In this regime damping dominates ( n( ) ( ) ) and we find a very small refrative index. Interestingly, propagation (nonzero n) is only possible due to absorption (see time averaged Poynting vetor below). Summary of normal modes a) undamped homogeneous waves and evanesent waves b) evanesent waves ) weakly damped quasi-homogeneous waves d) strongly damped quasi-homogeneous waves Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox Time averaged Poynting vetor of plane waves Sr ttm / (,) t (,) t (,) t dt, T tt / Er H r m m For plane waves we find: Er (,) t EexpikritEexpikr kr it Hr (,) t ker (,) t assuming a stationary ase E( t) E ( )exp( it) k n k ek' ek" r E Sr (, t ) exp re exp with e k' being the unit vetor along k and e k'' being the unit vetor along k..5 The Kramers-Kronig relation In the previous setions we have assumed a very simple model for the desription of the material's response to the exitation by the eletromagneti field. This model was based on quite strong assumptions, like a single harge whih is attahed to a rigid lattie et. Hene, one ould imagine that more omplex matter ould give rise to arbitrarily omplex response funtions if adequate models would be used for its desription. However we an show from basi laws of physis, that several properties are ommon to all possible response funtions, as long as a linear response to the exitation is assumed. These fundamental properties of the response funtion are formulated mathematially by the Kramers-Kronig relation. It is a general relation between ( ) (dispersion) and ( ) (absorption). This means in pratie that we an ompute ( ) from ( ) and vie versa. For example, if we have aess to the absorption spetrum of a medium, we an alulate the dispersion. The Kramers-Kronig relation follows from reality and ausality of the response funtion R of a linear system. That the response funtion is real valued is a diret onsequene from Maxwell's equations whih are real valued as well. Causality is also a very fundamental property, sine the polarization must not depend on some future eletri field. As we have seen in the previous setions, in time-domain the polarization and the eletri field are related as: t r r r r P (,) r t R( tt ) E (, r t ) dt P (,) r t R() E (, r t) d Reality of the response funtion implies:

30 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 59 R d d - i * i e e Causality of the response funtion implies: R y with for for Heaviside distribution for In the following, we will make use of the Fourier transform of Heaviside distribution: it i dtte P defined as integral only In Fourier spae, the Heaviside distribution onsists of the Dira delta distribution d( ) f f Dira delta distribution and the expression P(i/ ) involving a Cauhy prinipal value: i i i P d f( ) lim d f( ) d f( ) Cauhy priniple value As we have seen above, ausality implies that the response funtion has to ontain a multipliative Heaviside funtion. Hene, in Fourier spae (suseptibility) we expet a onvolution: i d Re d y y d y i e i P i y P d ( ) In order to derive the Kramers-Kronig relation we an use a small trik (this trik saves us using omplex integration in the derivation). Beause of the Heaviside funtion, we an hoose the funtion y for < arbitrarily without altering the suseptibility! In partiular, we an hoose: y y even funtion a) b) y y odd funtion Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 6 a) y y In this ase y y is a real valued and even funtion. We an exploit this property and show that i i y d ye d ye y is real as well Hene, we an onlude from equation (*) above that iy y P d Here P is a so alled prinipal value integral (G: Hauptwertintegral). * Now we have expressions for, and an ompute real and imaginary part of the suseptibility: * iy y iy y P d P d y i y P d Plugging the last two equations together we find the first Kramers-Kronig relation: P d. K-K relation Knowledge of the real part of the suseptibility (dispersion) allows us to ompute the imaginary part (absorption). b) y y The seond K-K relation an be found by a similar proedure when we assume that y y is a real odd funtion. We an show that in this ase i i y d ye d ye y is purely imaginary With equation (*) we then find that iy y P d (see (*)) and Again we an then ompute real and imaginary part of the suseptibility

31 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 6 iy y iy y P d P d y * P iy d and finally obtain P d. K-K relation The seond Kramers-Kronig relation allows us to ompute the real part of the suseptibility (dispersion) when we know its imaginary part (absorption). The Kramers-Kronig relation an also be rewritten in terms of the dieletri funtion, where one applies also the symmetry relation for : K-K relation for : ( ) ( ) ( ) ( ) and ( ) ( ) ( ) i ( ) ( ) ( ) P d, ( ) ( ) P d. dispersion and absorption are linked, e.g., we an measure absorption and ompute dispersion Example: ( ) ( ) ( ) Drude-Lorentz model.6 Beams and pulses - analogy of diffration and dispersion In this hapter we will analyze the propagation of light. In partiular, we will answer the question how an arbitrary beam (spatial) or pulse (temporal) will hange during propagation in isotropi, homogeneous, dispersive media. Relevant (linear) physial effets are diffration and dispersion. Both phenomena an be understood very easily in the Fourier domain. Temporal effets, i.e. the dispersion of pulses, will be treated in temporal Fourier domain (temporal frequeny domain). Spatial effets, i.e. the diffration of beams, will be treated in the spatial Fourier domain (spatial frequeny domain). We will see that: Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 6 Pulses with finite spatial width (i.e. pulsed beams) are superposition of normal modes (in frequeny- and spatial frequeny domain). Spatio-temporally loalized optial exitations deloalize during propagation beause of different phase evolution for different frequenies and spatial frequenies (different propagation diretions of normal modes). Let us have a look at the different possibilities (beam, pulse, pulsed beam) A) beam finite transverse width diffration plane wave (normal mode) beam A beam is a ontinuous superposition of stationary plane waves (normal modes) with different wave vetors (propagation diretions). 3 t E(,) r t E ( k)exp d k i k r B) pulse finite duration dispersion k stationary wave (normal mode) pulse A pulse is a ontinuous superposition of stationary plane waves (normal modes) with different frequenies. w k k k3 k4 k5 T p

32 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 63 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox u(, r ) u(, r ), u r r (, ) k u(, ). salar Helmholtz equation In the last step we inserted the dispersion relation (wave number k( )). In the following we often even omit the argument of the fixed frequeny. Er (,) t E( )exp i kr t d. C) pulsed beams finite transverse width and finite duration diffration and dispersion A pulsed beam is a ontinuous superposition of stationary plane waves (normal modes) with different frequeny and different propagation diretion 3 t Er (,) t E( k, )exp dkd i k r.7 Diffration of monohromati beams in homogeneous isotropi media Let us have a look at the propagation of monohromati beams first. In this situation, we have to deal with diffration only. We will see later that pulses and their dispersion an be treated in a very similar way. Treating diffration in the framework of wave-optial theory (or even Maxwell) allows us to treat rigorously many important optial systems and effets, i.e., optial imaging and resolution, filtering, mirosopy, gratings,... In this hapter, we assume stationary fields and therefore onst. For tehnial onveniene and beause it is suffiient for many important problems, we will make the following assumptions and approximations: ( ) ( ), optial transparent regime normal modes are stationary homogeneous and evanesent plane waves salar approximation Er (, ) E (, r) e E (, r) u(, r ). y y y exat for one-dimensional beams and linear polarization approximation in two-dimensional ase In homogeneous isotropi media we have to solve the Helmholtz equation Er (, ) Er (, ). In salar approximation and for fixed frequeny it reads.7. Arbitrarily narrow beams (the general ase) Let us onsider the following fundamental problem. We want to ompute from a given field distribution uxy (,,) in the plane z the omplete field uxyz (,, ) in the half-spae z, where z is our propagation diretion. The governing equation is the salar Helmholtz equation u(, r ) k u(, r ) To solve this equation and to alulate the dynamis of the fields, we an swith again to the Fourier domain. We take the Fourier transform 3 i u(, r ) U( k, )exp k( ) r d k whih an be interpreted as a superposition of normal modes with different propagation diretions and wavenumbers k( ) (here the absolute value of the wave-vetor k ). Naively, we ould expet that we just onstruted a general solution to our problem, but the solution is not orret beause of the dispersion relation: k k kx ky kz only two omponents of k are independent, e.g., kx, ky. Our naming onvention is in the following: k, k, k. x y z

33 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 65 Then, the dispersion relation reads: Thus, to solve our problem we need only a two-dimensional Fourier transform, with respet to transverse diretions to the propagation diretion z : u() r U(,; z)exp i xy dd. In analogy to the frequeny we all spatial frequenies. Now we plug this expression into the salar Helmholtz equation u( r) k u( r) This way we an transfer the Helmholtz equation in two spatial dimensions into Fourier spae d dz k U z d dz (, ; ), U z (, ; ). This equation is easily solved and yields the general solution i -i U(, ; z) U (, )exp (, ) z U (, )exp (, ) z, depending on (, ) k ( ). We an identify two types of solutions: A) Homogeneous waves, k, i.e., k real homogeneous waves B) Evanesent waves, k, i.e., k omplex, beause k z imaginary. Then, we have k=ki k, with k = ex e y and k = e z. k k' k'' evanesent waves α k β γ k Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 66 z the solution We see immediately that in the half-spae exp i z grows exponentially. Beause this does not make sense, this omponent of the solution must vanish U (, ). In fat, we will see later that U (, ) orresponds to bakward running waves, i.e., light propagating in the opposite diretion. We therefore find the solution: i i U(, ; z) U(, )exp (, ) z U(, ;)exp i (, ) U (, )exp (, ) z Furthermore the following boundary ondition holds: U(, ;) U(, ). In spatial spae, we an find the optial field for z by inverse Fourier transform: z u() r U(,; z)expi xy dd. α α ² + β ² > k² β k z i u( r) U(, )exp i, e xp xy dd. For homogeneous waves (real ) the red term above auses a ertain phase shift for the respetive plane wave during propagation. Hene, we an formulate the following result: Diffration is due to different phase shifts in propagation diretion for the different normal modes aording to their different spatial frequenies,. The initial spatial frequeny spetrum or angular spetrum at z forms the initial ondition of the initial value problem and follows from u ( x, y) u( x, y,) by Fourier transform: U(, ) u( x, y)exp x ydxdy, i As mentioned above the wave-vetor omponents, are the so-alled spatial frequenies. Another ommon terminology is diretion osine for the quantities / k, / k, beause of the diret link to the angle of the respetive γ

34 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 67 Sript "Grundkonzepte der Optik", FSU Jena, Prof. T. Pertsh, GdO3_Sript_4-5-8s.dox 68 pane wave. For example / k os x gives the angle of the plane wave's propagation diretion with the x -axis. Sheme for alulation of beam diffration We an formulate a general sheme to desribe the diffration of beams:. initial field: ( ). initial spetrum: (, ) by Fourier transform 3. propagation: by multipliation with exp i, z 4. new spetrum: U(, ; z) U(, )exp i, z 5. new field distribution: uxyz (,, ) by Fourier bak transform This sheme allows for two interpretations: ) The resulting field distribution is the Fourier transform of the propagated spetrum u() r U(, ; z) exp xy dd. i ) The resulting field distribution is a superposition of homogeneous and evanesent plane waves ('plane-wave spetrum') whih obey the dispersion relation (, )exp, u( r ) U i x y z d d. Let us now disuss the omplex transfer funtion H (, ; z) exp[ i (, ) z], whih desribes the beam propagation in Fourier spae. For z = onst. (finite propagation distane) it looks like: A) homogeneous waves k i z i exp,, arg exp, z Upon propagation the homogeneous waves are multiplied by the phase fator i B) evanesent waves exp k z k i i exp, z exp k z, arg exp, z Upon propagation the evanesent waves are multiplied by an amplitude fator < exp k z This means that their ontribution gets damped with inreasing propagation distane z. Now the question is: When do we get evanesent waves? Obviously, the answer lies in the boundary ondition: Whenever u ( x, y ) yields an angular spetrum U (, ) for k we get evanesent waves. Example: Slit Let us onsider the following one-dimensional initial ondition whih orresponds to an aperture of a slit: amplitude phase u (x) a for x u( x). otherwise -a/ a/ a sin a U ( ) FT u ( x) sin a x Obviously, (, ; ) exp, H z i z ats differently on homogeneous and evanesent waves: