Chapter 2 Classical propagation

Transcription

1 Chapte Classical popagation Model: Light: electomagnetic wave Atom and molecule: classical dipole oscillato n. / / t c nz i z t z k i e e c i n k e t z Two popagation paametes: n. Popagation of light in a dense optical medium t ex t p q p K m m S N Thee types of oscillatos:. bound electon atomic oscillato. vibational oscillato; 3. fee electon oscillatos.. Atomic oscillatos

2 . Popagation of light in a dense optical medium.. Atomic oscillatos If = esonant absoption Bee s law h = - e-adiated photon luminesce adiationless tansition If non-esonant tanspaent The oscillatos follow the diving wave but with a phase lag. The phase lag accumulates though the medium and etads the popagation of the wave font leading to smalle velocity than in fee space v =c / n. -- the oigin of n.. Vibational oscillatos K S 3 Hz Infaed spectal egion In a cystalline solid fom the condensation of pola molecules these oscillations ae associated with lattice vibations phonons. Classical model of a pola molecule an ionic optical medium..3 Fee electon oscillatos Fee electons K s = = Dude-Loentz model

3 . The dipole oscillato model.. The Loentz oscillato d x m m x m N m dt d x dx m m m x e t dt dt Light wave will dive oscillations at its own Fequency: t Solution; x t The gives: X ' X cos t eexp i t eexp it eexp it eexp i t ' it it it m X e imx e m X e With: e / m i X The macoscopic polaization of medium P: P esonant m e e Np x it The electic displacement D: D Thus P P m m m backgound P i P esonant esonant i

4 . The dipole oscillato model.. The Loentz oscillato low fequency limit: high fequency: Thus st Close to esonance: m st m st st 4 4 Fequency dependence of the eal and imaginay Pats of the complex dielectic constant of a dipole At fequencies close to esonance. Also shown is The eal and imaginay pat of the efactive index Calculated fom the dielectic constant.

5 . The dipole oscillato model.. Multiple esonance Take account of all the tansitions in the medium P Np m x i m D P. i Assign a phenomenological oscillato stength f to each tansition: f m whee f. i Fo each atom. Schematic diagam of the fequency dependence of the efactive index and absoption of a hypothetical solid fom the infaed to the x-ay spectal egion. The solid is assummed to have thee esonant fequencies with width of each absoption line has been set to % of the cente fequency by appopiate choice of the s.

6 . The dipole oscillato model..3 Compaison with expeimental data. n >> except nea the peaks of the absoption;. The tansmission ange of optical mateials is detemined by the electonic absoption in UV and the vibational absoption in IR; 3. IR absoption is caused by the vibational quanta in SiO molecules themselves.4 3 Hz m and Hz9. m; 4. UV absoption is caused by inteband electonic tansitionband gap of about ev theshold at 3 Hz5 nm ~ 8 m - ; 5. UV absoption depatue fom Loentz model; 6. n actually inceases with fequency in tanspaency egion the dispesion oiginates fom wings of two absoption peaks of UV and IR; 7. The phase velocity of light is geate than c in egion whee n falls below unity; 8. Goup velocity: d k g dk n dn / dk g dn dk c a Refactive index and b extinction co- fficient of fused silica SiO glass fom the Infaed to the x-ay spectal egion.

7 . The dipole oscillato model..4 Local field coection The actually atomic dipoles espond not only to the extenal field but also to the field geneated by all the othe dipoles local othe local P a P dipoles N e m N a N 3 a P 3 othe P 3 local P 3 a f dipoles i Clausius-Mossotti elationship Model used to calculate the local field by the Loentz coection. A imaginay spheical suface dawn aound a paticula atom divides the medium into neaby dipoles and distant dipoles. The field at the cente of the sphee due to the neaby dipoles is sunned exactly while the field due to the distant dipoles is calculated by teating the mateial outside the sphee as a unifomly polaized dielectic.

8 . The dipole oscillato model..5 The Kames-Konig elationships The discussion of the dipole oscillato shows that the efactive index and the absoption coefficient ae not independent paametes but ae elated to each othe. If we invoke the law of causality that an effect may not pecede its cause and apply complex numbe analysis we can deive geneal elationships between the eal and imaginay pats of the efactive index as follows: n P P ' d' ' n ' d' ' Whee P indicates that the pincipal pat of the integal should be taken. The K-K elationships allow to calculate n and and vice vesa.

9 . Dispesion This dispesion mainly oiginates fom the inteband absoption in the UV and the vibational absoption in IR Refactive index of SiO glass in the IR visible And UV egions Nomal dispesion : the efactive index inceases with fequency; Anomalous dispesion: the contay occus.

10 . Dispesion Pulse boadening t p Dispesion causes the vey shot pulse to boaden in time as it popagates though the medium. goup velocity dispesion GVD g d dk k n dn dk GVD d dk The Loentz model indicates that GVD is positive below an absoption line and negative above it. Thee is a egion of zeo GVD aound.3 m in silica. So shot pulses can be tansmitted down the silica fibe with negligible tempoal boadening at this wavelength.

11 . Optical anisotopy: biefingence The elationship of the P and P : thesusceptibility tenso. Px Py Pz 3 3 Chos sin g x y z to cystallin e axes : 3 x 3 y 33 z the pincipal 33 Cubic: 33 isotopic; Tetagonal hexagonal o tigonal: 33 uniaxial; Othohombic monoclinic o ticlinic: 33 biaxial. Re fo fative uniaxial cystal : n n index n and dielectic 33 cons tan t tenso