Ultrafast Optical Physics II (SoSe 2017) Lecture 2, April 21

Transcription

1 Ultrafast Optical Physics II SoSe 7 Lecture pril Susceptibility a Sellmeier equatio Phase velocity group velocity a ispersio 3 Liear pulse propagatio

2 Maxwell s Equatios of isotropic a homogeeous meia Maxwell s Equatios: Differetial Form mpere s Law Curret ue to free charges Faraay s Law Gauss s Law No magetic charge Free charge esity Material Equatios: Brig Life ito Maxwell s Equatios Polariatio Magetiatio

3 Derivatio of wave equatio No free charges No currets from free charges No magetiatio I the liear optics of isotropic meia without free charges Simplifie wave equatio: Wave i vacuum Source term Laplace operator: x y 3

4 Dielectric susceptibility a Helmholt equatio I a liear meium ielectric susceptibility is iepeet of optical fiel χ Ca be complex Refractive Iex Meium spee of light epeet o frequecy: 4

5 Susceptibility calculate usig Loret moel Plasma frequecy p Ne ε m / 5

6 Real a imagiary part of the susceptibility Q Real part ashe lie a imagiary part soli lie of the susceptibility of the classical oscillator moel for the ielectric polariability 6

7 Real a imagiary part of the susceptibility Example: EM-Wave polarie alog x-axis a propagatio alog -irectio: Physics otatio I geeral: Egieerig otatio Dispersio relatio: r c r 7

8 bsorptio a refractive iex Vs. wavelegth 8

9 9 Sellmeier equatios to moel refractive iex Normally there are multiple resoat frequecies for the electroic oscillators. It meas i geeral the refractive iex will have the form If the frequecy is far away from the absorptio resoace Q Ω >> Ω ~ χ Ω p r Ω i i i i i p i a λ λ λ

10 Liear propagatio of a pulse? j [ -j E e ] j [ ][ χ ] j [ Slowly varyig amplitue approximatio [ ] Neglectig iffractio e.g. isie a optical waveguie Fourier trasform jt E t t e c ] E E c <<

11 c c Liear pulse propagatio

12 Group velocity Vs phase velocity -j -j e e E ] [ p V t j g t j e V t e t t E / c V g c V p Group velocity: travellig spee of the pulse evelope. Phase velocity: travellig spee of the carrier wave uer the pulse evelope. Electric fiel a pulse evelope i time omai ] [ j -j e... Let s first tae a loo at the effect of the first two terms j

13 Group velocity Vs phase velocity I vacuum Urealistic Most commo case Possible Urealistic Urealistic apte from Ric Trebio s course slies 3

14 Use the chai rule : λ λ c / πc / λ πc Now λ π so : Recallig that : we have : or : Calculatig group velocity vs. wavelegth We more ofte thi of the refractive iex i terms of wavelegth so let's write the group velocity i terms of the vacuum wavelegth λ. v g / v g / λ πc πc λ c c π c λ λ λ π c c λ v / g λ apte from Ric Trebio s course slies

15 5 Group-velocity ispersio GVD What s effect of the 3 r term i the Taylor expasio of wave vector?... ] [ j ] [ V j g V g Group velocity becomes frequecy epeet. V g Group velocity ispersio GVD V j g e ] [ The pulse maitais its optical spectrum shape but acquires a quaratic spectral phase from GVD which will chage the pulse s temporal profile.

16 Group-velocity ispersio GVD g vg v πc g v λ λ c λ Positive GVD or ormal ispersio Negative GVD or aomalous ispersio > < v g v g < > Low frequecy travels faster High frequecy travels faster

17 Gaussia Pulse: Effect of GVD o pulse propagatio Pulse with Substitute: Gaussia Itegral: 7

18 Propagatio of istace: Expoet Real a Imagiary Part: -epeet phase shift iepeet o time FWHM Pulse with: etermies pulse with temporal quaratic phase Iitial pulse with: 8

19 Iitial pulse with: fter propagatio over a istace L: For large istaces: Magitue of the complex evelope of a Gaussia pulse t i a ispersive meium 9

20 v g Spectrum v g à 3 v g3 Dispersio Relatio Decompositio of a pulse ito wave pacets with ifferet ceter frequecy. I a meium with ispersio the wave pacets move at ifferet relative group velocity

21 Istataeous frequecy a chirp -epeet phase shift iepeet o time fter propagatio of L istace: etermies pulse with temporal quaratic phase E L t' L t'exp j t' exp[ jt' jφ L t'] Istataeous Frequecy: ist [ t' φ L t'] φ L t' t t' t' φ L t' t'

22 Liearly chirpe Gaussia pulse: positive chirp is t' φ L t' t' For positive GVD i.e. > lower frequecy travels faster a the istataeous frequecy liearly INCRESES with time. I aalogy to bir sous this pulse is calle a chirpe pulse or positively chirpe pulse. Time t

23 Liearly chirpe Gaussia pulse: egative chirp is t' φ L t' t' For egative GVD i.e. < higher frequecy travels faster. The istataeous frequecy liearly DECRESES with time. This pulse is calle a egatively chirpe pulse. Time t

24 Istataeous Frequecy: is t' φ L t' t' a temporal Phase a b istataeous frequecy of a Gaussia pulse urig propagatio through a meium with positive or egative ispersio 4

25 Trasform-limite pulse ~ E E texp jt t ~ E Et has a spectrum bawith of has a pulse uratio of ~ E t E exp jt π ν t Both are measure at full-with at halfmaximum FWHM. Ucertaity priciple: Time Bawith Prouct TBP ν t K umber epeig oly o pulse shape For a give optical spectrum there exist a lower limit for the pulse uratio. If the equality is reache we say the pulse is a trasform-limite pulse. To get a shorter trasform-limite pulse oe ees a broaer optical spectrum. 5

26 Temporal a spectral shapes a TBPs of typical ultrashort pulses Diels a Ruolph Femtoseco Pheomea

27 Some efiitios... c m m m λ λ c c v g λ λ π λ c c c v v v g g g g v Group velocity Group velocity ispersio GVD Uit: s /m Note: more ofte β is use to replace β a is GVD. Uit: s/m

28 GVD chages the pulse uratio a itrouces chirp v g λ λ vg v πc c g λ Positive GVD or ormal ispersio Negative GVD or aomalous ispersio < < > v g v g > Re faster positive chirp Blue faster egative chirp

29 Pulse travels through a ispersive bul meium t p t p Istataeous Frequecy Istataeous Frequecy time t time t Trasform-limite pulse Positive chirp

30 Group Delay & Group Delay Dispersio ϕ ϕ ϕ ϕ ϕ ϕ τ v g g Group elay i fs ϕ m m ϕ m ϕ τ g Group elay ispersio GDD i fs GDD > positive ispersio GDD < egative ispersio ϕ3 Thir orer ispersio TOD i fs 3 ϕ4 Fourth orer ispersio i fs 4 Group elay shift the time origi of the pulse evelope while GDD chages its shape.

31 Effect of absolute phase 3

32 Effect of group elay 3

33 Effect of positive orer ispersio

34 Effect of positive 3 r orer ispersio

35 Effect of egative 3 r orer ispersio 35

36 Effect of positive 4 th orer ispersio

37 Dispersio parameters for various materials 37

38 ~ ] [ ~ j I the time omai ~ ]! [ ~ j ~ ]! [ ~ j ]! [ t t j j t Liear propagatio equatio for pulse evelope from slie

39 ]! [ t t j j t t t ]! [ t t j j t t V t g V g t ' t ]! [ ' ' ' t t j j t I a frame of referece movig with the pulse at the group velocity: Liear propagatio equatio for pulse evelope

40 Effect of egative GVD GVD β 5 ps / m Iput pulse uratio:fs

41 Effect of positive GVD GVD β 5ps / m The output of last slie is tae as the iput here.

42 Real a imagiary part of the susceptibility Example: EM-Wave polarie alog x-axis a propagatio alog -irectio: I geeral: Dispersio relatio: r c r 4

43 Besies ispersio a meium may itrouce loss or gai Refractive iex gai a/or loss for: ~ χ Ω << Complex Loretia close to resoace : p Maximum absorptio: Half With Half Maximum liewith HWHM: 43

44 Real a imagiary parts: Complex wave umber i lossy meium: Reefie group velocity: e.g. at lie ceter: Chage i group velocity ca be positive or egative 44

45 bsorptio: For a wavepacet optical pulse with carrier frequecy Ω Parabolic loss or gai approximatio: Gai: HWHM gai bawith 45