Lecture 5 Op+cal resonators *

Transcription

1 Lecture 5 Op+cal resonators * Min Yan Op+cs and Photonics, KTH 12/04/15 1 * Some figures and texts belong to: O. Svelto, Principles of Lasers, 5th Ed., Springer.

2 Reading Principles of Lasers (5th Ed.): Chapter 5. Skip: 5.5.2, 5.5.3, Squeeze*: 5.5.1, /04/15 * Squeeze means to get basic physical feeling of the sec+on. 2

3 Laser Pump I in I out Mirror 1 Gain medium Mirror 2 Cavity field: Quality: stability, photon life+me not yet ac+ve General proper4es: Par+ally open Long (cm to <1m) 12/04/15 3

4 Contents Content Time 1. Resonator varieties Cavity Q Stability condition Wave-optics mode solutions Unstable resonators 5 Total: 80 12/04/15 4

5 Contents Content Time 1. Resonator varieties Cavity Q Stability condition Wave-optics mode solutions Unstable resonators 5 Total: 80 12/04/15 5

6 Resonator varie+es Plane- parallel (FP) Concentric (L=2R) Confocal (L=R) Ring Others: R<L<2R; different Rs o Stable o Unstable 12/04/15 6

7 Resonator varie+es Plane- parallel (FP) Concentric (L=2R) Confocal (L=R) 12/04/15 ν n =? q(z)=? TEM lm? Radius of curvature Beam spot size 7

8 More accurately 12/04/15 8

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25 More accurately Beam shape Phase front Resonant freq. Decay +me Field: E 2 R=20µm λ 0 =1µm Confocal 12/04/15 25

26 More accurately Beam shape Phase front Resonant freq. Decay +me Field: E 2 R=20µm λ 0 =1µm Confocal 12/04/15 26

27 Contents Content Time 1. Resonator varieties Cavity Q Stability condition Wave-optics mode solutions Unstable resonators 5 Total: 80 12/04/15 27

28 Photon life+me Light intensity aher m round- trips: Photons aher m round- trips: Phenomenological assump+on: γ : single- pass logrithmic cavity loss Time- dependence of field: By Fourier transform: FWHM of resonance peak (MHz range) 12/04/15 28

29 Quality factor Example R 1 =R 2 =0.98, T i =0, L=90cm, λ=630nm Δυ c =1.1MHz, Q= /04/15 29

30 Contents Content Time 1. Resonator varieties Cavity Q Stability condition Wave-optics mode solutions Unstable resonators 5 Total: 80 12/04/15 30

31 Stability condi+on Ray op+cs Aher one roundtrip Aher n roundtrips For overall ABCD matrix not to diverge 12/04/15 31

32 Two- spherical- mirror case Ray op+cs ABCD matrix for spherical mirror: for free space: Aher finding round- trip ABCD matrix, If one defines: 12/04/15 Stability condi+on: A: concentric B: confocal C: planar 32

33 Contents Content Time 1. Resonator varieties Cavity Q Stability condition Wave-optics mode solutions Unstable resonators 5 Total: 80 12/04/15 33

34 Wave op+cs Fresnel- Kirchoff integral 12/04/15 34

35 Modes: Infinite aperture Modes: TEM transverse modes ABCD law of Gaussian beam propaga+on q must repeat itself aher a round trip: q must be complex, hence Since AD- BC=1 Stability condi+on: (re- derived) 12/04/15 35

36 Modes: Two- mirror case Round- trip matrix: Since A=D, from red eq. in previous slide Implica4on: q 1 is purely imaginary equiphase surface coincides with mirror 1 Implica4on 2: beam spot size can be computed based on q 1. 12/04/15

37 Beam size q 1 Special case: symmetric mirrors (R 1 =R 2 =R) 12/04/15 37

38 Beam size: special cases Confocal (g=0): Near- plane (g=1- ε, ε=l/r small): Concentric (g=- 1+ ε, ε small): Example: L=1m, λ=514nm, Confocal: w c =0.4mm; w 0 =0.29mm Near- plane (R=10m): w c =0.61mm; w 0 =0.59mm Concentric (R=0.5m, ε=0.1): w c =0.61mm; w 0 =0.14mm 12/04/15 38

39 Mode frequencies [keep simple] Fresnel- Kirchoff integral for paraxial wave through a general ABCD system That is, if TEM lm modes then Aher a round trip in cavity q=q 1. With round- trip ABCD matrix which has a unit amplitude and a phase Total phase change for a round trip: which must be n2π: 12/04/15 39

40 Mode frequencies (special cases) For two- mirror cavi+es: Near- planar resonator Confocal resonator Concentric resonator 12/04/15 40

41 Finite aperture Purpose: modal discrimina+on Single- transverse- mode condi4on (approximately): Fresnel number For con- focal case: 12/04/15 41

42 Contents Content Time 1. Resonator varieties Cavity Q Stability condition Wave-optics mode solutions Unstable resonators 5 Total: 80 12/04/15 42

43 Unstable resonators: Hard- edge Unstable confocal Purpose: large mode area while less modes Round- trip loss M: field spot magnifica+on (dependent on g 1 and g 2 ) Disadvantage: Annular- ring beam shape (doughnut) High- order annular rings in beam exist Sensi+ve to perturba+on 12/04/15 43

44 Unstable resonators: Soh- edge Solu4on: Mirrors with radially- varying reflec+vity, with a profile in Gaussian or even super- Gaussian shape. 12/04/15 44

45 Contents Content Time 1. Resonator varieties Cavity Q Stability condition Wave-optics mode solutions Unstable resonators 5 Total: 80 12/04/15 45

46 One more slide Fiber cavity Microdisk cavity Q>10 8 Photonic crystal cavity 12/04/15 46 Microdisk SEM: D.K. Armani et al., Nature 421, p.925 (2003).