The laser oscillator. Atoms and light. Fabry-Perot interferometer. Quiz

Transcription

1 toms and light Introduction toms Semi-classical physics: Bohr atom Quantum-mechanics: H-atom Many-body physics: BEC, atom laser Light Optics: rays Electro-magnetic fields: Maxwell eq. s Quantized fields: photons The laser oscillator Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Introduction Quiz Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Longitudional modes Fabry-Perot interferometer Transmission through one mirror Er( r, t) E( r, t) Et( r, t) Perfect mirrors: T = 1 and =9%. Transmission through two mirrors Er( r, t) Perfect mirrors! E( r, t) Et( r, t) Interference between two travelling waves E e i(kz ωt) E e i(kz+ωt) Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3

2 Feedback Longitudional modes Transmission of FPI Longitudional modes E τ in 1 E in E out /τ 2 E out τ 1 + F 1 (ω) τ 2 + F 2 (ω) τ 1, τ 2 transmission coefficients of the two mirrors F 1 = exp(iωd/c) transport of the wave between the mirrors F 2 = r 1 r 2 exp(iωd/c) feedback Feedback: ( ) 1 Eout E out = F 1 (τ 1 E in ) + F 1 F 2 τ 2 τ 2 Thus E out = τ 1τ 2 F 1 [1 F 1 F 2 ] E in or E out = E int 2 1 r 2 e iφ Φ = 2ωd = 2kd. c Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Keep in mind! Longitudional modes Transmission in terms of iry function: I t = I T 2 (1 ) F sin 2 (Φ/2) F 4 = (1 ) Finesse F and quality factor Q: F = 2π = π F Φ FWHM 2 Free spectral range: Transmission =.2 =.5 = Phase φ/2 [π] = π 1 and Q = ν ν = c 2d = 2πd ν ν FWHM c(1 ) Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 nalysis of He-Ne laser FPI Longitudional modes dditional phase shifts at mirrors Optical path length d = ηd Optical path length in cavity ηl + (d L) ngle of incidence not 9 : φ = 2kd cos θ. Surface quality better than λ/1. Transmission always smaller than 1. Intensity in FPI can be much larger than input intensity. Wavelength: λ = nm Cavity mirror separation of.21 m. Free spectral range: ν = c/2d =.71 GHz. Doppler broadening of gain profile: 1.5 GHz. Number of modes: 2 3 Finesse FPI: F =3. Number of modes: n = 2d/λ 35.. Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3

3 Huygens principle Fresnel-Kirchhoff integral E(x,y, ) z E(x, y, z) Each point on a wavefront S is a source of spherical waves, where the superposition of all these spherical waves gives a new wavefront S in space. Christiaan Huygens, Traité de la Lumiére, Parijs, 169. aether: signal-carrying medium ethernet Kirchhoff diffraction integral (no polarization): E(x, y, z) = i dx dy e ikr E(x, y, ) cos ( n, r ) λ r In the paraxial approximation: r 2 = (x x ) 2 + (y y ) 2 + z 2. cos ( n, r ) 1. Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Fresnel diffraction Fresnel approximation (z x, y): r = z 1 + (x x ) 2 z 2 + (y y ) 2 z 2 z + (x x ) 2 2z + (y y ) 2 2z Fresnel diffraction integral: ( [ E(x, y, z) = ieikz ik (x x dx dy ) 2 + (y y ) 2] ) exp E(x, y, ) 2z or E(x, y, z) = E(x, y, ) h z (x, y ), the pulse response of free space : ( ( h z (x, y ) = ieikz ik x 2 + y 2) ) exp 2z Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Fraunhofer diffraction Fraunhofer approximation (z x, y, far-field): (x x ) 2 + (y y ) 2 = x 2 + y 2 + x 2 + y 2 xx + yy 2z 2z 2z 2z z Fraunhofer diffraction: ( E(x, y, z) = ieikz e ik(x2 +y 2 )/2z ik(xx dx dy + yy ) exp z Fourier optics: E(x, y, z) = (ν x, ν y, z)f [ E(x, y ) ] (ν x, ν y ) spatial frequencies and amplitude ( x (ν x, ν y ) =, y ) (ν x, ν y, z) = ieikz e ik(x2 +y 2 )/2z x 2 + y 2 xx + yy 2z z ) E(x, y, ) Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3

4 pproximations Fraunhofer diffraction by an aperture Fresnel: z 3 k [ (x x ) 2 + (y y ) 2] 2 8 Fraunhofer: z k ( x 2 + y 2) 2 Example: He-Ne laser λ=632.8 nm and a=1 cm. In that case we have z.23 m (Fresnel) and z 496 m (Fraunhofer). Lenses restores validity Fraunhofer diffraction: Fourier optics. perture D = 2a: E(x, y, ) = E x a, y a E(x, y, ) = x > a, y > a Fraunhofer diffraction: e ik(xx +yy )/z dx dy = aperture ( ) kρρ = 2π ρ J dρ z Standard integral: ( ) kρρ ρ J dρ = a 2 J 1(kaρ/z). z kaρ/z 2π dρ ρ e ikρρ cos(φ φ)/z dφ.5. J (x) J 1 (x) Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Fraunhofer diffraction by an aperture -.5 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, 214 x (π) 14 / 3 of laser resonantors Solution: E(ρ, z) = ie ikz e ikρ2 /2z 2πa2 Intensity (=iry pattern): I (ρ, z) = I (, z) ( 2J1 (kaρ/z) (kaρ/z) J 1 (kaρ/z) (kaρ/z). Zero in pattern: ρ = 1.22 D. G.B. iry, British astronomer (1835) Intensity 2J 1 (x)/x ) Position x [π] The larger the wavelength or the smaller the aperture, the more pronounced is the diffraction Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 d d d d Formal diffraction theory à la Fresnel: E(x, y, d) = dx dy K(x, y; x, y )E(x, y, ), ( K(x, y; x, y ) = i [ ik (x x λd exp ) 2 + (y y ) 2] ) 2d Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3

5 of laser resonantors of laser resonantors Mode of the resonator: E (x, y, ) = γe(x, y, ), d d d d For a round trip we have: E (x, y, ) = dx dy K(x, y; x, y )E(x, y, d) = dx dy K(x, y; x, y ) dx dy K(x, y ; x, y )E(x, y, ) = dx dy K (x, y; x, y )E(x, y, ), or d d d d γe(x, y, ) = dx dy K (x, y; x, y )E(x, y, ). The Schrödinger equation for laser cavities Intensity loss per transit only due to diffraction: K (x, y; x, y ) = dx dy K(x, y; x, y )K(x, y ; x, y ). I q+1 I q = γ 2 < 1. Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 of laser resonantors Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Higher order transverse modes Fraunhofer approximation: K (x, y; x, y ) = Ce ik(xx +yy )/z, or d d d d γe(x, y, ) = C dx dy e ik(xx +yy )/z E(x, y, ). E(x, y, ) is its own Fourier transform, or in its simplest form E, (x, y, ) e (x2 +y 2 )/w 2, w the waist of the laser beam. In general, we have solutions E m,n (x, y): E m,n (x, y, ) H m ( 2x/w)H n ( 2y/w)e (x2 +y 2 )/w 2. Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3

6 Fox-Li analysis in 1-D (Bell Sys. Techn. J. 4, 453 (1961)) Fox-Li analysis in 1-D (Bell Sys. Techn. J. 4, 453 (1961)) b b 2a Fresnel diffraction in 1-D (no paraxial approximation): Start: ( u q+1 (x 2 ) = eiπ/4 2 u q (x 1 ) e ikρ 1 + b ) dx 1, λ a ρ ρ ρ = b 2 + (x 1 x 2 ) 2. u (x 1 ) = 1 x 1 a and u (x 1 ) = x 1 > a. 2a Width mirrors: 2a = 5λ Distance mirrors: b = 1λ Fresnel number : N F = a2 bλ = 6.25 elative amplitude x=a/ * Number of round trips Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Fox-Li analysis in 1-D: Field profile Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Fox-Li analysis in 1-D: Damping of the cavity 1.2 elative amplitude first transit.2 after 3 transits Gaussian beam Position x [a] Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 mplitude center Number of round trips Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3

7 Losses in a resonator Mode profile in a planar cavity Loss rate per transit confocal, sym confocal, asym plane, sym plane, asym Fresnel number N F Fresnel number: N F = a2 bλ Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Mode profile in a confocal cavity 1 Intensity Position [a] Phase [π] Position [a] Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Fox-Li analysis in 2-D Intensity Position [a] Phase [π] Position [a] Fresnel diffraction in 2-D: u q+1 (r 2, φ 2 ) = i 2λ 2π b dr 1 r 1 dφ 1 u q (r 1, φ 1 ) e ik 2 = b r r 2 2 2r 1 r 2 cos(φ 1 φ 2 ). The distance b 1 is for confocal spherical mirrors: 2a ( 1 + b ) 1 b 1 = b 1 2 where i = b b 2 r i 2 i = 1, 2 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3

8 Summary Longitudinal modes ν = c 2d Transverse modes TEMmn Mode characteristics: Spatial dependence Frequency dependence Mode competition Modes and the gain profile Spectral hole burning Spatial hole burning Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3 Peter van der Straten (Nanophotonics) Photon physics 214 Lecture 5 March 6, / 3