1. Lasers An Application-Oriented Overview of the Current State-Of-The-Art in Industry and Research Dr. Dirk Lorenser, Spring 2009
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5. What is a Laser ? cavity losses l c gain g pump energy t Resonator + Gain Medium g > l c g = l c g < l c
6. What is a Laser ? cavity losses l c gain g pump energy t Resonator + Gain Medium g > l g = l g < l output coupling l out output beam total losses l = l c + l out
7. What is a Laser ? t Condition 1 gain = losses or round-trip gain G = 1 g > l g = l g < l
8. What is a Laser ? gain g Resonator + Gain Medium output beam a resonator has resonance frequencies ! L 2nL = q · n = "effective" refractive index inside resonator
9. What is a Laser ? Condition 2 2nL = q · or round-trip phase = q ·2 q = 1,2,3... n = refractive index inside resonator
10. What is a Laser ? Any oscillator (electronic, mechanical, optical...) has to meet the following conditions: Condition 1 round-trip gain = 1 Condition 2 round-trip phase = q ·2 A Laser is an Optical Oscillator Electronic Oscillator Optical Oscillator g amplifier feedback network oscillator output
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14. Coherence Properties of Laser Beams A light field is called coherent when there is a fixed phase relationship between the electric field values at different locations or at different times A Gaussian beam with perfect spatial and temporal coherence
15. Coherence Properties of Laser Beams A Gaussian beam with good spatial but bad temporal coherence (would not give good interference contrast in a Michelson interferometer with an arm length difference greater than the coherence length)
16. Coherence Properties of Laser Beams A Gaussian beam with OK temporal coherence but bad spatial coherence (has bad "beam quality", its irregular phase fronts do not make it possible to focus it down to a spot as small as can be obtained with a perfect Gaussian beam)
17. Stimulated Emission and Optical Gain Interaction of Photons with a two-level Atom E 2 (excited state) E 1 (ground state) E = h h h h before after Spontaneous Emission ("Fluorescence") Stimulated Absorption ("Absorption") Stimulated Emission
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20. Stimulated Emission and Optical Gain Interaction of Photons with a large number of two-level Atoms: Atomic Rate Equations energy population light field with photon density n E 2 E 1 N 2 N 1 collection of atoms with numbers N 2 in excited state and N 1 in ground state stimulated transitions (absorption/emission) spontaneous emission
21. Stimulated Emission and Optical Gain Atomic Rate Equations Spontaneous Emission ( 2 = 1/ 2 ) Stimulated Emission Stimulated Absorption photon density n
22. Stimulated Emission and Optical Gain Atomic Rate Equations K em = K abs !!! photon density n Stimulated Emission and Stimulated Absorption are two variants of the same physical mechanism !!!
23. Stimulated Emission and Optical Gain Atomic Rate Equations photon density n The Inversion N = N 2 – N 1 determines if there is net absorption or amplification of the incident light field
24. Stimulated Emission and Optical Gain Atomic Rate Equations photon density n N 2 N 1 N 2 < N 1 Net Absorption N 2 N 1 N 2 = N 1 Transparency N 2 N 1 N 2 > N 1 (N>0) Gain population inversion = gain !
25. Stimulated Emission and Optical Gain Creating a Population Inversion population E 2 E 1 N 2 N 1 Boltzmann's Law energy Because of Boltzmann's Law a population inversion is not possible in thermal equilibrium
26. Stimulated Emission and Optical Gain Creating a Population Inversion population E 2 E 1 N 2 N 1 energy A Population Inversion is always a nonequilibrium state which requires continuous or pulsed excitation ( pumping ) of the laser-active material pump
27. A Simple Laser Model The Coupled Cavity and Atomic Rate Equations optical resonator mode with photon number n Gain Medium with Inversion N 2 = 1/ 2 2 = upper-state lifetime C = 1/ C C = cavity lifetime (1) (2) output R pump
28. A Simple Laser Model Steady-State Solutions ( dN/dt = 0, dn/dt = 0) (1) (2) N, gain R n, P out R th N th I II threshold I below threshold ( n = 0) II above threshold ( n > 0) Inversion and gain are constant ("clamped") when laser is lasing gain = losses in an oscillator ! lasing laser "slope": Output Power increases linearly with pump rate above threshold
29. A Simple Laser Model Steady-State Solutions ( dN/dt = 0, dn/dt = 0) (1) (2) N, gain R n, P out R th N th I II threshold II above threshold ( n > 0) lasing when laser oscillation sets in (n>0) the circulating photons saturate the gain by depopulating the upper laser level due to stimulated emission: saturation with n=0 (e.g. when preventing laser operation by blocking the resonator) the inversion ("small signal gain") continues to rise proportionally to R: gain "saturation" N 0
30. A Simple Laser Model "Slope" of typical high-power diode lasers threshold threshold
31. A Simple Laser Model Dynamic Properties of the Simple Model "Spiking" Relaxation Oscillations steady-state numerical solution with LabView K = 1 C = 2 2 = 0.02 R = 0.1
32. Resonators and Laser Modes simple plane-plane (Fabry-Perot) Resonator refractive index n L
33. Resonators and Laser Modes simple plane-plane (Fabry-Perot) Resonator L gain g g( ) 0 refractive index n
34. Exercise: Longitudinal Modes What are the longitudinal mode frequencies of this resonator ? L refractive index n 1 L 1 L 2 refractive index n 2
35. Resonators and Laser Modes Homogenous vs. Inhomogenous Gain Saturation g( ) 0 gain total cavity loss l g > l intensity modes with g( ) > l start to lase
36. Resonators and Laser Modes Homogenous Gain Saturation ("the winner takes it all...") g( ) 0 gain total cavity loss l single-frequency laser ! intensity mode with highest gain wins
37. Resonators and Laser Modes Inhomogenous Gain Saturation g( ) 0 gain total cavity loss l multiple-longitudinal-mode laser (5 modes) intensity modes saturate gain independently
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39. Gain curve He Ne 1600 MHz 0 Resonator modes Gain curve Lasing threshold Lasing modes
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43. Resonators and Laser Modes Why the simple plane-plane Resonator is a bad practical choice: horrible beam quality (multiple transverse modes) extremely alignment-sensitive (mirrors must be exactly parallel) diffraction and "walk-off" losses
44. Resonators and Laser Modes Practical resonators typically use curved mirrors to obtain well-defined laterally confined Gaussian modes:
45. Resonators and Laser Modes Resonator Modes are electric field distributions that maintain their complex amplitude distribution after one resonator round-trip Resonator Modes are solutions of the Electromagnetic Wave Equation which satisfy the boundary conditions of the resonator: "Transverse Modes" This part determines the longitudinal modes belonging to a given transverse mode (n,m)
46. Beam Quality Gaussian TEM 00 Mode "Fundamental Transverse Mode" "Diffraction Limited " w 0
47. Beam Quality Higher-Order Hermite-Gaussian TEM nm Modes : 0 w 0 The M 2 Factor for non-Gaussian Beams
48. Beam Quality M 2 and Beam Parameter Product (BPP) BPP = Beam Waist Radius x Divergence Half-Angle [mm ·mrad] Gaussian TEM 00 Beam (M 2 = 1): Non-perfect Laser Beam with M 2 > 1: -> in practical calculations with non-Gaussian Beams with M 2 > 1 just replace λ with the "effective wavelength" M 2 λ and use normal Gaussian Beam Formulas or Software Programs !
49. Beam Quality Spot Sizes of focused Gaussian Beams replace λ with M 2 λ for non-diffraction limited beams with M 2 > 1 f = focal length w 1 = collimated input beam radius on lens w 0 = beam radius in focus Focusing with a lens Focusing with optics characterized by NA NA = numerical aperture (n ·sin( )) w 0 = beam radius in focus
50. Beam Quality Intensity of focused Gaussian Beams The Gaussian Intensity Distribution contains 86.5% of its total power within the central "spot" defined by the 1/e 2 beam radius w I 0 = peak intensity (in center of spot) w = beam radius (1/e 2 ) Gaussian Intensity Distribution: Watch out for factor of 2 when calculating the peak intensity from the power P and radius w of a Gaussian laser beam: P = total power in the laser beam w = beam radius (1/e2)
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53. Exercise: Focused Spot Sizes d = Optical Data Storage: Toward higher data densities using smaller wavelengths and higher NA focusing optics d = d =
58. Resonator Design Free-space Laser Resonators and Gaussian Beam Optics are nowadays designed with easy-to use Software based on the ABCD Matrix Formalism Excellent Freeware Program: Laser Canvas http://lamar.colostate.edu/~pschlup " LaserCanvas cavity modeling software, available from P. Schlup, Colorado State University, Fort Collins, CO 80523."
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