Introductory Chapter of my Lecture "Lasers" held at the University of Applied Sciences in Biel for Microtechnology Students in Spring 2009

1. Lasers An Application-Oriented Overview of the Current State-Of-The-Art in Industry and Research Dr. Dirk Lorenser, Spring 2009

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 = &quot;effective&quot; 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

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 &quot;beam quality&quot;, 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 (&quot;Fluorescence&quot;) Stimulated Absorption (&quot;Absorption&quot;) Stimulated Emission

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 (&quot;clamped&quot;) when laser is lasing gain = losses in an oscillator ! lasing laser &quot;slope&quot;: 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 (&quot;small signal gain&quot;) continues to rise proportionally to R: gain &quot;saturation&quot; N 0

30. A Simple Laser Model &quot;Slope&quot; of typical high-power diode lasers threshold threshold

31. A Simple Laser Model Dynamic Properties of the Simple Model &quot;Spiking&quot; 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 (&quot;the winner takes it all...&quot;)    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

39. Gain curve He Ne 1600 MHz  0  Resonator modes Gain curve Lasing threshold Lasing modes

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 &quot;walk-off&quot; 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: &quot;Transverse Modes&quot; This part determines the longitudinal modes belonging to a given transverse mode (n,m)

46. Beam Quality Gaussian TEM 00 Mode &quot;Fundamental Transverse Mode&quot; &quot;Diffraction Limited &quot;  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 &quot;effective wavelength&quot; 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 &quot;spot&quot; 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)

53. Exercise: Focused Spot Sizes d = Optical Data Storage: Toward higher data densities using smaller wavelengths and higher NA focusing optics d = d =

54. Exercise: Focused Spot Sizes d  1.1  m d  0.3  m d  0.7  m

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 &quot; LaserCanvas cavity modeling software, available from P. Schlup, Colorado State University, Fort Collins, CO 80523.&quot;