Ruby Laser Crystal structure of sapphire: -Al2O3 (aluminum oxide). The shaded atoms make up a unit cell of the structure. The aluminum atom inside the dashed hexagonal prism experiences an almost cubic field symmetry from the oxygen atoms on the prism. Schematic energy level diagram for ruby – Cr3+ ions in sapphire. Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Ruby Laser: Absorption Spectra Absorption coefficient and absorption cross-section as a function of wavelength for pink ruby. These absorption spectra are slightly different depending on whether the incident polarized light being absorbed is linearly polarized with its electric vector parallel, or perpendicular, to the c symmetry axis of the crystal. Detailed absorption spectrum of pink ruby in the 686 – 702 nm region showing the absorption peaks corresponding to the R1 and R2 components of the ruby laser transition. Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Ruby Laser Simple electrical circuit for driving a flashlamp Schematic energy level diagram of three- and four-level lasers Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Ruby Laser: Pumping Schematic arrangement of Maiman’s original ruby laser Elliptical reflector arrangement for optical pumping a laser crystal by a linear flashlamp Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Helium-Neon Laser: Pumping by Collision Calculated variation of energy transfer cross-section for a collision between two atomic species as a function of the energy discrepancy E∞. The probability of excitation transfer is linearly dependent on the cross-section Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Helium-Neon Laser: Energy Level Diagram Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Helium-Neon Lasers Schematic arrangement of the first gas laser. Typical schematic design of a modern laser. Taken from Lasers and Electro-Optics: Fundamentals and Engineering by Christopher Davis, Cambridge University Press, 1996

Introduction to Optical Electronics Quantum (Photon) Optics (Ch 12) Resonators (Ch 10) Electromagnetic Optics (Ch 5) Wave Optics (Ch 2 & 3) Ray Optics (Ch 1) Photons & Atoms (Ch 13) Laser Amplifiers (Ch 14) Lasers (Ch 15) Photons in Semiconductors (Ch 16) Semiconductor Photon Detectors (Ch 18) Semiconductor Photon Sources (Ch 17) Optics Physics Optoelectronics

Putting it all together Theory of Laser Oscillation Laser Amplification Medium + Optical Resonator = Laser

Population Difference Depletion of the steady-state population difference 1 2

Population Inversion Population Difference Steady-State Difference Saturation Time Constant* Four-Level Laser Three-Level Laser *What is the small-signal approximation?

Amplifier Nonlinearity Gain Coefficient Note: 0() is called the small-signal gain coefficient. Why?

Amplifier Nonlinearity Gain

Saturable Absorbers

Saturated Gain Coefficient small-signal region large-signal region small-signal: large-signal:

Gain Coefficient Inhomogeneously Broadened Medium  

Laser Amplification Medium

Optical Resonator Optical Resonator Resonator response I

Conditions for Laser Oscillations Gain Condition: Laser Threshold Phase Condition: Laser Frequencies

Exercise 15.1-1 Threshold of a Ruby Laser At the line center of the 0 = 694.3-nm transition, the absorption coefficient of ruby in thermal equilibrium (i.e., without pumping) at T = 300 K is (0) = - (0) ≈ 0.2 cm-1. If the concentration of Cr3+ ions responsible for the transition is Na = 1.58 x 1019 cm-3, determine the transition cross section 0 = (0). A ruby laser makes use of a 10-cm-long ruby rod (refractive index n = 1.76) of cross-sectional area 1 cm2 and operates on this transition at 0 = 694.3 nm. Both of its ends are polished and coated so that each has a reflectance of 80%. Assuming that there are no scattering or other extraneous losses, determine the resonator loss coefficient r and the resonator lifetime p. As the laser is pumped, (0) increases from its initial thermal equilibrium value of -0.2 cm-1 and changes sign, thereby providing gain. Determine the threshold population difference Nt for laser oscillation.

Saturated Gain Coefficient Laser Turn-On Time Steady State r Loss Coefficient () Gain Coefficient  (Photon-Flux Density)

Steady-State Population Difference Nt 2Nt s  Flux Density Photon N0 N Population Difference Pumping Rate

Output Flux Density vs. Transmittance Laser  Transmittance Output Photon-Flux Density

Characteristics of Laser Output Internal Photon-Number Density Output Photon Flux & Efficiency

Laser Oscillations B Resonator modes allowed modes

Exercise 15.2-1 Number of Modes in a Gas Laser A Doppler-broadened gas laser has a gain coefficient with a Gaussian spectral profile given by where is the FWHM linewidth. Derive an expression for the allowed oscillation band B as a function of D and the ration 0(0)/r where r is the loss coefficient. A He-Ne laser has a Doppler linewidth D = 1.5 GHz and a midband gain coefficient 0(0) = 2 x 10-3 cm-1. The length of the laser resonator is d = 100 cm, and the reflectances of the mirrors are 100% and 97% (all other resonator losses are negligible). Assuming that the refractive index n = 1, determine the number of laser modes M.

Homogeneously Broadened Medium

Inhomogeneously Broadened Medium Typical Doppler

Doppler Broadening Laser Line (atomic) Transverse Mode Polarization Brewster Window Polarization

Longitudinal Mode Selection Etalon d1 d Gain Resonator Modes Etalon Modes Laser Output

How to Pulse Lasers Modulator Modulator Peak Power Average Power t

Pulsed Lasers t Gain Switching t t Q-Switching t Modulated absorber Loss Pump t t Laser Output Q-Switching Modulated absorber t Laser Output Loss Gain

Gain Switched Laser

Q-Switching

Pulsed Lasers Cavity Dumping t Mode Locking Optical Modulator Gain Output Gain Loss Mirror Transmittance Cavity Dumping Optical Modulator Mode Locking

Mode-Locked Laser TF M = 5 M = 15 M = 25

Exercise 15.4-3 Demonstration of Pulsing by Mode Locking Write a computer program to plot the intensity I(t)=|A(t)|2 of a wave whose envelope A(t) is given by the sum Assume that the number of modes M = 11 and use the following choices for the complex coefficients Aq. Equal magnitudes and equal phases. Magnitudes that obey the Gaussian spectral profile |Aq| = exp[-1/2 (q/5)2] and equal phases. Equal magnitudes and random phases (obtain the phases by using a random number generator to produce a random variable uniformly distributed between 0 and 2.

(a) Equal magnitudes and equal phases. (b) Magnitudes that obey the Gaussian spectral profile and equal phases. (c) Equal magnitudes and random phases (obtain the phases by using a random number generator to produce a random variable uniformly distributed between 0 and 2.