Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240

Transcription

1 Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240 John D. Williams, Ph.D. Department of Electrical and Computer Engineering 406 Optics Building - UAHuntsville, Huntsville, AL Ph. (256) williams@eng.uah.edu Office Hours: Tues/Thurs 2-3PM JDW, ECE Summer 2010

2 Chapter 7: Conditions for Producing a Absorption and Gain Population Inversion Saturation intensity Development and Growth of a Laser Beam Exponential Growth Factor (Gain) Threshold Requirements Laser Chapter 7 Homework: 1, 2, 6, 10, 11, 12, 13 Cambridge University Press, 2004 ISBN-13: All figures presented from this point on were taken directly from (unless otherwise cited): W.T. Silfvast, laser Fundamentals 2 nd ed., Cambridge University Press, 2004.

3 Absorption and Gain Recall the conditions for stimulated emission and the current therein Frequency dependent decay rate: Cavity resonance: Change in intensity through the volume of the cavity

4 Absorption and Gain for Homogeneous Broadening A simplified version of the intensity differentiated about the path length is: Which can be solved to find the normalized intensity of the cavity for homogeneous broadening as: Using the relationships for stimulated emission in a cavity: where, g(v) in units of 1/m And the cross section of the stimulated emission in terms of 1/m 2 : Where we define N ul in 1/m 3

5 Absorption and Gain at Resonance At the resonance frequency: Where and Allowing one to determine the emission cross section simple from the relative radiative decay rates Thus for homogeneous broadening:

6 Inhomogeneous Broadened Radiative Transmission Let us now consider the effects of Doppler broadening The emission line shape should therefore be Gaussian

7 Inhomogeneous Broadened Radiative Transmission The population of the upper to lower transitions as a function of frequency due to Doppler broadening is; Yielding a gain coefficient of:

8 Inhomogeneous Broadened Radiative Transmission at Resonance Let us solve for the case where the emission frequency is at resonance M N is the mass number of the laser species z is the distance the beam has propagated in the medium

9 Optical Absorption The equations developed were important to spectroscopists and electrical engineers long before lasers came to be. z is the distance the beam has propagated in the medium Beer s Law where α is the absorption coefficient of the incident field Note that the absorption coefficient is the same as that for an EM field traveling through a dielectric medium 2 εµ σ α = 2πν 1+ 1 lossy dielectric 2 2πνε

10 Population Inversion Necessary condition for laser process The equation above decays to zero as a function of penetration depth z unless the upper state population density is larger than that of the lower state Such a condition generates an exponential growth in or gain in the medium as the radiation propagates Thermodynamic Equilibrium Energized system via pumping to generate gain

11 Saturation Intensity Sufficient condition for laser process Assume a steady state excitation to the upper atomic energy state Where R u is the pumping flux (number of excitations per unit volume), the normal decay transitional decay time constant τ u Sdf In the case where a cycled beam has a pulse duration, τ p, < τ u one can calculate the energy per unit area, or fluence, as:

12 Growth of a Beam in a Gain Medium with Homogeneous Broadening Sufficient condition for laser process

13 Growth of a Beam in a Gain Medium with Homogeneous Broadening Sufficient condition for laser process We define one gain length, l g, such that the numerical value below equal to one length:

14 Growth of a Beam in a Gain Medium with Homogeneous Broadening Sufficient condition for laser process A cylinder of length L provides sufficient length to provide gain and produce a laser spot size of radius d Also note that a spherical emitter would defeat the purpose of producing a focused coherent beam

15 Growth of a Beam in a Gain Medium with Homogeneous Broadening Sufficient condition for laser process Typical ratios require a L/da ratio of about 10 to 1000 The desired gain values range from 7 to 17 in order to reach saturation

16 Growth of a Beam for Doppler Broadening A similar analysis can be shown for Doppler broadened beams

17 Stimulated Emission Cross Section

18 Growth of a Beam in a Gain Medium with Homogeneous Broadening Sufficient condition for laser process

19 The use of Mirrors to Improve Divergence and System Size Sufficient condition for laser process

20 2 Mirror Laser Cavity Sufficient condition for laser process a 1 and a 2 are the fractional losses of the mirrors

21 Sufficient Gain Duration Sufficient condition for laser process Where m is the multiplicative gain length obtained by passing light back and forth across the medium between two mirrors m times This simplified solution assumes that the mirror reflectivity is 100% The distance between the mirrors is d - L

22 The use of Mirrors to Improve Divergence and System Size