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 35899

Ph. (256) 824-2898 email: williams@eng.uah.edu

Office Hours: Tues/Thurs 2-3PM

JDW, ECE Summer 2010

Chapter 7: Conditions for Producing a Laser

• Absorption and Gain

• Population Inversion

• Saturation intensity

• Development and Growth of a Laser Beam

• Exponential Growth Factor (Gain)

• Threshold Requirements

Cambridge University Press, 2004 ISBN-13: 9780521541053

All figures presented from this point on were taken directly from (unless otherwise cited): W.T. Silfvast, laser Fundamentals 2nd ed., Cambridge University Press, 2004.

Chapter 7 Homework: 1, 2, 6, 10, 11, 12, 13

Absorption and Gain

• Recall the conditions for stimulated emission and the current therein

Frequency dependent decay rate:

Change in intensity through the volume of the cavity

Cavity resonance:

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:

• And the cross section of the stimulated emission in terms of 1/m2:

g(v) in units of 1/m

Where we define ∆Nul in 1/m3

where,

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:

Inhomogeneous Broadened Radiative Transmission

• Let us now consider the effects of Doppler broadening

• The emission line shape should therefore be Gaussian

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:

Inhomogeneous Broadened Radiative Transmission at Resonance

• Let us solve for the case where the emission frequency is at resonance

MN is the mass number of the laser species

z is the distance the beam has propagated in the medium

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

where α is the absorption coefficient of the incident field

Beer’s Law

Note that the absorption coefficient is the same as that for an EM field traveling through a dielectric medium

−

+= 1

21

22

2

πνεσεµπνα lossy dielectric

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

Saturation Intensity • Sufficient condition for laser process • Assume a steady state excitation to the upper atomic energy state

• Where Ru is the pumping flux (number of excitations per unit volume) , the normal decay transitional decay time constant τu

• Sdf

• In the case where a cycled beam has a pulse duration, ∆τp , < τu one can calculate the energy per unit area, or fluence, as:

Growth of a Beam in a Gain Medium with Homogeneous Broadening

• Sufficient condition for laser process

Growth of a Beam in a Gain Medium with Homogeneous Broadening

• Sufficient condition for laser process

We define one gain length, lg, such that the numerical value below equal to one length:

Growth of a Beam in a Gain Medium with Homogeneous Broadening

• Sufficient condition for laser process

Also note that a spherical emitter would defeat the purpose of producing a focused coherent beam

A cylinder of length L provides sufficient length to provide gain and produce a laser spot size of radius d

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

Growth of a Beam for Doppler Broadening

• A similar analysis can be shown for Doppler broadened beams

Stimulated Emission Cross Section

Growth of a Beam in a Gain Medium with Homogeneous Broadening

• Sufficient condition for laser process

The use of Mirrors to Improve Divergence and System Size

• Sufficient condition for laser process

• Sufficient condition for laser process

2 Mirror Laser Cavity

a1 and a2 are the fractional losses of the mirrors

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

The use of Mirrors to Improve Divergence and System Size