Lasers PH 645/ OSE 645/ EE 613 - UAH - Engineering
Transcript of Lasers PH 645/ OSE 645/ EE 613 - UAH - Engineering
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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: [email protected]
Office Hours: Tues/Thurs 2-3PM
JDW, ECE Summer 2010
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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
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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:
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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,
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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:
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Inhomogeneous Broadened Radiative Transmission
• Let us now consider the effects of Doppler broadening
• The emission line shape should therefore be Gaussian
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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:
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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
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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
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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
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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:
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Growth of a Beam in a Gain Medium with Homogeneous Broadening
• Sufficient condition for laser process
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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:
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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
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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
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Growth of a Beam for Doppler Broadening
• A similar analysis can be shown for Doppler broadened beams
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Stimulated Emission Cross Section
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Growth of a Beam in a Gain Medium with Homogeneous Broadening
• Sufficient condition for laser process
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The use of Mirrors to Improve Divergence and System Size
• Sufficient condition for laser process
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• Sufficient condition for laser process
2 Mirror Laser Cavity
a1 and a2 are the fractional losses of the mirrors
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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
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The use of Mirrors to Improve Divergence and System Size