ECE 455: Optical Electronics Lecture #9: Inhomogeneous Broadening, the Laser Equation, and Threshold...

ECE 455: Optical Electronics

Lecture #9:

Inhomogeneous Broadening, the Laser Equation,and Threshold Gain

Substitute Lecturer: Tom Spinka

Tuesday, Sept. 22nd, 2009


Topic #1: Inhomogeneous Broadening


Homogeneous vs. Inhomogeneous Broadening

• There are two general classification of line broadening:

– Homogenous — all atoms behave the same way (i.e., each effectively has the same g(ν).

– Inhomogeneous — each atom or molecule has a different g(ν) due to its environment.

• What physical processes result in homogeneous broadening? Inhomogeneous broadening?


Inhomogeneous Broadening

• Excited atoms in glasses (or other materials with little long-range order) are inhomogeneously broadened because the “host” looks different at each site

• Excited atoms in crystals are generally homogeneously broadened because of periodicity to structure

http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Graphics/QuartzGlass.jpg


Inhomogeneous Broadening

• Each emitter produces a homogeneously-broadened lineshape

• For Example: Stark Broadening

n0

E

position

E

Composite lineshape

n

ß


Doppler Broadening

• In a gas, atoms are moving in all directions with velocities given by the Maxwellian distribution:


Doppler Broadening

• In the frame of the emitter, a homogeneously-broadened lineshape is generated, but in the lab frame (seen by an external observer) the homogeneously-broadened lineshape is frequency-shifted by the Doppler effect

A B C

y

x

Observer

n0 nB

vC

v

Av


Doppler Broadening

• The Doppler broadened profile is the weighted sum of the lineshapes arising from all possible velocities:

• What is the lineshape function, g(v), for a Doppler broadened transition?

Dndoppler

DnH

n


Doppler Broadening

• TWO ASSUMPTIONS:1. Homogeneous linewidth is small compared to the

inhomogeneous linewidth

2. Gas atom/molecule velocity distribution is Maxwellian

• Maxwellian velocity distribution:

• Relationships between velocity and frequency:


Doppler Broadening

• This is a Gaussian Distribution:


Doppler Broadening

• Doppler Broadening in a Nutshell:– Physical Origin: Movement of transitioning atoms/molecules in

the laboratory reference frame

– When It Is Important/Dominant: High Temperature, Small Mass

– Lineshape Function:

– Lineshape Function at Line Center:

– Full-Width at Half-Max (FHWM)


Broadening Examples (Cu-Vapor Laser)

• The copper vapor laseroperates at T = 1750 K!

http://www.spectronika.com/MaltaCVshowWeb.jpg http://omlc.ogi.edu/news/sep98/gallery_sep98/Bild2.jpg


Broadening Examples (Sodium Doublet)

t = 16 ns 32P3/2

588.9 nm

Ground (3s 2S½)

What is the pressure-broadened linewidth in He at 1 Torr and 400ºC?


Topic #2: Threshold Gain


Requirements for Laser Action

1. Gain Medium• Can be a gas (or plasma), liquid, or solid

• Must have an established population inversion N2 > N1

• When a medium is inverted, the “stimulated emission” process is stronger than the “absorption” process

2. Excitation Mechanism or “Pump”• “Pumping” is the process, sequence of processes, or method by which

one “excites” or promotes atoms into the upper laser level

• Pumping may well not be “direct” but via an intermediate level or set of levels

hn hn

2 2

1 1



3. “Seed” for Stimulated Emission• In some cases, the “seed” is supplied by spontaneous emission

• In other cases, the “seed” can be provided in another fashion, generally another laser. “Seeding” a laser can improve1. Timing jitter for pulsed lasers

2. Wavelength stability

3. Spectral purity (bandwidth)

2

1

steradians 4πintoemission

sspontaneoustimulatedemission



4. Optical Cavity or Resonator• This is necessary to:

1. Provide optical feedback to the gain medium

2. Define the spatial and longitudinal lasing modes

HighReflector PUMP

OutputCoupler

Gain Medium

ß ß


Gain and Amplification

• At the heart of the laser is its ability to “amplify” light within a particular frequency (wavelength) range– This requires a population inversion which can not exist under

equilibrium conditions

• In thermodynamic equilibrium, the relative population of available states is given by:



Pumping rates

(cm-3-s-1)

2

1

P2

P1

A21 t2

t1

• Consider the population inthe upper laser level as afunction of time:

Spontaneous Emission

Stimulated Emission

Absorption Pumping

Non-radiative Relaxation



• When the spectral bandwidth of the radiation field is small compared to the broadened transition:


• Now we introduce the photon intensity:

• Now we group terms:




• Lets try to regain the link to the physics in the “stimulated processes” term (2):

Stimulated EmissionCross-Section

# of Photons Crossing aunit Area per unit Time



• Put it all together! THIS IS IMPORTANT!!!

• The stimulated emission cross-section is purely aproperty of the material

SpontaneousEmission

StimulatedEmission Absorption

Pumping

Non-RadiativeRelaxation

StimulatedEmission

Cross-Section

# of Photons Crossing aunit Area per unit Time



• Consider the radiation field as it travels through a gain medium:– Keep in mind that this is at a single frequency

– Neglect spontaneous emission

– Propagation in the z-direction



• The result looks suspiciously like Beer’s Law …

Gain Coefficient(cm-1)


Threshold Gain

• What is threshold gain?

• Threshold Gain is the gain value that exactly compensates for losses occurring on a round-trip through the cavity

GainMedium

R2ReflectivityR1


Threshold Gain

Gain: g(n)

R2R1

Lg

I0

I0R2e2gLg

I0egLg

I0R2egLg

• Follow the photon intensity through a round trip through a generic cavity with a gain medium:

• Threshold Condition:


Threshold Gain Examples (He-Ne Laser)

25 cm

R1 = 100% R2 = 99.5%

This is an exceedingly low threshold gain. In general, mirror and other losses are much larger, and the corresponding threshold gain is much higher.


Threshold Gain Examples (Diode Lasers)

This case represents the other end of the spectrum. Threshold for diode lasers is 4 orders of magnitude greater than the He-Ne case.

400 μm

R1 = 95% R2 = 40%


Threshold Gain Examples (Absorption Cell)

Absorption cell; loss a = 3%-cm-1

Gain cell

40 cm 4 cmR2 = 0.4R1 = 0.99


Threshold Gain Examples (Absorption Cell)

Absorption cell; loss a = 3%-cm-1

Gain cell

40 cm 4 cmR2 = 0.4R1 = 0.99

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ECE 455: Optical Electronics Lecture #9: Inhomogeneous Broadening, the Laser Equation, and Threshold...

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