Line-Broadening Mechanisms & Nonradiative Decays

Homogeneous and Inhomogeneous Broadening

If a mechanism broadens the lineshape the same way for each atom, it is homogeneous.

If the mechanism distributes the resonance frequencies over a spectral range , it is inhomogeneous.

Now we are going to examine various physical mechanisms.

Measurement of a Lineshape

The total lineshape for a system can be easily measured through absorption spectroscopy.

White light Spectrometer

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Collision BroadeningCollision broadening results from atomic/molecular collisions in a gas, or from interaction with lattice phonons in a solid.

Collision broadening is homogeneous.

Collisions cause random phase jumps in the wavefunctions.

Alternatively, we can also think that the incoming electric field goes through phase jumps.

The new field is no longer monochromatic.

Collision Broadening (cont’d)

For a monochromatic wave, the absorption (and S.E.) rate was given by:

Since the field is now not monochromatic, we have a distribution of energy density ρover frequencies. The elemental transition rate for an infinitesimal range is:

The overall rate is found by integration:

Collision LineshapeWe now introduce the collision lineshape:

Using the property of the delta function, we get:

The lineshape should be normalized:

Collision LineshapeIf the average time between collisions is τc, then the lineshape is given by a Lorentzian:

The bandwidth is:

Collision Broadening: ExampleLet’s estimate collisional broadening for a He-Ne laser. In a gas at pressure p, and atomic mass M, collision time is given by:

For Ne at 0.5 atm, we find:

The frequency of the emission is (632 nm):

Therefore the light is still highly monochromatic.

Natural BroadeningNatural (or intrinsic) broadening originates from spontaneous emission and results from quantum nature of energy levels. The uncertainity principle sets a limit between lifetime and tansition energy.

Natural Broadening is homogeneous. It is also given by a Lorentzian, but with a new bandwidth and collision time:

We previously estimated lifetime for a visible transition as 10 ns. As a result, for that transition we have:

Broadening Due to ImpuritiesIf there is an impurity (inhomogeneity) in a symmetric medium, like glass, nearby atoms will experience a local electric field. This field will affect the electron energy levels, known as «Stark shift».

The result is an «inhomogeneous broadening». It will shift the resonance frequencies. The new distribution of resonant frequencies is given by a function g*.

Inhomogeneous distribution functions are typically Gaussian.

The bandwidth ∆ν*0 depends on the impurities.

Inmomogeneous broadening due to impurities is particularly strong in glasses.

Nd:Glass laser: The main laser transition is at 1.05 µm. The bandwidth due to impurities in glass becomes:

This broad bandwidth supports short pulse durations.

The Doppler EffectThe doppler effect tells us that the frequency of waves depends on the motion of source and observer.

The doppler effect also happens for electromagnetic waves.

Doppler BroadeningAssume that an atom/molecule is moving in z-direction with velocity vz.

If an electromagnetic wave of frequency ν is incident, the frequency seen by the atom is:

When the atom is moving away from the light source, the radiation frequency is red-shifted. When it is moving towards the source, frequency is blue-shifted.

Absorption happens when the doppler-shifted frequency coincides with the transition resonance frequency:

OR:

We can interpret this as a shift of resonance frequency:

Hence, the Doppler broadening is inhomogeneous.

Doppler LineshapeVelocity distribution in a gas is given by Maxwell distribution:

The velocity distribution is the same as resonant frequency distribution:

/

We again obtain a Gaussian lineshape with a bandwidth:

Doppler Linewidth of a He-Ne Laser

At 300 K, the doppler linewidth for Ne is:

You will calculate average velocities and this bandwidth in a HW problem.

Combination of Broadening Mechanisms

Homogeneous broadenings always give a Lorentzian.

Inhomogeneous broadenings always give a Gaussian.

Overall lineshape of different effects is given by a convolution:

If two homogeneous mechanisms overlap, the total lineshape is another Lorentzian with a bandwidth:

If two inhomogeneous mechanisms overlap, the total lineshape is another Gaussian with a bandwidth:

The proof is a HW problem.

Non-Radiative Decay

An excited state can also decay via non-radiative processes.

The detailed description is complicated.

We will only do a qualitative discussion.

We will also consider radiative and non-radiative processes combined.

Collisional DeactivationFor a gas or liquid, the energy is transferred to the colliding species as excitation and/or kinetic energy.

In solids, the energy is transferred to the lattice phonons or vibrational modes.

The deactivation can be represented as:

The process is particularly effective when mass of B is small. This is used in CO2 laser, where He atoms take the surplus energy.

Collisional ActivationThe reverse process can also take place i.e. species B can e excited by taking up the energy of A:

The change of population due to both processes is given by:

In thermodynamic equilibrium:

However, in case of strong pumping, the term on the LHS is much stronger.

Collisional Decay Rate

In case of strong pumping, we can neglect collisional activation.

Then we can write the rate equation as:

Or, by definign a nonradiative decay rate and lifetime:

Collisions may also excite/deactivate the colliding species.

During a collision, apart from kinetic energy, internal energies may also shift.

This process is particularly important for gas lasers like He-Ne.

Combination of Radiative and non-Radiative Processes

If both radiative and non-radiative processes take place, we can write:

Or, by defining an overall decay time:

Which has a solution:

Spontaneous Emission LifetimeSpontaneous emission results only from radiative processes. Hence the emission power in a certain volume V is:

As a result, spontaneous emission decays with overall lifetime.

To find the radiative lifetime, we define «fluorescence quantum yield» as (number of emitted photons)/(number of excited atoms):

We can measure the yield, and then determine radiative lifetime.

Saturation

Now we will consider emission and absorption in presence of strong EM wave.

In this case, the energy level populations differ strongly from equilibrium.

The consequent phenomenon is called «sarutation».

Saturation of Absorption: Homogeneous Line

Rearranging:

Steady state solutionAt steady state:

OR

Where, using :

We get the saturation intensity:

When the intensity reaches the saturation, the level population difference become ∆N = Nt / 2. Then the emission processes balances absorption. The absorption is hence «saturated».

Change of a homogeneous line with saturation

Remember the absorption coefficient:

For homogeneous lineshape:

And defining:

Absorption coefficient reduces with the same lineshape.

Unsaturated absorption coefficient

Change of a homogeneous line with saturation

Saturation of Gain: Homogeneous Line

As before, levels 3 and 1 decay rapidly, hence:

Where Rp is the pumping rate from g to 3.

At steady state:Note that this is twice the saturation intensity for absorption.

Change of a homogeneous line with saturation

As previous:

Gain coefficient reduces with the same lineshape.

Unsaturated gain coefficient

Now we will consider the change of gain profile, rather than absorption.

Inhomogneously Broadened Line

Saturating beam «burns a hole» in the gain profile.

The saturating beam only interacts with atoms with resonance frequency in the neighbourhood of light frequency.