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1.2 Population inversion - PowerPoint PPT Presentation


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E k. E i. 1.2 Population inversion. Absorption and Emission of radiation. Consider an atom or molecule with two energy levels, E i and E k A direct, radiative transition between these states would be associated with a photon of frequency ν : h ν = Δ E = E k – E i.

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slide1

Ek

Ei

1.2 Population inversion

Absorption and Emission of radiation

Consider an atom or molecule with two energy levels, Ei and Ek

A direct, radiative transition between these states would be associated with a photon of frequency ν: hν = ΔE = Ek – Ei

Molecule in energy levelEi

Consider:

What are the possible radiative transitions?

What is the probability of a transition taking place?

How does the number of photons change with the transition?

slide2

Absorption and Emission of radiation

Ek

Ei

  • Three possible transitions:
  • “Induced absorption”:
  • Molecule in Ei absorbs a photon and is excited to Ek

One less photon of energy hν

Probability of transition is: dPik/dt = Bik ρ(ν)

dPik/dtis the probability per second of a molecule absorbing a photon

Bik is the Einstein coefficient of induced absorption

ρ(ν) is the spectral energy density (the number of photons of frequency ν per unit volume)

slide3

Absorption and Emission of radiation

Ek

Ei

(b) “Spontaneous emission”:

Molecule in Ek decays spontaneously to Ei by emitting a photon in an arbitrary direction

One more photon of energy hν (arbitrary phase & direction)

Probability of transition is dPki/dt = Aki

dPki/dtis the probability per second of the excited molecule emitting a photon

Aki is the Einstein coefficient of spontaneous emission (or the spontaneous transition probability)

Spontaneous emission is not influenced by the presence of other photons in the medium

slide4

Absorption and Emission of radiation

Ek

Ei

(c) “Induced emission” (or “stimulated emission”):

A photon of appropriate frequency induces the transition from Ekto Ei

One more photon of energy hν. The new photon has the same frequency, phase, and direction at the original photon

Probability of transition is dPki/dt = Bkiρ(ν)

dPki/dtis the probability per second of the excited molecule emitting a photon

Bki is the Einstein coefficient of induced emission

slide5

Absorption and Emission of radiation

Relation between Bik and Bki:

The Einstein coefficients of induced absorption and emission are directly related through the degeneracy, gx, of each level x:

Bik = (gk/gi) Bki

In the case where each level has the same degeneracy (gi = gk), the Einstein coefficients of induced absorption and emission are identical

– in other words, the probability of induced emission is the same as that of induced absorption

How can we make practical use of induced emission?

slide6

Population inversion

Population inversion

A system with a population inversion is not in thermal equilibrium; populations of energy levels are not governed by the Boltzmann distribution

IF we can alter the population distribution so that more molecules are in higher energy levels rather than lower energy levels, this is called population inversion.

With a population inversion, photons passing through the gain medium will be amplified (by induced emission) rather than attenuated (by induced absorption).

slide7

Population inversion

A resonator or cavity (produced by the two mirrors) is used to achieve selective feedback of some of the cavity modes in the cavity – that is, photons that travel on the axis between the mirrors are preferentially amplified over photons going in different directions:

slide8

Threshold condition

Threshold condition

The probability of stimulated absorption and emission depends on the populations of the upper and lower states. With Ni molecules in level Ei and Nk molecules in level Ek , the intensity after distance z is:

I = I0e–α z

where I0 is the initial intensity and the absorption coefficient, α, is:

Here σ is the absorption cross-section and is related to Bik

α = [Ni – (gk/gi) Nk] σ

When a population inversion exists (Nk> Ni) the intensity after distance z is greater than the initial intensity (I> I0).

However, we also need to consider

other photon losses in the cavity

slide9

Threshold condition

Photons may be lost in the cavity owing to transmission through the mirrors, scattering from surfaces and particles, diffraction losses, and absorption by other materials in the cavity. If all of these losses contribute to a loss coefficient, γ, then the intensity owing to cavity losses after a round trip in the cavity is

I = I0e– γ

From above, if the cavity has length L, the round trip gain (considering only the population inversion) is then:

I = I0e–2αL

To compensate for cavity losses, the threshold condition for amplification in the cavity is then:

ΔN = [(gk/gi)Nk – Ni] > (γ / 2σL)

slide10

Generating a population inversion

“Pumping”: delivery of energy to produce a population inversion

E4

E3

rapid relaxation

E2

rapid relaxation

E3

pump

E2

pump

lasing

pump

lasing

E2

lasing

E1

rapid relaxation

E1

E1

Two-level system

Three-level system

Four-level system

Notes:

A true two-level system cannot produce a population inversion

Only E1 is populated at thermal equilibrium (E2 >> E1)

A three-level system must be pumped harder than a four-level system – that is, more molecules must be pumped into the excited level to produce lasing

Can we sustain a population inversion in a given laser?

slide11

Generating a population inversion

It is difficult to maintain a population inversion:

Lasers that maintain a population inversion indefinitely produce continuous output – termed CW (for continuous wave) lasers

Lasers that have a short-lived population inversion produce pulsed output – these are pulsed lasers

Pulsed lasers may be of three types: “normal” pulsed lasers, Q-switched lasers, and mode-locked lasers

Pumping can be achieved either:

1) optically – e.g., flashlamps (pulsed) or Hg arc lamps (CW operation)

2) electrically – e.g., electric discharge in a gas and in semiconductor lasers

slide13

Spectral characteristics of laser emission

The photon emitted between two levels is not perfectly monochromatic. The linewidth is affected inter alia by:

Natural lifetime (usually gives narrow linewidth)

Molecular motion (Doppler broadening)

Collisions (Pressure broadening, solvent effects)

This linewidth results in a gain profilefor the laser:

Only that part of the gain profile that is above the threshold can lase.

The gain profile must be considered together with the cavity modes to determine the laser spectrum