PHYS20312 Wave Optics - Section 5: Lasers
5.1 Absorption and Emission in Atomic Systems.
Spontaneous emission
dN 2 hν = −N2 A21 dt
− A21t ∴ N2 (t) = N2 (0)e
1 where N2 = number density of excited atoms A21= “Einstein A coefficient” Stimulated emission
Emitted photons have the same ν 2 dN hν 2 = −N B ρ dt 2 21 hν hν where energy density, ρ=I/c
1 (I= intensity, c= speed of light)
and B21= “Einstein B coefficient for stimulated emission” Absorption
dN 2 1 = −N B ρ dt 1 12
where hν B12= “Einstein B Coefficient for absorption”
1
But these three processes operate at the same time, hence the total rate equation for level 2:
dN 2 = −N B ρ − A N + N B ρ dt 2 21 21 2 1 12
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PHYS20312 Wave Optics - Section 5: Lasers
Now let’s consider what happens at thermal equilibrium
dN 1 2 = 0 dt 8πhν 3 1 2 Energy density corresponds to a black body, i.e. ρ ν = × ( ) 3 hν c exp(kT )−1
N 2 −hν 3 Thermal population of energy levels, i.e = exp( kT ) N1
∴Rate equation becomes
8πhν 3 A = 21 3 hν hν c (exp(kT )−1) B12 exp(kT )− B21
Comparing the left-hand side to the right-hand side, we deduce
B12 = B21 A 8 h 3 21 π ν = 3 B21 c
N.B. For a degenerate system, g1B12=g2B21, where gi= number of degenerate sub-levels in level i.
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PHYS20312 Wave Optics - Section 5: Lasers
5.2 Optical Gain and Population Inversion
2
I I+dI
1
dz Figure 1 Optical gain. Consider the change in intensity of a beam of light as it passes through a thin slice of material, as illustrated in Fig. 1:
dI = (rate of stim. emissions per vol. – rate of absorptions per vol.) × hν × dz I dI = (N 2 B21 − N1B12 ) hνdz c where spontaneous emission has been neglected. Since B21=B12 dI I = (N2 − N1 )B21 hν dz c Solving the differential equation,
z I(z)= I(0)eγ where the ‘gain coefficient’, �, is given by
hν γ = (N2 − N1 )B21 = (N2 − N1 )σ c where σ is the ‘stimulated emission cross-section’. For � > 0, the intensity of the light will increase as it passes through the material i.e. there will be ‘optical gain’.
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PHYS20312 Wave Optics - Section 5: Lasers
However, optical gain is not sufficient to obtain laser operation, which is the oscillation of light in a resonator. For oscillation we need:
1. Gain ≥ Losses, i.e. � ≥ � (where the loss coefficient, α >0) PLUS 2. Positive feedback
For � > 0, we require �! > �! but for any thermal population distribution �! < �!. We therefore need to excite or ‘pump’ the material, known as the ‘gain medium’, to get
�! > �! (typically by illuminating with another light source of by passing an electric current through the material). The condition of � = �! − �! > 0 is known as a ‘population inversion’.
Now the rate equation becomes
�! = −�!�!"� − �!"�! + �!�!"� + � where � is the ‘pump rate’ i.e. the rate of excitation to level 2.
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PHYS20312 Wave Optics - Section 5: Lasers
5.3 Optical cavities
Optical cavities, also known as ‘optical resonators’, provide the positive feedback required for oscillation. The simplest form is two reflecting surfaces aligned to reflect light back and forth (c.f. a Fabry-Perot etalon). The combination of optical gain with positive feedback required for a laser is achieved by placing a pumped gain medium inside an optical resonator.
Figure 2 Optical gain in a resonator.
Light circulating in a cavity without any optical gain will undergo losses due to absorption, scattering and transmission through the mirrors. However, as a population inversion is produced in the material by pumping then the resultant gain begins to compensate for the losses. When the gain equals or exceeds the losses then oscillation can occur and lasing begins.
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PHYS20312 Wave Optics - Section 5: Lasers
5.3 Threshold and Steady-State Operation.
Consider a beam of light, of initial intensity �!, as it undergoes one round-trip in a cavity, as shown in Fig 3.
Figure 3 Gain and loss during a cavity round trip.
After passing through the gain medium for the first time, it experiences net gain (i.e. gain exceeds the loss) and the intensity is:
�!exp � − � � and after subsequent reflection from the right-hand mirror it becomes:
�!exp � − � � �! It then passes back through the gain medium and at this point it is
�!exp 2 � − � � �! It is then reflected by the left-hand mirror so that when it returns to its starting position it is now:
�!exp 2 � − � � �!�! For steady-state operation, we require that there is no change after one round-trip i.e.
�! = �!exp 2 �!! − � � �!�! where �!! is the ‘threshold gain coefficient’ and is the gain required to just balance the total losses (such as absorption and scattering in the gain medium, and transmission through the mirrors). Re-arranging we get: 1 � = � − ln � � !! 2� ! !
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PHYS20312 Wave Optics - Section 5: Lasers
If the gain medium is pumped more so that the population inversion, �, increases such that � > �!! then the intensity will grow. However, a greater intensity increases the rate of stimulated emission which will cause � to drop until we are back at the situation of
� = �!!. In other words, there is a feedback mechanism that always keeps � at its threshold level,�!!, during steady-state operation; increasing the pump rate just means that the threshold population inversion is maintained at a higher optical intensity.
5.4 Four-level lasers.
The two-level model we have been using is not realistic; most real systems have more levels than this. A real material is often chosen as ‘good’ gain medium because it forms a ‘4- level laser system’, as illustrated in Fig 4.
Figure 4 A four-level laser system.
Here, pumping occurs from the ground level (level 0) to the ‘pump band’ (level 3), from where there is rapid relaxation to the ‘upper laser level’ (level 2). The lasing transition is between the upper laser level and the ‘lower laser level’ (level 1), from where there is rapid relaxation back to the ground level. The material is chosen such that the lifetimes of the
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PHYS20312 Wave Optics - Section 5: Lasers transitions from levels 3 to 2 and from 1 to 0 are short compared to the lifetime of the lasing transition, i.e.
�!" ≫ �!", �!"
This ensures that �! ≫ �! when pumped i.e. the population inversion is maximized and �~�!
In this case the rate equation becomes:
� = � − � �!"� + �!"
and in steady-state, � = 0 and � = �!!
� 1 1 �!! � = − �!" = � − �!! �!" �!!�!" �!" but � = �� and !!" = ! ! !! ℎ� � ∴ � = � − !! �!!� �!" or
� � = � � − �!! where the threshold pumping rate �!! �!! = �!" which is the rate of spontaneous emission at threshold and where the ‘slope efficiency’
ℎ� � = �!!�
Together �!! and � characterize the output from the laser as a function of pumping rate (known as the ‘power curve’), as illustrated in Fig. 5:
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PHYS20312 Wave Optics - Section 5: Lasers
Figure 5 The 'power curve' for a laser.
6.5 An example gain medium - Nd:YAG.
Yttrium aluminium garnet (Y3Al5O12 – “YAG”) is a colourless crystal that can be artificially grown. The YAG crystal provides a strong, transparent and thermally-conductive (i.e. good for getting rid of waste heat) host material for the lasing species – see Table 1 for its other properties. Nd:YAG is a YAG crystal doped with about 1% of neodymium atoms, which are added as an impurity during growth; Nd3+ replaces the Y3+ in the crystal lattice. The Nd3+ forms a 4-level laser systems (see Fig 6) with a 230 µs upper laser level lifetime and a large stimulated emission cross-section of σ=2.8x10-19 cm-2. Crystals of Nd:YAG are usually cut into rods or slabs and pumped either with high power arc-lamps or laser diodes – see Fig 7.
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PHYS20312 Wave Optics - Section 5: Lasers
Table 1 Properties of Nd:YAG (from ‘Solid State Lasers’ by W. Koechner).
3+ 2S+1 Figure 6. Energy level diagram for Nd . Energy level notation: LJ (S, L, J refer to total momenta). Figure from from ‘Solid State Lasers’ by W. Koechner.
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PHYS20312 Wave Optics - Section 5: Lasers
Figure 7 . Pump schemes for an Nd: YAG laser. A picture of an Nd:YAG rod is shown in the centre (from Wikipedia). Arc-lamp and laser diode based schemes are shown on the left and right, respectively. Images from ‘Solid State Lasers’ by W. Koechner.
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