LASER AMPLIFICATION FUNDAMENTALLY AN ELECTROMAGNETIC WAVE TRAVELLING THROUGH A

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Laser Amplification

Fundamentally, an electro-magnetic wave travelling through a medium may gain or lose energy depending on the state of excitation of that medium

Consider a small segment of medium, with refractive index  and cross-sectional area A, between z and z + z. The incident intensity is I(z) integrated over all frequencies. It is also assumed that the medium is composed of simple, two-level atoms of the type previously discussed. The population densities in the lower and upper states are n_m and n_n respectively. We will also assume propogation in one direction and therefore ignore non-directional spontaneous emission.

The change in total power in passing through the segment of medium is:

This may also be expressed in terms of the radiation density U and Einstein B coefficients:

(note that the radiation density is due to the incident power, not black body radiation which we may assume is negligible)

Now, the intensity is integrated over all frequencies. So, the intensity at a particular frequency, , is where is the spectral line profile function, defined such that:

The intensity can therefore be related to the radiation density by .

We can therefore write the rate of loss of intensity as: (substituting for U)

This function is clearly of the form which has as its solution

Hence, we can identify an absorption coefficient, K:

Taking into account the degeneracies of the states, and the relationship between the A and B coefficients, this relationship may also be written:

LASER AMPLIFICATION FUNDAMENTALLY AN ELECTROMAGNETIC WAVE TRAVELLING THROUGH A

This is known as the Füchbauer–Ladeberg formula.

Clearly, for a normal sample of atoms in thermal equlibruim, the lower state will have overwhelmingly the larger population. k() will therefore be positive, and absorption of the beam will occur. However, if the population of the upper state can be made such that k() is negative, amplification can occur. In this case, k() is referred to as the small-signal gain coefficient.

Example, The lifetime of the upper level of the 632.8 nm transition in the HeNe laser is 110^⁷ s. Calculate the degree of population inversion required to give a gain coefficient of 0.07 m^¹.

Assumptions and simplifications:

Assume the degeneracies of the upper and lower states are equal.
Assume the refractive index is 1.
Ignore any broadening of the spectral line except the natural width due to the lifetime.
A common simplification in such problems is to assume the line profile function is given by and .

The lifetime is 110^⁷ s, hence the Einstein A coefficient is 110⁷ s^¹. The wavelength is 632.8 nn, so the frequency is 0.5 THz. Given the relationship between natural lifetime and bandwidth, Hz^¹ at the wavelength of the transition.

Using the Füchbauer–Ladeberg formula derived above, the required density of population inversion is 7.8210¹¹ m^³ (c.f., density of atoms in a HeNe laser running at a pressure of 10 mmHg is 10²² m^³) i.e., there must be 7.8210¹¹ m^³ more atoms in their upper state than in the lower.

Threshold Condition for Laser Action

What conditions must be satisfied in the laser medium for laser action to occur?

Considering again a simple two-level system;

To get amplification at all, a population inversion (i.e., N_n > N_m)
A strong stimulating field is necessary, to ensure that stimulated emission is stronger than spontaneous.
I.e., , the stimulating energy density >> black body mode density.

(1) is difficult. E.g., for h = 5 eV, and temperature = 1000 K (kT = 0.086 eV)

In any case, a population inversion is a non-equilibrium situation, and in fact would require a negative temperature!

(2) can always be achieved, with a powerful enough lamp!

Assuming the above conditions can be satisfied, it is also necessary to achieve a threshold condition for laser action.

Consider a laser cavity containing a medium of length L with a gain of k per unit length and mirrors with reflectivities R₁ and R₂. All other sources of losses (e.g., diffraction) are ignored. Then, the gain in energy in the beam, after one round-trip in the laser cavity is:

, defining (in this, of course, other sources of loss could be included)

Now, to get amplification, we require G  1 i.e., . Now, k is a function of wavelength, so we will only amplify those frequencies such that .

Using LASER AMPLIFICATION FUNDAMENTALLY AN ELECTROMAGNETIC WAVE TRAVELLING THROUGH A (note – ““ because here we are dealing with gain coefficients rather than absorption coefficients)

the threshold condition becomes:

LASER AMPLIFICATION FUNDAMENTALLY AN ELECTROMAGNETIC WAVE TRAVELLING THROUGH A

If no atoms are excited, the absorption coefficient at the centre frequency of the transition, ₀ is where n₀ is the total number density of atoms. The threshold condition simplifies further to:

LASER AMPLIFICATION FUNDAMENTALLY AN ELECTROMAGNETIC WAVE TRAVELLING THROUGH A , giving a fractional population inversion,

e.g., for pink ruby, cm^¹. Take a 10 cm rod with silvered ends, ( )

Then,

(Note – g_m = g_n = 4 for the laser transition in ruby)

Bandwidth of a Laser

Any oscillator posesses a property known as the quality factor, Q. This may be defined in many equivalent ways, but for our purposes, we will define it in terms of the laser frequency and bandwidth:

Again, for any oscillator, the rate at which energy is lost has the form:

At what rate is energy lost in the laser cavity?

Suppose a group of photons (P₀ at time zero) is injected into a laser cavity, and suffers losses described by the quantity  defined earlier.

Then, the number of photons remaining in the cavity at time t is:

where  is the round-trip time of a photon in the cavity.

Hence, the characteristic time for a photon remaining in the cavity is

Comparing with the earlier expression involving Q, we can see that:

or,

Hence, from our original definition of Q, the bandwidth of the laser is simply:

*This result is quite astonishing! You might think that the bandwidth of a laser line should depend on the laser medium. It doesn’t – it depends on the laser construction. You might as well regard the laser medium as a coherent ensemble of atoms, molecules or whatever behaving in consert rather than individual objects.

Examples:

1, Suppose we have a passive (unpumped) laser cavity 0.5 m long, with 2% losses per round trip.

Hence,

The bandwidth is then simply:

2, In an ideal laser, all losses are replenished, hence and . In practice, and – a possible time and frequency standard.

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