Sunday, February 13, 2011

Mastering Physics: ± The Laser

One of the biggest impacts of the advances made in quantum physics on our everyday lives comes from the widespread use of lasers. (Laser stands for light amplification by stimulated emission of radiation.) Lasers can be found in everything from CD players to devices used to perform eye surgery. The concepts of definite energy levels within atoms and photons being released during energy transitions are central to both quantum physics and the operation of lasers.
The key to the operation of lasers is that a photon of the right energy can induce an atom in an excited state to emit a photon that has the same energy, phase, polarization, and direction of propagation as the initial photon. When this is the case, you get two photons for the price of one--which explains the origin of the term "light amplification." To induce such an emission a photon must have an energy equal to the difference between the atom's excited state and its unexcited (or ground) state. However, when such a photon encounters an atom in the ground state it will be absorbed and promote the atom to its excited state--in this case you lose a photon. To obtain a working laser, therefore, it is necessary for there to be more atoms in the excited state than in the ground state. This is called a population inversion. When a population inversion exists, the system will act as a net source of photons, because you get more photons than you initially put in.

Part A
Which of the following systems will act as net sources of radiation?
  1. 60% excited state, 40% ground state
  2. 30% excited state, 40% ground state, 30% other state
  3. 30% excited state, 20% ground state, 50% other state

Part B
According to the Maxwell-Boltzmann distribution law, the proportion of atoms in two particular states is given by the energy of the two states and the temperature T of the system. If E_ex and E_g are the respective energies of the excited and ground states then the ratio of the number of states in each will be
\frac{n_{\rm ex}}{n_{\rm g}} = \frac{e^{-E_{\rm ex}/{k_{\rm B}T}}}{e^{-E_{\rm g}/{k_{\rm B}T}}},
where k_B is Boltzmann's constant.

A researcher who has found a method of heating samples of gas to arbitrarily high temperatures proposes to achieve a population inversion simply by raising the temperature of a sample of gas. Will this researcher be successful?

Part C
After realizing the failure of the first technique, the researcher now proposes to raise the ratio of excited to unexcited atoms n_{\rm ex}/n_{\rm g} to only 0.8 and then will achieve the rest of the population inversion through other means. If the researcher wishes to create a laser with a wavelength of 500\;{\rm nm}, what temperature T must the sample be raised to?
Express your answer in kelvins to two significant digits.

  T  = 130000
  \rm K
As you can see, the temperatures needed to promote a sample to close to the ratio needed for a population inversion are very high and are practically unobtainable in a laboratory. Instead, use is made of other methods such as driving an electric current through a sample of gas or inducing population inversions in semiconductors.
The result that you achieved in B--that it is impossible to achieve a population inversion solely by raising the temperature--is of some significance to the behavior of the universe. If this were not the case, then it would be possible for huge systems (such as stars) to act like giant lasers, sending enormous amounts of coherent directed radiation throughout the galaxy whenever they reached the necessary temperature.

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