Resonators

• Electromagnetic Decay
  Place an eight-layer reflector (see exercise and fig. ) at x=0 and set the left boundary to fixed and the right boundary to absorbing. Set the initial wavefunction so that there are an integral number of wavelengths, , between the left boundary and the reflector. Plot the total energy as a function of time and explain the relationship of this plot to the quality factor, Q.
• Lasers 
    Use the previous exercise to create a laser. Do this by creating a segment that has gain and fills the cavity. Adjust the gain such that the total amplification in one pass exceeds the transmission loss through the reflector, where

  

  and L is the length of the cavity. Remember that g can be positive or negative and that g will be reduced as the total energy in the system approaches 10 if you have saturation turned on by using [Pref] .

  a.
  Calculate the threshold gain and see what happens to the initial sine wave when you create a segment filling the region with index and threshold gain. Try gain values above and below the threshold value.
  b.
  Initialize the laser configuration with a Gaussian wavepacket inside the resonating cavity. Set the gain to . Is lasing established at a single frequency? Why or why not? Look at an FFT of the output detector reading, , with the number of FFT points set to 1024 for maximum frequency resolution. Actual laser cavities will exhibit similar phenomena. They will lase simultaneously in a number of modes.

    
  Figure: Laser configuration for exercise 36. =low index of refraction, =high index of refraction.
  

  Quantum Decay
    Create a standing wave to the left of a narrow (but high) potential barrier. Plot the total probability as a function of time. Although the decay looks like it might be exponential, and many authors use this system as a model for nuclear decay, it is not. It is a power law for long times. Both tunneling and packet spreading are occurring![11,12]