Steps and Barriers

• Electromagnetic Step
    Load WAVE from DOS and set both boundaries to Absorbing . First create a segment by clicking on the green segment button in the bottom left-hand corner and then clicking inside the wave graph. Now double-click inside the segment to bring up the inspector, and edit the parameters as follows:
  • X Position
  • Width of Segment
  • Index of Refraction
  • Gain = 0
  • Color of Segment = 2
  • Able to Drag = FALSE (i.e., Not checked.)
  Close the inspector by clicking OK. Use [Init]|[Gaussian] to create a right-going Gaussian pulse with the following attributes:
  • Amplitude =1
  • Width
  • Position

  a.

  Calculate theoretical reflectance for a light wave of normal incidence on glass with .

  b.

  Create detectors to monitor the reflected and transmitted wave and enable a detector analysis graph. How does reflectance of the peak compare to the prediction of theory? Note the inversion of the reflected pulse. Why is the pulse inside the medium narrower than the original pulse?

  c.

  Measure the speed of the incident and transmitted pulses. Is this what you expected?

  d.

  Change the initial condition so that the pulse is a left-going pulse starting inside the medium. How does reflectance of the peak compare to the prediction of theory? Is the reflected pulse inverted? Why does the transmitted pulse have a larger amplitude than the incident pulse?

  e.

  Note the continuity of the electric field at the boundary between two media.

  f.

  Calculate the [Time Average] of the square of the incident, transmitted and reflected Gaussian pulses (you will find this option in the menu brought up by the attributes button of the detector analysis graph). Make sure you have the Sqr(Amp) radio button checked in the Average Inspector panel. How do these average values compare to the prediction of theory? Hint: Is the transmitted energy the square of the electric field?

• Electromagnetic Barrier
    For what frequencies (between 1 and 15 Hz) of normally incident light will reflection from a sliver of glass () with thickness be maximized? For what frequencies will transmission be maximized? Check your answers through simulation and determine the reflectance at the maxima.
• A One-Layer Antireflection Coating  
    Near the left edge of the wave graph, place a source producing a right-traveling sinusoidal wave with a frequency of 10 Hz. Put a glass segment () at , extending to the right boundary. Create a one-layer antireflection film on the left face of the glass segment. The thickness of the film should be , where is the wavelength of the incident wave inside the coating (adjusted for ).

  a.

  Compare the simulated reflectance to the theoretical prediction given by

  

  b.

  What is the reflectance of the wavelength, , for which the film was created? What is the reflectance for , , and ?

  c.

  Change the antireflection medium to magnesium fluoride (). Observe the resultant reflectances for , , , and .
  

• Quantum Step
    Select the Schrödinger equation and create a segment with a potential energy of in the region . Set the initial wave to be a Gaussian with the following conditions:
  • Amplitude =1
  • Width
  • Position
  • Momentum k=100
  Create two detectors, one inside the segment and one outside the segment at . The detectors measure probability current and can be used to determine the tansmission and reflection coefficients. The boundaries should be set to Absorbing . Select a [Graph-2]|[Detector Readings]

  analysis graph. Run the program.

  a.

  Notice that the Gaussian pulse breaks apart into transmitted and reflected components reminiscent of the classical wave simulation. Notice also that the spatial frequency of phase oscillations increases as the potential energy decreases, in keeping with an increase in kinetic energy. Click the blue attributes button on the detector graphs and perform a time integral of the detector readings. Since the detectors read current, J,

  

  the integrals represent the probability of a particle passing a given detector.

  b.

  Increase the potential step to by double-clicking on the segment and editing the inspector. Run the program and estimate the reflection and transmission coefficients from the pulse heights and widths. Notice that the phase wavelength decreases inside the higher-potential segment.

  c.

  Increase the potential to . The pulse is almost entirely reflected by the step, although it does penetrate the forbidden region. Use the integral of the probability current to show this.

  Answer (partial): Atomic units simplify many of the relationships between wave and particle properties. A particle with momentum in a zero-potential region has energy and wavelength . If this particle-wave propagates into a region where , then the kinetic energy increases to and the momentum becomes . Use the coordinate reader to check the wavelength: hold the mouse down inside the wave graph and move it between two phase maxima.

• Quantum Antireflection Coating
    Create a double potential step by adding a second segment of width d to the left of a potential step or . Set the potential and width of this new segment so that the following conditions are satisfied:

  

  Run the program. Notice that the reflection of the quantum wave packet is almost zero. Use detectors and current integrals to determine the reflection and transmission probabilities.