Homework on Optical Computing (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material. You should show all of your calculations (neatly) and justify all of your answers for full credit.

You want to use a Fabry Perot interferometer as a logic device using Gallium Arsenide lasers (l = 840 nm = 0.840 microns) for input.

1. Your cavity is tailored to be 0.827 microns long. What are the three longest wavelengths of laser light that will produce standing waves inside this cavity?
The allowed wavelengths for a standing wave within the cavity are determined by the boundary condition that the amplitude must be maximal (an antinode) at each of the cavity boundaries. These allowed wavelengths are given by the expression l = 2L, L, 2/3 L
l = 1.654, 0.827, 0.551 microns

2. Which of these three wavelengths is most likely to be provided by a Gallium Arsenide laser? (Hint: Typical indices of refraction are between 1.3 and 2.)
The wavelength of the laser in air, l₀, is 0.840 microns. Therefore the wavelength in the cavity will be given by l = l₀ / n. For an index between 1.3 and 2 only the smallest of the three normal wavelengths, 0.551 microns, could be produced.

3. What should the index of refraction be when the cavity is resonating (producing standing waves)?
Again, the expression, l = l₀ / n, is used. Solving for n, l₀ /l = n
0.840 microns /0.551 microns = 1.53

4. The cavity achieves this index when the intensity of the light entering the cavity reaches 3 megawatts per square centimeter. The output of the cavity will then reach 10,000 megawatts per square centimeter. For lower input intensities, the output is about 1 megawatts per square centimeter. What is the size of this resonant effect in decibels (use resonant output over non-resonant output)?
The decibel unit represents ten times the log of the ratio between a quantity and a reference quantity or, alternately, the difference between two logarithms. 10 log₁₀ (10,000/1) =10 [ log₁₀ (10,000) - log₁₀ (1) ] = 40 dB

5. You have two input signals of 0.8 megawatts per square centimeter each, in addition to the steady-state beam. What should the intensity of the beam be if the devices is to operate as an AND? What if it is to operate as an OR?
The gate formed by the interferometer “turns on” when the index of the material becomes such that the cavity supports a standing wave equal to the wavelength of the incident beam. This index is a function of the intensity of the incident light.
An AND gate requires both inputs to be high to produce a high output. Therefore, the sum of the input signals and the steady state must be equal to the critical intensity.
0.8 MW/cm² + 0.8 MW/cm² + x = 3 MW/cm² Solving for x one finds that the steady state should be at 1.4 MW/cm²
An OR gate requires only a single input to be high in order to produce a high output. Therefore, the sum of a single input single and the steady state must be equal to the critical intensity.
0.8 MW/cm² + x = 3 MW/cm² Again, solving for x one finds the steady state should be at 2.2 MW/cm²

1.	Your cavity is tailored to be 0.827 microns long. What are the three longest wavelengths of laser light that will produce standing waves inside this cavity? The allowed wavelengths for a standing wave within the cavity are determined by the boundary condition that the amplitude must be maximal (an antinode) at each of the cavity boundaries. These allowed wavelengths are given by the expression l = 2L, L, 2/3 L l = 1.654, 0.827, 0.551 microns
2.	Which of these three wavelengths is most likely to be provided by a Gallium Arsenide laser? (Hint: Typical indices of refraction are between 1.3 and 2.) The wavelength of the laser in air, l₀, is 0.840 microns. Therefore the wavelength in the cavity will be given by l = l₀ / n. For an index between 1.3 and 2 only the smallest of the three normal wavelengths, 0.551 microns, could be produced.
3.	What should the index of refraction be when the cavity is resonating (producing standing waves)? Again, the expression, l = l₀ / n, is used. Solving for n, l₀ /l = n 0.840 microns /0.551 microns = 1.53
4.	The cavity achieves this index when the intensity of the light entering the cavity reaches 3 megawatts per square centimeter. The output of the cavity will then reach 10,000 megawatts per square centimeter. For lower input intensities, the output is about 1 megawatts per square centimeter. What is the size of this resonant effect in decibels (use resonant output over non-resonant output)? The decibel unit represents ten times the log of the ratio between a quantity and a reference quantity or, alternately, the difference between two logarithms. 10 log₁₀ (10,000/1) =10 [ log₁₀ (10,000) - log₁₀ (1) ] = 40 dB
5.	You have two input signals of 0.8 megawatts per square centimeter each, in addition to the steady-state beam. What should the intensity of the beam be if the devices is to operate as an AND? What if it is to operate as an OR? The gate formed by the interferometer “turns on” when the index of the material becomes such that the cavity supports a standing wave equal to the wavelength of the incident beam. This index is a function of the intensity of the incident light. An AND gate requires both inputs to be high to produce a high output. Therefore, the sum of the input signals and the steady state must be equal to the critical intensity. 0.8 MW/cm² + 0.8 MW/cm² + x = 3 MW/cm² Solving for x one finds that the steady state should be at 1.4 MW/cm² An OR gate requires only a single input to be high in order to produce a high output. Therefore, the sum of a single input single and the steady state must be equal to the critical intensity. 0.8 MW/cm² + x = 3 MW/cm² Again, solving for x one finds the steady state should be at 2.2 MW/cm²