97.472 Problem Analysis

Week 0 (January 3-4 )

  1. Prove Snell’s Law at a planar boundary using Fermat’s Principle.

    2. Week 1 (January 7-11)

  2. A prism having refractive index n has a small apex angle a. Show that paraxial rays entering one face of the prism at angle of incidence q ,with respect to the normal, exit the opposite face at an angle approximately given by qd = (n-1)a.
  3. Prove Equation (2.5) of the notes for paraxial rays.
  4. Determine the NA and acceptance angle from air of a silica glass fiber if the refractive index of the core is 1.475 and
    1. the cladding has index 1.46
    2. the cladding is stripped so that the core is surrounded by air.
    3. Repeat if the light enters the fiber through index matching gel having index 1.4.
  1. Tiny glass balls are often used as lenses to couple light into and out of optical fiber. The fiber end face is located at distance f from the sphere. For a sphere of radius a=1mm and refractive index 1.48, determine the distance f such that a ray parallel to the optical axis at height y=0.7mm is focused into the fiber core.
  2. An optical fiber has core diameter 100mm and NA=0.36 . The output light from the fiber is collimated by a lens with focal length 25 mm. What is the diameter of the collimated beam and what is the beam divergence?

    3. Week 2 (January 14-18)

  3. A plano-concave lens with f= -50 mm and a plano-convex lens with f=250mm are separated by 200 mm with the plane surfaces facing each other. A laser beam with diameter 0.66 mm and beam divergence 1.4 mrad is incident on the concave surface. What is the diameter and divergence of the output beam from the convex surface? What does this lens combination accomplish?
  4. The light from a Nd:YAG laser at wavelength 1.06 mm is a Gaussian beam of 1W optical power and beam divergence 2qo = 1mrad. Determine the beam waist radius, the depth of focus, the maximum intensity, and the intensity on the beam axis at distance z=100 cm from the beam waist.
  5. An argon-ion laser produces a Gaussian beam of wavelength l=488nm and waist radius Wo=0.5mm. Choose a single lens to focus the light to a spot of diameter 100 mm. What is the shortest focal-length lens that may be used?

    6. Week 3 (January 21-25)

  6. A 25mm diameter lens with focal length 5 cm is stopped down to aperture diameter 50 mm. A collimated laser beam which has been expanded to 10 mm is incident on the lens. If the wavelength is 635 nm, calculate and plot (use MathCAD or MATLAB) the diffraction pattern at the focal plane. What is the diameter of the focused spot? What is the F/# of the lens?
  7. Determine the electric field magnitude at the center of a Gaussian beam in air (a point on the beam axis at the beam wais) if the beam is 1W and the beam waist radius is Wo=0.1mm .
  8. TM polarized light in a medium with index n1 is incident at angle q1 on a plane boundary to a medium having index n2 where n1<n2. Show that the Brewster angle qB where no reflection occurs for a plane wave is given by tan qB=n2/n1.
  9. A plane wave crosses a plane boundary between GaAs (n=3.6) and air. Calculate and plot (use MathCAD or MATLAB) the reflection coefficient and reflectance for angle of incidence q1 from 0-90 degrees for
    1. TE polarization
    2. TM polarization.

Week 4 (January 28- February 1)
  1. A fiber has core radius 25 mm, core index n1=1.48 and D=0.01.
    1. If l=1320 nm, what is the V-number?
    2. How many modes propagate in the fiber?
    3. What percentage of the power flows in the cladding?
    4. Repeat (a)-(c) if the index difference is reduced to D=0.003.
  1. Find the core radius necessary for single-mode operation at 1320 nm of a step-index fiber with n1=1.480, and n2=1.478. What are the NA and maximum acceptance angle for this fiber?
  2. Use MATLAB or MathCAD to plot E(r)/Eo for 0<r/a<3 for for a fiber having V=1.0, 1.4, 1.8, 2.2, 2.6, and 3.0.

    3. Week 5 (February 4- February 8)

  3. Plot the refractive index profiles from n1 to n2 as a function of radial distance r £ a for GRIN fibers having a=1,2,4,8,and ¥. Assume the core radius is 25 mm , n1=1.48 and D=0.01
  4. Calculate the number of modes at 820 nm and 1300 nm in a GRIN fiber having parabolic index profile, 25 mm core radius n1=1.48 and D=0.02. How does this compare to step-index fiber?
  5. Sunlight impinges on a transmission grating that is formed with 5000 lines per cm. Does the third-order spectrum overlap the second-order spectrum? Take red to be 780 nm wavelength and violet to be 390 nm wavelength.

    6. Week 6 and 7 (February 11- March 1)

  6. Derive the expression for the group index Ng and show that vg=co/Ng.
  7. It is desired to use multimode fiber with core diameter 50mm. together with a source that has very narrow linewidth at 1300nm. What is the BL product for the following:
    1. stepped index with n1=1.480, D=0.0135
    2. GRIN with n1=1.480, D=0.0135, a=2.
  1. A single-mode silica fiber has Dmat=22 ps/(nm-km) and Dwg= -6 ps/(nm-km) at 1550 nm. Compare the bit rate possible over 10 km of fiber for
    1. an LED operating at 1550 nm with linewidth 45nm
    2. a laser diode operating at 1550 nm with linewidth 5 nm.
  1. Determine the core radius of multimode SI fiber with NA=0.1 if the number of modes is 5000 when the wavelength is 0.87 mm. If the core refractive index is n1=1.445, the group index N=1.456 and D is approximately independent of wavelength, determine the pulse spread for a 2 km fiber.