Appendix: The Fresnel Equations
89
Problems
2.1 Suppose a certain wave is specified by y = 8 cos 2π (2t − 0.8z), where y is
expressed in micrometers and the propagation constant is given in μm
−1 . Find
(a) the amplitude, (b) the wavelength, (c) the angular frequency, and (d) the
displacement at time t = 0 and z = 4 μm. [Answers using Example 2.1: (a)
8 μm; (b) 1.25 μm; (c) 4 π; (d) 7.512 μm.]
2.2 Let E 0x = E 0y = 1 in Eq. (2.7). Write a software program to plot this equation
for values of δ = (nπ)/8, where n = 0, 1, 2, …. 16. What does this show about
the state of polarization as the angle δ changes?
2.3 Show that any linearly polarized wave may be considered as the superposition
of left and right circularly polarized waves that are in phase and have equal
amplitudes and frequencies.
2.4 Light traveling in air strikes a glass plate at an angle θ 1 = 33°, where θ 1 is
measured between the incoming ray and the normal to the glass surface. Upon
striking the glass, part of the beam is reflected and part is refracted. (a) If the
refracted and reflected beams make an angle of 90° with each other, show that
the refractive index of the glass is 1.540. (b) Show that the critical angle for
this glass is 40.5°.
2.5 A point source of light is 12 cm below the surface of a large body of water
(n water = 1.33). Show that the radius of the largest circle on the water surface
through which the light can emerge from the water into air (n air = 1.000) is
13.7 cm.
2.6 A right-angle prism (internal angles are 45°, 45°, and 90°) is immersed in
alcohol (n = 1.45). Show that the refractive index of the prism must be 2.05
if a ray incident normally on one of the short faces is to be totally reflected at
the long face of the prism.
2.7 Show that the critical angle at an interface between doped silica with n 1 =
1.460 and pure silica with n 2 = 1.450 is 83.3°.
2.8 Consider the power reflection for light incident normally at the interface
between an optical fiber end and air. If the refractive indices of the fiber and
air are n fiber = 1.48 and n air = 1.000, show that the reflectance R = 0.037 and
the transmittance is T = 0.963.
2.9 Consider a step-index fiber having n 1 = 1.48 and n 2 = 1.46. (a) Show that the
numerical aperture is 0.243. (b) If the outer medium is air with n = 1.00, show
that the acceptance angle θ A for this fiber is 14°.
2.10 Suppose a certain multimode step-index optical fiber has a core diameter of
62.5 μm, a core index of 1.48, and an index difference = 0.015. (a) Show
that at 1310 nm the value of V is 38.4. (b) Show that the total number of modes
is 737.
2.11 A given step-index multimode fiber with a numerical aperture of 0.20 supports
approximately 1000 modes at an 850 nm wavelength. (a) Show that the diameter of its core is 60.5 μm. (b) Show that the fiber supports 414 modes at
1320 nm. (c) Show that the fiber supports 300 modes at 1550 nm.
89
Problems
2.1 Suppose a certain wave is specified by y = 8 cos 2π (2t − 0.8z), where y is
expressed in micrometers and the propagation constant is given in μm
−1 . Find
(a) the amplitude, (b) the wavelength, (c) the angular frequency, and (d) the
displacement at time t = 0 and z = 4 μm. [Answers using Example 2.1: (a)
8 μm; (b) 1.25 μm; (c) 4 π; (d) 7.512 μm.]
2.2 Let E 0x = E 0y = 1 in Eq. (2.7). Write a software program to plot this equation
for values of δ = (nπ)/8, where n = 0, 1, 2, …. 16. What does this show about
the state of polarization as the angle δ changes?
2.3 Show that any linearly polarized wave may be considered as the superposition
of left and right circularly polarized waves that are in phase and have equal
amplitudes and frequencies.
2.4 Light traveling in air strikes a glass plate at an angle θ 1 = 33°, where θ 1 is
measured between the incoming ray and the normal to the glass surface. Upon
striking the glass, part of the beam is reflected and part is refracted. (a) If the
refracted and reflected beams make an angle of 90° with each other, show that
the refractive index of the glass is 1.540. (b) Show that the critical angle for
this glass is 40.5°.
2.5 A point source of light is 12 cm below the surface of a large body of water
(n water = 1.33). Show that the radius of the largest circle on the water surface
through which the light can emerge from the water into air (n air = 1.000) is
13.7 cm.
2.6 A right-angle prism (internal angles are 45°, 45°, and 90°) is immersed in
alcohol (n = 1.45). Show that the refractive index of the prism must be 2.05
if a ray incident normally on one of the short faces is to be totally reflected at
the long face of the prism.
2.7 Show that the critical angle at an interface between doped silica with n 1 =
1.460 and pure silica with n 2 = 1.450 is 83.3°.
2.8 Consider the power reflection for light incident normally at the interface
between an optical fiber end and air. If the refractive indices of the fiber and
air are n fiber = 1.48 and n air = 1.000, show that the reflectance R = 0.037 and
the transmittance is T = 0.963.
2.9 Consider a step-index fiber having n 1 = 1.48 and n 2 = 1.46. (a) Show that the
numerical aperture is 0.243. (b) If the outer medium is air with n = 1.00, show
that the acceptance angle θ A for this fiber is 14°.
2.10 Suppose a certain multimode step-index optical fiber has a core diameter of
62.5 μm, a core index of 1.48, and an index difference = 0.015. (a) Show
that at 1310 nm the value of V is 38.4. (b) Show that the total number of modes
is 737.
2.11 A given step-index multimode fiber with a numerical aperture of 0.20 supports
approximately 1000 modes at an 850 nm wavelength. (a) Show that the diameter of its core is 60.5 μm. (b) Show that the fiber supports 414 modes at
1320 nm. (c) Show that the fiber supports 300 modes at 1550 nm.