90
2 Optical Fiber Structures and Light Guiding Principles
2.12 A certain step-index fiber has a 25 μm core radius, n 1 = 1.48, and n 2 = 1.46.
(a) Show that the normalized frequency at 820 nm is V = 46.5. (b) Show
that 1081 modes propagate in this fiber at 820 nm. (c) Show that 417 modes
propagate in this fiber at 1320 nm. (d) Show that 303 modes propagate in this
fiber at 1550 nm. (e) Verify that the percent of the optical power flows in the
cladding for the different wavelengths is 4.1% at 820 nm, 6.6% at 1320 nm,
and 7.8% at 1550 nm.
2.13 Consider a fiber with a 25 μm core radius, a core index n 1 = 1.48, and =
0.01. (a) Show that at 1320 nm the parameter V = 25 and the number of modes
M = 312. (b) Verify that 7.5% of the optical power flows in the cladding. (c) If
the core-cladding difference is reduced to = 0.003, show that the number
of modes M = 94 and 13.7% of the optical power flows in the cladding.
2.14 Suppose a certain step-index fiber has a 5 μm core radius, an index difference
= 0.002, and a core index n 1 = 1.480. (a) By calculating the V number,
verify that at 1310 nm this is a single-mode fiber. (b) Verify that at 820 nm
the fiber is not single-mode because V = 3.514. (c) With the result from (b),
verify by observation from Fig. 2.20 that the fiber supports the LP 01 and LP 11
modes at 820 nm.
2.15 Consider a 62.5 μm core-diameter graded-index fiber that has a parabolic
index profile (α = 2). Suppose the fiber has a numerical aperture NA = 0.275.
(a) Show that the V number for this fiber at 850 nm is 63.5.
(b) Verify that 1008 modes propagate in the fiber at 850 nm.
2.16 Consider a 50 μm core diameter graded-index fiber that has a core index n 1
= 1.480 and a cladding index n 2 = 1.465. (a) Using the exact expression for
the index difference given in Eq. (2.39), show that = 1.008%. (b) Using the
approximation for given in the right-hand side of Eq. (2.39), show that =
1.014%. This shows that the approximation is quite accurate.
2.17 Calculate the number of modes at 820 and 1300 nm in a graded-index fiber
having a parabolic-index profile (α = 2), a 25 μm core radius, n 1 = 1.48, and
n 2 = 1.46. How does this compare to a step-index fiber? [Answer: At 820 nm
for the graded-index fiber, M = 543 and at 1300 nm, M = 216. For a step-index
fiber, at 820 nm, M = 1078 and at 1300 nm, M = 429.]
2.18 Calculate the numerical apertures of (a) a plastic step-index fiber having a
core refractive index of n 1 = 1.60 and a cladding index of n 2 = 1.49, (b) a
step-index fiber having a silica core (n 1 = 1.458) and a silicone resin cladding
(n 2 = 1.405). [Answer: (a) 0.58; (b) 0.39.]
References
1. See any general physics book or introductory optics book; for example: (a) D. Halliday, R.
Resnick, and J. Walker, Fundamentals of Physics, 11th edn. (Wiley, 2018); (b) E. Hecht, Optics,
Pearson, 5th edn. (2016); (c) K. Iizuka, Engineering Optics (Springer, Berlin, 2019)
2 Optical Fiber Structures and Light Guiding Principles
2.12 A certain step-index fiber has a 25 μm core radius, n 1 = 1.48, and n 2 = 1.46.
(a) Show that the normalized frequency at 820 nm is V = 46.5. (b) Show
that 1081 modes propagate in this fiber at 820 nm. (c) Show that 417 modes
propagate in this fiber at 1320 nm. (d) Show that 303 modes propagate in this
fiber at 1550 nm. (e) Verify that the percent of the optical power flows in the
cladding for the different wavelengths is 4.1% at 820 nm, 6.6% at 1320 nm,
and 7.8% at 1550 nm.
2.13 Consider a fiber with a 25 μm core radius, a core index n 1 = 1.48, and =
0.01. (a) Show that at 1320 nm the parameter V = 25 and the number of modes
M = 312. (b) Verify that 7.5% of the optical power flows in the cladding. (c) If
the core-cladding difference is reduced to = 0.003, show that the number
of modes M = 94 and 13.7% of the optical power flows in the cladding.
2.14 Suppose a certain step-index fiber has a 5 μm core radius, an index difference
= 0.002, and a core index n 1 = 1.480. (a) By calculating the V number,
verify that at 1310 nm this is a single-mode fiber. (b) Verify that at 820 nm
the fiber is not single-mode because V = 3.514. (c) With the result from (b),
verify by observation from Fig. 2.20 that the fiber supports the LP 01 and LP 11
modes at 820 nm.
2.15 Consider a 62.5 μm core-diameter graded-index fiber that has a parabolic
index profile (α = 2). Suppose the fiber has a numerical aperture NA = 0.275.
(a) Show that the V number for this fiber at 850 nm is 63.5.
(b) Verify that 1008 modes propagate in the fiber at 850 nm.
2.16 Consider a 50 μm core diameter graded-index fiber that has a core index n 1
= 1.480 and a cladding index n 2 = 1.465. (a) Using the exact expression for
the index difference given in Eq. (2.39), show that = 1.008%. (b) Using the
approximation for given in the right-hand side of Eq. (2.39), show that =
1.014%. This shows that the approximation is quite accurate.
2.17 Calculate the number of modes at 820 and 1300 nm in a graded-index fiber
having a parabolic-index profile (α = 2), a 25 μm core radius, n 1 = 1.48, and
n 2 = 1.46. How does this compare to a step-index fiber? [Answer: At 820 nm
for the graded-index fiber, M = 543 and at 1300 nm, M = 216. For a step-index
fiber, at 820 nm, M = 1078 and at 1300 nm, M = 429.]
2.18 Calculate the numerical apertures of (a) a plastic step-index fiber having a
core refractive index of n 1 = 1.60 and a cladding index of n 2 = 1.49, (b) a
step-index fiber having a silica core (n 1 = 1.458) and a silicone resin cladding
(n 2 = 1.405). [Answer: (a) 0.58; (b) 0.39.]
References
1. See any general physics book or introductory optics book; for example: (a) D. Halliday, R.
Resnick, and J. Walker, Fundamentals of Physics, 11th edn. (Wiley, 2018); (b) E. Hecht, Optics,
Pearson, 5th edn. (2016); (c) K. Iizuka, Engineering Optics (Springer, Berlin, 2019)