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5 Optical Power Coupling
Problems
5.1 Analogous to Fig. 5.2, plot and compare the emission patterns from a Lambertian source and a source with an emission pattern given by L(θ) = L 0 cos
3
θ.
Assume both sources have the same peak radiance L 0 , which is normalized
to unity in each case.
5.2 Consider light sources where the emission pattern is given by L(θ) = L 0
cos
m
θ. Plot L(θ) as a function of m in the range 1 ≤ m ≤ 20 at viewing angles
of 10°, 20°, and 45°. Assume all sources have the same peak radiance L 0 .
5.3 A laser diode has lateral (ϕ = 0°) and transverse (ϕ = 90°) half-power beam
widths of 2θ = 60° and 30°, respectively. Show that the transverse and lateral
power distribution coefficients for this device are T = 20.0 and L = 4.82.
5.4 An LED with a circular emitting area of radius 20 μm has a Lambertian
emission pattern with a 100-W/(cm
2 - sr) axial radiance at a 100-mA drive
current. (a) How much optical power can be coupled into a step-index fiber
having a 100-μm core diameter and NA = 0.22? (b) How much optical power
can be coupled from this source into a 50-μm core-diameter graded-index
fiber having α = 2.0, n 1 = 1.48, and = 0.01? [Answer: (a) P LED; step =
191 μW; (b) P LED; graded = 159 μW]
5.5 A GaAs optical source that has a refractive index of 3.600 is closely coupled
to a step-index fiber that has a core refractive index of 1.465. If the source
size is smaller than the fiber core, and the small gap between the source and
the fiber is filled with a gel that has a refractive index of 1.305, verify the
following parameters (see Example 5.4): (a) The reflectivity at the sourceto-gel interface R s-g = 0.219; (b) The reflectivity at the gel-to-fiber interface
R g-f = 3.34 × 10
–3 ; (c) The total transmission through the gel is T = 0.778
(d) The power loss from the source into the fiber is L = 1.09 dB.
5.6 Consider a Lambertian LED source with a 50-μm diameter emitting area.
(a) If this source is connected to an optical fiber that has a 62.5-μm core
diameter with NA = 0.18, show that the coupling efficiency is 3.24% = −
14.9 dB. (b) If a spherical microlens is used to improve the coupling efficiency,
show that the maximum magnification is M max = 1.56. (c) Show that with
this lens the coupling efficiency is 5.06% = −13.0 dB.
5.7 The end faces of two optical fibers with core refractive indices of 1.485 are
perfectly aligned and have a small gap between them. (a) If this gap is filled
with a gel that has a refractive index of 1.305, show that the reflectivity at a
gel-fiber interface is 4.16 × 10
–3 ; (b) Show that the power loss at a gel-fiber
interface is L = −10 log (1 − R) = 0.018 dB. (c) If the gap is very small,
show that the power loss in decibels through the joint when no index-matching
material is used is 0.17 dB. Note that n = 1.0 for air.
5.8 Consider the three fibers having the properties listed in Table 5.1. Use
Eqs. (5.23) and (5.21) to verify the entries in this table for connector losses
(in decibels) due to the indicated axial misalignments.
5.9 Consider Eq. (5.35) that gives the coupling loss for two fibers with unequal
core radii. Show that the coupling losses in decibels as a function of α R / α E for
5 Optical Power Coupling
Problems
5.1 Analogous to Fig. 5.2, plot and compare the emission patterns from a Lambertian source and a source with an emission pattern given by L(θ) = L 0 cos
3
θ.
Assume both sources have the same peak radiance L 0 , which is normalized
to unity in each case.
5.2 Consider light sources where the emission pattern is given by L(θ) = L 0
cos
m
θ. Plot L(θ) as a function of m in the range 1 ≤ m ≤ 20 at viewing angles
of 10°, 20°, and 45°. Assume all sources have the same peak radiance L 0 .
5.3 A laser diode has lateral (ϕ = 0°) and transverse (ϕ = 90°) half-power beam
widths of 2θ = 60° and 30°, respectively. Show that the transverse and lateral
power distribution coefficients for this device are T = 20.0 and L = 4.82.
5.4 An LED with a circular emitting area of radius 20 μm has a Lambertian
emission pattern with a 100-W/(cm
2 - sr) axial radiance at a 100-mA drive
current. (a) How much optical power can be coupled into a step-index fiber
having a 100-μm core diameter and NA = 0.22? (b) How much optical power
can be coupled from this source into a 50-μm core-diameter graded-index
fiber having α = 2.0, n 1 = 1.48, and = 0.01? [Answer: (a) P LED; step =
191 μW; (b) P LED; graded = 159 μW]
5.5 A GaAs optical source that has a refractive index of 3.600 is closely coupled
to a step-index fiber that has a core refractive index of 1.465. If the source
size is smaller than the fiber core, and the small gap between the source and
the fiber is filled with a gel that has a refractive index of 1.305, verify the
following parameters (see Example 5.4): (a) The reflectivity at the sourceto-gel interface R s-g = 0.219; (b) The reflectivity at the gel-to-fiber interface
R g-f = 3.34 × 10
–3 ; (c) The total transmission through the gel is T = 0.778
(d) The power loss from the source into the fiber is L = 1.09 dB.
5.6 Consider a Lambertian LED source with a 50-μm diameter emitting area.
(a) If this source is connected to an optical fiber that has a 62.5-μm core
diameter with NA = 0.18, show that the coupling efficiency is 3.24% = −
14.9 dB. (b) If a spherical microlens is used to improve the coupling efficiency,
show that the maximum magnification is M max = 1.56. (c) Show that with
this lens the coupling efficiency is 5.06% = −13.0 dB.
5.7 The end faces of two optical fibers with core refractive indices of 1.485 are
perfectly aligned and have a small gap between them. (a) If this gap is filled
with a gel that has a refractive index of 1.305, show that the reflectivity at a
gel-fiber interface is 4.16 × 10
–3 ; (b) Show that the power loss at a gel-fiber
interface is L = −10 log (1 − R) = 0.018 dB. (c) If the gap is very small,
show that the power loss in decibels through the joint when no index-matching
material is used is 0.17 dB. Note that n = 1.0 for air.
5.8 Consider the three fibers having the properties listed in Table 5.1. Use
Eqs. (5.23) and (5.21) to verify the entries in this table for connector losses
(in decibels) due to the indicated axial misalignments.
5.9 Consider Eq. (5.35) that gives the coupling loss for two fibers with unequal
core radii. Show that the coupling losses in decibels as a function of α R / α E for