228
5 Optical Power Coupling
power P in the fiber by
P =
2π
0
a
0
p(r ) r dr dθ
(5.25)
For an arbitrary index profile, the double integral in Eq. (5.25) must be evaluated
numerically. However, an analytic expression can be found by using a fiber with a
parabolic index profile (α = 2.0). Using Eq. (2.40), the power density expression at
a point r given by Eq. (5.24) becomes
p(r ) = p(0)
1 −
r
a
2
(5.26)
Using Eqs. (5.25) and (5.26), the relationship between the axial power density
p(0) and the total power P in the emitting fiber is
P =
πa
2
2
p(0)
(5.27)
To calculate the power transmitted across the butt joint of the two parabolic gradedindex fibers with an axial offset d, consider the diagram shown in Fig. 5.10. The
overlap region must be considered separately for the areas A 1 and A 2 . In area A 1 the
emitting fiber limits the numerical aperture, whereas in area A 2 the numerical aperture
of the receiving fiber is smaller than that of the emitting fiber. The vertical dashed
line separating the two areas is the locus of points where the numerical apertures are
equal.
To determine the power coupled into the receiving fiber, the power density given
by Eq. (5.26) is integrated separately over areas A 1 and A 2 . Because the numerical
aperture of the emitting fiber is smaller than that of the receiving fiber in area A 1 ,
all of the power emitted in this region will be accepted by the receiving fiber. The
Fig. 5.10 Core overlap region for two identical parabolic graded-index fibers with an axial
separation d, where points x 1 and x 2 are arbitrary points of symmetry in areas A 1 and A 2
5 Optical Power Coupling
power P in the fiber by
P =
2π
0
a
0
p(r ) r dr dθ
(5.25)
For an arbitrary index profile, the double integral in Eq. (5.25) must be evaluated
numerically. However, an analytic expression can be found by using a fiber with a
parabolic index profile (α = 2.0). Using Eq. (2.40), the power density expression at
a point r given by Eq. (5.24) becomes
p(r ) = p(0)
1 −
r
a
2
(5.26)
Using Eqs. (5.25) and (5.26), the relationship between the axial power density
p(0) and the total power P in the emitting fiber is
P =
πa
2
2
p(0)
(5.27)
To calculate the power transmitted across the butt joint of the two parabolic gradedindex fibers with an axial offset d, consider the diagram shown in Fig. 5.10. The
overlap region must be considered separately for the areas A 1 and A 2 . In area A 1 the
emitting fiber limits the numerical aperture, whereas in area A 2 the numerical aperture
of the receiving fiber is smaller than that of the emitting fiber. The vertical dashed
line separating the two areas is the locus of points where the numerical apertures are
equal.
To determine the power coupled into the receiving fiber, the power density given
by Eq. (5.26) is integrated separately over areas A 1 and A 2 . Because the numerical
aperture of the emitting fiber is smaller than that of the receiving fiber in area A 1 ,
all of the power emitted in this region will be accepted by the receiving fiber. The
Fig. 5.10 Core overlap region for two identical parabolic graded-index fibers with an axial
separation d, where points x 1 and x 2 are arbitrary points of symmetry in areas A 1 and A 2