Biophotonics_Concepts_to_Applications

As Sect.2.4describes, from Snell’s law the minimum or critical angleφcthat
supports total internal reflection at the core-cladding interface is given by

sin/c¼

n 2

n 1

ð 3 : 1 Þ

Rays striking the interface at angles less thanφcwill refract out of the core and
be lost in the cladding, as the dashed line shows. By applying Snell’s law to the air–
fiber face boundary, the condition of Eq. (3.1) can be used to determine the
maximum entrance angleθ0, max, which is called theacceptance angleθA. This
calculation yields the relationship

n sinh 0 ;max¼n sinhA¼n 1 sinhc¼ n^21 n^22

 1 = 2

ð 3 : 2 Þ

whereθc=π/2−φc. If a ray enters thefiber at an angleθ 0 less thanθA,itwillbe
totally internally reflected inside thefiber at the core-cladding interface. Thus the
angleθAdefines anacceptance conefor incoming light. Rays that fall outside of the
acceptance cone (for example the ray indicated by the dashed line in Fig.3.3)will
refract out of the core.
Equation (3.2) also defines thenumerical aperture(NA) of a step-indexfiber for
meridional rays:

NA¼n sinhA¼ n^21 n^22

 1 = 2

n 1

ffiffiffiffiffiffi

2 D

p

ð 3 : 3 Þ

The approximation on the right-hand side of Eq. (3.3) is valid because the
parameterΔis much less than 1. Because it is related to the acceptance angleθA, the
NA is used to describe the light acceptance or gathering capability of a multimode
fiber and to calculate source-to-fiber optical power coupling efficiencies. The NA
value generally is listed on opticalfiber data sheets.

n 2 Cladding

Refracted ray lost in the cladding

n 2 Cladding

Propagating rays are

captured in the

acceptance cone

Non-captured rays

outside of the

acceptance cone

Fig. 3.3 Ray optics picture
of the propagation mechanism
in an opticalfiber

58 3 Optical Fibers for Biophotonics Applications