Biophotonics_Concepts_to_Applications

3.1.2 Modal Concepts.

Whereas the geometrical optics ray approach gives a simple picture of how light is
guided along afiber, mode theory is needed to explain concepts such as mode
coupling, signal dispersion, and coherence or interference phenomena. Figure3.4
shows the oscillatingfield patterns of three of the lower-order transverse electric
(TE) modes. The mode order is equal to the number of times thefield value is zero
within the guiding core, as is illustrated by the solid dots in Fig.3.4. As is shown by
the plots, the electricfields of the guided modes are not completely confined to the
core but extend partially in the cladding. Thefields oscillate harmonically in thefiber
core and decay exponentially in the cladding region. The exponentially decaying
field is referred to as anevanescentfield. Thefields of the low-order modes are
concentrated near thefiber axis and do not penetrate far into the cladding region. The
fields of the higher-order modes are located more strongly near the core edges and
penetrate farther into the cladding.
As the core radius a shown in Fig.3.1is made progressively smaller, all the
higher-order modes except thefundamental mode(the zeroth-order linearly polarized
mode designated by LP 01 ) shown in Fig.3.4will gradually get cut off and conse-
quentlywill not propagate in thefiber. Afiber in which only the fundamental mode can
propagate is called asingle-modefiber. An important parameter related to the cutoff
condition is theV number(also called thenormalized frequency)defined by

V¼

2 pa

k

n^21 n^22

 1 = 2

¼

2 pa

k

NA

2 pa

k

n 1

ffiffiffiffiffiffi

2 D

p

ð 3 : 4 Þ

where the approximation on the right-hand side comes from Eq. (3.3). The V
number is a dimensionless parameter that designates how many modes can prop-
agate in a specific optical fiber. Except for the fundamental LP 01 mode, a
higher-order mode can exist only for values of V greater than 2.405 (with each

Evanescent tails of the modes extend into the cladding

Cladding n 2

Cladding n 2

Core n 1

Exponential decay

in the cladding

Harmonic

variation

in the core

Fundamental

LP 01 mode

Second-order

mode

Third-order

mode

Zero

crossings

Fig. 3.4 Electricfield distributions of lower-order guided modes in an opticalfiber (longitudinal
cross-sectional view)

60 3 Optical Fibers for Biophotonics Applications