2.4 Modes in Circular Waveguides
59
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. 2.19 Electric field distributions for several of the lower-order guided modes in a symmetricalslab waveguide
but, instead, they extend partially into the cladding. The fields vary harmonically
in the guiding region of refractive index n 1 and decay exponentially outside of this
region. For low-order modes the fields are tightly concentrated near the center of the
slab (or the axis of an optical fiber), with little penetration into the cladding region.
On the other hand, for higher-order modes the fields are distributed more toward the
edges of the guide and penetrate farther into the cladding region.
Solving Maxwell’s equations shows that, in addition to supporting a finite number
of guided modes, the optical fiber waveguide has an infinite continuum of radiation
modes that are not trapped in the core and guided by the fiber but are still solutions
of the same boundary-value problem. The radiation field basically results from the
optical power that is outside the fiber acceptance angle being refracted out of the
core. Because of the finite radius of the cladding, some of this radiation gets trapped
in the cladding, thereby causing cladding modes to appear. As the core and cladding
modes propagate along the fiber, mode coupling occurs between the cladding modes
and the higher-order core modes. This coupling occurs because the electric fields of
the guided core modes are not completely confined to the core but extend partially
into the cladding (see Fig. 2.19) and likewise for the cladding modes. A diffusion
of power back and forth between the core and cladding modes thus occurs; this
generally results in a loss of power from the core modes.
Guided modes in the fiber occur when the values for β satisfy the condition n 2 k < β
< n 1 k. At the limit of propagation when β = n 2 k, a mode is no longer properly guided
and is called being cut off . Thus unguided or radiation modes appear for frequencies
below the cutoff point where β < n 2 k. However, wave propagation can still occur
below cutoff for those modes where some of the energy loss due to radiation is
blocked by an angular momentum barrier that exists near the core-cladding interface
[17]. These propagation states behave as partially confined guided modes rather
than radiation modes and are called leaky modes [5, 6, 12, 13]. These leaky modes
can travel considerable distances along a fiber but lose power through leakage or
tunneling into the cladding as they propagate.
59
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. 2.19 Electric field distributions for several of the lower-order guided modes in a symmetricalslab waveguide
but, instead, they extend partially into the cladding. The fields vary harmonically
in the guiding region of refractive index n 1 and decay exponentially outside of this
region. For low-order modes the fields are tightly concentrated near the center of the
slab (or the axis of an optical fiber), with little penetration into the cladding region.
On the other hand, for higher-order modes the fields are distributed more toward the
edges of the guide and penetrate farther into the cladding region.
Solving Maxwell’s equations shows that, in addition to supporting a finite number
of guided modes, the optical fiber waveguide has an infinite continuum of radiation
modes that are not trapped in the core and guided by the fiber but are still solutions
of the same boundary-value problem. The radiation field basically results from the
optical power that is outside the fiber acceptance angle being refracted out of the
core. Because of the finite radius of the cladding, some of this radiation gets trapped
in the cladding, thereby causing cladding modes to appear. As the core and cladding
modes propagate along the fiber, mode coupling occurs between the cladding modes
and the higher-order core modes. This coupling occurs because the electric fields of
the guided core modes are not completely confined to the core but extend partially
into the cladding (see Fig. 2.19) and likewise for the cladding modes. A diffusion
of power back and forth between the core and cladding modes thus occurs; this
generally results in a loss of power from the core modes.
Guided modes in the fiber occur when the values for β satisfy the condition n 2 k < β
< n 1 k. At the limit of propagation when β = n 2 k, a mode is no longer properly guided
and is called being cut off . Thus unguided or radiation modes appear for frequencies
below the cutoff point where β < n 2 k. However, wave propagation can still occur
below cutoff for those modes where some of the energy loss due to radiation is
blocked by an angular momentum barrier that exists near the core-cladding interface
[17]. These propagation states behave as partially confined guided modes rather
than radiation modes and are called leaky modes [5, 6, 12, 13]. These leaky modes
can travel considerable distances along a fiber but lose power through leakage or
tunneling into the cladding as they propagate.